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A Coupled Euler/Navier-Stokes Algorithm for 2-D Unsteady Transonic Shock/Boundary-Layer Interaction
by Steven R. Allmaras
GTL Report #196 March 1989 A Coupled Euler/Navier-Stokes Algorithm for 2-D Unsteady Transonic Shock/Boundary-Layer Interaction by Steven R. Allmaras
GTL Report #196 March 1989
This research was initially supported by a grant from Rolls-Royce, PLC, supervised by Dr. P. Stow. Subsequent research was funded by Air Force Office of Scientific Research grant F49620-78-C-0084, supervised by Dr. J. Wilson and Dr. H. Helin. During this portion of the research, the author was personally supported by an AFRAPT Traineeship in conjunction with Pratt & Whitney Engineering Division, Hartford, CT. A Coupled Euler/Navier-Stokes Algorithm for 2-D Unsteady Transonic Shock/Boundary-Layer Interaction by Steven Richard Allmaras
Submitted to the Department of Aeronautics and Astronautics on 14 February, 1989 in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Computational Fluid Dynamics
Abstract
This thesis presents a coupled Euler/Navier-Stokes algorithm for solving 2-D un- steady transonic flows. The flowfield is described by a Defect formulation, where sepa- rate Euler and Navier-Stokes algorithms are used on overlapping grids and are coupled through wall transpiration fluxes. The work is separated into three major contributions. The first contribution is a new algorithm for the solution of the 2-D unsteady Euler equations. The algorithm incorporates flux-splitting to capture shocks crisply and with minimal oscillations. To reduce numerical errors, grid independent second order accu- racy is achieved for both steady and unsteady flows. This is done by a formulation in which both solution averages and gradients are stored for each cell. The scheme allows no decoupled modes; hence, no explicitly added artificial is necessary. The second contribution is a Thin-Shear-Layer Navier-Stokes algorithm for viscous regions. The algorithm uses two-point differencing across the boundary layer, which is second order accurate for both inviscid and viscous terms on nonsmooth grids. Flux- splitting is used for the streamwise discretization to capture shocks. The spatial dis- cretization of this scheme also admits no decoupled modes and does not require added artificial dissipation. A semi-implicit time integration is employed, which allows a time step determined by the streamwise grid spacing only. The algorithm uses a dynamic coordinate rescaling to evolve the viscous grid to the changing boundary layer thickness. The final contribution of the work is a explicit relaxation procedure for coupling the Euler and Navier-Stokes algorithms together. The coupling is through boundary conditions-specified outer edge values for the viscous solution and wall transpiration fluxes for the Euler solution. Computational results are presented for a series of duct geometries. The test cases are used to demonstrate the accuracy of the Euler algorithm, the Navier-Stokes algo- rithm, and the fully coupled algorithm. Results are compared with analytic theory, experimental results, and other computational methods.
Thesis Supervisor: Michael B. Giles Title: Assistant Professor of Aeronautics and Astronautics
2 Acknowledgements
I'm very tired, and I have this one last page left. Thinking back on my 5 1/2 years here at M.I.T., including a summer at NASA Ames and another at Pratt & Whitney, I see so many new people that I've met. Many of them contributed directly to this research, but many more have helped with their friendship. I have space to mention only a few by name here, but I'd like to thank them all. In fact, if you've stopped to read this page, chances are that you are one of them. Thank you.
I would like first to thank my advisor and good friend, Mike Giles, who was always around and willing to discuss new ideas, even when they weren't so good. I also appre- ciate all the time he spent editing this thesis. My thanks also go to my other committee members, Profs. Murman and Covert, for all their helpful suggestions. I thank my of- ficial readers Mark Drela, John Dannenhoffer, and Bob Ni, for their comments on this thesis. I've learned so much from Mark and John over the years. Bob also gave me my introduction to industry's way of doing things.
I want to also mention some of the gang here, past and present: my office mate, Mark Turner-it sounds morbid, Mark, but thanks for sharing the suffering (and the high points); Jeff Bounds, my roomate for so many years; Itwacvrts KaAAWrEp7s, who also goes by various English aliases; Dana Linquist, who did a fine job of editing; Cathy Mavriplis; Tom Roberts; Rich Shapiro; Bernard Loyd; Eric Ducharme; some of the newer types, Rob Plumley, Pete Silkowski, Bill Steptoe, and Sharon Newman; and Matthew T. Scott, who gave me more with his friendship than he probably realized.
Yes, I had my share of computer headaches. My thanks go to Bob Bruen who went out of his way to make things run a little smoother. I'd also like to thank Bob Haimes for his help.
During this long journey I've always had support and encouragement from my family: Dad, Mom, my sisters Joan, Kathy, Mary and Colleen, and my two brothers-in-law, Michael and Mark. I can't thank them enough.
And finally to Jesus Christ, who was always by my side, even when I didn't see him.
This research was initially supported by a grant from Rolls-Royce, PLC, supervised by Dr. P. Stow. Subsequent research was funded by Air Force Office of Scientific Re- search grant F49620-78-C-0084, supervised by Dr. J. Wilson and Dr. H. Helin. During this portion of the research, I was personally supported by an AFRAPT Traneeship in conjunction with Pratt & Whitney Engineering Division, Hartford, CT.
3 To my brother, Anthony, the engineer of the family ...
4 Contents
Abstract 2
Acknowledgements 3
Nomenclature 16
1 Introduction 19
1.1 Euler Algorithm ...... 20
1.2 Thin-Shear-Layer Navier-Stokes Algorithm ...... 22
1.3 Viscous/Inviscid Coupling ...... 24
1.4 Overview of Thesis ...... 26
2 Governing Equations 29
2.1 2-D Navier-Stokes Equations ...... 29
2.1.1 Boundary Conditions ...... 33
2.1.2 Nondimensionalization ...... 33
2.1.3 Turbulence Modeling ...... 36
2.2 Asymptotic Analysis ...... 37
2.2.1 Euler Equations ...... 38
2.2.2 Thin-Shear-Layer Navier-Stokes Equations ...... 38
2.2.3 Prandtl Boundary Layer Equations ...... 42
2.3 Composite Solution: Defect Formulation ...... 43
5 2.4 Farfield Boundary Conditions ...... 46
2.4.1 Inlet Boundary Conditions ...... 49
2.4.2 Exit Boundary Conditions ...... 50
2.5 Parametric Vector ...... 51
3 Euler Algorithm 54
3.1 Objectives ...... 54
3.2 Flux-Splitting ...... 54
3.3 First Order Spatial Discretization ...... 58
3.4 Second Order Spatial Discretization ...... 60
3.4.1 Gradient Equations ...... 62
3.4.2 Gradient Equations: First Order Discretization ...... 67
3.4.3 Gradient Equations: Second Order Discretization ...... 69
3.5 Time Integration ...... 75
3.5.1 Stability Analysis ...... 75
3.6 Farfield Boundary Conditions ...... 78
3.7 Inviscid Wall Boundary Conditions ...... 79
3.7.1 Velocity Reflection Treatment ...... 79
3.7.2 Characteristics Treatment ...... 80
3.8 Euler Timing Study ...... 85
4 Thin-Shear-Layer Navier-Stokes Algorithm 90
4.1 Objectives ...... 90
4.2 Defect Formulation Revisited ...... 91
4.3 Coordinate Rescaling ...... 92
6 4.4 Spatial Discretization ...... 93
4.4.1 Cross-streamwise Discretization (ir-direction) . 94
4.4.2 Streamwise Discretization (C-direction) .. . . 96
4.4.3 Artificial Dissipation ...... 97
4.5 Time Integration ...... 97
4.5.1 Stability Analysis ...... 98
4.6 Discrete Defect Equations ...... 99
4.7 Boundary Conditions ...... 101
4.7.1 Wall Boundary Conditions ...... 101
4.7.2 Edge Boundary Conditions ...... 101
4.7.3 Inlet/Exit Boundary Conditions . . . 103
4.7.4 Leading Edge Boundary Conditions . 104
4.8 Scaling Parameter ...... 106
4.9 Cebeci-Smith Turbulence Model ...... 110
4.10 Newton Solution Procedure ...... 112
4.10.1 Newton's Method ...... 113
4.10.2 Discrete Equation Linearization .. . . 114
4.10.3 Jacobian Matrix ...... 124
4.10.4 Gaussian Elimination ...... 127
4.11 Similarity Profiles ...... 128
4.11.1 Initial Profile Guess ...... 130
4.12 Viscous Grid Generation ...... 132
5 Euler/Navier-Stokes Coupling 134
5.1 Interpolation of Edge Solution ...... 136
7 5.2 Wall Transpiration Fluxes ...... 137
5.2.1 Modification of Euler Wall Boundary Conditions ...... 139
5.2.2 Numerical Integration of Mass Defect Equation ...... 140
5.2.3 Proper vs. Approximate Transpiration Fluxes ...... 141
5.3 Numerical Coupling Procedure ...... 144
6 Discussion of Results 148
6.1 Results for Euler Algorithm ...... 148
6.1.1 Subsonic Circular Bump ...... 149
6.1.2 Transonic Circular Bump ...... 151
6.1.3 Supersonic Circular Bump ...... 152
6.1.4 Order of Accuracy Study ...... 153
6.1.5 1-D Shock Tube ...... 155
6.1.6 Unsteady Quasi-1-D Laval Nozzle ...... 157
6.2 Results for TSL Navier-Stokes Algorithm ...... 183
6.2.1 Blasius Similarity Solution ...... 183
6.2.2 Turbulent Flat Plate ...... 184
6.3 Results for Coupled Euler/Navier-Stokes Algorithm ...... 192
6.3.1 Subsonic Compression Duct ...... 192
6.3.2 Oblique Shock Impinging on a Laminar Boundary Layer ..... 193
6.3.3 Transonic Diffuser ...... 197
7 Conclusions 226
7.1 Euler Algorithm ...... 226
7.2 Thin-Shear-Layer Navier-Stokes Algorithm ...... 228
8 7.3 Coupling Procedure ...... 229
7.4 Recommendations for Future Research ...... 231
7.4.1 Euler Recommendations ...... 231
7.4.2 Navier-Stokes Recommendations ...... 233
7.4.3 Coupling Recommendations ...... 235
A Cell Metrics for Euler Algorithm 244
B Stability Analysis: Euler Algorithm 247
C Stability Analysis: TSL Navier-Stokes Algorithm 252
D Steady-State Acceleration 256
D.1 Euler Algorithm ...... 256
D.2 TSL Navier-Stokes Algorithm ...... 261
E Test Case Descriptions 265
F Code Listing 273
9 List of Figures
2.1 Control Volume ...... 30
2.2 Laminar Boundary Layer on a Flat Plate ...... 39
2.3 Inlet and Exit Farfield Boundaries for a Typical Geometry ...... 47
3.1 Flux-Splitting for a Discontinuous Solution in 2-D ...... 57
3.2 Euler Grid with Solution and Geometry Storage Locations ...... 59
3.3 Constant Solution Approximation Within Cells (First Order) ...... 59
3.4 Linear Solution Approximation Within Cells (Second Order) ...... 61
3.5 Interpolation for Midpoint Rule Evaluation of Fluxes ...... 61
3.6 Interpolation for Two-Point Gauss Quadrature of Fluxes ...... 71
3.7 Effect of First-Stage Integration Constant on Stability Boundary ... . 77
3.8 Numerical Entropy Rise Through a Stagnation Point ...... 81
3.9 Normal Vectors for Actual Wall vs. Linear Wall Segment ...... 85
3.10 Linear Wall Face Normals Used at Gauss Points ...... 86
3.11 Analytic Wall Geometry Normals Used at Gauss Points ...... 87
4.1 Viscous Grid Notation (computational space) ...... 94
4.2 Laminar Separation Bubble in a Subsonic Expansion Duct ...... 108
4.3 Predicted Edges of Viscous Grid Through Separation, (Eq. 4.33), Based on Fixed Grid Solution ...... 109
4.4 Actual Edges of Viscous Grid Through Separation ...... 109
4.5 Block Jacobian Matrices for Local Equations ...... 124
10 4.6 Modified Block Jacobian Matrices at Boundaries ...... 126
4.7 Block Jacobian Matrices for Global Equations ...... 126
5.1 Composite Grid Topology for Euler/Navier-Stokes Coupling ...... 135
5.2 Flow Chart for Numerical Coupling Procedure ...... 145
6.1 Subsonic Circular Bump: M = 0.5, T = 10%, 64 x 16 grid, Character- istics Procedure ...... 160
6.1 Subsonic Circular Bump: M = 0.5, r = 10%, 64 x 16 grid, Character- istics Procedure ...... 161
6.2 Subsonic Circular Bump: M = 0.5, r = 10%, 64 x 16 grid, Velocity Reflection Procedure ...... 162
6.3 Transonic Circular Bump: M = 0.675, r = 10%, 64 x 16 grid ..... 163
6.3 Transonic Circular Bump: M = 0.675, r = 10%, 64 x 16 grid ..... 164
6.4 Transonic Circular Bump: M = 0.675, r = 10%, 64 x 16 grid, Solution Interpolated to Nodes for Plotting ...... 165
6.5 Supersonic Circular Bump: M = 1.4, r = 4%, 64 x 16 grid ...... 166
6.5 Supersonic Circular Bump: M = 1.4, r = 4%, 64 x 16 grid ...... 167
6.6 Order of Accuracy Study: Computational Grids ...... 168
6.7 Order of Accuracy Study: Mach Number Distribution for 64 x 16 Grid 169
6.8 Order of Accuracy Study: Effect of Grid Resolution on Total Pressure Error ...... 170
6.9 Order of Accuracy Study: Randomized Computational Grids ..... 171
6.10 Order of Accuracy Study: Mach Number Distribution for 64 x 16 Ran- domized Grid ...... 172
6.11 Order of Accuracy Study: Effect of Grid Resolution on Total Pressure Error for Smooth and Randomized Grids ...... 173
6.12 1-D Shock Tube Problem: Solution at t = 0.14492 ...... 174
6.12 1-D Shock Tube Problem: Solution at t = 0.14492 ...... 175
11 6.13 1-D Shock Tube Problem: Solution for Second Order Godunov Scheme of Van Leer [831 ...... 176
6.14 Laval Nozzle: Grid Comparison of Steady Pressure Distribution .... 177
6.15 Laval Nozzle: Unsteady Distribution of Pressure Due to a Sinusoidal Exit Pressure Oscillation of 15% Amplitude and Reduced Frequency of 2 178
6.16 Laval Nozzle: Grid Comparison of Unsteady Pressure Distribution Due to a Sinusoidal Exit Pressure Oscillation: Phase = 0 ...... 179
6.17 Laval Nozzle: Grid Comparison of Unsteady Pressure Distribution Due to a Sinusoidal Exit Pressure Oscillation: Phase = 90* ...... 180
6.18 Laval Nozzle: Grid Comparison of Unsteady Pressure Distribution Due to a Sinusoidal Exit Pressure Oscillation: Phase = 1800 ...... 181
6.19 Laval Nozzle: Grid Comparison of Unsteady Pressure Distribution Due to a Sinusoidal Exit Pressure Oscillation: Phase = 270* ...... 182
6.20 Blasius Similarity Solution: Effect of Grid Refinement on Profiles ... 186
6.21 Blasius Similarity Solution: Effect of Grid Refinement on Solution Error 187
6.22 Turbulent Flat Plate: Computational Grid, 30 x 48 ...... 188
6.23 Turbulent Flat Plate: Skin Friction Distribution ...... 189
6.24 Turbulent Flat Plate: Minimum Grid Spacing Along Plate ...... 190
6.25 Turbulent Flat Plate: Velocity Profile at x = 4.08 ...... 191
6.26 Oblique Shock/Boundary Layer: Experimental Geometry (from [8]) . . 194
6.27 Transonic Diffuser: Experimental Geometry (from [71]) ...... 198
6.28 Compression Duct: Composite Grid and Flowfield ...... 206
6.29 Compression Duct: Velocity Profiles Using TSL Navier-Stokes Equa- tions (Profiles at Stations Indicated in Fig. 6.28a) ...... 207
6.30 Compression Duct: Velocity Profiles Using Defect Equations (Profiles at Stations Indicated in Fig. 6.28a) ...... 208
6.31 Oblique Shock/Boundary Layer: Coarse Computational Grid (Euler 64 x 16, Viscous 59 x 15) ...... 209
12 6.32 Oblique Shock/Boundary Layer: Fine Computational Grid (Euler 128 x 32, Viscous 123 x 31) ...... 210
6.33 Oblique Shock/Boundary Layer: Coarse Grid Density Contours (A = 0.005) ...... 211
6.34 Oblique Shock/Boundary Layer: Fine Grid Density Contours (A = 0.005)212
6.35 Oblique Shock/Boundary Layer: Comparison of Separation Bubbles . 213
6.36 Oblique Shock/Boundary Layer: Comparison of Wall Pressure and Skin Friction Distributions ...... 214
6.37 Oblique Shock/Boundary Layer: Coarse Grid Density Contours (A = 0.005) for Unmodified Coupling Procedure ...... 215
6.38 Oblique Shock/Boundary Layer: Coarse Grid Wall Pressure and Skin Friction Distributions for Unmodified Coupling Procedure ...... 216
6.39 Transonic Diffuser: Grid and Mach Number Distribution for Entire Dom ain ...... 217
6.40 Transonic Diffuser: Grid and Contour Plots for Diffuser Section .... 218
6.40 Transonic Diffuser: Grid and Contour Plots for Diffuser Section .... 219
6.41 Transonic Diffuser: Wall and Core Distributions for Diffuser Section . 220
6.41 Transonic Diffuser: Wall and Core Distributions for Diffuser Section . 221
6.42 Transonic Diffuser: Unsteady Static Pressure on Upper Wall at z/h* = 5.4222
6.43 Transonic Diffuser: Variation in Amplitude (rms) of Unsteady Static Pressure Along Upper Wall ...... 223
6.44 Transonic Diffuser: Solution of Defect Equations on Coarse Grid (Euler 48 x 8, Viscous 48 x 32) ...... 224
6.45 Transonic Diffuser: Solution of TSL Navier-Stokes Equations on Coarse Grid (Euler 48 x 8, Viscous 48 x 32) ...... 225
7.1 Embedded Grid ...... 232
A.1 Notation for Metric Integration Along a Face ...... 246
13 D.1 Effect of Gradient Update Under-Relaxation on Stability Boundary of Second Order Scheme (a1 = 0.17) ...... 257
14 List of Tables
3.1 Comparison of CPU Time (in psec) Per Cell Per Time Step ...... 88
D.1 Effect of Relaxation and AF on Steady-State Convergence: Subsonic Circular Bump on 32 x 8 Grid ...... 261
D.2 Effect of Relaxation and AF on Steady-State Convergence: Transonic Circular Bump on 32 x 8 Grid ...... 262
D.3 Effect of AF on Steady-State Convergence: Turbulent Flate Plate With a 30 x 48 Grid ...... 264
15 Nomenclature
A cell area
Az,, Ay , Ayy, second area moments about cell centroid A, B, C, G block Jacobian matrices C speed of sound c, cp specific heats at constant volume, pressure C! skin friction coefficient C vector of characteristic variables e static internal energy per unit mass E total energy per unit mass F inviscid flux vector, see (2.3) F* split flux vectors, see (3.6) G inviscid flux vector, see (2.3) h static enthalpy per unit mass H total enthalpy per unit mass inviscid flux vector, H = FI + Gj iJ computational coordinates (streamwise, normal to wall) 1, j J* Cartesian unit vectors 1-D Riemann invariants k coefficient of thermal conductivity LEUL, LTSL, LDEF analytic operators (Euler, TSL Navier-Stokes, Defect) Lh L-sL> LDEF discrete operators (Euler, TSL Navier-Stokes, Defect) M Mach number A unit vector normal to boundary or face p static pressure P total pressure; Legendre polynomial Pr Prandtl number q enthalpy flux qg, qy Cartesian heat transfer components velocity vector, q= ut + vj R perfect gas constant Re Reynolds number R viscous flux vector, see (2.4); vector of Newton residuals R, Rx, Ry Euler residual vector (average, z and y-gradients) 8 entropy, arclength from inlet unit vector tangential to boundary or face S Sutherland's constant, entropy S viscous flux vector, see (2.4) and (2.39) t time T static temperature U, V Cartesian velocity components UTr wall shear velocity U state vector of conservation variables U, UX, UY Euler conservation state vectors (average, z and y-gradients) WI, W2, W3, W4 Roe's parametric variables, see (2.74)
16 W state vector of Roe's parametric variables, see (2.74) W, WX, Wy Euler parametric state vectors (average, z and y-gradients) z, y Cartesian position components X vector of Newton unknowns x) location of wall ratio of specific heats, y = cp/c,, 6 boundary layer thickness boundary layer displacement thickness (incompressible) A change or difference At time step size ch discrete error 1 vertical coordinate (computational space) FIB Blasius' similarity variable boundary layer momentum thickness (incompressible) A Courant-Friedrichs-Lewy (CFL) number; characteristic A diagonal matrix of characteristics A coefficient of molecular viscosity 1/ coefficient of kinematic viscosity i/i, '2 smoothing coefficients in scaling parameter definition, see (4.37) streamwise coordinate (computational space) p static density A coefficient in scaling parameter definition, see (4.33) r shear stress; time coordinate (computational space) IrXZ, 'my) ryy components of shear stress tensor frequency control volume, boundary of control volume viscous grid scaling parameter
subscripts
0 inlet stagnation conditions C cell centroid e edge of viscous grid; Euler parameter energy energy conservation equation extrap value extrapolated from computational domain interior gi Gauss points ghost value evaluated at ghost cell or point computational coordinate indices J edge of viscous grid (j = J) I, r left, right interpolated solution 1, t laminar, turbulent m face midpoint mass mass conservation equation mom momentum conservation equation
17 n normal to boundary or face ref reference conditions 8 tangential to boundary or face t time derivative spec value specified v viscous parameter w wall x z-component or x-derivative y y-component or y-derivative upper, lower faces of viscous cell (j + 1, j)
superscripts
dimensional ( ) unit vector; inviscid solution ( )' nondimensional quantity in Sutherland's law + wall variable left, right interpolated solution h discrete quantity (n) quantity at nth multi-stage level n quantity at nth time or iteration level
18 Chapter 1 Introduction
Transonic flows at high Reynolds numbers are often dominated by the interaction of shocks with boundary layers. This interaction can also be naturally unsteady. Examples of such flows are engine inlets, where the shock and boundary layer in the diffuser
section of the inlet can interreact with the compressor face in an oscillating manner. These oscillations are typically small but depending on geometries and flight conditions, they may become large, seriously degrading performance. Pressure oscillations of 20% magnitude have been observed experimentally [201.
Today prediction of 2-D unsteady, strongly interacting flowfields is becoming possi- ble with the tools of CFD. Because of the strength of the shock/boundary layer interac-
tion and extent of viscous regions, these flowfields are presently simulated by algorithms
solving the full Navier-Stokes equations everywhere. In principle, all relevant physics will
be present in such solutions, but this is not necessarily the case. Within boundary lay-
ers significant flow variations exist, which must be resolved to be accurately predicted.
This requires local grid refinement, which in turn requires a priori knowledge of the boundary layer thickness--something that is part of the solution. Furthermore, these
Navier-Stokes solvers typically use the same discretization throughout the flowfield. This has the advantage of simplicity in implementation, but it has several drawbacks. For example, viscous effects are calculated everywhere even though they are negligible
throughout most of the flowfield. This results in wasted computing effort. Another
drawback is that flowfields often include different features, such as shocks, boundary layers, acoustic waves, which are predicted by the same discretization. Each of these could be more efficiently solved by different algorithms.
The objective of the present thesis is the development of an alternative method for
19 the solution of strongly interacting flowfields, such as those involving unsteady shocks. The method developed in this thesis is a coupled 2-D Euler/Navier-Stokes algorithm.
Separate but coupled algorithms are developed to solve the inviscid and viscous regions of the flowfield. Both the Euler and Navier-Stokes algorithms are designed to accurately predict the important physics of their respective regions.
The thesis consists of three major contributions: a new Euler algorithm for the outer inviscid flow, a new Thin-Shear-Layer Navier-Stokes algorithm for viscous regions, and a procedure for coupling the two together. Each of these three contributions are briefly described in the following sections.
1.1 Euler Algorithm
In the outer inviscid region, the 2-D Euler equations are used to model the flow since strong shocks and rotational flow may be present. Correct prediction of the Rankine-
Hugoniot shock jump relations necessitates a conservative discretization. In addition it is desirable to capture moving shocks crisply, without pre- or post-shock numerical oscillations. There may also be low amplitude acoustic waves or weak shocks which physically propagate without being dissipated. In numerical calculations these may be dissipated or dispersed by excessive numerical errors, either in the form of truncation error or explicitly added artificial dissipation. Thus, it is important to minimize these errors.
Many Euler algorithms are in current use; most of these solve the unsteady Eu- ler equations in a time-marching manner, but were originally developed and tuned for steady calculations. Some of the more popular use central-differencing or partial up- winding, and may be separated into two classes: cell-centered and cell-vertex schemes.
Examples of the cell-centered algorithms include those of Pulliam and Steger [651; and Jameson, Schmidt and Turkel [43]. These schemes have been formulated assuming a smooth, logically rectangular grid. They are spatially second order accurate on such
20 grids, but when used on nonsmooth (i.e. realistic) grids, their order of accuracy is re- duced, increasing numerical errors. In fact, Turkel [80 has analytically shown stringent restrictions on grid smoothness for achieving second order accuracy with a cell-centered central-differencing scheme in 1-D.
Cell-vertex algorithms include those of Ni [62]; Hall [38]; and Jameson [44]. In contrast to the cell-centered schemes, these cell-vertex schemes are second order accurate for steady-state regardless of grid smoothness [34]. For unsteady flows, grid assumptions reduce the spatial accuracy to first order for nonsmooth grids.
A feature of all these algorithms is that they must use explicitly added artificial dissipation or smoothing to capture shocks and constrain neutrally stable decoupled modes. Unfortunately, the added smoothing can reduce accuracy and robustness in several ways. First, the smoothing introduces user adjustable constants. Second, the smoothing to capture shocks is often different from that used to control decoupled modes. As a result, switches must be formulated to turn each on only where desired. Third, Lindquist [54,55] has shown for cell-vertex methods that care must be taken in
constructing the smoothing operators to avoid reduced accuracy for nonsmooth grids. Furthermore, the smoothing operators need to be modified near boundaries; if poorly
done, this can lead to divergence or nonphysical solutions [9]. Finally, the artificial
dissipation is often constructed and tuned for steady-state. Because of this, numerical
oscillations near shocks, that are small for steady-state, can become undesirably worse
for unsteady flows.
Another class of algorithms gaining popularity are those that use flux-splitting, a form of upwinding. Members of this class include the schemes of Chakavarthy [19]; Bun- ing and Steger [14]; and Anderson, Thomas, and van Leer [7]. These methods attempt
to use the underlying physics of the Euler equations more directly in their formulation.
They have the advantage that they are often able to capture shocks without explicitly
added artificial dissipation. They also do not allow decoupled modes. However, these
schemes are cell-centered and suffer reduced accuracy on nonsmooth grids. In addition, they tend to be more dissipative than the central-differencing schemes on smooth grids.
21 This thesis presents a new algorithm for the solution of the 2-D unsteady Euler equations. The algorithm incorporates upwinding to capture shocks crisply and with minimal oscillations. To reduce numerical errors, grid independent second order accu- racy is achieved for both steady and unsteady flows. This is done by a formulation in which both averages and gradients are stored for each cell. The scheme allows no decoupled modes; hence, no explicitly added artificial dissipation is necessary. Thus, the present scheme retains many of the advantages of previous upwind methods without their poor accuracy on nonsmooth grids.
1.2 Thin-Shear-Layer Navier-Stokes Algorithm
In boundary layers, where viscous effects are important, the flow exhibits large vari- ations across the boundary layer (normal to the surface). Streamwise variations often remain comparatively small. As a result, the streamwise viscous stress gradients may be neglected and the region modeled by the Thin-Shear-Layer Navier-Stokes equations.
An exception to this situation is when shocks impinge on the boundary layer. Then streamwise flow variations are large, with near step discontinuities in the outer part of the boundary layer and complex compression structures lower in the boundary layer. If macroscopic changes are desired, rather than these fine structures, then the streamwise viscous stress gradients may still be neglected as long as the equations are solved in a conservative formulation.
The large flow variations across the boundary layer place special requirements on numerical algorithms. To best predict these variations, high accuracy is important for
the discretization normal to the surface. This is true not only for inviscid terms but
also for viscous terms. In addition, resolution of these large variations can result in very small grid spacing normal to the surface, especially for turbulent cases. Schemes using explicit time integration are limited by numerical stability to a time step proportional to this small spacing. Thus, the choice of time integration becomes important. Finally,
the thickness of the boundary layer itself becomes a numerical problem. Although a part of the solution, it must be known to specify an appropriate grid. Accurate
22 knowledge of the thickness cannot be known a priori in many flows, such as those involving shock/boundary layer interaction or large separation.
Typically, Navier-Stokes algorithms are developed from existing Euler algorithms by adding discretized viscous terms. Examples are the schemes of Beam and Warm- ing [10]; Swanson and Turkel [78]; Rai [66]; Davis, Ni, and Bowley [26]; and Thomas and Walters [79]. In all of these schemes, the viscous stresses are discretized by 3-point central-differencing assuming a uniform grid. For the highly stretched grids (i.e. large cell size variation) used for viscous simulations, these fail to be second order accurate. Those schemes that do not use flux-splitting for the inviscid terms [10,26,78 require artificial dissipation to damp decoupled modes. Construction of these operators is par- ticularly crucial for viscous regions since they can contaminate solutions by masking physical viscosity [2,26}.
An extension of the Euler algorithm presented in this thesis to Navier-Stokes would suffer from some of these same disadvantages. Even with the solution averages and gra- dients stored for each cell, unconditional second order accuracy for the viscous stresses would not be possible. Furthermore, the scheme uses explicit time integration. For these reasons, this thesis also presents a separate TSL Navier-Stokes algorithm designed to meet the requirements of viscous regions.
The main objectives of this development have been second order accuracy for both inviscid and viscous terms across the boundary layer, a time integration which uses practical time steps, and a means of adapting the grid to the changing boundary layer thickness. The first objective is met by a discretization across the boundary layer which is similar to that used in the Keller Box scheme [48] for the solution of the Boundary
Layer equations. It is second order accurate for both inviscid and viscous terms on highly stretched grids. To capture shocks in the outer flow, upwind differencing is used in the streamwise discretization. The resulting spatial discretization has no unconstrained decoupled modes and needs no added artificial dissipation.
To meet the second objective, the algorithm incorporates a semi-implicit time in- tegration. Discretization normal to the surface is integrated implicitly and that in the
23 streamwise direction is integrated explicitly. Hence, the solution can be integrated at a time step depending on the streamwise grid spacing only. For a given streamwise station and time step, the implicit system is solved by Newton's method.
The third objective is realized by a dynamic coordinate transformation. The coordi- nate normal to the wall is rescaled by the unsteady local boundary layer thickness, and the transformed equations solved on a fixed grid. A similar adaptive procedure has been used by Carter [16] and Drela [28] for the solution of the Boundary Layer equations.
1.3 Viscous/Inviscid Coupling
Separation of the flow into viscous and inviscid regions requires coupling or commu- nication between the different regions to properly solve the entire flowfield. Hence, the third and final major contribution of this thesis is a procedure for coupling the Euler and TSL Navier-Stokes algorithms together.
Previous methods for viscous/inviscid coupling using the Euler and Navier-Stokes equations have been zonal approaches. The viscous and inviscid regions are separated from one another and coupling is through information transfer near the interface. Two variants of the zonal approach exist: patched grids and slightly overlapping grids.
In patched grid approaches, such as Rai [66,67] and Nakahashi [61], a Navier-Stokes grid is constructed about the body to a predetermined point, and then an Euler grid constructed abutting it. Coupling between the grids is by conservative evaluation of the fluxes along the interface. A drawback of this approach is placement of the interface.
If the interface is fixed, then it must be placed well outside viscous regions. However, this requires a priori knowledge of boundary layer thicknesses. On the other hand, if the interface is allowed to move, then the entire domain must be regridded.
In overlapping grid techniques, such as Denek et al [27], coupling between the dif- ferent grids is through solution interpolation in the overlap region. This requires both solutions to be physically realistic in the region of overlap. Hence, for Euler/Navier-
24 Stokes coupling the entire overlap must be well outside any viscous regions. In this approach a conservative formulation near the interface is very difficult. A further com- ment is that flowfields calculated by Rai [66] using both patched and overlapping grids show noticeable numerical noise in overlap regions, and Rai seems to have abandoned using overlapping grids in favor of patched grids for similar test cases in later papers.
In all the references listed above, the same inviscid discretization has been used in each region. If different algorithms are used in each region and the solution matched at the interface, then differing truncation errors will cause a numerical boundary layer in the vicinity of the interface. Therefore, special treatment (such as smoothing) near the interface would be required to avoid these nonphysical boundary layers.
For these reasons, an alternative form of coupling is used this thesis. The flowfield is solved on overlapping grids with the Euler grid extending to the body surface. That portion of the Euler solution within the viscous region is computed but is not physi- cally meaningful. Coupling between the Euler and Navier-Stokes solutions is through boundary conditions-specified outer edge values for the viscous solution and wall tran- spiration fluxes for the Euler solution. Coupled in this manner, the Euler grid remains fixed while the viscous grid evolves with the changing boundary layer thickness. Thus, a priori knowledge of boundary layer thicknesses is not required to solve the flow accu- rately.
This form of coupling is closer to viscous/inviscid interaction techniques solving the Boundary Layer equations within viscous regions. Examples of algorithms using the Potential equations for the inviscid flow are those of Le Balleur [49,50,51] and Carter [17]. Algorithms modeling the inviscid flow by the Euler equations have been developed by
Johnston and Sockol [45]; Whitfield et al [88]; Bussing and Murman [15,60]; Savant and Wigton [72]; and Drela and Giles [30].
These coupling methods are based on different but equivalent viewpoints of the boundary layer's effect on the outer inviscid flow. Lighthill [53] has shown that the primary effect of the boundary layer is a displacement of the inviscid streamlines away from the surface of the body. For the solution of the outer inviscid flow, this effect can
25 be produced by physically enlarging the body by the local displacement thickness of
the boundary layer. Thus, one coupling method is to construct an inviscid grid outside
the displacement surface, where a zero normal mass flux condition is specified. This
method was used by Drela and Giles [30], where the grid was solved as part of the solution. For use with a conventional Euler or Potential solver, this coupling method has the disadvantage that the grid must be regenerated as the displacement surface changes.
Lighthill also showed that the displacement effect can equivalently be produced by a distribution of sources along the surface of the body; this results in transpiration or blowing at the surface. Thus, the inviscid flow is solved to the body surface, where transpiration fluxes are specified; these are obtained from an integration of the viscous
equations across the boundary layer. An advantage of this method is that the inviscid
grid is fixed and need be generated only once. This coupling method was used by all others listed. It is also the method of choice in this thesis. Thus, this thesis repre- sents the first extension of coupling through transpiration fluxes to the solution of the Euler/Navier-Stokes equations.
A feature of the present coupling is the use of the Defect formulation of Le Balleur [50,51]. An important aspect of this approach is the solution of the TSL Navier-Stokes equa- tions written in the form of Defect equations within viscous regions. With different algorithms solving the inviscid and viscous equations, the discrete Defect equations subtract off the relative truncation error. This allows the inviscid and viscous solutions
to match smoothly at the edge of the viscous grid. In principle, it also makes a con- servative formulation at the interface possible; in this it has a potential advantage over other overlapping grid techniques.
1.4 Overview of Thesis
Chapter 2 presents the governing equations for fluid motion, including nondimen- sionalization and an asymptotic analysis to simplify them for the flowfields of present
26 interest. The resulting equation sets are the Euler and Thin-Shear-Layer Navier-Stokes equations. Boundary conditions and a parametric state vector used in both the Euler and Navier-Stokes algorithms are also discussed.
The development of the Euler algorithm is presented in Chapter 3. A first order upwind scheme, similar to existing methods, is first discussed. It is then extended to second order accuracy by a new approach of using both cell averages and gradients. An explicit time integration is then presented, along with a stability analysis for both the first and second order discretizations.
Chapter 4 develops the TSL Navier-Stokes algorithm. A dynamic coordinate trans- formation is defined. Then the different streamwise and cross-stream spatial discretiza- tions are developed. This is followed by the semi-implicit integration for evolving the viscous solution in time. Minor modifications of the discrete equations are then discussed for implementing the Defect formulation. This chapter concludes with a discussion of the Newton solution procedure for the implicit system resulting from the semi-implicit time integration; it includes equation linearization and direct solution of the linearized system by Gaussian elimination.
Chapter 5 describes the procedure used to couple the Euler and Navier-Stokes al- gorithms together. This includes interpolation of the Euler solution for edge boundary conditions on the viscous solution, and integration of the viscous equations across the boundary layer for wall transpiration fluxes. This chapter concludes with a flowchart of the numerical coupling procedure.
Chapter 6 presents a series of numerical test cases to demonstrate the capabilities of the Euler algorithm, the TSL Navier-Stokes algorithm, and the coupling procedure.
Two-dimensional channel flow test cases for Euler solvers are first presented. Then an order of accuracy study is conducted to numerically demonstrate the second order accuracy of the Euler algorithm on smooth and randomized grids. Its unsteady shock tracking capabilities are demonstrated for two test cases. The first is a 1-D shock tube problem, where results are compared to the analytic solution; and the second is a quasi- 1-D Laval nozzle, where it is compared to the central-differencing scheme of Jameson.
27 The second order spatial accuracy of the cross-stream discretization in the TSL algorithm is verified using Blausius' similarity solution for laminar flat plate flow. This is followed by a turbulent flat plate case verifying the correct implementation of the turbulence model; results are compared with experiment.
The fully coupled Euler/Navier-Stokes algorithm is then demonstrated for both steady and unsteady flows using three test cases. The first test case is a laminar boundary layer in a subsonic compression duct. It is used to demonstrate the nu- merical improvements resulting from the solution of the Defect equations. The second is a steady oblique shock impinging on a flat plate laminar boundary layer. Surface
pressure and friction coefficients are compared with experiment. The third and final
test case is a transonic diffuser with turbulent boundary layers. The algorithm is used
to detect self-induced unsteadiness in the shock/boundary layer interaction found in experiments.
Finally, Chapter 7 gives conclusions and recommendations for future research.
28 Chapter 2 Governing Equations
The equations governing the motion of a fluid are presented in this chapter. No at-
tempt is made to derive the equations; detailed derivations can be found elsewhere [87,73,52,6]. Instead the governing equations will be stated briefly along with major assumptions
made. Subsequent analysis, including nondimensionalization and asymptotic analysis
to simplify the governing equations, will be presented in more detail since they are more
pertinent to material covered in this thesis. This chapter will conclude with additional
analysis of inviscid characteristics for boundary conditions and representation of the
inviscid conservation and flux vectors by a parametric vector.
2.1 2-D Navier-Stokes Equations
The motion of a compressible, two-dimensional (2-D) fluid is governed by the Navier-
Stokes equations. These represent the conservation of mass, momentum and energy in a fixed control volume, such as that shown in Figure 2.1. In differential form the equations
are given by, aP+Pv 0 mass: + (2.1a) at ax a1y
x-momentum: apu+ 8pu2 + +pta - a + a-,, (2.1b) at ax a y ax ax ay 8pv apuv apv2 ap aTzy aTyy y-momentum: + + a+ - a + ai, (2.1c) at ax a y ay ax ay apE apuH apvH a energy:at + (2.1d) a + a =-- qz + u zz2 + vTy)
+ a-(qy + uTzy + Vryy)
29 na
Figure 2.1: Control Volume
In these equations p is density, p is pressure, E is total internal energy, H is total enthalpy, and uf + vi is the velocity vector (u and v are the Cartesian components in the x and y directions, respectively). Viscous effects are represented by the shear stress tensor (7',,, ry, Ty1 ) and the heat conduction vector qgS + qyj.
The governing equations may also be written in a more convenient vector form, aU aF aG 8R aS + - + -- a x& ay y a+(2.2) where the conservation vector U and inviscid flux vectors F and G are defined as, P Pu ( pv
U (P , F = = , (2.3) PVPut jpt) 2+ p pE k puH I pvH j and R and S contain the viscous terms, 0 0
R = , S =I. (2.4)
-qz+ uriz+vrzy -qp+u-rx+v ry) The Navier-Stokes equations may also be written in integral form,
U A+ (F-+ Gj) - A da = (RA +Sj) - n da, (2.5)
30 where the integral is over the control volume fl bounded by the curve 8; n^ is the unit outward normal to fl. The integral and differential forms are equivalent for continuous solutions; one is obtained from the other by the divergence theorem. The integral form is more general, since it is valid for discontinuous solutions. It will be used as the starting point for the finite volume algorithm of Chapter 3. The differential form will be used for further analysis and the starting point for the finite difference algorithm of Chapter 4.
In writing the Navier-Stokes equations, several assumptions have been made. The first is that the density is high enough that the fluid may be approximated as a contin- uum rather than discrete molecules. Other assumptions are that no chemical reactions, mass diffusion, or heat addition through radiation occur. Inclusion of these effects re- quires major modifications to the numerical algorithms presented in this thesis. A fur- ther assumption is the absence of body forces, including gravity, electromagnetic forces, three-dimensional rotation effects, etc. Typically, body forces appear in the equations as source terms (undifferentiated), which are easily modeled. Thus, modification of the present theory to include these effects is minor.
Apart from these assumptions, Equations (2.1) are quite general, since the viscous stresses and heat transfer have been left undefined. To solve the equations, these need to be specified. In addition, an equation of state relating the thermodynamic variables (p, p, E and H) is required.
The fluid is assumed to be a Newtonian fluid in which the shear stresses are linearly dependent on the fluid strain rates; it is also assumed to satisfy Stokes' hypothesis. With these assumptions, the components of the stress tensor may be written as, c') 2 au av 2 au a V rc = 21s- -1A - + -- =-- 2 , (2.6a) ix ay ax 3 (ax yY 3 z a v 2 09U av 2 JA20 - 0U ,r,, = 2pA ay 1 a--- + -Y = - Y2 a , (2.6b) 09U av r., = A (--Y + -), (2.6c)
where p is the coefficient of molecular viscosity. Consistent with a Newtonian fluid is Fourier's law, which states that the heat transfer is linearly dependent on temperature
31 gradients,
qz = -k-T q, = -k---, (2.7) 5 - ay where T is temperature and k is the coefficient of thermal conductivity.
The set of equations is closed by an equation of state relating the thermodynamic variables, as well as relations for yu and k. Here a perfect gas is assumed,
p = pRT, R = gas constant. (2.8)
Assuming further that the gas is calorically perfect (i.e. the specific heats c, and c,, are
constant), then the static and total internal energy are given by,
e = cT, E = e + 1(u2 + v 2), (2.9)
and the static and total enthalpy by,
h = cPT, H = h+ (u2 +v 2 ). (2.10)
Further useful relations are,
pH = pE + p, (2.11)
p = (y -1) PE - 1 p(u2 + V2)], (2.12)
P i pH - p(s2 + V2)], (2.13) l 1 2 1
where y = cp/c, is the ratio of specific heats (-y = 1.40 for air) and R = c, - c,.
For a perfect gas the coefficients of molecular viscosity and thermal conductivity are functions only of the temperature. Sutherland's law for the molecular viscosity is employed, y T ' 3 2 Tref+ S -- = (-32( (2.14) Aref Tref T+S ' where S is Sutherland's constant (S = 110K for air), Tref is some reference temperature and pref is the molecular viscosity measured at that temperature. For air Sutherland's
law is accurate to within 2% over a wide range of temperatures [87, p 28-29]. For most
gases the thermal conductivity is approximately proportional to the molecular viscosity.
Here this relation is assumed exact, k = A(2.15) kref pref
32 2.1.1 Boundary Conditions
Boundary conditions are necessary to define a unique solution. For the physical problems presented in this thesis, conditions at two different types of boundaries need be considered. The first are solid wall boundaries. Because of friction, fluid in contact with a wall must not move relative to the wall; this is the no-slip condition. For a stationary wall this gives,
at wall: u=0, v=0. (2.16a)
Only stationary walls are considered in this thesis. In addition, the fluid in contact with the wall must have the same temperature as the wall or the same heat transfer as the wall, at wall: T = Tw or (q + qyj) -h = qw. (2.16b)
Only adiabatic walls (qw = 0) are considered in this thesis.
The second type of "boundary" is the flow far from the wall or body, where the flow approaches some known uniform or stagnation conditions. The treatment of these farfield boundary conditions for numerical computations is presented in Section 2.4.
2.1.2 Nondimensionalization
Nondimensionalization is performed on the Navier-Stokes equations before they are analyzed. From this process, certain nondimensional groups emerge whose magnitudes can be used to characterize the behavior of solutions.
In the following, dimensional quantities are denoted with a bar ( ), and nondimen- sional quantities are left naked. Using this notation, variables in all preceding equations should have been barred but were not for the sake of clarity. All dependent and indepen- dent variables are nondimensionalized by a reference length Lref, density Pref, velocity cref, molecular viscosity coefficient jiref, heat conduction coefficient kref, and specific
33 heat constant ZPref:
X= i y= t=- Lref Lre' Lref/Cref
Pu U Pref Cref Cref (2.17) pEH P= , E= - H - 2 > PrefCref Cref cre
T= ___ p= k= k . Cref/CPref Aref kref The shear stresses ( y, and f) are nondimensioned consistent with their definitions
(2.6), and the heat transfer terms (T- and T) are nondimensioned consistent with the viscous work terms in the energy equation.
= q = - . (2.18) Areferef /Lref' Arefrefflref
In addition, the nondimensional specific heats and gas constant become,
__ ce 1 R _ '-l c,=_ =1, cV-._ - , R= - . (2.19) CPref CPref 'CPref y
When these definitions are substituted into the Navier-Stokes equations (2.1), two nondimensional parameters emerge. The first is the Reynolds number,
Re = prefiref Lref (2.20) Pref which results from comparison of the groups used to nondimensionalize inviscid and viscous terms in each equation. The second is the Prandtl number,
Pr = PrefCPref (2.21) kref which results from comparison of the definition of the heat transfer (2.7) and the group used to nondimensionalize it.
The resulting nondimensional Navier-Stokes equations are modified from (2.1) only by a factor of 1/Re multiplying the viscous terms,
aU cl F aG 1 8R 1 aS --- + axo-- + aiy - = Reax---- + ReayR y. (2.22)
34 The nondimensional vectors U, F, G, R and S are unchanged from (2.3) and (2.4). With the exception of the heat transfer definition (2.7) and Sutherland's law (2.14), the rest of the equations previously listed retain their forms when nondimensionalized. The nondimensional heat transfer definition is modified by the presence of the Prandtl number,
_ -8_ k OT q.=-k- =& q= ---- , (2.23a) ay Pr az __ -87 k 8T k g- k--.a = (2.23b) ay Pr ay It will be more convenient to write the heat transfer in terms of p and H instead of k and T. Thus, using (2.10),
qz = --- [-- u - V - (2.24a) Pr up( ; _X - -xax]
qy = ------u - . (2.24b) Pr pcP 5-y - yUY_ y From (2.15) and (2.19), the group k/psc is unity and will be dropped from this point forward.
Sutherland's law (2.14) poses a problem due to the presence of 9, which becomes a user input in nondimensional form along with the Reynolds and Prandtl numbers. If nondimensionalized by the group ir2f/Zref, it becomes awkward to compute. Thus, in this equation T and 9 are chosen to be nondimensionalized by the reference temperature
Tref,
A ='= _ , '(2.25) Aref T ref Tref where the prime indicates a different nondimensionalization from that used in the gov- erning equations. Substitution into Sutherland's law gives,
yA T'/ (2.26) pAref Tref T'O+ S' where pref = 1 and Tr'ef = 1. The relation of Tref and T' to the rest of the variables is by the choice of reference conditions. In this thesis, inlet plenum or stagnation conditions are used, where eref is the stagnation speed of sound,
Pref = O, Cref = o, Tref = , (2.27) iref = i, kref = 0, CGref = O
35 The speed of sound is given by,
=C = . (2.28)
Therefore, using the perfect gas relation the reference temperature becomes,
-21 -2 Tref = Ce = Iref (2.29) ,yR Y - 12Eref
and the nondimensional temperature becomes, h T'= (-y - 1)T = (-I - 1). (2.30) cp
2.1.3 Turbulence Modeling
The Navier-Stokes equations in their present form admit turbulent behavior for high Reynolds number flows. Turbulence is unsteady stochastic flow characterized by a wide
range of length scales, and resolution of all these scales for practical geometries is be-
yond present day computing resources. To make the problem of turbulence manageable, the governing equations are statistically averaged, and mean rather than instantaneous
quantities solved for. The resulting equation set is known as the Reynolds averaged (or
more precisely, mass or Favre averaged for compressible flows) Navier-Stokes equations.
They are modified from the original equations by apparent stresses, resulting from the averaging of nonlinear terms, which must be modeled through empirical correlations.
Here the Boussinesq approximation is used; these apparent stresses are assumed pro- portional to the mean strain rates and temperature gradients, similar to the molecular viscosity (2.6) and heat transfer (2.7). The essence of the approximation is a redefinition
of A and k, A = JI + Pt, k = k, + kt, (2.31)
where pj and k are the laminar or molecular viscosity and heat conduction coefficients used previously, and At and kt are their turbulent counterparts. The turbulent Prandtl number,
Prt = AtpCP (2.32) kt
36 is assumed constant; the value Prt = 0.9 is used here. The resulting Reynolds averaged
shear stress tensor and heat conduction are given by,
2 1 u 8v\ = -(pt + lt) 2 -- , (2.33a) 3 ax ay
ryy= (ML+jp) 2 gy )' (2.33b)
5+ -9 , (2.33c) 'r, = (Al + At) -
( p_ [ 8H au av1 qy = *P r+ Pr-- )a oy - Uu ay - v ay]. (2.33e)
In future equations if the ( ) subscript is missing, the molecular values are assumed. Section 4.9 describes the model used for the turbulent or eddy viscosity coefficient.
2.2 Asymptotic Analysis
Aerodynamic problems are often characterized by high Reynolds numbers (Re
106). Hence, they may be mathematically classified as singular perturbation problems,
because the highest order derivatives are multiplied by a small parameter. Solutions
exhibit "boundary layer" behavior where the importance of the viscous terms (highest
order derivatives) is confined to thin regions near boundaries. Within these layers the
viscous terms become the same order as the inviscid convection terms, and the solution
gradients become large. Outside these layers, viscous effects are negligible, and solution
gradients are fairly small. Furthermore, certain viscous terms typically dominate over
others within these boundary layers.
Since this discovery by Prandtl in 1904, a great deal of research effort has been
expended deriving and solving simplified versions of the Navier-Stokes equations. In this section some of this analysis is repeated to derive simplified equation sets.
37 2.2.1 Euler Equations
Consider the flow outside of the boundary layers. The solution gradients are small, thus the viscous stresses are negligible compared to the inviscid terms. Formally, this can be analyzed by taking the limit of Re -+ oo. The resulting equations are known as the Euler equations: aU OF OG + -- + a= at ax ay 0, (2.34a) or
df UdA+ (F+ G) -d 8 =o, (2.34b) where P Pu ( pv
2 U = P F P + = . (2.34c)
kpEj puH ) pvH j
This is a system of first order equations, whereas the full Navier-Stokes equations are a system of second order equations. A consequence of this is loss of boundary conditions, in particular the no-slip and heat transfer constraints at a wall. Only the boundary condition of no mass flux through the wall may be retained,
Solid wall: (us + VJ) A = 0, (2.35) where n is the normal to the wall; this is also referred to as a slip condition. Neglecting shear stress and heat transfer, boundary conditions far from the body (i.e freestream conditions) are unchanged from the full Navier-Stokes equations. They are treated in detail in Section 2.4.
2.2.2 Thin-Shear-Layer Navier-Stokes Equations
For many geometries, properties within a boundary layer are changing slowly in the streamwise direction even though variations normal to the wall are large. Thus, viscous stresses normal to the wall dominate those in the streamwise direction. This
38 behavior is better illustrated by considering steady incompressible laminar flow over a
flat plate as shown in Figure 2.2. At the leading edge of the plate the flow stagnates, and a boundary layer of thickness 3(Y) (dimensional) begins to grow. Far from the plate
the flow is undisturbed with velocity U, parallel to the plate and total enthalpy Hm. In the immediate vicinity of the leading edge, the full Navier-Stokes equations must be used because streamwise changes are large. However, streamwise changes are small away from the leading edge, allowing an order of magnitude analysis.
U 0
U
Figure 2.2: Laminar Boundary Layer on a Flat Plate
Assume p, p and k are constant or negligibly changing and order 0(1); thus, they can be dropped from the following analysis. At the stagnation point u = 0, and far
downstream (outside the boundary layer) u approaches Uoo. Therefore, one may choose
some length 7 over which u changes by order 0(1). From the mean value theorem, this gives au/ax 0(1). At the plate surface v = 0 and approaches some nonzero value at
the edge of the boundary layer. From the incompressible mass equation, this nonzero value must be order O(9/z) if av/ay is to balance au/ax. Thus, if 6 = IL is the
nondimensional thickness of the boundary layer, then the following general assumptions
can be made: U ~ 0(i), V ~ O(6), H ~0(1), (2.36)
and (1)a, O(1). (2.37) a a Gx ta y Given this information, the magnitude of each term in the Navier-Stokes equations may
39 be estimated (these estimates are displayed under each term). au Ow, mass: - + - = 0 Ox ay (2.38a) 1 1
au &u lap v [4a2U a 2u 1 a2 v1 x-momentum: - + V + -- = -- -- + 23+8b axy P a Re 3 y 3zozay] (2.38b) 1 1 ? 1 1/62 1
ov CV lap v [02v 4 0 2v 1 a2U y-momentum: U + V - + -- = - + - y2+ a oy p ay Re ax 2 3 ay 3+ xiyj (2.38c) 6 6 ? 6 1/6 1/6 OH OH v 1 a2H a u 8 av1 energy: u-+v-= Ox ay Rel.PrlaX2 (U) _-(X-a Ox5']a 1 1 1 1 62 1 rO2H a au ( vl + P -yA (2.38d) 1/62 1/62 1
+4 8a ( au) 2 6 U v +0 a { V U a (V v) +3 ax axz 3 c8x 49y CIX ( y + ax- ax 1 1 1 62 + au a ( )+ 4 o v 2 au
1/62 1 1 1
From the x-momentum equation the boundary layer thickness may be estimated. For the largest of the viscous terms to balance the inviscid terms, 6 0O(1/vf/N).
The incompressible Thin-Shear-Layer (TSL) Navier-Stokes equations result if all terms up to order 0(6) are retained. All inviscid terms are retained including the pressure gradients whose magnitudes are unclear. Assuming Pr 00(1), this results in neglecting some normal and all streamwise viscous derivatives, with the exception of the y-momentum equation. In the y-momentum equation, the viscous terms 1/Re a 2v/y 2 and 1/Re a2u/axay are order 0(6), the same order as the inviscid convective terms.
This leaves two options: either retaining or neglecting both terms. Retaining both is the safe assumption, but this results in only one streamwise viscous stress in the entire equation set. Furthermore, experience shows that they are generally smaller than
40 predicted by the above analysis. Thus, the choice made in this thesis is to neglect both for the sake of efficiency, leaving only the inviscid y-momentum equation.
A similar asymptotic analysis has been performed for the incompressible Reynolds averaged equations in Ref. [6]. Using the Boussinesq approximation, the same terms are retained as in the present laminar analysis. Therefore, only the results will be given here. In addition, the presence of compressibility should not change the overall picture. Thus, the compressible, Reynolds averaged form of the Thin-Shear-Layer Navier-Stokes equations are, au 8F OG 1as (2.39a) 8t-- + --8z + --8iyy = ----Re 8y' (2.y9 where
Pp pu PVp 0)
U = , F = ,+P G =, S = . (2.39b) PV PUV pv 2 + p 0
pE puH , pvH q U, F and G are the same inviscid conservation and flux vectors as the full Navier-Stokes equations (and the Euler equations); and S is the redefined viscous vector where, au - =(+'0 Yg-, T =P + 1 (2.39c) q= -- + p 1-- - + (1--- U-. (Pr Prt ) y Pr Prt ay*
The discussion on the y-momentum equation has been belabored because many references (e.g. [6]) retain 8 2 v/8y 2 but neglect a2u/axay when writing the compressible version of the TSL equations. This happens for the following reason: For incompressible flow the mass equation may be used to simplify the y-momentum viscous terms, giving,
2 ar a-y ( ar2 v 82(2 + 8x ay (x2 + 2 (incompressible). (2.40)
An order of magnitude analysis retains only the 8 2 v/8y 2 , and the result is written directly for the compressible equations without recalling that the process leading to (2.40) cannot be done when compressibility is allowed.
41 2.2.3 Prandtl Boundary Layer Equations
A slightly more simplified equation set is obtained if only those terms of order 0(1)
are retained in each equation. These Prandtl Boundary Layer equations differ from the
above only in the y-momentum equation. Here, only the pressure gradient is retained,
y-momentum: - 0. (2.41) ay
Hence, the pressure is assumed constant through the boundary layer and known from the outer inviscid flow.
In practice, solutions of the TSL equations (2.39) generally indicate that pressure
is nearly constant through the boundary layer, and the other inviscid terms in the y- momentum equation are small. However, this is a significant simplification for unsteady flows as well as steady supersonic edge flows. Removal of the unsteady and convective
terms in the y-momentum equations destroys acoustic wave propagation within the boundary layer. By fixing the pressure within the boundary layer to be that of the
outer inviscid flow, all cross-stream waves are eliminated, and all streamwise waves are dominated by those in the outer inviscid flow. Thus, for supersonic edge flow all information propagates downstream. An application where this limitation is quite important is shock/boundary layer interaction. The Boundary Layer equations do not permit any information to propagate ahead of the shock even though the flow is subsonic near the wall. Thus, they fail to predict much of the physics of the interaction. Another
difference from the TSL equations is that the Prandtl Boundary Layer equations do not asymptote to the Euler equations far from the wall, where viscous terms become
negligible. This raises questions for coupling to an outer inviscid solution modeled by the Euler equations.
For these reasons, the Prandtl Boundary Layer equations are not used in this thesis, except for initialization. They have been presented here mainly to point out differences from the TSL Navier-Stokes equations.
42 2.3 Composite Solution: Defect Formulation
As identified in the previous section, the flow within the boundary layer or vis- cous region is described by the TSL Navier-Stokes equations (2.39). Far from the wall, these equations asymptote to the Euler equations (2.34) describing the outer inviscid flow. Because of this, the TSL equations could be used to describe the entire flowfield, but this is computationally inefficient since the viscous terms are known to be negligi- ble outside the boundary layer. This section presents an alternate formulation of the problem, involving a composite solution using both the Euler and TSL Navier-Stokes equations. The subject of viscous/inviscid coupling is treated in greater detail in Chap- ter 5. The analytic formulation of the problem is presented here, since it is needed for the development of the TSL algorithm in Chapter 4.
Probably the most obvious means of obtaining a composite solution is to solve the TSL equations from the wall to the edge of the boundary layer and patch to a solution of the Euler equations in the outer flow. In this approach, the domain is divided into two regions and the solution in each represented by different equations sets. The final solution is then a patch of the viscous and inviscid solutions. For reasons explained in Chapter 1, the flowfield is described instead by the Defect formulation of
Le Balleur [50,51] in this thesis. In this approach, two equation sets, both applicable throughout the entire domain, are used to describe the flowfield; the composite solution is the sum of the two solutions.
As noted earlier, the entire flowfield is described by the TSL Navier-Stokes equations (2.39), aU 8F 8G I as +--+- - R - =(2.42) at ax 8 Re ay' along with the wall boundary conditions (2.16),
y=0 : u=0, v=0, q=0, (2.43) and appropriate conditions far from the wall. In the Defect formulation these equations are split into an inviscid contribution described by the Euler equations (2.44a) and a
43 viscous contribution described by the Defect equations (2.44b),
at +U'zax'y -- +' - = 0) (2.44a) a a A 1 as(.4b --U -U+-(F - F) + -( - G) + =0.(.4b at( - xU)+Y5y Re y Far from the wall, the viscous solution U asymptotes to the inviscid solution U,
U -+ as y -+ oo. (2.45)
Outside the boundary layer, the Defect is zero, leaving only the Euler equations.
Within the boundary layer, Eq. (2.44a) results in an inviscid solution which has no physical significance. Le Balleur refers to this as the pseudo-inviscid solution. It does not satisfy the inviscid solid wall boundary conditions (2.35); instead, the fluxes at the wall are driven by the viscous contribution. They may be obtained by integrating the
Defect equations across the boundary layer, subject to the constraints (2.35) and (2.45), A 1 a8 0Aa 0 G = G. - S.+- (F - F)dy+ ( - U)dy. (2.46) Re + o at o These are referred to as wall transpiration fluxes and are treated in greater detail in
Chapter 5.
Analytically, this approach is no different if the Defect equations (2.44b) are replaced by the full TSL equations within the boundary layer. The real advantage of the Defect equations becomes apparent when the equations are solved numerically. When different algorithms are used to solve the Euler and TSL Navier-Stokes equations, the viscous solution U and the inviscid solution U may not match outside the boundary layer. The source of this mismatch is differing truncation error between the two algorithms. On the other hand, if the viscous algorithm is used to solve the Defect equations within the boundary layer, then this difference in truncation error is subtracted off. This allows
the viscous and inviscid solutions to match each other. Therefore, solution of the Defect
equations may be viewed as an elegant means of truncation error manipulation. To see this, the truncation error is analyzed using operator notation.
Let the operator LTSL denote the analytic TSL Navier-Stokes equations (2.39) and
LEUL the analytic Euler equations (2.44a). Let U be the analytic solution of the TSL
44 equations with no-slip boundary conditions, and U be the associated pseudo-inviscid solution,
LTSL(U) = 0, LEUL(R) = 0. (2.47)
By construction, U asymptotes to U far from the wall. Similarly, assume two different numerical operators, LhIDEUL and LhTSL' for the solution of the Euler and TSL equations, respectively, and let U" and Uh be their solutions with appropriate boundary conditions,
Lrs,(U ) = 0, LEUU) 048)
The discrete viscous solution Uh will asymptote to some inviscid solution far from the wall, and it is desired that this inviscid solution be U". However, this is not necessarily the case. from the wall, the analytic solution U of the TSL equations is also a solution of the Euler equations, and vice versa,
LEUL(U) -+ 0 as y -+ oo, (2.49)
LTS,($) = 0. (2.50)
On the other hand, the exact solution Uh of the discretized TSL operator does not asymptote to an exact solution of the discretized Euler operator,
L h(Uh) -* 4h 0 as y -+ oo. (2.51)
Nor is the inviscid solution ih an exact solution of the discretized TSL operator,
LrsL(U") = # 0. (2.52)
As stated, the reason is differing truncation errors between the two numerical operators or algorithms. Because of this, it is not possible for Uh to match U outside the boundary layer, except in the limit of infinite grid resolution. A computational example of this is shown in Fig. 6.29.
Fortunately, this situation can be rectified. The key is to construct an altered discrete viscous operator which allows Uh (the inviscid solution) as an exact solution. This is simply done by subtracting off the error incurred when the inviscid solution Uh
45 is substituted into the old viscous operator. This altered viscous operator, called the discrete Defect operator, is given by,
LDEF TSL( L (Uh) = 0. (2.53)
It is satisfied by the altered viscous solution Uh. Far from the wall, U will asymptote
to an inviscid solution, and by construction this inviscid solution will be Uh. Hence, solution of the Defect equations is a means of subtracting off the undesired difference
in truncation error between the discretized Euler and TSL operators. A computational
example of this is also shown in Fig. 6.30.
This conclusion can alternately be obtained by noting that U - U = 0 is both an
analytic solution of the Defect equations (2.44b) and an exact solution of any properly
discretized form of the Defect equations.
Note that the original discrete viscous solution Uh is not a solution of the discrete
Defect operator . Nor is the altered viscous solution Uh an exact solution of the
discrete TSL operator L -
2.4 Farfield Boundary Conditions
In external aerodynamics, the flow approaches some uniform freestream conditions
far from the body. Similarly, in internal aerodynamics, the flow far upstream and
downstream approaches some known uniform or stagnation conditions. In numerical
calculations of such flows, the domain must be truncated some finite distance from the
region of interest, as depicted in Figure 2.3. For reasons of efficiency, these truncated
edges of the domain, referred to as farfield boundaries, should be placed relatively close
to the region of interest. This raises the question of appropriate boundary conditions to
apply. The solution at the farfield boundaries is not known a priori since it is affected
by the flow inside the computational domain; thus, boundary conditions must evolve
with the calculation. Ideally, these boundary conditions should result in a physically
relevant solution and not cause non-physical reflection of waves passing out of the do-
main. Hedstrom [40] has shown this can be done by performing a characteristic analysis
46 of the one-dimensional (1-D) equations normal to the boundary. The number and type of conditions specified is determined by those characteristics entering the domain.
Inlet Exit
NJJ'11 ----7 7 ......
Figure 2.3: Inlet and Exit Farfield Boundaries for a Typical Geometry
This section derives the characteristics for the 1-D Euler equations and discusses their use in developing boundary conditions. Within the context of a 1-D flow assump- tion, the analysis is also appropriate in a boundary layer if the TSL equations are used, since no streamwise viscous terms are present.
Consider the flow at the inlet or exit station (assumed at constant z) where variations in y are assumed zero. Defining the substantial or particle derivative as, D = a + - (2.54) Dt at a'(4
the Euler equations (2.34) may be written in the form, Dp au_ mass: -- + p- (2.55a) Dt z = 0, Du 1 ap z-mom: D- + -= 0, (2.55b) Dt paz Dv y-mom: = 0, (2.55c) DE uap pau energy: + -+ -- = 0. (2.55d) Dt p X p az
The y-momentum equation is already in the desired form, stating that v is constant
along the characteristic dz/dt = u; this may also be considered a vorticity wave. The
second desired relation involves the change in entropy s, given by the second and third
laws of thermodynamics, T ds = de + p d(1/p). (2.56)
47 Rewritten in terms of the substantial derivatives and using (2.9), the change in entropy along a particle path is,
D8 DE Du pDp_ T -- - u- - --- = 0 (2.57) Dt Dt Dt p2 Dt where the right-hand-side vanishes by substitution from the Euler equations (2.55).
Thus, entropy is constant along a particle path (dz/dt = u characteristic). Entropy itself is cumbersome to work with, but by using the perfect gas relations (2.8) and (2.9), changes in entropy can be shown to be proportional to changes in p/p'f. Hereafter, this quantity will be used in place of entropy.
The nonlinear Euler equations do not, in fact, decouple into characteristic form due to entropy dependence in the final two relations. But the equations may approximately be decoupled by assuming the flow is locally isentropic (which is more restrictive than
(2.57)) in these final two relations. If the process is isentropic, then changes in the speed of sound (2.28) can be written in terms of either p or p alone,
1 1 1 - dc = - -dp - -dp) dp= - dp. (2.58) 2c (p P 2pc 2-pc
Substitution into the Dp/Dt term in the mass equation gives, 2 Dc 8u c+ C - 1 Dt 8zau = 0, (2.59) and substitution into the Op/8z term in the z-momentum equation gives, Dts 2 8c D+ 2C = 0. (2.60) Dt -- 1 ax
Adding and subtracting these equations gives the final two relations,
- + (u c) -] U 2 = 0. (2.61)
These relations define the Riemann invariants Ji which are constant along dz/dt = u c characteristics. Combining (2.55c), (2.57) and (2.61) results in the characteristic form,
--t +A axa - , (2.62a)
48 where 2 U+ C
2 U - 2C) A=diag(u+c,u-c,t,u), C= ~ 7 - . (2.62b)
p'Y V
This information is used to construct the solution at the inlet and exit planes. The four characteristics are evaluated based on information extrapolated from the domain interior and separated into right and left-moving waves. For supersonic flow (u > c) all are right-moving, and for subsonic flow (u < c) three are right-moving and one left- moving. Those characteristic variables corresponding to characteristics entering the domain are used to enforce specified boundary conditions.
2.4.1 Inlet Boundary Conditions
At the inlet, right-moving characteristics enter the domain. Thus, four boundary conditions are specified for supersonic inflow, and three are specified for subsonic in- flow. For supersonic flow, the inlet solution is completely evaluated based on the inlet freestream conditions, M> 1 U = Uspec. (2.63)
For subsonic inflow, entropy, total enthalpy and v velocity are specified. The Riemann invariant J- corresponding to the characteristic u - c is evaluated based on information extrapolated from the interior,
M< 1 8 s Sepec H = Hapec V = VsPec (2.64)
J~ = Jetrap J+ = J+( Hapec, Vspec, Jextrap)- This last relation is obtained by first writing u and c in terms of the Riemann invariants,
U = - [J+ + J-] , C = [J+ - J]. 2 4 (2.65)
49 Next, total enthalpy is written in terms of u, v and c using (2.13) and (2.28), c2 u2 +v 2 H = -+ .(2.66) -Y - 1 2 Substituting for u and c and solving for J+ gives the desired result,
J+ = y-3)J-~ + 4 (_Y+ 1) H - 1V2] 'Y- 1 (J-)2 . (2.67) y1+11 U V L 2J 2''J- Once the characteristic variables are evaluated, the primitive variables can be recovered as follows: u and c are obtained from (2.65); density is obtained by eliminating pressure
from entropy (8 = p/p7 ) using (2.28),
p = 2 ],1 ;(2.68)
and pressure is given by (2.13).
2.4.2 Exit Boundary Conditions
At the exit, right-moving characteristics leave the domain. Thus, no boundary conditions are specified for supersonic outflow, and only one is specified for subsonic outflow. For supersonic flow, the exit solution is obtained completely by extrapolation,
M> 1 : U = Uextrap. (2.69)
For subsonic flow at the exit, pressure is specified. Entropy, v velocity and the Riemann invariant J+ are extrapolated,
M<1 : P = p(O)pe 8 = 8extrap V = Vextrap (2.70)
J+ = xtrap
J~ = J~(p(t).pe, 8 extrap, Je+xtrap).
In the present work the specified pressure is allowed to be a known function of time.
The last relation is obtained by first writing c in terms of p and a,
VP(2.71)-(,-). Y (./p)
50 Substituting for c from (2.65) and solving for J- gives,
J~ = J+ _ _Y _ -- 1 )Y( (q)'. (2.72)
The primitive variables are recovered from the characteristic variables similar to the inlet: u and c from (2.65), p from (2.68), H from (2.66), and pressure from (2.13).
Specific details of the implementation of these farfield boundary conditions are pro- vided in Sections 3.6 and 4.7.3. Two special cases of transient reverse flow within a boundary layer at the inlet and exit planes are discussed in Section 4.7.3.
2.5 Parametric Vector
The inviscid fluxes F and G are nonlinear functions of the conservation variables U, (pu) 2 (Pu (pU) + P 2 p p F(U) =U pu + (2.73a) puv (pu) (pv) p puH (pi) [PE + p]
(pv) (pu)(pv) puv I p 2 (2.73b) G P=2 + (pto) p pvpv +pIH (p)[pE+p] / p where pressure is, 1 (pU)2 + (pV)2 P =( - 1l) pE -. (2.73c) Ip 2 P I
51 An alternate and useful convention is to introduce Roe's parametric vector [691,
p1/ 2 \ , p1/2ts I w21 W = 2 W (2.74) p 1/2 V W3 p112H j W41 Roe's parametric vector has the property that all of the components of U, F and G may be written as quadratic functions of the components of W. For example, the mass flux pu is given by,
Pu = (p1/2)(pl/2U) = wIw 2, (2.75) and the pressure by,
' 1 r(p1/2)(P1/2H) - (/2 )2 + (p1/2V)2 (2.76) ' H)" - 2' ( /( J)
_'7-l [W1W4 - -1((W2)2+ (W3) 2)]
The remainder of the conservation and flux vectors are given by, P WJWi I
IL W1W3 Pu w&2 F(W) = = ( 2 W 2 +P , (2.77b)
puH w2W 4
PtLV tV2 W 3 pvv(W) ( W3W3W W4 (2.77c)
puv tv3 w2
Roe's parametric vector is used extensively in the Euler and Navier-Stokes algo- rithms developed in Chapters 3 and 4. An operation that is required in the Euler
52 algorithm is recovery of the parametric variables given the conservation variables. This operation can be done analytically starting with density and working towards total internal energy:
2 p =- Wi -. 1P1/2
PU = W1W2 W1 (2.78)
PV = W1W 3 W3 -3 P- W1
PE = 1WW 4 + y~IW2 + W W41pE+ -y 2- wi 2 W \2 +W32) I
Another operation is recovery of the derivativesi of the parametric variables given the
derivatives of the conservation variables. This caa also be performed analytically, as- suming U and W are known,
(p)x = 2w,(wl). (WOZ (P)z 2w 1
(pu)x = WI(W 2)z+ w2(w1). -+ (W 2 )2 =- [(Pu)x - W2(w1)] W1
(pv)z = w1 (w3 )z + ws(w 1 ). -+ (W )z = -[(Pv)x - W3(wi)] 3 W1
(pE)2 = 1[wi(w4).+ w4(wl)z] - (W4) - [4y(pE)z - W4(wi):
+ ' [W2 (W 2 )z + w3(Ws)z] - ( -1) (W2 (W 2 )z + W3(W3)x)] (2.79) where ( ). denotes a derivative.
Equations (2.77a-c) are a compact means of evaluating the vectors U, F and G. In Chapter 3 some notation for this functionality will be useful. In general, each component of U, F and G is a coefficient matrix multiplying the vector of all possible quadratic pairs wmwn. This is cumbersome, so the following convection is adopted to denote this functionality,
U(W), F(W), G(W) = W.Wn. (2.80)
Do not confuse this with a dot product notation.
53 Chapter 3 Euler Algorithm
In this chapter, the 2-D unsteady Euler algorithm is developed. Prior to the discus- sion of the algorithm, a short review of flux-splitting is first presented. The development of the algorithm begins with the discussion of a first order accurate flux-split scheme, similar to existing methods. This discretization is then extended to second order ac- curacy in a formulation where both solution averages and gradients are stored for each cell. In subsequent sections, equations are derived and discretized to obtain these av- erages and gradients. An explicit time integration and numerical stability analysis are then presented. Next, a the implementation of wall and farfield boundary conditions is discussed. The chapter ends with a CPU timing study of the present algorithm.
3.1 Objectives
The Euler algorithm is developed with two main objectives. The first is to capture unsteady shocks crisply and with minimal numerical oscillations. The second objective is to reduce numerical errors on nonsmooth grids by achieving grid independent second order accuracy for both steady and unsteady flows.
3.2 Flux-Splitting
The different versions of the Euler algorithm described in this chapter make use of flux-splitting, which is treated more or less as a "black box" in the development of the spatial and temporal discretizations. For this reason the flux-splitting formulation used is described in this section separate from the development of the rest of the algorithm.
54 In the Euler algorithm to be discussed, the flux is evaluated at points in the flow where the approximation to the solution is discontinuous (i.e., at faces). For one- dimensional flow the physically correct flux F is obtained from the exact solution to
Riemann's initial-value problem, 8U 8F U,., z > 0 -+ -X = 0, U(X, 0) =.(3.1) at ful Ugz< Physically, two solution states are separated by a membrane at z = 0. When the membrane bursts, the flow is described by the interaction of a contact discontinuity, an expansion fan and possibly a shock wave. The exact nonlinear nature of these interac- tions is utilized in schemes developed by Godunov [36] and van Leer [83]. These are computationally very expensive since the solution to Riemann's problem for arbitrary U, and U must be obtained iteratively. More efficient algorithms have been developed which are based on approximate solutions to Riemann's problem; several are described and compared by van Leer et al [85]. For example, two methods developed by Roe [69] and Osher [64] retain the essential characteristics of the motion of finite discontinuities
(i.e., shocks and contact discontinuities) but not that of expansion fans. When used in conjunction with first order upwind differencing, these formulations allow stationary shocks and contact discontinuities in 1-D flow.
A still more approximate class is known as flux-vector-splitting. These methods retain only the characteristics of shock motion by distinguishing between the domain of dependence for subsonic and supersonic flow. The flux at a given point is split into two contributions representing backward and forward (or left and right) moving waves. For supersonic flow only one of the components is present. Two widely used members of this class are the splitting techniques of Steger and Warming [77] and van Leer [84]. Steger and Warming split the fluxes based on the eigenvalues of the flux Jacobian 8F/8U. The resulting splitting has two main disadvantages. The first is that the split fluxes are not continuously differentiable at sonic and stagnation points, resulting in noticeable errors; and the second is that stationary shocks in 1-D are not allowed. An improvement is the splitting of van Leer which gives continuously differentiable split flux components and stationary shock structures. Van Leer's flux-splitting is used to evaluate fluxes in the present Euler algorithm and is described in more detail.
55 The flux-vector-splitting of van Leer starts by separating the flux F(U) into right moving (F+) and left moving (F-) waves,
F(U) = F+(U) + F-(U), (3.2) where the flux Jacobian aF+/8U has non-negative eigenvalues and aF-/8U has non- positive eigenvalues. These split fluxes are constructed to satisfy the following five restrictions:
1. Fi(U) and 8F+/8U must be continuous. 2. F (U) must reduce to F(U) for supersonic flow, that is
F+=F, F-=O if M>+1 F+=O, F-=F if M< -1
3. The components of F+ and F- must mimic the symmetry of F with respect to Mach number:
F+(M)= F~(-M) if Fk(M)= Fk(-M)
4. 8F*/8U must have one eigenvalue vanish for subsonic flow. 5. F (M), like F(M), must be a polynomial in M of the lowest possible degree.
The first and fourth restrictions constitute the major improvements over the splitting of
Steger and Warming. The fourth restriction makes F (U) degenerate for subsonic flows, making it possible to construct stationary shock structures. The resulting subsonic split fluxes are given by,
FM , = L(1 M)2,
F+Fmom = Fa 3 [(_ 7 - 1)Mly.c, 2 (3.3)
F ergy F*s 2 ( - 1)M 2]2 enry - 2a(-y02 - 1)
Since F+ represents right moving waves, proper domain of dependence is preserved if it is evaluated based on the left solution U1 . Likewise, F- is evaluated based on U,.. Hence, F = F+(UI) + F-(U). (3.4)
56 AA
U1 Ur
Figure 3.1: Flux-Splitting for a Discontinuous Solution in 2-D
Evaluation of the split flux vectors in this manner is sometimes referred to as a MUSCL scheme [7,83].
In two-dimensional flow, wave interaction becomes much more complex, and no analog of the Riemann problem exists. The only practical option is to assume a locally 1-D Riemann problem, such as that depicted in Fig. 3.1. Given a face oriented at some angle to the x-axis with a solution jump across the face, the flux per unit length H - A through the face is desired. Here A and n4 are the unit tangential and normal to the face, and H = Ft + Gj. Using van Leer's flux-vector-splitting, the splitting is based on
the component of the Mach number normal to the face. The normal Mach number Mn
and the tangential Mach number M, are defined by,
Mn = (VIC)- , M, = (VIC)- ', (3.5)
where q = us + vJ is the velocity vector. The 2-D split fluxes with face-normal momen-
tum components are given by,
2 (i A)mass = PC(l Mn)
H n ab ( - 1)Mn 2 (HH a - )n-nmom = (HJ )mas. [ L (3.6) (i - 94)s-mom = (iii - 9)maa. cM,
2 22 (H - )energy = ( -)m.c 2 - 1) +
The corresponding z and y-momentum components are obtained by rotating the coor-
57 dinates,
(1i - )x-mom = (N- -J ])s-mom[f+ ( )n-mom([- A](
(iii - )y-mom = (ii -n)s-mom[j - ] + (Ni - )n-momki - ].
Following the same approach as for 1-D, the split flux vectors are evaluated based on the left and right solutions,
=+(Ul) -A+~ n. (3.8) H A- r
The third restriction listed above removes any possible ambiguity in the flux calculation
caused by choosing the normal vector A opposite to that depicted in Fig. 3.1.
3.3 First Order Spatial Discretization
The first order spatial discretization is described initially to introduce general nota- tion and concepts and help distinguish what is necessary to extend the scheme to second
order accuracy. In practice this first order scheme is not used because it has very poor accuracy.
The domain of interest, such as the interior volume of a duct or the external volume
of an airfoil, is subdivided into a large number of control volumes or cells. Each cell is an n-sided polygon, where typically the polygons are either triangles or quadrilaterals.
All test cases presented in this thesis contain quadrilateral cells, but the analysis of this chapter is valid regardless of the number of sides or faces per cell.
Figure 3.2 shows a typical layout of quadrilateral cells. Within each cell the solution is assumed constant, U(x, y) = U + O(A), where U is some average over the cell, or alternatively, the solution at the center of the cell; A is some dimension of the cell (either Ax or Ay). This constant solution approximation within cells is illustrated for the 1-D case in Fig. 3.3. Substitution of this piecewise constant approximation into the
integrated Euler equations (2.34) over each cell results in the semi-discrete equations, d _ (AU) + A -idAs = 0, H = FA+ Gj, (3.9) faces
58 where A is the cell area, Ni.- is the flux and As the length of each face of the cell. The value of the flux vector on each face is obtained using flux-splitting, where the left and right interpolated solutions are the cell averages in the two adjacent cells. In supersonic flow, it reduces to first order upwind differencing.
0 0 0 o Solution: U, UX, 0 Uy 0 Geometry: 0 z, y
Figure 3.2: Euler Grid with Solution and Geometry Storage Locations
U(z)
X Figure 3.3: Constant Solution Approximation Within Cells (First Order)
This scheme is only first order accurate because the interpolations to each face are only first order accurate; it is also highly dissipative as shown in Section 6.1.4. However, it has in its favor the fact that shocks can be captured without oscillations or overshoots using no additional dissipation operators. Using the van Leer split fluxes, steady-state shocks are captured in general with one or two internal cells [84].
Laid out in this manner, this scheme belongs to the class of cell-centered schemes
59 along with the central-difference scheme of Jameson, Schmidt and Turkel [43] and the flux-split scheme of Anderson, Thomas and van Leer [7]. In fact, this scheme is equiva- lent to the first order option in the scheme of Ref. [7]. Alternatively, central-differencing results if the flux at each face is based on the average of the cell center solutions of the two adjacent cells (the scheme then becomes second order accurate for sufficiently smooth grids).
3.4 Second Order Spatial Discretization
The second order extension of the above scheme originated from a Godunov-type scheme by van Leer [82] for the linear 1-D wave equation. In this section it is extended to the Euler equations in two dimensions. Within each cell the solution is approximated by a linear Taylor series expansion,
U(z,y) = U + (z-X)Ux + (y- y)UY + O(A 2 ) (3.10) where U is some average over the cell and (Ux, U~) is some average gradient over the cell. The cell centroid (z, y,) is that point about which the first area moment is zero; it is defined as, ff xdA ff dA XC =, ye (3.11) dA ff eA Y J This linear solution approximation within cells is depicted in Fig. 3.4 for the 1-D case.
Expanding about the centroid of the cell has the property that the gradient terms vanish when U(z, y) is integrated over the cell area. Thus, U is the integrated solution average over the cell, ffU dA = AU+ 0(d 4 ). (3.12)
This integral appears in the time term of the Euler equations. Hence, when the linear approximation (3.10) is substituted into the Euler equations, the resulting semi-discrete equations have the same form as (3.9).
60 U(X)
I I I I I I I I I I I I I I
I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I zc z Figure 3.4: Linear Solution Approximation Within Cells (Second Order)
If U is known to at least second order accuracy and the gradients (UX, Uy) known to at least first order, then the approximation of U(z, y) at any point within the cell is second order accurate. This is true, in particular, for the interpolation of U to points along each face of the cell. Therefore, the line integral of the fluxes around the cell can be evaluated to second order truncation error regardless of cell shape. In other words, with both average and gradient information, the spatial discretization of the semi-discrete equations (3.9) can be made second order accurate for nonsmooth grids. b 8
n
eU (U, Ux, UU U at (X, y0/)i
U, UX,- U-Y), at (XC, yc),
a Figure 3.5: Interpolation for Midpoint Rule Evaluation of Fluxes
The simplest means of evaluating the flux on each face is by the midpoint rule. Using the average and gradient information in the two adjacent cells, left and right solutions
61 are interpolated to the face midpoint as shown in Fig. 3.5,
(U) = U1 + (Xmn-c)UX + (ym-yc)UY,
(U), = Ur + (,-nXcr)Uxr + (ym-yc,)Ur,, where (xrn, ym) is the midpoint of the face, (x0 , yc) is the centroid of the left cell, and
(Zc, yC), is the centroid of the right cell. The flux through the face is then,
3 f - A ds = Hm(Ui, Ur) -iA8 + 0(A ), (3.14) where Hm(Ul, U,) is the flux at the midpoint of the face calculated with the split flux formulas (3.6). In later sections of this chapter, methods other than the midpoint rule will be considered for calculating the flux along each face.
3.4.1 Gradient Equations
The semi-discrete equations (3.9), hereafter referred to as the average equations, describe the evolution of the cell averages U. The cell gradients (Ux, Uy) must also be calculated. Analytically, U, Ux and Uy in the Taylor series expansion (3.10) should be equated with the solution and its derivatives at the centroid,
U= U(ze, ye), Ux = , U =a . (3.15) Ix a
In practice, UX and Uy need only satisfy this requirement to first order accuracy, leaving much latitude in the choice of computational methods. Those considered vary greatly in computational effort, robustness and overall solution quality.
The simplest method for computing the gradients is by central-differencing of the cell averages U. This makes use of the divergence theorem, 1 ff dA +1 Ud U ~X =(3.16)AA UY 4 1f dA= IUdx.
In the evaluation of the line integral, the solution on each face is taken as the average of U on the two adjacent cells. Thus,
U = + ~ (U, + U1)f1Ay, U = - 2(U,. + U1) Azj, (3.17)
62 where Ax and Ay are defined consistent with counter-clockwise integration. This method has the disadvantage of allowing upstream propagation of information in su- personic flow; this counteracts the use of flux-splitting in the average equations. Fur- thermore, this method for calculating UX and Uy requires smoothing since it admits sawtooth modes in U. Also some smoothing is probably required near shocks.
A second option is to update the gradients by minimizing the solution jump at faces.
For a uniform grid in 1-D (a 2-D version was not considered), the update formula becomes,
n+ 1 (- Ax n (-n +Ax-n -x U+ - Ux + i ~ x- (3.18) Ax 2k" 1 2 1
For steady flows, this may be a simple and efficient technique, but like central differ- encing it allows upstream propagation in supersonic flow. It may also need additional smoothing or limiting near shocks.
The method chosen to calculate the gradients in this thesis is to evolve them in time along with the cell averages using the first moments of the Euler equations. This method is computationally intensive compared to the previous two, but if properly discretized this method yields an algorithm possessing several desirable and unique features. For example, the gradients have the same domain of dependence as the averages. Thus, no information propagates upstream in supersonic flow. In addition, shocks are cap- tured without oscillation in one cell using no added smoothing or limiting (although an overshoot in the gradient at the shock does occur). Furthermore, the algorithm has very little dissipation, resulting in high accuracy and good propagation of unsteady weak acoustic waves. Finally, no unconstrained modes exist; thus, no explicit back- ground smoothing (along with user input knobs and additional nonphysical boundary conditions) is required.
The key to the present method is reinterpretation of the linear approximation (3.10) as a truncated expansion in terms of Legendre polynomials, 0o 00 U(X, y) = E E Umn Pmn(Z, y), (3.19) m=O n=O
63 where Pmn(x, y) are the generalized 2-D Legendre polynomials (m and n are the expo-
nents of z and y, respectively, in the highest order term in each polynomial, and m + n is the order of each polynomial); the first three polynomials in this series are,
Poo(X, y) = 1, Pio(z, y) = z - z., Poi(z, y) = y - yC, (3.20)
and in the notation of (3.10), U = Uoo, Ux = U1j and Uy = U01 . This series is identical to a Taylor series expansion up to and including linear terms. The Legendre polynomials satisfy the following orthogonality relation:
ffPab(Z, Y) Pmn(X, y) dA = 0, a + b 0 m + n (3.21)
In words, the product of two polynomials of different order integrated over the control
volume is zero. As is easily seen, Poo, P10 and Po0 satisfy this constraint. The next order polynomials are given by,
P o(z, y) 2 2 = (zX-z) + a20(z-zc) + b2o(y-ye) + C20
Po2(z, Y) = (y-y)2 + ao2 (z-z) + bo2(y-ye) + co 2 (3.22)
P1(x, y) = (Z-Zc)(y-y) + ai(x-x,) + bi1(y-yC) + C11, where the constants a, b and c are determined from orthogonality with Poo, P10 and
P0 1 . For the present analysis, their values are not important, only the fact that they can always be determined from the geometry of the cell.
Interpretation of (3.10) as a series in Legendre polynomials rather than a Taylor series expansion about (zr, yC) affects the definition and hence calculation of the coeffi- cients U, UX and Uy. In this sense, the present algorithm is a departure from typical finite difference or finite volume methods. As an expansion in Legendre polynomials, the coefficients are determined using the orthogonality relations (3.21). U is the average solution over the control volume, and UX and Uy are given by the first moments of the solution about the centroid.
JU dA = f[U + (X- ze)UX + (y - Y)Ur + - ]dA = AU, (3.23a)
ff(X- ze) U dA = ff(X - z) [U + (X - XC) Ux + (y - yC) UY + -- -1 dA
64 = AzzUx + AzUy, (3.23b)
ff(y - yc)U dA = ff(y - y) [U + (z - XC)Ux + (y - yO)U + ... ]dA
= AzyUx + AyyUy, (3.23c) where A is the cell area, and A,., A., and A., are the second area moments about the cell centroid,
A = ff (z- x)2 dA, A., ff (y yc)2 dA, A2. = (x -x) (y - y) dA. (3.24)
Appendix A gives metric formulas for cells of arbitrary shape, including the area, first and second area moments, and location of the cell centroid.
As noted by van Leer [82] for the 1-D case, equating UX and Uy to the first moments of the solution results in a linear approximation which minimizes the integrated least square error over the cell. This is easily shown as follows: Define the integrated error as, & = U(z, y) - [U+ (z-z )Ux+ (y - y)] ]2dA. (3.25)
The stationary point of if is obtained by setting the partial derivatives with respect to
U, UX and Uy to zero,
= 2 ff {U(z, y) - [U + (z-xe)Ux + (y - y)Ury] dA = 0, at r e= -2 f (X-xc){U(xy) - [U + (z-z)Ux + (y-y)Uy dA = 0, (3.26)
ae = -2ff (y-iy){U(Xy) - [U + (-zc)Ux + (y-y)U7]} dA = 0.
This stationary point leads directly to the relations (3.23).
The next step in the process is to construct equations to evolve the gradients in time. Backtracking, U is defined as the average of the solution over the cell, and it is evolved in time using the integrated Euler equations. Likewise, from Eqs. (3.23b-c), UX and Ur are equated to the first moments of the solution over the cell; thus, they should be evolved in time using the first moments of the Euler equations.
For completeness, the equations for the averages (the integrated Euler equations) are given first before the moment equations are derived. The integral form of the Euler
65 equations is, SffUdA + fHA - ds = 0, H = FA + Gj. (3.27)
Upon substitution of the Legendre series into the time term, only the cell average U remains as shown in (3.23a). For the present, the flux integral will be left unaltered. Thus, the average equations become,
(3.28) AU] = -f - n^ d8 .
The first moments of the Euler equations about the z and y-axes are obtained by multiplying the Euler equations in differential form by (y - y,) and (z - z), respectively, and integrating over the cell volume. More specifically, for the first moment about the y-axis, (z-z) + V dA = 0. (3.29)
The time term can be rewritten as,
( - e) dA =d (z-xj)UdA, (3.30) JJ at -dt and the flux term as,
ff(z-zC)V - A dA = ffV - [(z - x)A] - F dA (3.31) = + (z-z )H.- n^d - ff FdA.
The first moments of the Euler equations are then,
dff(x- )U dA +f (x-zx)N-Ad - ff FdA = 0, dff (y-yc)UdA + (y-yc)H - Ads - GdA = 0.
Substitution of the Legendre series (3.10) into the time term then gives the equations for the evolution of the cell gradients,
AxUx + AxUy] = - f(z -x)- A ds + F dA, (3.33)
AyUx + AIUy] = - (y - y)NH - A ds + G dA.
Not surprisingly, there are different approaches to discretizing the right-hand sides of the average (3.28) and gradient (3.33) equations. Two will be discussed in detail. The
66 first is relatively inexpensive to compute and results in first order truncation error in the moment equations (but still second order for the average equations). This method was ultimately abandoned because it admits sawtooth modes in the gradients which must be smoothed. In addition, shocks cannot be captured without some form of gradient smoothing or limiting. The second method, the one used for all test cases presented in this thesis, is computationally more expensive and results in second order truncation error in the moment equations for steady flow (first order for unsteady). It is more rigorous and robust in that no unconstrained modes exist and no additional smoothing or limiters are needed near shocks.
3.4.2 Gradient Equations: First Order Discretization
In the first discretization method, the line integrals of the fluxes in (3.28) and (3.33) are evaluated along each face of the cell using the midpoint rule. It was previously described for the average equation fluxes in Section 3.4 (see Fig. 3.5),
J A- n ds = im- A8 + O(L8), J(x- z)H -fids = (Z -Zc)Hm - A8A + O(A 3 ), (3.34) J (y- yo)N -n ds = (ym-y,)N -1, A8, + O(A83 ).
Here (xm, y,,,) are the coordinates of the face midpoint, and n - A is the flux at the face midpoint calculated using flux splitting. Two points should be noted here. The first is that the split flux vector calculated for the average equations is also used for the gradient equations. Thus, the expensive operation of calculating the split fluxes need not be repeated for the calculation of the gradients. The second point is that evaluation of the "moment" fluxes by the midpoint rule results in (relative) first order truncation error even if H, - n were known exactly. To see this assume H - A is linear along the face, then the moment flux (z-z )NA - n is quadratic and the integral becomes, fn d8 = + 1X~~lfAA a O8( i! (X-XC)N- 0d= (XM-X A - n.6 + -A8A a , H- Alinear (3.35)
Therefore, the midpoint rule ignores the variation of 14 along the face for the moment fluxes.
67 The remaining area flux integrals in the moment equations are evaluated assuming
F(U) and G(U) are linear over the cell, 8F -8F - F(U(z, y)) = F(U) + (x- x)aU Ux + (y-y) j U, + 0(A), SCe (3.36) G(U(x,y)) = G(U) + (Z-Zc)gj Ux + (y-ye) Uy + 0(A 2).
Then their integrals become,
//F(U) dA = AF(U) + 0(A 4 ),
ffG(U)dA = AG(U) + O(A 4 ).
Hence, the final discretized average and gradient equations are,
d[AU] = - Hm-A8,
AxUx + AUy = - Z(Zm c)lfm - A8 + Aj(U), (3.38)
dA.Ux + AyyU = - j(yim-yc)Ifm -iAs + AG(U).
The x and y moment equations result in a coupled system for the cell gradients Ux and U.
As stated previously, the midpoint rule evaluation of the flux integrals results in first order truncation error in the moment equations and, hence, first order accurate gradients. But the average equations are still discretized to second order accuracy for nonsmooth grids, giving second order accurate averages. Thus, the solution within each cell is known to second order accuracy.
This scheme has two major drawbacks. The first is that in the vicinity of shocks, the gradients grow unbounded; therefore, they must be limited or turned off near shocks. Allmaras and Giles [51 present a technique for doing this. It will not be discussed further here, except to say that the technique requires a switch to sense shocks, and such switches typically also turn on near sonic lines and stagnation points. Hence, the technique degrades the solution unnecessarily in these regions.
A second drawback is that unconstrained sawtooth decoupled modes in the cell gra- dients are admitted by the discretized equations. Since the gradients are only first order,
68 this can be controlled by inexpensive constant-coefficient second difference smoothing.
These sawtooth modes result from neglecting the tangential variation of the flux along each face in the moment equations. Thus, they can also be constrained by a two-point integration along each face as discussed in Ref. [5]; this also gives second order truncation error at the expense of doubling the number of flux evaluations. However, if two-point integration is used and the discretization of other terms is done more carefully, then both the sawtooth modes and problems near shocks can be completely eliminated. This is the subject of the next section.
3.4.3 Gradient Equations: Second Order Discretization
There are two major points of departure between the discretization of the last section and that of the present section. The first is a two-point integration of fluxes and moment fluxes along each face to reduce truncation error and constrain sawtooth modes in the gradient equations. The second is more careful attention to nonlinear consistency which becomes important near shocks.
If the conservation variables U are assumed linear within each cell, then it is in- consistent to assume that F(U) and G(U) are also linear within cells and along faces, since they are nonlinear functions of U. Normally, this is of no concern since the error is second order for smooth flow. However, it makes a great deal of difference in the present scheme near shocks. The easiest method of maintaining nonlinear consistency is by the use of Roe's parametric vector W [69]. As discussed in Section 2.5, each of the components of U, F and G are quadratic functions of the components of W. Hence, the formulation begins by assuming that W rather than U is linear within each cell,
W(X, y) = W + (z-z,)Wx + (y-y,)WY + 0(d2 ) (3.39)
Since U(W), F(W) and G(W) are all quadratic functions, it will be necessary to expand integrals of quadratics in W. To simplify the analysis, this functionality will be denoted by, U(W), F(W), G(W) = WWn. (3.40)
69 Do not confuse this with a dot product notation; it refers to the form of the vectors given in (2.77).
Since U(x, y) is now represented by the product of two linear Legendre series, the time terms in the average and gradient equations must be reexamined. The discretized time term in the average equations (integrated Euler equations) becomes,
dff UdA= d fJ[Wm+ (x-x)Wx.+ (hyc)Wym+ ]*
[Wn + (x-ze)WXn + (y-y)WYn + --- ] dA ______(3.41) =id t AWmWn + AzzWXmWxn + AyyWymWyn
6 + Azy(WXmW-n + WmWXn)] + 0(A )
For exact W, Wx and Wy this relation is fourth order accurate (relative order since
A O(A 2 )) due to the orthogonality of the Legendre polynomials of differing order.
In practice, nonlinear consistency in the time terms is not important; therefore, the gradient terms may be neglected, giving second order accuracy (Az ~ 0(A')),
dJJU dA = +[AWmWn] + 0(A 4). (3.42)
This is equivalent to assuming U is linear within each cell. Similarly, the time terms in the gradient equations (moment of the Euler equations) become,
df ~~ d d (x-- x)U dA = (z-X) [Wm [Wn] dtJJX c dti dA
= d [A.. (WmWxn + WxmWn) (3.43)
+ Azy(WmWYn + WYmWn)] + 0(A 5 ),
(y- y)U dA = (Y--YO[Wm[ Wn] dA
dt_ dt dt LAzy (WmWxn + WXmWn) (3.44)
+ Ay, WmW yn + WYmWn)] + 0(A).
Again, nonlinear consistency is not important in these equations, but neglect of the next order terms (products of the gradient terms) results in first order truncation error. Thus,
70 for unsteady flow the gradients on each cell will be first order accurate. Equations (3.43) and (3.44) are equivalent to (3.38) if the gradient terms Ux and Uy are interpreted as,
Ux = W.Wxn + WXmW., Uy = W.Wyf + WYmW.. (3.45)
b
9+
(U, U (U+), at (xe, y,)j 0 (U, UX,_ UY),. (U,) (U at (z, yc)r
a Figure 3.6: Interpolation for Two-Point Gauss Quadrature of Fluxes
The flux terms in the average and gradient equations are,
nf ds, f(x-z,)H- fids, (y - y') - ds. (3.46)
Rewritten in terms of the parametric vector W, the first is quadratic along each face, and the second and third are cubics. These flux integrals are evaluated using a two-point Gauss quadrature [22] (which is exact for cubics) along each face. Figure 3.6 shows a typical face and the location of the Gauss points (X9,, y,:,), given by,
X, = [(z + za) (,b Xa) 2 F3(3.47) [ (ya ) where (Xa, Ya) and (xb, yb) are the endpoints of the face. Two-point Gauss quadrature consists of evaluating the integrand at the two Gauss points and averaging these con- tributions. Denoting the flux at the Gauss points as H', , then for a given cell and face
71 of that cell the integrals are,
fb -fids = (s nA~+ + (n .)4)
(z-X )1- Adsn)(- = (zg+-ze)(N -)g+ + (Z,_ -. -),, (3.48) (ya- e)N -A de= 2[(y+-)(i -.^)+ + (y,_- yC)( - A),,
where A = A - Ax/Asj is the outward normal of the face consistent with counter- clockwise integration.
In actual computation of these flux and moment flux integrals, each face is visited once and the following procedure used: The left and right solutions (Wj)l and (W,,), are interpolated to each of the Gauss points from the two adjacent cells using the linear approximation (3.39) within each cell (see Fig. 3.6),
M,1);= WI + (a-xg-XC)Wxi + (yg+-yC1)Wy1, (3.49) (Wg)r Wr + (Xg gxcr)W Xr + (yg a-ycr)WYr,
where (Xe, Yc) is the centroid of the left cell and (xe, y),. that of the right cell. Next, the split flux vectors are computed at each Gauss point based on these interpolated solutions. Finally, the flux and moment flux integrals are evaluated and distributed to the residuals of the adjacent cells. In the following equations, the face normal is taken to be the outward normal for the left cell (the outward normal for the right cell is equal and opposite).
~8= I + ( gth)g+ + ( '-. A)g_
Ri 1 = Rx+ [(g+-z)(i- a)+ + (z,_ -zc)(A - i),_
Ry = Rx; + -[(yx+i-i)(H-)+ + ( H ) sH ku-r = !F I + (3.50) R, = R, - (2 -n) + + (H -n)_
Rxr = Rxr - 2(z+- z)(H - ),+ + (,z9. ,)(H )
RY, = WY, - -[ (Yg+-,)( -r),Jg+ + (Yg - cr)( - A)_g
This results in an efficiently vectorized process.
72 In the previous method of discretization, the integrals of F and G over the cell area in the moment equations were evaluated assuming F and G are linear (3.37). Here, these integrals are evaluated in a nonlinearly consistent manner by expanding the quadratic functions in W,
JJFdA = ff[Wm + (X-Xc)WXm + (y-yc)Wym + O(A2),
[W. + (x-x)wx. + (y-y)Wy_ + O(A2)] dA
_(3.51) [_ ___ = [AWmWn + AxzWxmWXn + AYYWymWYn
+Ay(WXmWYn + WYmWXn)] + 0(A'6 )
and similarly for the integral of G over the cell area. For exact W, WX and Wy
this relation is fourth order accurate (relative order since A ~ 0(A 2 )) due to the orthogonality of the Legendre polynomials of differing order. Note that the gradient products (multiplied by the second area moments) are second order relative to the
average products (multiplied by the cell area).
The final semi-discrete system is obtained by replacing the terms in (3.28) and (3.33) by their discretized versions. The time derivative terms are replaced by (3.42), (3.43)
and (3.44); the flux and moment flux integrals are replaced by the two-point Gauss quadratures (3.48); and the integrals of F and G over the cell area are replaced by (3.51).
The truncation error in the average equations is second order on nonsmooth grids.
Thus, the cell averages are second order accurate for both steady and unsteady flows. The gradient equations are discretized to first order truncation error for unsteady flow
and second order for steady-state. Therefore, the gradients are first order accurate for
unsteady flow and second order for steady-state. The final result is that the solution is
known to second order accuracy within each cell.
This formulation has two main advantages over that of Section 3.4.2. The first is that
no gradient smoothing or limiters are needed to capture shocks; and the second is that
no decoupled modes are admitted in the gradients. An additional advantage is second
order accurate gradients for steady flows. The main disadvantage is the extra computing
73 required in the two-point Gauss quadrature of the flux integrals (this essentially doubles the computational work).
The fact that this formulation does not require any shock or background smoothing also makes it more rigorous. Smoothing operators contain user adjustable constants, and they require additional, nonphysical boundary conditions. They also contain switches which must be formulated to turn them on only where desired, such as near shocks. Furthermore, smoothing of the gradients allows upstream propagation of information in supersonic flows.
The ability of this discretization to capture shocks, as opposed to that of Sec- tion 3.4.2, rests in the nonlinear consistency between the line and area integrals, (3.48) and (3.51), in the moment equations. If the gradient products multiplied by the second area moments in (3.51) are neglected, then divergence occurs at shocks. These terms are below truncation error for unsteady flow, and neglecting them corresponds to a lin- ear evaluation of that integral. Note that it is nonlinear consistency, rather than Roe's parametric vector W, that is essential here for capturing shocks. Roe's parametric vec- tor has been used because it has been found to be the most efficient means of obtaining this nonlinear consistency.
Alternative nonlinear consistent discretizations are possible. For example, the for- mulation of Section 3.4.2 can be modified so that it also converges at shocks. This is done by modifying the F and G integrals (3.37) so they are nonlinearly consistent with the midpoint evaluation of the flux and moment flux integrals (3.34). The modification of (3.37) is to interpolate the conservation variables U to the midpoint of each face, eval- uate F and G, and average the results. This modification has been only been attempted on fairly smooth rectangular grids. Simple averaging of the face midpoint values may not give desireable results on more irregular grids (of course, this is an academic point since Roe's parametric vector is a superior approach to the problem).
74 3.5 Time Integration
For purposes of analysis in this section, the semi-discrete form of the average and gradient equations of Section 3.4.1 may be written as,
d --
d dtt AU = -R (3.52) SA:ZUX + AUr=_ (.)
-[AU + AyyUy = -RY, where U, Ux and Uy are given by,
U = WmWn, UX = WmWxn + WxmWn, Uy = WmWyn + WrymWn. (3.53)
These semi-discrete equations, defined for each cell, have the form of a very large, coupled system of nonlinear ordinary differential equations (O.D.E.s). In the present algorithm they are integrated in time using an explicit 3-stage Runge-Kutta scheme.
Given the nonlinear O.D.E.,
(3.54) ddt f(U), the solution u(t) is integrated one time step by,
U(o) = Un 0 U(1) = uCO) + aAtf(u( ))
u(2) = u(o) + 1/2Atf(u(1)) (3.55)
U(3) = U(o) + Atf(U( 2 ))
Un+1 = U(3).
This scheme is second order accurate in time independent of the value of the first-stage
integration coefficient a. Thus, a is a free parameter which can be used to maximize
the time step At.
3.5.1 Stability Analysis
The stability of the present Euler algorithm is analyzed in Appendix B using the linearized 1-D Euler equations rather than the more traditional scalar wave equation.
75 This choice of equations has been prompted by complications arising from the use of flux-splitting and coupling between the average and gradient equations. In supersonic flow, all characteristics move in the same direction, and the linearized Euler equations completely decouple into the separate characteristic equations. As a result, the stability restriction for the linear Euler equations is the same as that for the wave equation. In subsonic flow, the use of van Leer's flux-splitting reduces the maximum CFL number from that obtained for supersonic flow. This reduction is not easily obtained from analysis of the wave equation.
In the analysis of Appendix B, the CFL number for a given Mach number and first-stage integration constant a is defined as,
At A = (IuI+ c)--.Ax (1-D) (3.56)
Figure 3.7a shows the effect of the first stage integration constant a on the stability boundary for the first order scheme using cell averages only (Section 3.3). For supersonic flow the maximum CFL is A = 2 at a z 0.17, but the stability boundary is sensitive to the choice of a. For subsonic flow the maximum CFL decreases to A = 1.9 at M = 0. As a comparison, the maximum CFL for a 3-stage central-differenced scheme (with a = 1/2) is A = 2 independent of Mach number.
The stability boundary for the full second order scheme using cell averages and gradients (Section 3.4.1) is shown in Fig. 3.7b. For supersonic flow, the maximum CFL number is A = 0.59 at a s 0.17 and decreases as M -+ 0. Again, the maximum CFL is sensitive to the choice of a. A linear curve fit for the maximum CFL is then given by
0.51+ 0.08M M < 1 Amax = (3.57) 0.59 M > 1
The same stability constraint is used in 2-D, where the CFL number is redefined as,
|A|+c |v(+c\ A =At (uIc+ ). (2-D) (3.58)
76 2.0 - (Note: vertical scal es different)
A. UNSTABLE
1.5 -
M > 1
M = 1 /2
M=0
1.0 -
STABLE
0.5 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
a) First Order Scheme (Averages Only) 0.7-
0.6-
A
0.5 - UNSTABLE
M > 1
0.4- M = 1/2
M=O
0.3-
0.2-
STABLE n. I 0.0 0.1 0.2 0.3 0.4 0.5 a 0.6 0.7 0.8 b) Second Order Scheme (Averages and Gradients)
Figure 3.7: Effect of First-Stage Integration Constant on Stability Boundary
77 The update of the solution is slightly complicated by the use of the parametric vector
W. Given the updated conservation variables U, Uix and Uy at time level n + 1, the updated values of W, WX and Wy need to be recovered from (3.53). Fortunately, this can be performed analytically as shown in Section 2.5. The averages W are obtained from U by (2.78). Once W is known, Wx is obtained from Ux and Wy from Uy by (2.79).
3.6 Farfield Boundary Conditions
Prior to the calculation of fluxes and residuals in each stage of the Runge-Kutta integration, boundary conditions are performed at the inlet, exit, and walls of the domain. These boundary conditions are implemented through the use of ghost cells outside the computational domain. In this manner, the calculation of residuals for cells along the boundaries is no different than that for interior cells, leading to more systematic coding and efficient vectorization.
Boundary conditions at the inlet and exit planes are performed using the charac- teristic analysis of Section 2.4. For a face on the inlet or exit plane (assumed to be at constant ), these boundary conditions are performed independently at each of the two Gauss points on the face. The current solution is extrapolated to one of the Gauss points using the linear solution approximation (3.39) within the cell adjacent to the bound- ary (see Fig. 3.6). This extrapolated solution is used to determine the face normal Mach number (M = u/c), and hence, which characteristics are right and left-moving.
Those characteristic variables corresponding to characteristics leaving the domain are also evaluated based on this extrapolated solution. The remainder of the characteris- tic variables are determined from appropriate specified conditions. These characteristic variables represent a corrected solution at the boundary, from which corrected primitive or parametric variables are recovered as outlined in Sections 2.4.1 and 2.4.2. When per- formed for both Gauss points, this procedure results in the corrected boundary solutions
W, , which are then stored as the ghost cell solution. The information is stored in the average and y-component of the gradient, with the centroid of the ghost cell located at
78 the midpoint of the face,
Wghost = 1(W9+ + W ,_) WXghost = 0, WYghost = W9+ - W-. (3.59)
The ghost cell solutions are used to calculate fluxes through the farfield boundary faces by the same procedure as for all internal faces.
3.7 Inviscid Wall Boundary Conditions
As stated in the previous section, wall boundary conditions are performed at each
stage of the Runge-Kutta time integration, prior to the calculation of fluxes and resid- uals. The wall boundary conditions are also implemented through the use of ghost cells.
At an inviscid (or slip) wall the mass flux through the wall is specified. For a solid wall it is zero, but for coupling with a viscous solver using a transpiration flux it is nonzero. In this section two different procedures are described for implementing this boundary condition.
3.7.1 Velocity Reflection Treatment
The first procedure is a simple velocity reflection technique. The solution is extrap- olated to the two Gauss points on the wall face using the linear approximation (3.39)
in the cell adjacent to the wall. At each point, the velocity vector is transformed into
wall-normal coordinates, and the extrapolated normal velocity thrown out in favor of the specified value.
(qs)ghost = ' (e9extrap Ughost (q) ghost [A (qn)ghost (3.60)
(qn)ghost =(n)spec Vghost (qa)ghost I + (qn)ghost[ ' AI
The remainder of the extrapolated solution (i.e., the density and total enthalpy) are assumed correct. This information then defines the ghost solution at the two Gauss
79 points of the face. These solutions are stored in the average and x-component of the
gradient of the ghost cell, with the centroid of the ghost cell located at the face midpoint,
- 1 Wg - W9 .. 1 Wghost = - (W9+ + W ,_) WXghost - 9 ,z/V WYghost = 0- (3.61) 2 Ax/- 3 This is only a means of storing the information; it says nothing of the solution gradients at the wall.
This procedure is more or less standard practice for cell-centered schemes. It enforces the normal velocity constraint and allows all other components of the solution at the wall to float to values predicted by the Euler algorithm. This procedure for implementing the wall boundary condition works well for most cases, but in certain situations it
gives unsatisfactory results. An example is inviscid flow (i.e., solid walls-no coupling) with stagnation points. This boundary condition treatment results in a nonphysical entropy rise through the stagnation points; this numerically generated entropy is mostly contained within the first cell row adjacent to the wall. Since the present algorithm has low numerical dissipation, the generated entropy is convected downstream of the stagnation points without being dissipated. In addition, the magnitude of the entropy rise through a stagnation point tends to remain fairly constant as the grid is refined.
As a result, flows with stagnation points are not well predicted because of this entropy layer along the surface.
A case in point is the subsonic circular bump flow presented in Section 6.1.1. Fig-
ure 3.8 shows a blowup of the grid at the leading edge stagnation point, along with a plot
of the entropy error through the stagnation point. The rise in the entropy through the stagnation point is approximately 0.53%. Section 6.1.1 presents further computational
results using this first wall boundary condition procedure.
3.7.2 Characteristics Treatment
Computational results suggest that specifying only the normal velocity at a solid wall
is somehow insufficient. The theory of 2-D characteristics shows that there are actually
two additional boundary conditions that must be specified there. Two flow properties,
80 0.4 - ..---t-
0.2
0.0 i 0. 6 0.8 1.0 1.2 1.4
a) Grid Near Stagnation Point 0.008-
Velocity Reflection Treatment ------Characteristics Treatment
0.006-
/ / - - S - Sinlet Siniet
0.004
0.002
- 7/ 0
-0.002 + 0. 3 0.8 1.0 1.2 1.4 x b) Entropy Error on Lower Wall (S -+ p/pt)
Figure 3.8: Numerical Entropy Rise Through a Stagnation Point
81 namely entropy and vorticity (or perhaps total enthalpy), are convected along the solid wall. When numerically implementing the wall boundary conditions, these entropy and' vorticity conditions are typically neglected. They are neglected because they are approximately satisfied by the discretized conservation equations, and because they only have an indirect influence on the fluxes at the wall. For a solid wall (q, = l-q= 0), the flux vector at the wall is given by,
pqn 0
pq+pt nA| - n~ (H n pun+IA= = I .] (3.62) * pvqn + P[ A- A] P[ A. A] pHqn 0 J This means that only the pressure contributes to the flux at the wall. Therefore, the only influence of the entropy and vorticity at the wall is through their effect on the wall pressure.
This suggests that better solutions may be obtained if more care is taken in the implementation of wall boundary conditions to ensure that entropy and vorticity are properly convected along the wall. To test this hypothesis, the following procedure has also been implemented for shockless flows:
In Section 2.4, a characteristic analysis of the 1-D Euler equations was performed.
In a similar manner, the 2-D Euler equations may be written in the form, Dp+V.jo(36a mass: Dp + PV - q= 0, (3.63a) Dt
momentum: -- + -Vp = 0, (3.63b) Dt p DE 1 p energy: D + -(qj- V)p + -V - q= 0, (3.63c) Dt p p where D/Dt is the particle derivative, D a(q ) a a+ a (3.64) -= -~t+ (W- V) =- + U- + v -. (.4 Dt at z y
As in Section 2.4, these equations may be manipulated to show that entropy S is convected unchanged along a particle path, DS= 0. (3.65) Dt
82 In computations, the quantity p/p"' is used in place of the entropy. In addition, a simple rewriting of the energy equation gives,
DHI _18p -H- = --- (3.66) Dt p at(
An equation may also be written for the change in vorticity along a particle path.
However, in the present algorithm, it is easier to work with the convection of the total enthalpy along the wall rather than the vorticity.
The solid wall is always a particle path. Hence, Eq. (3.65) states that entropy is always constant along the wall, equal to its value specified at the inlet,
SW(8) = SW(0), (3.67) where s is the wall arclength measured from the inlet. The total enthalpy is also specified at the inlet. Its value at any point along the wall is obtained by integrating (3.66) from the inlet,
Hw (8) = Hw(0) + lp d . (3.68) fo Pat Note that in steady-state, the total enthalpy is constant along the wall.
In this second wall boundary condition procedure, the ghost solutions at the two Gauss points of each wall face are obtained in a manner similar to that used for the farfield conditions. At each Gauss point, zero normal velocity, entropy (3.67), and total enthalpy (3.68) are specified. In accordance with characteristic theory, one piece of in- formation is taken from the extrapolated solution; this is the 1-D Riemann invariant J+ normal to the wall (n is the normal vector pointing out of the computational domain):
Solid Wall: qnghost = nspec = 0
Sghost = [SIspec = (8) (3.69)
Hghost = [H]spec = (8)
J+ghost = IJ+Iextrap = [- + I extrap
At each time step, the integral in (3.68) is lagged one time step and the integrand is
evaluated based on the average solution in the cells adjacent to the wall.
83 Figure 3.8b also shows the entropy error through a stagnation point using this sec- ond procedure. Entropy errors in the immediate vicinity of the stagnation point are reduced by a factor of two compared to the results of the velocity reflection proce- dure. Downstream of the stagnation point, the entropy errors reduce almost to zero. This constitutes a significant improvement over the previous wall boundary condition procedure.
Unfortunately, this characteristic boundary condition treatment is not valid through shocks. Across a moving shock, both entropy and total enthalpy have jump discontinu- ities satisfying the unsteady Rankine-Hugoniot relations. A correct implementation of the entropy and total enthalpy boundary conditions for shocks has not been determined yet.
In the subsonic circular bump case of Section 6.1.1 and the sin2z bump case of Section 6.1.4, this second boundary condition procedure is used for both the upper and lower walls. In the transonic case of Section 6.1.2, this second procedure is used on the lower wall from the inlet to the point where the flow first becomes supersonic. The velocity reflection procedure is used downstream of this point. The entire upper wall remains subsonic, so this second procedure is used. All other test cases presented in this thesis, including all viscous flows, use the velocity reflection wall boundary condition procedure.
Wall-Normal Vector Treatment
In the implementation of the wall boundary conditions, another area where care should be taken is the definition of the unit normal vectors at the Gauss points of the wall faces. The definition of these wall normals makes a difference regardless of whether the velocity reflection or characteristic procedures are used. The outward normal should not be taken as the normal to the wall face (n = Ay/As S - AZ/As J), but rather something closer to the actual normal to the wall geometry at those points, as shown in
Fig. 3.9. The algorithm is accurate enough to sense the difference between a wall made up of linear segments (each the length of a single cell) and a curved wall. The effect on
84 the solution is typically small but noticeable, yet becomes very pronounced near sonic regions as shown in Fig. 3.10 and 3.11. Both figures show density contours for a 2-D transonic Laval nozzle using different normals for the upper wall (see Section 6.1.1 for a discussion of plotting procedures). The case in Fig. 3.10 was computed using the normals for the linear wall faces, while Fig. 3.11 was computed using the actual wall geometry to calculate the normals at the Gauss points. The difference in the solutions is greatest at the throat. This can be explained by the fact that transonic flows are known to be very sensitive to surface geometry perturbations.
SActual wall
L Linear wall face
Figure 3.9: Normal Vectors for Actual Wall vs. Linear Wall Segment
For all test cases presented in this thesis, the actual analytic wall geometry is used to define the wall normal vectors at Gauss points. For a more general and flexible procedure, the wall geometry could be spline fitted and the spline coefficients stored as discussed in Dannenhoffer [24]. The wall normals could then be calculated from this spline information.
We are unaware of other finite volume methods in existence which require such care in the implementation of wall boundary conditions, and we believe this to be a positive statement on the accuracy of the current algorithm.
3.8 Euler Timing Study
Different versions of the present Euler algorithm have been coded (see Appendix F), and computational results are presented in Chapter 6. In this section, computational
85 1
x
0 -2 -1 i 2 z a) Density Contours (A = 0.01)
1.0-
0.9-
up e 0.8 - rlower
0.7- lower
0.6- I 0.5 - "pper
0.4- -1 0 1 2 x b) Density on Upper and Lower Walls
Figure 3.10: Linear Wall Face Normals Used at Gauss Points
86 1
6 o\ \ \ / I 0 -2 -1 0z i 2 a) Density Contours (A = 0.01)
1.0 -
0.9-
p up 0.8-
lower
0.7- lower
0.6-
0.5- upper
0.4 1 -2 -1 01 x 2 b) Density on Upper and Lower Walls
Figure 3.11: Analytic Wall Geometry Normals Used at Gauss Points
87 speed of the algorithm is compared to two existing codes. The first is UNSFLO [33], coded by Giles and Haimes; it uses Ni's Lax-Wendroff algorithm [62]. The second is a
MacCormack algorithm [59], coded by Shapiro [76].
Each code was timed on both scalar and vector/parallel computers. The codes were run on dense grids (e.g., 128 x 32 cells) to minimize the influence of boundary condition treatment and other overhead. This gives a more accurate assessment of the computational time spent during the interior discretization in each algorithm. Table 3.1 gives timing results for both the first order (Section 3.3) and second order (Section 3.4.3) versions of the present scheme, as well as the two mentioned above. The scalar machine results are more representative of the number of floating point operations required by the interior discretization of each scheme. However, the numbers should be viewed as approximate, since each is subject to the different coding styles and data management structures. Note also that a great deal more effort was expended on the UNSFLO and MacCormack codes to efficiently vectorized them for the Stellar GS-1000.
Table 3.1: Comparison of CPU Time (in p.sec) Per Cell Per Time Step
D.E.C. Stellar MicroVax II' GS-1000b
Present (1st order) 5.2 0.23
Present (2nd order) 17.4 0.73
UNSFLO 3.0 0.10
MacCormack 1.7 0.028
a scalar machine b vector/parallel machine
The second order version of the present scheme takes approximately three times longer on the scalar machine than the first order version. The two main reason for this is increase in CPU time is evaluating twice the number of split flux vectors and solution of 12 equations per cell compared to four. Compared to UNSFLO, the first
88 order version runs about 60% slower, while the second order version is five to six times slower per time step.
This straightforward comparison of CPU times may not be completely fair to the present algorithm. For a given grid, it is more accurate than the first order accurate
MacCormack algorithm (see Sec. 6.1.4 for an accuracy study of the present algorithm). For unsteady shocked flows it is also quite likely to be more accurate than Ni's algorithm
(see Secs. 6.1.5 and 6.1.6 for unsteady shock studies). Thus, if the accuracy of the present algorithm allows it to be run on a grid with half the density in 2-D, then the CPU time will be cut by a factor of eight.
An additional point made by Shapiro [76] is that the present algorithm is well suited for the massively-paralleled supercomputers currently in development. These comput- ers consist of a very large network of microprocessors working in parallel, where each processor performs the calculations for a single cell. It turns out that clock speeds are often limited by data transfer between processors, rather than floating point operations on each processor. While the present algorithm is floating point intensive, it has rel- atively little communication between cells. At a given stage of the Runge-Kutta time integration, the residuals for each cell can be obtained by a single pass visiting all the faces. In this process, each quadrilateral cell exchanges information with four other cells (i.e., a five-cell stencil). For comparison, a given cell exchanges information with eight cells during the residual calculation in the flux-split scheme of Anderson et al [7].
MacCormack's cell-centered scheme, as well as Jameson's cell-centered scheme [43], use a five-cell stencil to calculate the fluxes through a given cell. However, these schemes require smoothing, which involves a fourth-difference term. This must be evaluated by a single-pass nine-cell stencil or a two-pass five-cell stencil. Thus, efficient coding of these schemes' smoothing operators becomes an important issue in their implementation on these new supercomputers.
89 Chapter 4 Thin-Shear-Layer Navier-Stokes Algorithm
In this chapter, the Thin-Shear-Layer Navier-Stokes algorithm is developed. The Defect formulation, introduced in Chapter 2, is further discussed and an outline of the approach used to discretize it is given. This approach begins by describing a dynamic co- ordinate transformation of the governing equations. The spatial approximation is next developed, where different discretizations are used across the boundary layer and in the streamwise direction. This is followed by presentation of the semi-implicit time integra- tion. Numerical implementation of boundary conditions are discussed next. The Newton procedure used to solve the implicit system at each streamwise station is described; this includes equation linearization and matrix inversion using Gaussian elimination. The chapter ends with discussion of the solution initialization and grid generation.
4.1 Objectives
The TSL Navier-Stokes algorithm is developed with three major objectives. The first is high accuracy across the boundary layer by second order accurate discretization on stretched grids for both inviscid and viscous terms. The second objective is a time integration technique where the time step is not limited by numerical stability to be proportional to the small grid spacing normal to the wall. The third objective is a means of adapting the viscous grid to the changing boundary layer thickness.
Before presentation of the algorithm, a few comments are in order. This algorithm is developed from the start with the intent to use it only in viscous regions and to couple it with an outer inviscid solver. As will be discussed in Chapter 5, the form of coupling used in this thesis is new for Euler/Navier-Stokes algorithms. To reduce
90 unnecessary geometric complexity in the coupling, the TSL Navier-Stokes equations (2.39) are discretized on a grid with one set of coordinate lines parallel to the y-axis, rather than on a body normal grid. Errors in the TSL equations are introduced by this assumption, particularly in the definition of the shear stress and heat transfer. However, these errors are assumed small for the duct cases considered in this thesis, since their walls are nearly parallel to the x-axis.
4.2 Defect Formulation Revisited
The Defect equations (2.44b) are discretized in this chapter assuming the pseudo- inviscid solution U is known. Its discrete approximation U" is given by the solution of the Euler algorithm of Chapter 3.
From the presentation of the Defect formulation in Section 2.3, two approaches to the discretization of the Defect equations become apparent. The first is to discretize the equations as written and solve for a viscous defect U - U given a known pseudo- inviscid solution U. The second approach emerges from the discrete operator analysis
(2.53). A discretized TSL operator or algorithm is first developed. Then the discrete
Defect operator is constructed by taking the difference of the TSL operator operating on the viscous solution U and the known inviscid solution U. This suggests rewriting the Defect equations as,
aG ias1U 8 a$ 8. a a a G+ + = 0. (4.1) at ax ay Reay at ax ay
For historical reasons, this second approach has been chosen to discretize the Defect equations. The spatial and temporal discretization of the TSL Navier-Stokes equations are presented in Sections 4.4 and 4.5; these discrete operators are then used to construct the discretization of the Defect equations in Section 4.6.
In fact, the algorithm presented in this chapter was originally used to solve the TSL equations. In the course of the research project, unsatisfactory mismatches between the viscous and inviscid solutions at the edge of the viscous grid were obtained on more
91 difficult test cases. This was eventually attributed to differences in truncation error, and it was at this point that the advantages of the Defect equations became apparent. After deciding that a proper discretization of the Defect formulation was beyond the scope of this thesis (see Section 5.2.3), a simplified discretization was developed and is presented in Section 4.6. For coarser grids, results were found to improve substantially. Examples of these improvements are presented in Chapter 6.
4.3 Coordinate Rescaling
The extent of the viscous region is not known a priori; hence, the computational grid cannot be specified. Since it changes both along the wall and in time, some form of adaptation must be used to determine the extent of the viscous region and the grid. The technique used here has been used in the Boundary Layer schemes of Carter [16] and Drela [28]. The y-coordinate is rescaled by the local boundary layer thickness, and the resulting transformed equations are solved on a fixed grid in computational space.
Define the coordinate transformation to computational space,
t, = X, tj = y zt) (4.2)
where A(x, t) is a scaling parameter which depends on the local boundary layer thickness (it will be defined in Section 4.8), and j(z) is the location of the stationary wall. Wall curvature effects are accounted for by the inclusion of j in the definition of t7. Derivatives with respect to x, y and t then become,
a _ a + ata _a A2 \a aaa aA A
ay- ayoa(4.3)a,7 a' a ar a 7a a 7At a at at ar at a7 ara a2' where aA aA . (4.4) Ax x At t dx
92 The transformed TSL Navier-Stokes equations (2.39) can then be written as,
au t~au + [F_(zq+ j)a]+ ---- 1- - -s02 ar A an I a A A an A aq ReAaa (4.5)
or in conservation form as,
(AU) + (AF) + G - 8-(. .F-dn CT a at, Re zz~ . (4.6)
The corresponding form of the Defect equations (4.1) is,
(AU) + (AF) + - ( e S - (A. + Q.)F - AtnU (4.7)
0.17U)]At+9Z)(AxiiG')+-+-[-(FP U
Hereafter, the time coordinate r will be replaced by t to avoid confusion with the shear stress.
4.4 Spatial Discretization
In boundary layers or any other shear layers, gradients of viscous stresses across the layer are as important as inertial forces. Therefore, in discretizing the Navier-Stokes
equations, care should be taken to approximate the viscous terms as accurately as the inviscid terms.
As stated in Section 1.2, most of the Navier-Stokes algorithms in current use do not
satisfy this objective. They use 3-point central-differencing assuming a uniform grid to discretize viscous terms. For the grids commonly used in viscous calculations, where
cell size varies rapidly across the boundary layer, this discretization fails to be second
order accurate. Furthermore, only cell-vertex schemes can claim second order accuracy for inviscid terms.
An extension of the Euler algorithm of Chapter 3 to Navier-Stokes by adding dis- cretized viscous terms would also have accuracy problems. Even with the averages and gradients stored for each cell, grid independent second order accuracy would not be
possible for viscous terms. As a result, a different spatial discretization is developed
93 in this section. In the present algorithm, both inviscid and viscous terms across the boundary layer are discretized to second order accuracy for nonsmooth grids.
4.4.1 Cross-streamwise Discretization (q-direction)
The Keller Box scheme [48] is a highly successful algorithm for the solution of the Boundary Layer equations. In this scheme, second order accuracy is obtained for both inviscid and viscous terms across the boundary layer (ti-direction). The key to this scheme is rewritting the governing equations as a system of first order equations and using two-point differencing to evaluate all i7-derivatives. Such is the approach taken in the present algorithm.
j= J i e e e
o Solution: W, T, q
j+1 o o o . Geometry: C, v1 e e e e e e ~1 e e e e 0 e LJ i- 1 - -, / / / , / / 1 / T-T71 , i i+1
Figure 4.1: Viscous Grid Notation (computational space)
The TSL Navier-Stokes equations are solved as a system of six first order equations as written in (4.6), where the vectors U, F, G and S are given by (2.39b) and (2.39c). The primary unknowns are taken to be the inviscid solution, given by Roe's parametric vector W = ( p1/2, 1/2ts, p1/ 2V, p1/2 H )T, as well as the shear stress r and enthalpy flux q. These six unknowns are stored at face midpoints as shown in Fig. 4.1. With unknowns stored in this manner, two-point differencing is used to evaluate ti-derivatives on each cell. For example,
aG G+ - 2 G_ + O(A17 ), (4.8) ani,+1/2 A17
94 where the convention ( ) + = ( ) ,i+1 and ( )_- = ( )i,, is used throughout this chapter. This discretization is second order accurate for nonsmooth spacing in the n-direction.
Derivatives with respect to t and C will remain undefined for the present, except to say
that aU/at is evaluated at the center of the cell using the midpoint rule in 17 (which is
also second order accurate),
au a U+ + 0(A 2 +ui 7 ). (49) at i,j+1/2 at \ 2 The resulting semi-discrete TSL Navier-Stokes equations in conservation form on each
cell are given by,
[ ) + (AF) + 1 (G+ - G _) - 1-(S+ - S-) (4.10)
- Azi(t+F+ - vi-F ) - j2x(F+ - F-_) - At;(ti+U+ - n-U_)] = 0,
where Ai is the scaling parameter, and A,; and Ais are its streamwise and time deriva- tives (4.4); all of these are evaluated at the current streamwise station. The A. deriva-
tive is given by,
Azi = 1 2 - A.. 11 2 .Ai+ (4.11)
The scaling parameter at the left and right faces of the cell is obtained by interpolating
the values at the midpoints of the two adjacent cells. For example,
_ A Ai + ACA+ 1 Adel = Ci+1/2 - 6,-1/2, (4.12) 'dr +Ai Ad, = C43/2 - 6+1/2-
The definitions of shear stress r and enthalpy flux q on each cell complete the system.
They are discretized in the same manner as the conservation equations,
Tr+ + r- (A + pt) + = 0, (4.13a)
q+ +q- p g H+ - H_ [ 1 1)O 1 1 ]U2 -U2 1-+g10. + _ + - = 2 (Pr Prt AA 2 Pr Prt AiA17 (4.13b)
Here the molecular viscosity y at the cell center is the average of the face midpoint
values given by Sutherland's law (2.26), + +p=' 3 /2 1+S' 2T +I (4.14)
and the turbulent viscosity Pt at the cell center is evaluated as shown in Section 4.9.
95 4.4.2 Streamwise Discretization (c-direction)
The C-derivative in the conservation equations is evaluated by differencing the fluxes at the midpoints of the left and right faces of the cell (the subscript j + 1/2 is assumed in the following analysis),
2 8(AF) (AF)i+1/ 2 - (AF)j-1/ 2 + O('d,7 ). (4.15) ae AC+OA .(.5
This formula is second order in Ai1 due to the midpoint rule. The accuracy in Ae depends on the evaluation of the fluxes at the faces Fi 1/ 2. These fluxes are evaluated using flux-splitting to crisply capture shocks in the outer regions of the boundary layer,
Fii1/2= F+(W 1 /2) + F-(W 1 2 ), (4.16) where F are the van Leer split flux vectors defined in Section 3.2. W+ is the solution at the face interpolated from the right, and W- that interpolated from the left. As in the Euler algorithm, it is these interpolations which determine the accuracy of the discretization.
Expanding about the center of the cell, the interpolated solution at the midpoint of the left and right faces of the cell is,
Wi / = Wi + O(A 2 2 4.17) 1 2 2 a (
The first term W, is simply the average of the values at j and j + 1,
2 Wj= 1(Wei+1 + W,,,) + O(A,7 ). (4.18)
Unfortunately, the second term 49W/8e creates both accuracy and stability problems. Since the solution is stored at face midpoints, a cell contains no 8/8 information; it must be obtained elsewhere. Options include approximation of aW/8e by a similarity assumption if the outer edge gradients are known, or by one-sided differencing as done in Ref. [7]. The first is order 0(6 A) accurate at the wall, where 6 is the boundary layer thickness, and the second is order O(Ad 2) accurate only for sufficiently smooth grids. For both of these options, stability problems occur with the time integration scheme of
96 the next section. For this reason the gradient term is dropped completely, giving a first order interpolation,
+ Wid) i+= 21( ++ W/2 = (Wi+1,+1+ Wi+1,J), W 1 b 1 W_1/2,j/2 =2 (Wi-1,J+1 + Wi-1 ,d), W- 1/ 2 ,s1 / 2 = (Wij++ Wi,j). (4.19)
These interpolations are O(AC, A? 2) accurate.
4.4.3 Artificial Dissipation
Given sufficient grid density to resolve the gradients of the boundary layer solu- tion, the present spatial discretization admits no decoupled sawtooth modes. Hence, no added artificial dissipation is required. This is an important advantage over most other Navier-Stokes algorithms which do admit decoupled modes. The reason is because construction of artificial dissipation operators requires great care for Navier-Stokes algo- rithms (even more so than in Euler algorithms). If poorly constructed, these operators can contaminate solutions by masking physical viscosity as shown in Refs. [2,261.
The use of flux-splitting in the streamwise direction also eliminates any need for added artificial dissipation to capture shocks.
4.5 Time Integration
Resolution of the large gradients within the viscous region can result in very small grid spacing normal to the wall, particularly for turbulent cases. To avoid impractical time step restrictions on a fully explicit scheme resulting from this small spacing, the equations are integrated in time using a semi-implicit formulation. Discretization nor- mal to the wall (si-direction) is evaluated implicitly, whereas streamwise discretization
(C-direction) is evaluated explicitly. As will be shown in the stability analysis of Ap- pendix C this allows the solution to be integrated using a time step At determined by the streamwise resolution Ax only.
97 In the present implementation, the equations are integrated using a single-stage
backward Euler for v7-derivatives and forward Euler for C-derivatives to advance the solution from time t (level n) to t + At (level n + 1). Thus, streamwise flux derivatives
(4.15) in the conservation equations are evaluated explicitly using the solution at level n. The remaining spatial derivatives in the conservation equations are evaluated completely at the new time (level n + 1), giving
1 [n+( U+ +U_ )n+ 1 A U+ + Uy] + 1 [(F+n +(AF ,/2 - S)-1/2 Idt 2 \ 2 d 1 [C1(S + (c+ - G_)n+1 - (S+ - S-)n+1 - A,?(q+F+ - t-F_)n+l
- 9zj(F+ - F _)n+1 - At (1+U+ - t-U-)n+ = 0. (4.20)
In these equations, the time derivative of the scaling parameter is given by,
Ati = * A (4.21)
In a similar manner, the shear stress and enthalpy flux definitions (4.13) are evaluated completely at the new time level. These equations form an implicit nonlinear system
for the evolution of the solution in time. At each streamwise station and time step, this system is solved by Newton's method as described in Section 4.10.
The single-stage backward Euler integration has the advantages of simplicity and
robustness over other methods. Unfortunately, it suffers from the fact that it is only first order accurate in time. Other implicit integration methods were considered but discarded. For example, Crank-Nicholson, which is a second order trapezoid rule in
time, was not used because it allows sawtooth modes in time that are damped only
by nonlinearities. Three-point backward differencing in time [101 was not used because
it requires an additional level of solution storage compared with the Euler or Crank- Nicholson methods.
4.5.1 Stability Analysis
The stability of the present semi-implicit time integration is analyzed in Appendix C
using a von Neumann stability analysis of a linear 2-D advection/diffusion equation.
98 From the analysis, the amplification factor for a given time step may be written as the product of a factor depending only on the C-discretization and another depending only on the ti-discretization. The implicit tl-discretization is integrated in time with a single stage backward Euler; hence, its amplification factor is always less than one. In other words, this component of the algorithm is unconditionally stable. The time integration for the 6-discretization is a single stage forward Euler, which is unconditionally unstable for all spatial discretizations except first order upwind differencing. For this spatial discretization the constraint on the CFL number is,
At A = (|ul + c) AX < 1. (4.22)
Thus, the stability of the viscous time integration depends only on the streamwise spacing Ax. Note also that this stability constraint is less restrictive than that for the second order version of Euler algorithm in Chapter 3 (see Section 3.5.1).
Other spatial discretizations in the C-direction, such as the two suggested in the previous section, require either a three-point backward difference in time or a multi- stage time integration [57] for stability. For reasons of robustness and computing time, these alternative streamwise discretizations and time integration schemes were not im- plemented.
4.6 Discrete Defect Equations
The previous two sections have developed the discrete TSL Navier-Stokes opera- tor (4.20) where the explicit streamwise discretization is given by (4.15), (4.16) and (4.19). In this section, this operator is used to construct the discretization of the Defect equations (4.1).
A proper discretization of the Defect equations would result by taking the difference of (4.20) operating on the viscous solution Uh and the known pseudo-inviscid solution
O. This requires the inviscid solution to be interpolated to every cell of the viscous grid. This has not been done. When the present Euler and TSL Navier-Stokes algorithms
99 were originally coded, only the inviscid solution and its gradients at the edge of the
viscous grid were used in the viscous scheme. Modification of the code to interpolate
the inviscid solution to each point in the viscous grid was deemed too cumbersome for
the expected gains. Instead, a simplified discretization of the Defect equations has been constructed by assuming a linear inviscid profile through the boundary layer; this profile is based on the solution and gradients at the edge of the boundary layer.
The Euler algorithm of Chapter 3 is spatially second order accurate even for non- smooth grids. The TSL algorithm developed in this chapter is spatially second order accurate in the q-direction but only first order in the streamwise direction. Thus, the major difference in truncation error is the first order streamwise discretization of the present TSL algorithm. Hence, this is also the major source of solution mismatch at the edge of the viscous grid. This component of the relative truncation error is largely eliminated by substituting the inviscid edge solution into the discrete TSL equations of the outermost viscous cell and subtracting the result from all cells at each station.
Hence, the simplified discrete Defect operator on each cell is given by,
U)], [LTSL(Uh)] -[LS(J] ,, 4.3 [LE(Uh h ILDEF TSL(U/2 i,j+1/2 ITS' (U )] ,J -1/2 '
where j = 1 to J - 1, and LTsL is the operator notation for (4.20). This is a proper discretization of the Defect equations only for the outermost cell.
The pseudo-inviscid solution at the edge is known at both the n and n + 1 time
levels in terms of the parametric vector and its gradients:
We, (4.24) 8x e' y Thus, W is known at the upper and lower faces of the outermost cell by interpolation,
=a wk(4.25a)
W k = ^k _ ;_=," -() (4.25b)
where k = n or n + 1, and At, = 17 - 1-1. The conservation and inviscid flux vectors
are given by (2.77),
U+ = U(W+), F+ = F(W+), $+ = G(W+), (4.26a)
U- = U(W ), = F(W ) . = G(W_). (4.26b)
100 Substitution of these values into the discrete TSL equations (4.20) then gives the pseudo- inviscid component of the simplified discrete Defect operator (4.23).
4.7 Boundary Conditions
Boundary conditions must be specified at the wall, outer edge of the viscous grid, and inlet and exit planes of the duct.
4.7.1 Wall Boundary Conditions
At each streamwise station, containing (J - 1) cells across the grid, there are 6J unknowns (neglecting global unknowns to be defined later) and 6(J -1) equations. This leaves 6 boundary conditions to be specified to close the numerical system. At the wall the no-slip conditions (2.16) are specified:
u(C,, t) = 0, v(C,0,t) = 0, (4.27)
H (C, 0, t) = H or q(C ,0,t) = qw.
All viscous test cases in this thesis are run with an adiabatic wall condition (qw = 0).
4.7.2 Edge Boundary Conditions
The outer edge of the viscous grid is a nonphysical boundary as are the inlet and exit planes. Three conditions need to be specified to close the numerical system at each streamwise station.
Physically, the number and type of quantities to be specified at the edge is not completely clear. The pressure p is imposed on the boundary layer by the outer inviscid flow, except perhaps at separation, and should be specified in some form. The normal velocity produces a displacement effect on the outer inviscid flow and should not be
101 specified. From an equations standpoint, at least one boundary condition should involve
the shear stress or enthalpy flux. From the theory of 1-D characteristics, if the flow normal to the boundary is inflow then three quantities should be specified. Conversely, for outflow only one condition should be specified. Physically, the edge of the boundary layer is always an inflow since boundary layers grow, entraining more and more fluid.
However, the scaling parameter defining the outer edge in the present scheme is not physical, and outflow occurs near separation and shocks. Regardless, the applicability of 1-D characteristics to the present situation is questionable. Whereas the flow is nearly normal to the inlet and exit planes, the flow is nearly parallel to the edge boundary
(typically, the entry angle is less than one degree). Hence, significant errors may result in this analysis by neglecting the large component of flow parallel to the boundary.
In the end, it is not clear what the three edge boundary conditions should be. After much experimentation, the following conditions seem to be adequate for most situations:
p(, 17j) = PC, r(c, t1, t) =0, H((,r,,t) = fh, (4.28) where p, and H, are the pressure and total enthalpy of the outer inviscid flow interpo- lated to the edge of the viscous grid. The shear boundary condition is fairly safe because it is a natural result of the Navier-Stokes equations far from the wall. Specification of enthalpy flux q instead of the condition on H also works well.
A combination that gives poor results is specifying both pressure and the edge velocity u,. Results indicate that the problem is overspecified with this combination.
The symptom is a numerical boundary layer at the outer edge of the viscous grid in certain circumstances. If not adequately resolved, this results in odd/even decoupled modes in the shear stress profile and on rare occasions divergence.
To complicate matters, a streamwise divergence phenomenon was found in the test case of Section 6.3.2, involving an oblique shock impinging on a flat plate boundary layer. The cause was eventually traced to an interaction between the edge pressure boundary condition and the coupling (to be discussed in Chapter 5). For unknown reasons, this phenomenon disappears if the incoming Riemann invariant (normal to the edge boundary) is specified instead of pressure.
102 In general, matching of the viscous and inviscid solutions is more satisfactory if the pressure is specified instead of the Riemann invariant. Hence, all test cases presented in this thesis use (4.28) unless otherwise stated.
4.7.3 Inlet/Exit Boundary Conditions
In the present viscous algorithm, two options exist for starting a boundary layer. The first is to specify a leading edge somewhere on the upper or lower walls, where the boundary layer starts from zero thickness; the treatment of this case is discussed in Section 4.7.4. The second option is to specify a desired boundary layer profile of given thickness at inlet of the duct. In this case inlet boundary conditions need to be implemented.
As in the Euler algorithm, boundary conditions at the inlet are treated using the characteristic analysis of Section 2.4 to set the solution at a row of ghost points. There are, however, two major differences from the procedure of the Euler algorithm. The first is the location where the boundary conditions are applied. Here they are applied at inlet cell vertices. The second is that reverse flow must be allowed for. Initial transients in a steady case or transients in an unsteady case may include pressure waves moving upstream out the inlet. These pressure waves may be strong enough to locally separate the boundary layer. Thus, three cases must be considered at the inlet: supersonic inflow, subsonic inflow and subsonic reverse flow. The first two are treated as shown in Section 2.4.1. For the special case of reverse flow within the boundary layer at the inlet, the outgoing Riemann invariant J-, entropy and v velocity are all extrapolated.
The pressure is specified based on the desired inlet profile.
M < 1, u < 0 : 8ghost Sextrap
Vghost Vextrap
ghost extrap (4.29
Pghost Pspec
ghost +(8ghost, Pghost, Jghost)'
103 Likewise, exit boundary conditions are enforced by setting the solution in a row of ghost cells outside the exit plane. The boundary conditions are applied as discussed in Section 2.4.2 starting at the edge of the viscous grid and working towards the wall. For supersonic outflow all quantities are extrapolated. For subsonic outflow pressure is specified, and the outgoing Riemann invariant J+, entropy and v velocity are extrap- olated. If the inviscid edge flow is subsonic then the specified pressure pspec is a user input. If the edge flow is supersonic, then the specified pressure is taken from the lowest supersonic point in the boundary layer.
The special case of transient reverse flow must also be dealt with at the exit. In general, reverse flow at the farfield boundaries is not good and can lead to nonphysical results. This is particularly true at the exit since nothing is really known of the fluid entering the domain. Thus, the reverse flow boundary conditions are only intended to continue a calculation past a separation bubble at the exit. For reverse flow the outgoing Riemann invariant J+ is extrapolated, v is specified to be zero, and pressure and entropy are taken as the last calculated ghost point values. J+ M< 1) u< J+ghost extrap
Vghost = 0
Pghost = last calculated pghost (4.30)
8ghost = last calculated 8 ghost
Jghost J (8ghost, Pghost, Jghost)-
4.7.4 Leading Edge Boundary Conditions
For some test cases it is desirable to simulate a leading edge, such as that shown in Fig. 2.2, rather than prescribe an inlet viscous profile. As noted in Section 2.2, the full Navier-Stokes equations must be used in the immediate vicinity of the leading edge to obtain an accurate solution there. This is not the goal here; instead, the present intent is to integrate past the leading edge to obtain an accurate solution farther downstream. To accomplish this, certain minor assumptions and modifications are required.
The actual location of the leading edge is assumed to be at a face or constant-C
104 grid line (say i - 1/2). At this face the scaling parameter A is set equal to zero; all streamwise coordinate lines emanate from this point.
Unfortunately, the outer edge boundary conditions (4.28) do not work for a leading edge. These boundary conditions must be modified for the first streamwise station of cells (i.e. that grid row with the leading edge point at its upstream face). At this first streamwise station, pressure, u velocity and total enthalpy are specified at the outer edge,
p((; 17,)0 _= Pe, u(,,ti,,t) = t 6, H(e, t7,) = H6. (Leading edge)
(4.31)
The only difference between these modified boundary conditions and the normal conditions used everywhere else is the specification of the edge velocity. It is necessary to specify the edge velocity for this first cell to initialize the starting boundary layer. The pressure and total enthalpy conditions alone do not convey any velocity informa- tion. Furthermore, no velocity information is obtained from the upstream faces, since they have all collapsed onto the leading Also note that these modified edge boundary conditions (4.31) for the first streamwise station are the same as those used to generate similarity profiles in Section 4.11.
The final modification involves the definition of the scaling parameter; all equations and parameters mentioned in this paragraph refer to the next section. The curvature of the boundary layer edge is typically greatest at the leading edge, and the smoothing used in (4.37) to evolve Ai seems to overly flatten or stunt the growth of the boundary layer thickness at the leading edge, causing divergence. This is remedied by turning the smoothing coefficient off (vI = 1, V 2 = 0) for this first station. Using the definition (4.33) also works at the leading edge.
105 4.8 Scaling Parameter
As stated in Section 4.3, the governing equations are transformed by scaling the y-coordinate with the local boundary layer thickness. Since the edge of the boundary
layer is not well defined, relations for the scaling parameter A(z, t) must be chosen empirically. Here it is defined in terms of the incompressible (or kinematic) displacement
and momentum thicknesses,
6 * (I dy, 6k= - ) dy , (4.32) 0 Ue 0 Ue Ue
where ue is the x-component of velocity at the edge of the boundary layer.
As suggested by Drela [31], one method is to express the scaling parameter as a
linear combination of SZ* and 6k,
+ *, (0 <- O<1) (4.33) A (X, t) = H k
where H is the value of the shape parameter for incompressible flat plate flow (H = 2.6 for laminar and H = 1.2 for turbulent). Defined in this manner, A is independent of
a for flat plate flow. Transforming (4.33) into computational space (q = y/A) leads to
the relation,
-1 - 1 = ali + ak* (4.34) H where
6* = 6* /A = 1 - - dq, Af U (4.35) i= Ok/A f -- (-- dY. fo Ue U ) These integrals are evaluated by the midpoint rule on each cell.
Figure 4.2 presents a typical laminar separation bubble in a subsonic expansion duct. The results were obtained using a fixed viscous grid with the outer edge well outside the boundary layer. Density contours and streamlines in Fig. 4.2 illustrate the growth of the boundary layer through separation. Based on this fixed grid solution, Eq. (4.33) can be used to predict the variation in the scaling parameter through separation. From this an estimate can be made of the final grid position if it were allowed to move. This
106 predicted edge of the viscous grid is shown in Fig. 4.3 for various values of a. The bottom curve corresponding to o = 1 was used by Drela [28]; it predicts a dip through separation, where the boundary layer is definitely growing. From Fig. 4.3, a value of o = 0.4 appears to best match the variation in the boundary layer through separation.
Unfortunately, this definition of the scaling parameter is not adequate. The primary reason is that (4.33) results in nearly discontinuous grids when shocks are encountered, leading to solution divergence. Other reasons are due to the first order streamwise dis- cretization used in Section 4.4.2. If it is used to solve the TSL equations (rather than the Defect equations), experience has shown it to be extremely sensitive to grid smooth- ness in both z and y. As a result, the grids produced by (4.33) tend not to be smooth enough to accurately predict regions of rapid streamwise change, such as separation.
Solution of the Defect equations reduces this sensitivity somewhat by subtracting off the truncation error in the outer boundary layer.
An additional problem with (4.33), as well as its replacement to be discussed, is the weighting factor o. A curious divergence has been observed at separation when A is too heavily weighted on 6*; the grid tends to grow unbounded. Computational experience has shown that this limits a to values above 0.8.
Clearly, resolution of most of these problems lies in smoothing the scaling parameter. After much experimentation, the most successful technique is to use an analogy with the 1-D heat equation to evolve the scaling parameter in time,
2 = v(Aeq - A) + v2 8 , (4.36)
where the coefficients v, and v 2 control the relative amount of smoothing as well as the relaxation of A in time. The previous definition (4.33) is used as the source term Aeq, and explicit second differences used for the dissipation term 82A/8X 2 . Thus,
1 +1= A' +v 1A1+[] + * l +V2 j+1 - 2A! +Al._ , (4.37)
A von Neumann stability analysis [68, p. 189] requires v2 < 1/2 for stability. All viscous test cases presented in this thesis use the values v, = 0.1, v2 = 0.45 and o = 0.8.
107 0.06
0.05-
0.04 -
0.03-
0.02
0.01-
0 z a) Density Contours
0.06
0.05
0.04 -
0.03 -
0.02-
0.01 -
0 -4 -20 246 8
b) Flow Streamlines Figure 4.2: Laminar Separation Bubble in a Subsonic Expansion Duct
108 0.15 - a = 0.0
0.2
Yedge 0.4
0.10- 0.6
0.8
1.0 0.05
-4 -2 0 2 4 6 8 z
Figure 4.3: Predicted Edges of Viscous Grid Through Separation, (Eq. 4.33), Based on Fixed Grid Solution
U.09UA~ -
Yedge (4.37) o= 0.8, -Eq. Vi = 0. 1, V2 = 0.45 0.07 ------Eq. (4.3 3) o = 0. 8
0.05-
vertic al scale doubled from Fig. 4.3 0.03 , , -4 -2 0 2 4 6 8 z Figure 4.4: Actual Edges of Viscous Grid Through Separation
109 To illustrate the smoothing effect of (4.37) on the grid, two more computations of the subsonic separation bubble case have been performed using dynamic grids. In the first run, Eq. (4.33) was used to evolve the grid, while in the second, Eq. (4.37) was used. Figure 4.4 shows a comparison of the two actual viscous grid edges obtained using (4.33) and (4.37) with o- = 0.8. The first point to note is that the grid produced by (4.37) is much smoother through separation. The second point to note is that the actual edge obtained with (4.33) in Fig. 4.4 is different than that predicted from the fixed grid solution in Fig. 4.3. The reason for the difference is that the solution is sensitive to grid variations, and this changes the local values of 9 k and 6k*, and hence A.
4.9 Cebeci-Smith Turbulence Model
The turbulent viscosity ut is approximated by the Cebeci-Smith 2-layer algebraic turbulence model. This model assumes local equilibrium between production, diffusion and dissipation of turbulence. This is an incorrect assumption for shock/boundary-layer interactions where significant nonequilibrium history effects are present. Models such as the k-E or the Johnson-King [461 models do account for such effects, but have not been validated for unsteady flows.
The purpose of a turbulence model in this thesis is to demonstrate the extension of the present Navier-Stokes algorithm to turbulent flows and to demonstrate the compu- tational savings of a semi-implicit approach. For these reasons the use of an algebraic model is justified, and the question of unsteadiness and incorporation of more accurate turbulence models is beyond the scope of this thesis.
The Cebeci-Smith model is composed of inner (law-of-the-wall) and outer (law-of- the-wake) regions. In the inner region, the turbulent viscosity is given in dimensional
110 form by,
= y 11 - e-+/A+ , = 0.40, A+ = 26, (4.38) + I?-iy- F Y 'W__) U Pui where ( )w refers to wall values. All quantities are nondimensionalized as in Sec-
tion 2.1.2, except the wall shear velocity i, which is nondimensionalized by the velocity
vrefCref/Lref to eliminate factors of Re from its definition. The resulting nondimen- sional inner model is,
pa(i) = p12 Re, 1 = xy 1 - e-V+/A+] =uTr/- irtu + Re, ur = . Uw Pw
In the outer region, the turbulent viscosity written directly in terms of nondimensional quantities is,
(A)= apu 6k*yRe, a = 0.0168 (4.40)
where 6* is the incompressible displacement thickness, and -y is Klebanoff's intermit- tency factor,
Y = 1+ 5.5(y/6)6 1. (4.41)
In this formula 8 is the boundary layer thickness, which is not well defined; it is approx- imated to be at q = 12. Thus, y/6 = t/12.
Starting from the wall, the inner model is used up to y,, where the outer model first
exceeds the inner; the outer model is used from there,
t= A W (4.42) p1 ), yI/ yc.
The turbulent viscosity is evaluated at cell centers, averaging the solution from the upper and lower faces as follows:
= (T+) + A(T-) r+ + r- p= w1 +w-, p = 2 , r = 2 (4.43)
111 The derivative au/ay is evaluated using two-point differencing,
au-= U_ .- (4.44) ay Ar
No transition model or mechanism for changing from laminar to turbulent boundary
layers is incorporated into the present algorithm.
For each cell at a given streamwise station, the turbulent viscosity is functionally dependent on the solution of the upper and lower faces of the cell and on the wall kine- matic viscosity vu, the wall shear velocity uT, the incompressible displacement thickness
6,*, and the scaling parameter A. These last three parameters, which will be referred to as global unknowns, play a special role in the Newton solution procedure of the next section since changes in them have an immediately global affect on the entire profile.
For reasons discussed in Section 4.10.2, vL,, is discarded as a global variable.
At separation the wall shear r,,, vanishes, resulting in a poorly behaved numerical system. Robustness is recovered if a shear velocity based on the local shear stress and density,
U' = ( , (4.45)
is used instead of that based on the wall values. This modification eliminates u, as a
global variable, but it will be retained as one in the analysis of the next section anyway.
Where this modification affects the linearization of the turbulence model will be noted in Section 4.10.2.
4.10 Newton Solution Procedure
At each streamwise station, a time step consists of the solution of the following nonlinear system:
112 Unknowns: W, r, q at face midpoints 6J
A, 6* u 3
Equations: Conservation on cells 4(J - 1)
r and q defns. on cells 2(J - 1) Boundary conditions 6 A, 6* and u, defns. 3
Here, J is the total number of faces and (J-1) is the total number of cells. This system is solved by a Newton linearization procedure with direct solution (Gaussian elimination) of the linear matrix equation at each iteration. The general solution procedure is similar to that used by Giles [32] and Drela [29].
4.10.1 Newton's Method
Newton's method for the solution of the nonlinear scalar equation f(x) = 0 is as follows: Given the solution at some iteration k, for which f(xk) # 0, the next guess xk+1 is obtained by setting the linear Taylor series expansion for f(Xk+l) equal to zero.
f(zk+1) = f Xk+ 6zk) t f (k) + fI(Xk)SXk = 0, (4.46a)
6zk = - (zk)/fI(k), (4.46b)
Xk+1 = k + (4.46c)
The procedure is stopped when Szk decreases below a desired tolerance. For an initial guess x 0 sufficiently close to the exact solution, this procedure converges quadratically
(i.e. if Ek =exact - zk is the solution error at the kth iteration, then ck+1 - (ek)2)_
For a system of nonlinear equations F(X) = 0, where X is a vector, Newton's procedure is similarly given by,
F(Xk+l) = F(Xk +Xk) ~ F(X) + (F)6Xk = 0, (4.47a)
Xk= aF )k]-- F(Xk), (4.47b)
113 Xk+1 = Xk+6Xk. +4 (4.47c)
The term OF/OX is the Jacobian matrix whose entries (OF/OX)., are the partial derivative of the mth equation F. with respect to the nth unknown X,
(OF _ O. a~m (4.48) ,=) . 09X m,n 49Xn
These entries are evaluated at the kth iterate. Because OF/OX is a matrix, a linear system must be solved to obtain the changes 6Xk.
In the present system, F(X) represents all the equations listed above, and X all the unknowns in the system. It will be convenient to divide the system into local and global equations and unknowns. Local unknowns are taken to be the unknowns at each face midpoint,
Xj = (p1/2, P1/2u, P 1/ 2 v, p1/2H, r, q)T
= (wI, w 2, W 3 , W4, r, q)T, and local equations are the conservation equations on each cell, shear stress and enthalpy flux definitions on each cell, and boundary conditions. Global unknowns XG are taken to be the scaling parameter A, incompressible displacement thickness 6k* and wall shear velocity u,. These variables occur in many of the local equations. Correspondingly, the global equations are the definitions of these three parameters.
4.10.2 Discrete Equation Linearization
The first task in the Newton linearization procedure is construction of the Jacobian matrix OF/OX at each iteration. In the analysis that follows extensive use of the chain rule is made.
Conservation Equations
At a given cell, the mass, momentum and energy conservation equations depend on the local unknowns at the two faces bounding the cell and the global unknowns. The
114 Jacobian entries are first derived for the TSL Navier-Stokes equations (4.20) and then for the Defect equations (4.23).
For purposes of linearization, the discrete TSL Navier-Stokes equations (4.20) can
be written as,
1 [,,,,U,+ U_ "+1 U aF " At .2 An + 1 [C /S1
+ (G+ - 0 _)n+1 - -(S+ - S-)n+' - An(t+F+ - i-F_)n+1 (4.50)
- jz(F+ - F _)"+l - At(t,+U+ - T-U_)n+] =0,
where the time derivative of the scaling parameter is,
(An+ - An) At = . (4.51) At
To obtain the Jacobian entries corresponding to the local unknowns X+ and X_ at the new time level (n + 1), the equations are first differentiated with respect to the
conservation variables U and flux vectors F, G and S at level n + 1,
at An+I A ' at An ~ At aG+ A' AG- At'
at AZ __ at AZ - 1 (4.52) -~1 A_ A 1 2iF + at 1 a _ 1
at 1 __ +1 as+ AqRe' as_ AtRe'
Then each of the vectors U, F, G and S are differentiated with respect to the unknowns
X . The parametric variables W = (p1/ 2, P1/2u, P1/2v, pu/2 H)T were chosen as primary
unknowns for consistency with the Euler algorithm of Section 3.4.3 and because each
entry in U, F and G is composed of quadratic functions of the components of this vector. This results in a very systematic linearization process. For example, the z-momentum
115 components of U, F, G and S are,
U 2 = Pts = WIw2 ,
2 2 F2 =pu +p=w + hiW[l - W 2 +W 2)], 2 4 2 ( 3 (4.53)
C 2 = PUV = W2W 3 -
S 2 = r
Thus, the x-momentum derivatives with respect to W and r are
8U2 8F 2 8G2 - + W4, =W2, aw1
8U2 8 22 8G2 = Wi, = 2W2 - 3 , aw2 aw2 aw2
8F _ l1 au - 0, 2 (4.54) W3 8w3 a 8U 2 8F2 -1 8G2 - 0 = 0, + Wi, 8w4 8w4 - 8w 4 852 = 1. 8r The final Jacobian entries are constructed using the chain rule. 8 8e 8U+ 8e 8G+ 8e 8s+ + F+ (4.55a) 8x+ aU+ 8x+ 8F+ 8X+ aG+ X+ + 8S at- 86 8F. 86C.- 8G_5-- + e as_ + F X + 8G.8. (4.55b) aX- 8U_ 8X_ as_ aX-
The conservation equations also depend on the scaling parameter An+', both directly
and indirectly through At, with the corresponding Jacobian entries,
1 (U+ + U_) n+1 (4.56) L 2J - (T7+U+ - r;_U_)f+1 .
The discrete Defect equations (4.23) may be written in terms of the discrete TSL
equations (4.50) as,
eD8F = e(U) - 6 (u) = 0. (4.57)
The inviscid component of the Defect equations is weakly dependent on the scaling parameter An+l through the definition of WV"+l; therefore, this is the only Jacobian
116 entry which must be modified from the TSL equations. Using partial derivatives already
defined, the Jacobian entry contribution from the inviscid solution is,
___ _) 8U..uF-- aF 8a&8G...] OW_.. +F ... W+8G+ ow _ xn+" (4.58) where oW_.. n+1 =-At . (4.59) The final Jacobian entry for An+1 is the difference of the viscous and inviscid compo- nents, a&DMP = at(U) a e(tr) (4.60) aA (A4A
Shear Stress Definition
Like the conservation equations, the shear definition (4.13a) on a cell depends on the local unknowns X+ and X_. It also depends on all the global unknowns (for turbu- lent cases). The shear definition is evaluated completely at the n + 1 time level, so
superscripts will be dropped and all quantities assumed to be evaluated at n + 1,
A (r+ + r-) - (_ + _) + - U- A+ ~, (4.61) 22(4.61)
Jacobians with respect to r are directly given by,
-T = O-- =-. (4.62) 49r+ ar- 2 To obtain the Jacobians with respect to the rest of the local unknowns, the equation is
first differentiated with respect to u and the molecular and turbulent viscosities, ae ae 1 au+ Ou- At (4.63) ae ae 1 U+ -- 1A_ a-U- aOA+ Jt.._2 At1 OpJt In terms of parametric variables, the u velocity and its Jacobians are,
P 1/29 W2 49t W2 IA= - -- - = - --au -. (4.64) 1/2 t-W 0,Ow2) 49W2 W1 From Sutherland's law (2.26), the molecular viscosity is given by,
T, S/ 2 Ta + S' To = -- ,O T0 and S' are constants (4.65a)
117 [ 1~~'2+v2) !!y)[4 _1 Wt2 2+ w3 2l T'= (-1) 1H - 2 i+ 2 2 W (4.65b) hence, introducing temperature T' as an intermediate variable,
8p 3/2 aT ' IT' T'+S''1 2' - (-1) [-4+ (W22+ w32
(T' - W)2 (4.65c) CIW2 b 1) 12j 8T' [_ W3] 2 a W3 (3 - 1) W1 8T2
aw 4 1. Derivatives of the turbulent viscosity with respect to the local unknowns, the molecular viscosity and the global unknowns are given in Section 4.10.2. The final Jacobian entries are given by combining derivatives with the chain rule,
86 86 au+ 8 8 8 ap+ T 86 8 /It 8p+ (4.66a) 8X+- 8u+8X~+ +T 8)X+ 8pt \)X+ 8ap+ aX+ '
86 at 8u 86at 872' 8p( 8put (4.66b) 8X au- 8X_ 8p8T'. X_ apt \X_ 8p-. aX)
Jacobian entries with respect to the global variables are given by,
8 6 8 _ apt 1 aA t E + 2 (4.67)
at 8 /Ie 8p
Enthalpy Flux Definition
The enthalpy flux definition (4.13b) is functionally equivalent to the shear stress defi- nition,
+p )H1(q++q-)- 1 + - 2 (Pr AV7 2 \(Pr.Prt8 (4.68)
118 Following the same procedure as for the shear stress definition:
ae at A
aq+ - 2 a a _ aH+ aH_ \Pr Prt} At ae 1 1 +At (4.69) 12 -2 1. ( 1 ) I(1- P)] A17'
ap = 1 [( _) H+- H_ 2 \Pr} Arq 2 Pr) A17 a_ _ H+ - H- U2 - U2 8 _ I (I _ lit \Prt) Arl 2 \ PrtI })
P1/2H w 4 8H H 8H H / 2 H 1 W1 -w- W 8iW4 =+W4 2 U 2 (4.70) 2 (P1/2 2 = W2 2 8U 2 U U =1 = 8u 1 2 - =+2-. p ' - - w+ Wi 8w2 w Thus, combining derivatives by the chain rule,
8t a6 8H+ 8f 8u 2 8t ap 8T ae ( ae ,(4.71a) ax+ 8H+ X+ apaT 8X+ +p (aX+ +p+8X+) _e a6 8H_ 8e 8p 8T + aut2 8Xax-. + ae ( 8X-.. 8H8X_ ap 8T L 8X- + 5A, \aXog _ + a Xt .(4.71b)
Jacobian entries for the global variables are,
ac a at + 1 1(++ q-), aA -apAt 8A+ at b_ apt 8 it (4.72)
at 8t 8up
Cebeci-Sxnith Turbulence Model
At a given cell, the inner model is functionally dependent on the solution at the bounding faces W . In addition it is dependent on various global parameters, namely, the wall kinematic viscosity v,,, = (li/p)., the wall shear velocity uT, and the scaling parameter
A. In functional form,
lt(i) = lit()(W+, W,., W )WUT, A). (4.73)
119 The linearization process is most convenient starting with the definition of y+ and working towards pyte', using the chain rule. y+ is dependent only on v. and u,
ay+ Y + ++ -- W =+ -. (4.74)
The mixing length 1 is functionally dependent on A through y = qA and on vw and u, through y+, 1 - e-+/A+] I = y (4.75)
The Jacobian with respect to A is given by, al a_ay - --j? [1 - (1 - C1A ay aA y+/A+)e-y+/A+ (4.76)
and the Jacobians with respect to y+, vw and u, are,
Cty _y+/A+ al - al ay+ al al 8 y+ =+ e (4.77) ay+au. avw a+ 8aw, u
Discretization only occurs in the final definition of the turbulent viscosity,
y Re L- (w1+ 1 -.). + - Re. (4.78)
It is functionally dependent on vw and u, through 1, thus,
8 (i) 8! (i) 8 W 0P 81al -2 - Re 8 (4.79) al A Ia,7 81 8VW' 891 89U,
Functional dependency on the face solutions W is through the discretization of p and
8t/8. Taking Jacobian with respect to u+ and u- gives,
8pt apW 1 sgn ( . (4.80) au+ au- A \at )
Then using the chain rule,
apl(i)W W2+ ( + 8, a + awl+ 8u+ awl+ ap awl+ au+ (Wj+)2 P 8,4') a,4') au. W2- + ap8 apw + aw1_ -u aw1-_ 8u (4.81) ap ( 1+ aw + 8u+ 8tv + au+ 2 2 + 1) 8,4t') _j~i au+ 8pt (w2) + . aW2 8U- W2+. au-
120 Functional dependence of pt on A is both direct and indirect through 1,
ap' a,4p a,4 81 p12 8u pl \u\ at ) +) 8a - Re+2 - Re-. (4.82) aA aA I at 9 2 9t A 09Y I a
For a correctly linearized Newton system, the wall kinematic viscosity v. should be kept as a global variable with its definition as a global equation. However, for the low heat transfer cases presented in this thesis, v does not vary substantially across the boundary layer. As a result, changes in v,., in the Newton process may be approximated by changes in the local values of v with no adverse effects on the convergence of the entire procedure. Thus, v, is dropped as a global variable (as was done in Ref. [28]) to aid computational efficiency. Its changes are approximated by local v changes as follows:
\ 2{ ++~ ) (4.83) )x((6,++ +,p_) _ (P+ +P b1+ _ + 6~wl- \wl+wl- 2w2 1- 2wl+w2-
Thus, the Jacobian entries for IAand wi are modified,
(a a p+ ) toApi) aveM+ _ a,4yal.\21w1 ( 1 491A,+) a~w 491A, av, (2w,+w / (4.84) opaa,4' ~'- a4') Wpa,4ta ap a,4')Wv. a,4') _ (A+ +pP- y p+ p (awl+ 8W1+ + V 4p+ awl+ a8V wl+W-- and likewise for the ( ) - entries.
In the next section, the matrix system resulting from the Newton procedure is solved by Gaussian elimination. Global variables are treated as additional right-hand- side vectors in the solution of the matrix system. Thus, dropping vw as a global variable reduces the number of right-hand-sides and speeds up the solution of the system.
If the shear velocity u' based on the local shear stress and density is used rather than that based on the wall values, then u, is eliminated as a global variable. Its changes are written in terms of changes in the local shear and density. With a local '(y) defined by,
(y)I, -' r(y) = r+ T- , p() = wi+i-, (4.85)
121 the Jacobians of pt with respect to w, and r are modified by,
_ _a,4 m au' _a, ali (tL Ur (r+) au' r+ au'- \2 r,' It W (4.86) ______amt814) at ')_ u, + a~ (awi+ aw,+ 5-u- 51,+ aw,+ a' u,(2 wj+
and likewise for the ( ) - entries. The wall shear velocity is retained as a global variable in the analysis of this section, even though it ceases to be a global variable when this modification is used.
The outer model for the turbulent viscosity is functionally dependent on the local density and the incompressible displacement thickness 8k*,
(0) =,4o) (w1+, w1 -, 6)- (4.87)
Because of the approximation y/6 = tj/12 in Klebanoff's intermittency factor, the scal-
ing parameter A does not enter the outer model. By direct differentiation, apf") * = apu' Re, (4.88)
and by application of the chain rule,
amt!*) _ u(*)a0 _ ap4)(aue6kZyRe)(wi)F (4.89) 8w1, p 8w1,
Global Equations
The definition of the scaling parameter A (4.37) contains a sum over all the cells and
can be written as,
e = (A'+1 _j 1 [k +1 - - ,2AZ+ 1- 2A6 +A! 1 ], (4.90)
where the discrete equations for the incompressible displacement and momentum thick-
nesses in computational space are given by, J-1 J-i = (6k*)j+1/2 = z - 617l+1/2, (4.91a) j=1 j=1 Ue j+1/2j
J-1 J-1 ( E 9 A/+1/2. (4.91b) Ok = E(,Ajk)j+1/2 - j=1 j=1 ( ) j+1/2 Hej+1/2
122 Within each cell the velocity u is taken as the average of the upper and lower face midpoint values. The edge velocity ue is approximated by the viscous velocity at the edge to avoid possible difficulties due to mismatches with the outer inviscid flow. Thus,
Uj+1/ 2 = 2 +1 Ue = UJ. (4.92)
1 Linearization begins by taking Jacobians of & with respect to A+ , R, and jk,
= 1 - [Otk - tH* - 1 (4.93a) at Vn(o ae-g o ~= -v 1 __ ,-V +1 . (4.93b) _0_ a* H_
Next, 6 and Ok are linearized with respect to u3 , uj+1 and ue,
a(__ _)_+1/2_ a(As;*)+1/ 2 - 1 (4a+1/2 au u+i (4.94a)
8( ~x~+1o2 ( ~k)j+1/ 2 6 9j+1/ 2 - - 1 - 2 -- )j1/2 j+ (4.94b) au, auj+1 2 Ue/j+i/ 2 tL
a(A =),+1i+ -U) 2 j+1/2, (4.94c) a(.duk ,+1/2
[1=- 2 ( (...12U2 Af1 1+ 1/ 2 - (4.94d) aue Ue / 2J )+1/j+ 1 2 Combining derivatives by the chain rule gives the final Jacobian entries with respect to the local unknowns at each node,
aeraela(A6*I-/ _(___)_+_/ -- a j-1/2 + (4.95a) axLa6,I au, auJ +a, (I( A)j-1/2 +a ( d~k~11/2 auj aek au, auj ax,'
_a8t at at -1 ( a*))+1/2 a 8(Ak),+1/2 _u_ X~X, + +i E au, ]X .(4.95b)
For turbulent flows, the Cebeci-Smith model introduces two global unknowns, the incompressible displacement thickness b* and the wall shear velocity ur, into the defi- nition of the turbulent viscosity. The incompressible displacement thickness is given by the equation,
=Z*A-6, (4.96)
123 *
*
*
*
*
*
* *
*
*
*
* *
*
*
*
hence, at_ ac 8E ~ - = -1 (4.97) and at a6,* au- at Sae ate ak al (4.98) aw , 1 a *aui aw12 ' aW2, 46* aU. 49 23
The wall shear velocity u,. is defined by,
Uern, (4.99) hence, at = ae 2ewIi 1$ =+ (w11 ) (4.100) au, aw11 (W11)2> 09ri 2w,1I4ri[ (ri)
4.10.3 Jacobian Matrix
r W1 W2 W3 W4 T q @ - 1 W1i w2 W3 W4 q @ j * * *0 0 0 * * * 0 0 0 mass 1/2 0 * * * 0 * * 0 X-mom 1/2 * * 0 0 0 0 y-mom 1/2 0 0 0 0 0 0 0 * energy 1/2 0 0 0 0 0 0 * 0 shear 1/2 0 0 0 0 0 0 0 * enth flux 0 1/2 Bi Ai
W1 r A bk 0' W2 W W 4 q Oj+1 0 0 0 0 0 0 * 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 * * * * 0 * 0 0 * * * 0 0 * Ci Ui
Figure 4.5: Block Jacobian Matrices for Local Equations (* denotes possible nonzero entries)
Since the governing equations on each cell depend only on the unknowns at the
upper and lower faces of the cell (neglecting global unknowns), the Jacobian matrix
124
*
* *
* *
*
*
*
* * * * *
*
*
*
*
*
*
*
*
* could be written as a block bidiagonal matrix. Such a matrix is very easily inverted. However, the presence of wall and outer edge boundary conditions disrupts this block bidiagonal structure. To remedy the situation, these boundary conditions are placed in their natural positions (the diagonal blocks multiplying the wall and outer edge face changes) and the governing equations shifted downward in the matrix. This results in a block tridiagonal matrix. The matrix structure is completed with the inclusion of additional block multipliers for the global variables and a block row for the global
equations. The full system can be written schematically as,
A 1 C1 U1 - 6X 1 R
B2 A2 C 2 U 2 6X 2 R2
B, Ai C, U, 6X _ R? (4.101)
B j-1 A j-1 C j-1 Uj_1 6Xj-1 Rj_1 Bj A, U, 6X, Rj
G 1 G 2 - G G J . 6X. RG J where 6X, = (Vp1/2, 6p1/2u, 6P1/ 2v, 6p1/ 2 H, 6 r, 6q)T (4.102)
6 = (SwI, 6w 2 , 6W3 , 6w4 , r, 6q)T
are the changes in the local unknowns at each node, and
6XG = (6A, 66k, *26.)T (4.103)
are the changes in the global unknowns.
In the matrix structure, R, are the residuals in the local equations, and RG are the
residuals in the global equations. The block Jacobian matrices Aj, B, and C, contain entries for the local equations on each cell; the structure of these block Jacobians is shown in Fig. 4.5. Because the local equations are shifted, Bi and C, are only half populated. The Jacobian matrices in the j = 1 block row are modified by the presence of the wall boundary conditions. Likewise, the j = J block row is modified by the outer edge boundary conditions. The structure of these modified blocks are shown in
125 Fig. 4.6. Finally, the block Jacobians for the global equations have the structure shown in Fig. 4.7.
aj=1 @j=2 w r q WI W) W)4 wIw2 w 4 2 W3 r q A 6k U, 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 b.c. u=0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 b.c. v=0 0100 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 00100 b.c. q=0 * * * * 0 * 0* * 0 0 energy 0j=3/2 0000 * * * * * * * * * 0 *0 * * * shear Oj=3/2 * * * * * * * * 0 * 0* * * * enthflux@j=3/2
A1 C1 U 1
OJ-1 oJ a~~w wi - w~ sai q WIl W2~ W3 W)4 r q A6t UT *I *2 *3 *4 * 0 * * * 0 0 0 * 0 0 mass OJ-1/2 * ~ *0* * 0 * * * * *0 * 0 0 X-mom OJ-1/2 * * * * 0 0 * * * * 0 0 * 0 0 y-mom 0 J-1/2 0 0 0 0 0 0 **0 0 *0 * 0 0 b.c. r=0,U=uue 0 0 0 0 0 0 *0 0* 0* * 0 0 b.c. q=0,H=He 0 0 0 0 0 0 * * * * 0 0 * 0 0 b.c. p=Pe B, A, U,
Figure 4.6: Modified Block Jacobian Matrices at Boundaries
a j= 1 0 j = 2 to J A k* U, Wl W) 2 W)3 W4 T q WI1W2w3 W4 r q **0 0 0 0 **0 0 0 0 * 0 0 A defn. **0 0 0 0 **0 0 0 0 * * 0 ai defn. *0 0 0 * 0 0 0 0 0 0 0 * 0 * u, defn.
Figure 4.7: Block Jacobian Matrices for Global Equations
126 4.10.4 Gaussian Elimination
The linear system (4.101) can be more conveniently solved if it is rewritten as two systems,
A 1 C1 6X ' R ' U1
B 2 A2 C2 6X 2 R2 U 2
B, Aj C, 6X = Rj - U, 6XG
B,_1 A,-1 CJ-1 X,_ 1 Rj-1 UJ_1 B, A, J 6XJ J R, . U J (4.104a) 5X 1
6X 2
t5X, [G1 G2 -- G - G-1 G, Ga] =-R (4.104b)
6X,_1 6Xj
The first system is a standard block tridiagonal matrix equation with four right hand sides. Its solution is the local changes 6Xj in terms of changes in the global unknowns
6 XG. The solution of this system consists of a forward sweep to eliminate the B, blocks and invert the A, blocks, transforming the matrix into an upper triangular; and a backward sweep to solve for the changes SXj:
a) Forward sweep: (j = 2 to J)
1 1 C' = [A 1]~ C1 Rlt = [A1 ] -1R U" = [A1 ]- U1 (4.105a)
A' = Aj - BC'._ 1 R'. = R, - BjR'_1 U' = U, - BjUg_1 (4.105b) C'. = [A']-'Ci R! = [A']- 1 R'. U' = [A']-'U'
b) Backward sweep: (j = J - 1 to 1)
127 R= R' U = U" (4.105c)
R= W1 - C'Rfi+1 U =U - C' Ui+i (4.105d)
The solution of the first system produces changes of the form,
6X, = -R - U, 6XG (4.106)
Substitution into the second system (4.104b), which consists of the global equations, then gives a matrix equation involving only the global changes.
G G - -... - -... - GU]XG (4.107)
= -RG + G1N+..+ Gjij +..+ Ggi .
Solution by Cramer's rule (since the matrix is 3 x 3) gives the global variable changes
6XG, which when substituted back into (4.106) give the local variable changes. These are then added to the local and global variables to update the solution.
Computational efficiency is increased in two ways during the forward sweep of the block tridiagonal solution. The first is by using the sparseness of the B, blocks (they have three nonzero rows) in the matrix multiplications BjC'._ 1 , BjR'! 1 and BjU _1. The second is calculation of [A']-1 C, [A']'R and [A'--lU directly by L-U decompo- sition [23, p. 152-156] of the matrix A', with ten right-hand-sides (six for C, one for R and three for U). During the L-U decomposition of the first and last blocks (j = 1 and j = J), partial pivoting is used to avoid possible zero diagonal entries. This procedure was coded from a routine obtained from Haimes [37].
4.11 Similarity Profiles
Similarity boundary layer profiles are used for generating an inlet boundary layer profile and initialization of the viscous solution. These profiles are obtained by an assumed self-similar solution of the steady compressible Prandtl Boundary Layer equa- tions for a flat plate. The equations are obtained from the TSL Navier-Stokes equations in transformed coordinates (4.5) with the following assumptions:
128 1. Steady flow => a/at = 0
2. U(C, t7) is constant along lines of constant Y1 => 8F/8 = 0
3. Pressure is constant => p = p,
For turbulent flows, the second of these assumptions is not completely correct. The streamwise derivative aF/ac will be small but nonzero for a flat plate. Thus, an exact self-similar turbulent boundary layer cannot exist. However, exact self-similar profiles are not the objective here. Since these similarity profiles are used only for inlet boundary conditions and solution initialization, approximate self-similar profiles are adequate.
Furthermore, it should be noted that the effect of small errors in an inlet profile will
quickly decay further downstream of the inlet due to dissipation.
With the three assumptions listed above, the governing equations become,
- G - -- S - Ar;F =0, (4.108a) at? Re
Si (4.108b)
+ ++1 1- . (4.108c) qP= + + p A Pr Prt ar, 2A Pr Pr 8ar7 The Cebeci-Smith turbulence model is used to define pit for turbulent similarity profiles.
Wall and outer edge boundary conditions are,
=O V =O q =O( 4.1 09 ) 9= : u=0, v=0, q=0(419 17 0 U
17 =1r: U=Ue, H=He
The system is closed by specifying the rescaling parameter A and its derivative A,.
The rescaling parameter itself is defined similar to the normal scheme (4.33),
- 1 -- 1 =OrG + H 8*, o = 0.8. (4.110) H
The streamwise derivative A, must come from a similarity assumption, such as the
power law, A ~ X, = AZ = MA/X. (4.111)
129 This parameter is important because it determines the physical size of the boundary layer. For an incompressible boundary layer on a flat plate (Blasius solution), the displacement and momentum thicknesses are given by [87, p. 264],
6* 1.7208 x 0.664x M =1/2. (4.112)
This value of m is used for all compressible laminar profiles. For an incompressible turbulent boundary layer on a flat plate, the growth of the boundary layer thickness can be approximated by the relations [87, p. 495],
7 6 1 Rep. ~ 0.018 Re,6/ , Ree ~ 0.0142 Re, 7, 6/ m = 6/7. (4.113)
Likewise, this value of m is used for all compressible turbulent profiles.
These equations are discretized in the same manner as the TSL Navier-Stokes equa- tions of Section 4.4 and solved by the same Newton procedure with two notable excep- tions. First, if the initial profile guess (to be discussed shortly) is poor, then divergence can result. This is remedied by freezing the global variables (A, 6* and u,) for the first two Newton iterations to allow the profile to partially converge and then converging the entire system. Second, partial pivoting is used during the L-U decomposition of all blocks in the solution of the block tridiagonal matrix because zero diagonal entries may exist.
4.11.1 Initial Profile Guess
An initial guess for the boundary layer profile is needed to start the Newton proce- dure for the generation of similarity profiles. For both laminar and turbulent profiles, this is obtained using approximate profile shapes and empirical correlations for the wall shear stress and the incompressible displacement and momentum thicknesses.
130 Laminar
For an incompressible laminar flat plate flow, the empirical correlations are taken from White [87, p. 2641,
_ 1.7208z O0.664 x v-R -* e . (4.114) rw 1 Ok cf=- - . PeUe 2 Re z The u velocity profile is assumed to be a hyperbolic tangent matching rw and u,
[i_- e-,A U(I) = 1, a = 2 1 + e-C'l ' pUe (4.115) au pu, 2ae-'* ) y A (1 + e-a) 2
Pressure and total enthalpy are assumed constant through the boundary layer, and v velocity and enthalpy flux q are assumed zero. This then gives the following relation for the density variation through the boundary layer,
An)= P (4.116) 7-1 [H, - -lu(n)2 ly 2J
Turbulent
For an incompressible turbulent flat plate flow, the empirical correlations for 6*, 0 and cf are also taken from White [87, p. 495,498],
6 7 1 Re6 . = 0.0180 Re / 6* = x Re. = 0.0180 z Re,- , Re,
Ree = 0.0142 Ret6 / 7 -> 8 = = 0.0142 z Re,-1/ 7, (4.117) Re. TW 1 0.455 Cf = 1/2Pee 2 Re [ln(0.06Re.)]2
The u velocity profile is a composite of law-of-the-wall and law-of-the-wake relations,
U+= min y+, In y+ + 5.5 + 2.5 sin(2 2 ], (4.118) where the wall variables y+ and u+ are defined by
y =.u- + = . (4.119) VWtL Ur
131 The shear stress profile is assumed to be Gaussian,
r(q) = r, exp ]. (4.120)
As in the laminar case, pressure and total enthalpy are assumed constant, and v velocity and enthalpy flux are assumed zero.
4.12 Viscous Grid Generation
In computational space, the viscous grid is rectangular with stretching normal to the wall (v-direction). Since solution gradients are typically larger near the wall than in the outer region of the viscous grid, the grid is stretched away from the wall. Algebraic stretching is used for laminar flows,
77i = 17 _J , j=ltoJ (4.121) where typical values for the coefficients are j7 = 14 and 8 = 1.5.
Grid generation for turbulent flows is much more complicated since the solution is more grid sensitive. Experience has shown that good predictions of wall shear re- quire the first node off the wall to be well within the viscous sublayer (AyI < 20 or Aymin 5 0.026 for Re, ~ 106). With this constraint on Ay at the wall, exponential stretching is used to avoid unnecessary resolution in the outer region of the grid, exp (a ) - 1 17i = ( exp(a) - 1 = 1 to J (4.122) where Y1 = 14 is again typical. The difficulty lies in finding an a to satisfy the Ay+n constraint (which in turn depends on the wall shear). The procedure is as follows:
Using the definitions of y+, u, and c1 , the normal coordinate in computational space q can be written in terms of y+,
A7 . (4.123)
From this formula, a very crude estimate of Ar? corresponding to y = 10 can be made using the turbulent relations for Rea = Ree(Re,) and c1 = cf(Re.), an average
132 i for the boundary layer, and the following approximations: an average i, based on
M = 1, A = 0, !~ 1, and piv ~ p,. The resulting relation is,
=-Aqmin mi "n (4.124)
where Re2 = 2fi/i. The value of a which gives t72 = Amin is then obtained by a Newton procedure,
F(a) = Aqmin exp(a) - 1] - 7 [exp ( ) - F'(a) = Aimin exp(a) - rtj 1 exp ( ) (4.125)
tak =k+1 - ak = -F(ak)IF' (ak).
133 Chapter 5 Euler/Navier-Stokes Coupling
Separation of the flowfield into inviscid and viscous regions requires a procedure for coupling the respective solutions. In this chapter, a procedure is described for coupling the Euler algorithm developed in Chapter 3 to the Thin-Shear-Layer Navier-Stokes algorithm developed in Chapter 4.
Many procedures exist for viscous/inviscid coupling; Chapter 1 presents a short review of some of the previous work in this area. For solution of the Prandtl Boundary Layer equations in conjunction with the Potential or Euler equations, the technique of inviscid wall transpiration fluxes or distributed sources has proven very successful. The present thesis represents the first extension of this coupling technique to the solution of the TSL Navier-Stokes equations, both for steady and unsteady flows. The fundamental issues addressed in this extension are the matching of the inviscid and viscous solutions with this equation set and the performance of the technique for boundary layers which may have substantial thickness. To better isolate these issues, geometric complexity is removed by making simplifying assumptions on the Euler and viscous grids. These simplifications are that j grid lines are vertical, even for curved walls, and coincide for the two grids as shown in Fig. 5.1.
In review, the flowfield is solved using the Defect formulation of Le Balleur [50,51j, described in Section 2.3. At each point in the flowfield two equation sets are used to represent the solution. The first are the Euler equations and the second are the TSL
Navier-Stokes equations written in the form of Defect equations. The Defect equations reduce to an identity in the outer inviscid portion of the flow. Thus, they are solved to the edge of the boundary layer and outer boundary conditions imposed by the solution of the Euler equations. Within the boundary layer, the solution of the Euler equations
134 I I I I
o Solution: W, WX, Wy 0 - Geometry: z, y __-4- 0
a) Euler Grid
o Solution: W, T, q
* Geometry: C, q;
b) Viscous Grid
I I I I I
y U 1/X c) Composite Grid Figure 5.1: Composite Grid Topology for Euler/Navier-Stokes Coupling
135 has no physical significance. This pseudo-inviscid solution does not satisfy the zero mass flux condition at the wall. Instead, it satisfies boundary conditions driven by the viscous solution; these are referred to as wall transpiration conditions. Hence, coupling the two solutions involves determining outer edge boundary conditions for the viscous equations and boundary conditions at the wall for the Euler equations. These boundary conditions are presented in Sections 5.1 and 5.2, and the numerical coupling procedure for integrating the equations in time is presented in Section 5.3.
5.1 Interpolation of Edge Solution
The outer boundary conditions for the viscous solver are discussed in Section 4.7.2. To implement these, the inviscid solution is needed at the edge of the viscous grid. This solution is interpolated from the inviscid grid in a rather straightforward manner. The assumption that j grid lines coincide for the Euler and viscous grids reduces the geo- metric complexity of finding the edge of the viscous grid; it also makes the interpolation essentially one-dimensional.
At each streamwise station i, the inviscid cell containing the edge of the viscous grid (i.e. the midpoint of the outermost constant-it face) is first found. Because the description of the Euler solution is discontinuous from cell to cell, a weighted average of the extrapolated solutions from the nearest two cell centroids is used. This maintains a continuous inviscid edge solution when the edge of the viscous grid traverses an inviscid grid line. The procedure is as follows: From a search of the inviscid grid at station i, the edge of the viscous grid (x,, y,) is located between two cell centroids, denoted
(e,) Y) and (X,, y,)(2). Using the average and gradient information in each cell, two extrapolated solutions are obtained,
W -= W1 +(z -z ))Wx +(ye-ye))Wy), (5.1a)
2 W( ) - + (X, -(2))Wx + (yy(2))y 2), (5.1b) where W is Roe's parametric vector (2.74). These two extrapolated solutions are
136 weighted by factors based on the distance from (x,, y,),
2 We = W(l) + (1 - w)W( ), (5.2a)
2 8 Wx + (1 - w)Wx , (5.2b) awl W1 e = + (1 ) . (5.2c)
The weighting factor w is a cosine function,
SY - y(1) w= 1[+icos r 2) _ ) (53)
A cosine weighting is better than simple linear weighting because it is less suscepti-
ble to limit cycle behavior when the edge crosses the location of a cell centroid and
interpolation switches from the centroids at j - 1 and j to those at j and j + 1.
5.2 Wall Transpiration Fluxes
The wall transpiration fluxes provide the displacement effect of the boundary layer on the outer inviscid solution. As first suggested by Johnston and Sockol [45], these are
derived by integrating the viscous Defect equations (2.44b) across the boundary layer,
subject to the constraint that the viscous solution U asymptotes to the inviscid solution
U. Written in Cartesian coordinates for a flat plate, the transpiration fluxes are,
GW = GW - Sw + (F -IF) dy + ( - U) dy, Re ax Jo at0f (.4 where ( ), denotes a flux evaluated at the wall. Applying the wall no-slip and heat
transfer boundary conditions (2.16) to the viscous wall fluxes Gw and Sw then gives
the transpiration fluxes for each equation,
A 00a* PV x (W - pu) dy + 5j( -p) dy, (5.4a)
2 a f 2 - u a 00 LP ( = J(i2 a)+ (3 - p) dy + 5 fo(PAU- - pu) dy - Re (5.4b)
(^^2 +0 p8w+afo 00 (P02 + = Pt + f(3i P)w - puv) dy + f(P - pv) dy, (5.4c)
pH = puH)HpuH +- (^ A) a t0(p- pE) dy. Re (5.4d)
137 These wall transpiration fluxes may be easier to understand by examining the viscous
and inviscid profiles through the boundary layer. For example, Figs. 6.29 and 6.30 show
the velocity profiles for a typical laminar boundary layer. For incompressible, steady- state flow, the mass transpiration flux for steady-state is the integrated area between the viscous and inviscid u velocity profiles.
From an equations standpoint alone, the problem is readily solved. Regardless of the boundary conditions imposed at the outer edge, the four transpiration fluxes insure
the correct inviscid solution outside the boundary layer and a smooth matching of the
viscous and inviscid solutions. However, the problem becomes more complex when the
physics of the equations involved is analyzed. The unsteady Euler equations describe
acoustic and convective wave propagation. Thus, it is physically incorrect to specify
all four transpiration fluxes at the wall. The proper choice of boundary conditions has been addressed by Murman and Bussing [60] using a 1-D characteristic analysis normal
to the wall. For the usual case of flow into the domain, three conditions are specified, and for flow out of the domain only one is specified. The conclusions of Ref. [60] are that the mass transpiration flux (5.4a) should always be specified, and for inflow the x-momentum (5.4b) and energy (5.4d) transpiration fluxes should also be specified.
In the solution of the Prandtl Boundary Layer equations (Section 2.2.3), three con-
ditions are specified at the outer edge (the pressure is one by default). Therefore, specification of the mass transpiration flux for the Euler solution is enough to insure solution matching at the edge of the boundary layer. This means that the two addi-
tional conditions specified for inflow are somewhat superfluous. These correspond to the convective characteristics (2.55c) and (2.57). Lighthill has explained this for the case of steady flow. The primary effect of the boundary layer on the outer inviscid flow is a displacement of the streamlines away from the surface of the body. For the solution of the outer inviscid flow, this effect can be produced by physically thickening the body by the displacement thickness 6* of the boundary layer. It can be equivalently produced by a distribution of sources on the surface of the body. In this case, the streamline located
at the displacement thickness of the boundary layer divides the outer flow from all fluid injected at the surface. Hence, properties of this injected fluid that are convected along
138 streamlines, such as the entropy and vorticity, have little effect on the outer inviscid solution.
Outer edge boundary conditions for the TSL Navier-Stokes equations are discussed
in Section 4.7.2. The basic conclusion of the analysis is that the number and type of boundary conditions to specify are not clear. Hence, it is also not clear which transpi-
ration fluxes to specify for the solutions to match at the edge of the viscous grid. As a
result, the approach taken in this chapter is based on the physical argument that the pri- mary effect is the displacement given by the mass transpiration flux; any other specified
conditions have little effect on the outer flow. Therefore, only the mass transpiration flux is used for coupling to the Euler solution.
5.2.1 Modification of Euler Wall Boundary Conditions
Once obtained, the mass transpiration flux condition must be implemented in the wall boundary conditions for the Euler algorithm. In this thesis, only flows without
stagnation points in the inviscid portion of the flowfield are considered. Hence, the velocity reflection procedure of Section 3.7.1 is used for implementing the wall boundary
conditions. The modifications from a solid wall to a transpiration flux condition are
actually quite minor. For the solid wall condition, the ghost cell mass flux ^,, normal
to the wall is set equal to zero. Here, this ghost cell mass flux is equated to the
transpiration mass flux obtained from (5.4a),
(f34)ghost = in)transpiration- (5.5)
All other components of the ghost cell solution are taken from the solution extrapolated
to the wall from the interior. Hence, the tangential momentum and energy transpiration
fluxes are not specified for inflow, but are allowed to float to values predicted by the Euler algorithm.
From a characteristic analysis, specifying only the mass transpiration flux seems ill- posed, but experience has shown that it gives the best performance in terms of a smooth inviscid profile through the boundary layer. The reason why this works may be linked
139 to the fact that the mass transpiration flux is typically very small (e.g., (O/t), ~ 10-3); hence, little fluid is actually injected at the wall. Stated in other words, the transpiration condition is very close to the solid wall condition which has only one boundary condition specified. Because of this, any possible ill-posedness may be overshadowed by numerical dissipation.
5.2.2 Numerical Integration of Mass Defect Equation
To integrate the mass Defect equation across the viscous grid, it is first transformed into the viscous computational space (4.2),
A A 1 a 00 a Gw - j w = Gw - W-Sw - jxFw + A(F - F) dij + -U) dv. (5.6)
These complete transpiration fluxes are displayed here, but only the mass equation is actually used. In the present work, the same discretization developed in Chapter 4 for the solution of the Defect equations is used to numerically integrate the mass Defect equation across the boundary layer. Hence, at a given streamwise station, the discrete counterpart of (5.6) is,
(- X A)n+1 = (C - -zF)
J-1
+ E [(An6F)i+1/ 2 - (A"F);_-1/2] +1/2 (5.7a)
,at .= * i +1/2*
The equations are written here in vector form, but only the mass equation is used. Consistent with Section 4.6 the pseudo-inviscid profile is assumed to be linear and given by the edge solution and its gradients. Thus, the Defect integrands 6F and 6U are given by,
6Fj+1/2 = FJ-i/2 - Fj+1/ 2, 6Uj+1/2 = UJ-1/ 2 - U+1/ 2. (5.7b)
Since the discretization of these equations is the same as that used to integrate the
Defect equations one step in time, all terms are known. In particular the split flux vectors at i 1/2 are known and do not need to be recomputed.
140 In the present coupling procedure, the time derivative term in (5.7) has been found to have a destabilizing effect for the more difficult cases attempted. In these circumstances, this term is neglected and a quasi-steady assumption for the mass transpiration flux is used. This assumption is minor since the time term is often physically small (although it can cause unbounded numerical growth).
5.2.3 Proper vs. Approximate Transpiration Fluxes
As stated in Section 4.6, a proper discretization of the Defect equations results if the pseudo-inviscid solution is interpolated to each point in the viscous grid and substituted into the discrete viscous equations on each viscous cell. It is proper in the sense that no approximations in the pseudo-inviscid profile through the boundary layer are made. Likewise, this is also the proper way to numerically integrate the mass Defect equation across the boundary layer to obtain the mass transpiration flux. This has not been done in the present research. Instead, a linear pseudo-inviscid profile given by the edge solution and its gradients is assumed.
The distinction between proper and approximate discretization of the Defect equa- tions and mass transpiration flux is important. When approximations are made in the pseudo-inviscid profile, complete matching of the inviscid and viscous solutions at the edge of the boundary layer cannot be guaranteed. Furthermore, these mismatches will remain in the limit of infinite grid resolution. A complete matching is only possible for finite grids, when the Defect equations are properly discretized.
The distinction is also important when considering conservative formulations at the edge of the viscous grid. If the viscous and inviscid solutions do not completely match at the edge, then it is not possible to maintain conservation. There will be some mismatch in the Defect equations between the viscous and inviscid fluxes through the edge of the viscous grid. This is true of the present discretization. On the other hand, it is possible in principle to construct a conservative formulation at this grid interface, if a proper discretization is used (resulting in completely matched solutions). In this case, the Defect equations result in zero flux difference at the edge of the viscous grid.
141 The assumption of a linear or even constant pseudo-inviscid profile through the boundary layer is typical of viscous/inviscid coupling techniques appearing in the liter- ature. In practice a linear assumption is actually quite good. Only for cases of thick or rapidly changing boundary layers in conjunction with severe geometries or shocks is there any appreciable variation of the pseudo-inviscid profile. Numerically, nonlinear variations can only occur for large boundary layer thicknesses measured in terms of the number of inviscid cells in the boundary layer. For example, if the boundary layer is completely contained within a single inviscid cell, there will be essentially no difference between a proper and approximate discretization of the Defect equations.
A proper discretization has not been done in the present research for two reasons. The first reason, is because interpolation of the pseudo-inviscid solution to each point in the viscous grid was deemed too cumbersome for the expected gains in accuracy, as discussed in Section 4.6. The second reason is that evidence suggests that the procedure may be susceptible to numerical instabilities.
Prior to the use of the discrete Defect equations (4.23), an attempt was made to properly integrate the mass Defect equation. The inviscid fluxes calculated from (3.48) were stored and then integrated across the boundary layer to determine the mass tran- spiration flux. This was possible since streamwise stations coincide for the viscous and inviscid grids. The resulting coupling procedure had poor stability, which degraded as the number of inviscid cells within the boundary layer increased. Odd/even decoupling modes often appeared in the transpiration fluxes and in the inviscid solution within the boundary layer. Smoothing of the transpiration fluxes gave only minor improvement to the stability. A possible explanation for this poor stability could be that (5.4a) predicts the value of transpiration flux which gives a matched solution at the edge, regardless of the pseudo-inviscid profile across the boundary layer. Thus, if the profile is not phys- ically realistic, neither will the transpiration flux be. To reiterate, this was attempted without solving a proper discretization of the Defect equations, and it is not clear what effect that would have on the determination of the wall transpiration fluxes and the pseudo-inviscid profile. Regardless, this experience suggests that care should be used in attempting a proper discretization of the Defect equations and determination of the
142 wall transpiration fluxes.
Similar attempts at such an integration of the Defect equations across the boundary layer seem to have not been made by previous researchers. Part of the reason for this is that all past work on viscous/inviscid interaction using transpiration fluxes has in- volved solution of the Prandtl Boundary Layer equations in various forms. In situations where there are appreciable variations in the pseudo-inviscid profiles, the validity of the Boundary Layer equations themselves becomes questionable.
Using the edge solution to describe the pseudo-inviscid profile has other consequences when the boundary layer is several inviscid cells thick. The first is inaccuracies in pre- diction of wave propagation in unsteady flows. Waves in the inviscid solution travelling normal to the wall are only seen by the viscous solution when they pass the edge of the viscous grid. Thus, false reflections off the wall are possible for higher frequency waves, whose wavelengths are small compared to the boundary layer thickness. How- ever, this is not a problem for the type of channel flows considered in this thesis. They are dominated by longitudinal waves, which are correctly predicted.
A second consequence of approximating the pseudo-inviscid profile by the edge solu- tion is a possible unstable feedback mechanism between the edge solution and the wall transpiration fluxes. If the boundary layer is several inviscid cells thick, then distur- bances at the wall take a finite time to reach the edge of the boundary layer. On the other hand, the transpiration fluxes react instantly to disturbances at the edge.
In practice symptoms of this unstable feedback mechanism have definitely been observed for supersonic flows. This subject is discussed in more detail for the specific case of an oblique shock interacting with a laminar boundary layer in Section 6.3.2. In that case the instability appears as a streamwise oscillatory divergence rather than a divergence in time; it is eliminated by a seemingly small change in the outer boundary conditions on the viscous solver. For subsonic and transonic flows it is not clear whether this feedback mechanism has been observed. Interactions between different elements of the code in Appendix F are sufficiently complex that it is often difficult to pin-point the cause of observed instabilities.
143 In principle, this unstable feedback mechanism could be prevented by using the inviscid solution interpolated to the wall for the outer edge boundary conditions on the
viscous solver. Then there would be no time lag for disturbances at the wall to be
felt by the edge solution. It would produce somewhat less satisfactory matching at the
outer edge, but this would be outweighed by the expected gains in stability. However, numerical experiments have shown this not to be the case. In fact, this is far less stable than using the inviscid solution interpolated to the edge of the viscous grid. The reason for this may be linked to the fact that the inviscid solution within the displacement
thickness of the boundary layer is nonphysical. It is also nonunique in that the entropy
and vorticity of the incoming fluid at the wall may be specified. If this is the case, its rectification is very difficult. It means that the inviscid solution interpolated to the wall must be obtained from the inviscid flow outside the displacement thickness- a procedure which involves the original feedback loop. That this modification of the coupling is unstable also lends evidence to suggest that problems may occur in a proper discretization of the Defect equations.
5.3 Numerical Coupling Procedure
Evolution of the flowfield in time is by means of an explicit coupling procedure between the Euler and TSL Navier-Stokes algorithms. At each time step the inviscid
and viscous solutions are integrated separately. Communication between the two is through the viscous outer edge boundary conditions (Section 5.1) and inviscid wall transpiration fluxes (Section 5.2). The procedure is schematically presented in Fig. 5.2. Given the transpiration fluxes at time level n, the Euler algorithm of Section 3.4.3 is integrated one time step to produce the updated inviscid solution Un+. Based on the position of the viscous grid at the previous time step, the inviscid edge solution 1 Ue+ and its gradients are interpolated from the updated inviscid solution using (5.2a).
With the updated edge solution, the TSL Navier-Stokes algorithm of Chapter 4 is then integrated one time step; this gives the updated viscous solution Un+ 1 . Finally, the mass Defect equation is integrated across the boundary layer by (5.7) to produce the
144 Euler time step 1 Un - un+
Interpolation of edge solution
A n+1
Viscous time step Un Un+1
Un+ 1
Mass transpiration flux
I(p)"l+1
Figure 5.2: Flow Chart for Numerical Coupling Procedure
145 updated mass transpiration flux at the wall for the next time step. This procedure is not iterated at each time step to converge the edge solution and transpiration fluxes.
Hence, these quantities are first order accurate in time, but this is not detrimental since the viscous solver itself is only first order accurate in time.
For steady-state calculations, the implicit approximate factorization techniques dis- cussed in Appendix D are applied to the changes in the Euler solution prior to interpo- lation of the edge solution. They are also applied to the changes in the viscous solution after calculation of the transpiration fluxes.
This form of explicit coupling is typical of viscous/inviscid interaction techniques involving solution the Boundary Layer equations. These may be contrasted to implicit coupling techniques, such as the Euler/Integral Boundary Layer algorithm of Drela and
Giles [30], where the entire flowfield is solved as a single system. The explicit methods have the advantage that the inviscid and viscous schemes are isolated from one another, allowing different schemes to be interchanged. This increases flexibility and eases ex- tension to more complex interaction problems. Unfortunately, these advantages over implicit methods are gained at the cost of reduced stability and robustness. Typically, the coupling procedure is the limiting factor in obtaining converged solutions.
Techniques exist to improve the stability of explicit coupling. For steady-state cal- culations, the inviscid solution is often updated more frequently than the viscous so- lution [51,88,72]; this is essentially under-relaxation of the viscous solution. Another technique is under-relaxation of the edge solution and transpiration fluxes. This may also be used for unsteady flows if the under-relaxation is uniform throughout the do- main, but it results in degraded time accuracy. In general, these techniques are con- structed and used without the aid of a stability analysis. An exception is the analysis of Le Balleur [49] for his Potential/Integral Boundary Layer algorithm. He assumed an exact solution of the steady linear Prandtl-Glauert equation coupled with a simplified one-equation model for the Integral Boundary Layer equations.
The present coupling procedure uses no under-relaxation techniques. Furthermore, the stability of the present coupling is very difficult to analyze for several reasons. Nor-
146 mal Fourier analysis is possible only for constant coefficient systems, and the boundary layer is by its very nature nonlinear; therefore, linearizations of the governing equa- tions have variable coefficients. Normal Fourier analysis also does not take into account boundary conditions, which are the real issues here. Hence, stability could only be analyzed as a generalized eigenvalue problem, and is deemed beyond the scope of this thesis.
147 Chapter 6 Discussion of Results
The Euler and Thin-Shear-Layer Navier-Stokes algorithms developed in the previous three chapters have been coded (see Appendix F), and are demonstrated by a series of test cases in the present chapter. The first section presents inviscid test cases demon- strating the Euler algorithm of Chapter 3; the second presents test cases for the TSL
Navier-Stokes algorithm of Chapter 4 without coupling to an outer inviscid solver; and the final section presents cases using the fully coupled Euler/Navier-Stokes schemes.
6.1 Results for Euler Algorithm
In this section, the Euler algorithm using the spatial discretization developed in
Section 3.4.3 is demonstrated by six numerical test cases. The first three are 2-D channel test cases showing the overall quality of results obtained for subsonic, transonic and supersonic flowfields. The fourth test case is a subsonic 2-D channel used to verify the spatial order of accuracy. The fifth test case is a 1-D shock tube problem, whose results are compared to the analytic solution. The final test case is an unsteady quasi-1-D Laval nozzle with forced exit pressure oscillations showing the shock tracking capabilities of the algorithm.
All numerical results presented in this section and in Section 6.3 were obtained using the second order scheme of Section 3.4.3. To reiterate, the scheme stores both averages and gradients for each cell. These are evolved in time using a finite volume discretization of the Euler equations and the first moments of the Euler equations.
Nonlinear consistency is maintained and two-point Gauss quadrature is used to evaluate fluxes on each face. The discretization results in second order accuracy for both averages
148 and gradients for steady flow (gradients are spatially first order for unsteady flow). All test cases were run using the actual analytic wall normals at each Gauss point as discussed in Section 3.7.2.
Three test cases presented in this section use the characteristic procedure (Sec. 3.7.2) to implement the solid wall boundary conditions. The subsonic bump case (Sec. 6.1.1) and the subsonic sin2 x bump case (Sec. 6.1.4) use this procedure on both the upper and lower walls. The transonic circular bump case (Sec. 6.1.2) uses this procedure on the entire upper wall. It is also used on the lower wall from the inlet to the point where the flow first becomes supersonic. The velocity reflection procedure (Sec. 3.7.1) is used downstream of this point. All other cases presented in this section and in Section 6.3 use the velocity reflection procedure. With the exception of the supersonic circular bump case, none of these flows have stagnation points in the inviscid portion of the flowfield.
6.1.1 Subsonic Circular Bump
The first test case is a 20% circular arc cascade at zero incidence with an inlet
Mach number of Miniet = 0.5. This case may equivalently be described as a 10% thick circular arc bump in a channel, and is sometimes referred to as the subsonic Ni bump after Ni [62] who was one of the first to use it. Results are presented for this case in
Fig. 6.1 for a 64 x 16 grid, shown in Fig. 6.1a. The grid was algebraically generated with constant Az along the bump and algebraic stretching away from the bump. The flowfield is completely subsonic with stagnation points at the leading and trailing edges of the bump, as shown by the Mach number contours (Fig. 6.1b) and upper and lower wall distributions (Fig. 6.1e). For this case overall quality of the solution is indicated by symmetry of the flowfield about the bump and the amount of total pressure production.
The distribution of total pressure change (1 - P/Piniet) within the flowfield is shown in
Figs. 6.1d and g, where the stagnation or total pressure is given by,
P = p 1+ 2 lM2] (6.1)
Because the flowfield is completely subsonic and uniform entropy is specified across the inlet, there should be no total pressure variations in the flowfield. Thus, all total
149 pressure changes are numerical iii nature. As the results indicate, most of the total pressure errors occur in the vicinity of the leading and trailing edges of the bump, where the rapid variation of the flow is not adequately resolved.
To show the effects of the wall boundary condition procedure on the solution, this test case was rerun using the velocity reflection procedure (Sec. 3.7.1). Selected plots of the resulting solution are shown in Fig. 6.2. The solution is noticeably inferior on the lower wall compared to the plots in Fig. 6.1, which were obtained with the charac- teristic boundary procedure (Sec. 3.7.2). The total pressure wall plot (Fig. 6.2c) shows approximately 1% change in P through both the leading and trailing edge stagnation points. The contour plots show this change is mostly confined to the first row of cells adjacent to the wall. This results in a noticeable numerical boundary layer. For the characteristic procedure solution, the total pressure change in the immediate vicinity of the stagnation points is approximately 0.5% and decreases to zero away from the stagnation points. No numerical boundary layers are seen in the corresponding contour plots. Additional comparisons of these two results are given in Section 3.7.2.
The contour and wall plots presented in Fig. 6.1, as well as those in Fig. 6.2, were generated using a somewhat unorthodox plotting procedure. Typically, solutions of cell-centered schemes are graphed by first interpolating (e.g., averaging) the solution to nodes and then plotting the resulting bilinear solution over each cell. This results in a visually continuous flowfield. In the present scheme, interpolation to nodes is not necessary because the averages W and gradients WX and Wy already describe a linear distribution within each cell. It is this linear distribution within each cell that is plotted in Fig. 6.1. As stated in the development of the Euler algorithm, the numerical description of the solution is discontinuous from cell to cell, and this can be seen in Fig. 6.1. This manner of plotting has been chosen because it more closely represents the actual discrete solution. The resulting plots are visually less pleasing (because of the minor discontinuities) than if the solution were interpolated to nodes prior to plotting. However, they convey more information about the properties of the algorithm, especially the values of the flow gradients. In particular they reveal the complete absence of unconstrained decoupling modes. They also show a remarkably
150 smooth solution, even though there are no explicit constraints in the algorithm on the continuity of averages and gradients throughout the flowfield. In the plots of Fig. 6.1, the only regions where the solution discontinuity is easily noticed are near the leading
and trailing edge stagnation points, which are not adequately resolved.
This plotting procedure is used to display all results obtained with the Euler al- gorithm of Chapter 3, including the inviscid portion of Euler/Navier-Stokes flowfields presented in Section 6.3. A comparison of plots obtained with a more normal plotting procedure is presented in the next test case.
6.1.2 Transonic Circular Bump
The second test case is the same geometry at a inlet Mach number of Miniet = 0.675. The grid is the same as that used for the subsonic case and is shown in Fig. 6.3 along with the resulting solution. With the higher inlet Mach number a supersonic region develops on the surface of the bump and terminates in a shock. This shock is captured with only one internal cell as the contour and wall distribution plots indicate. Furthermore, there are no pre- or post-shock oscillations in the discrete solution, but there is an overshoot in the gradient of the cell containing the shock. Methods using explicit smoothing to capture shocks usually smear the shock over three or four cells and produce oscillations in the process.
The dip in the solution immediately downstream of the shock in Figs. 6.3e and f is not a numerical oscillation. It is a Zierep singularity in the Euler equations resulting from an incompatibility between the normal shock relations and the zero mass flux condition on a curved wall. This singularity is better resolved at this grid density by the present scheme than other schemes, such as Ni's scheme [62] and Jameson's cell- centered central-differencing scheme [3], because the shock is not smeared.
Since the flowfield is inviscid, total pressure losses should only occur at the shock.
Figures 6.3d and g indicate that the majority of total pressure loss occurs at the shock, but numerical losses also occur elsewhere on the bump. The difference in total pressure
151 losses through the leading and trailing edge stagnation points is the result of differ- ent wall boundary condition procedures; the characteristic procedure was used for the leading edge and the velocity reflection procedure for the trailing edge.
From Fig. 6.3g, there is a 3.72% jump in total pressure loss across the shock; this translates into a total pressure ratio across the shock of 0.9628. Based on a pre-shock
Mach number of 1.377 (obtained from Fig. 6.3e), the normal shock relations [52, p. 62]
predict a total pressure ratio of 0.9637 or 3.63% total pressure loss. The difference is approximately 2% relative loss, indicating very good agreement.
Also worth mentioning is the fact that the solution is smooth through the sonic
point. Like the example presented in Section 3.7, the gradients near the sonic point in this case are greatly affected by the proper choice of outward unit normals in the implementation of solid wall boundary conditions.
This case is another reason for the choice of plotting procedure. As a comparison, this same solution has been replotted in Fig. 6.4 using straight-forward interpolation
to nodes. The resulting flowfield appears completely continuous. However, the plotting procedure has resulted in apparent smearing of the shock over at least two cells with overshoots. Away from the shock and stagnation points, the two plotted solutions appear essentially identical.
6.1.3 Supersonic Circular Bump
The third test case is a 4% thick circular arc bump in a channel with a supersonic inlet Mach number of Minit = 1.4. This flowfield is dominated by oblique shock gen- eration, reflection and interaction. The purpose of this case is to show the capture of oblique shocks by the present algorithm. The grid for this case is 64 x 16 cells and is shown in Fig. 6.5a; it was generated similarly to those of the other circular bump cases. The solution obtained on this grid is shown in Figs. 6.5b-g.
The flow in the inlet section of the channel is uniform, and an oblique shock forms
152 at the leading edge of the bump. This leading edge shock is weakened and curved by expansion waves emanating from the surface of the bump. A second shock forms at the trailing edge of the bump. The leading edge shock reflects off the upper wall (forming a normal Mach stem), intersects the trailing edge shock, and reflects off the lower straight wall downstream of the bump. The two shocks coalesce prior to the exit. These flow features are also shown by Shapiro [751 using an adaptive method to refine the grid near the shocks.
The wall distribution and contour plots illustrate that oblique shocks are not cap- tured with one internal cell as opposed to normal shocks. This is because flux-splitting at a given face is based on the local Mach number normal to the face, rather than the Mach number normal to the inclined shock. As a result subsonic splitting errors may allow information to propagate upstream in the direction normal to the shock. This is probably also the cause of post-shock oscillations visible in the upper and lower wall distribution plots.
This degradation of captured shocks inclined to the grid is the result of 1-D splitting based on the grid directions. Thus, it should occur for all flux-splitting formulations and is not restricted to the van Leer formulation used in this thesis.
6.1.4 Order of Accuracy Study
The fourth test case is an study of the spatial order of accuracy of the present scheme. The geometry is the same as the preceding cases, except the circular bump is replaced by a 10% thick sin 2 x bump which blends smoothly into the straight portions of the lower boundary. An inlet Mach number of Miniet = 0.5 is specified resulting in a completely subsonic flowfield.
Figure 6.6 shows the grids used to determine the effect of grid resolution on numerical losses. Each grid was generated using the same procedure as the previous test cases.
Figure 6.7 displays the solution on the 64 x 16 grid. It shows that the smooth geometry eliminates the stagnation points at the leading and trailing edges of the bump and their
153 associated large truncation errors. Because the flowfield is completely subsonic and uniform entropy is specified at the inlet, all losses in the flowfield are purely numerical.
In this study, the root-mean-square of total pressure loss over the domain is used as a measure of these numerical losses; it is defined as,
p ' (6.2) Pinlet 2 = [inle t
where I and J are the number of cells in x and y, respectively, and Pi,, is evaluated based on the average solution of each cell. Total pressure loss has been found in the past to be a good indicator of accuracy for Euler solvers.
Figure 6.8 shows the effect of grid resolution on numerical losses for two versions of the present scheme and Jameson's cell-centered central-differencing scheme [43]. The slope of each line in Fig. 6.8 is the order of accuracy of that scheme. The slope for the present scheme using averages only (Section 3.3) is -0.70. Analytically, this version of
the scheme is first order accurate, but in this norm it is not even first order. The second order version of the present scheme using both averages and gradients (Section 3.4.3) has a slope of -2.00, verifying that it is indeed spatially second order accurate. For
comparison, the central-differencing scheme (implemented by Allmaras and Baron [3,4])
is also plotted. Even though the grids are fairly smooth, this scheme fails to obtain
second order accuracy, since the slope of its line is only -1.41.
In the development of the present algorithm, no assumptions of grid smoothness were made. Thus, the scheme should remain second order accurate for grids which
are less smooth than those in Fig. 6.6. This expectation is confirmed in the following pathological test. Figure 6.9 shows four grids obtained by randomly perturbing all the interior nodes of the grids in Fig. 6.6. Each node was randomly perturbed in both x and y independently. For each node the new x position was allowed to move up to half
the minimum distance to the nearest two nodes,
X ,*O"= z,j + (1/2 - ai,) min [IXi, - Z;il,j1, tXi+1, 1 - zij1, (6.3)
where a, 3 is a random number uniformly distributed between 0 and 1. Displacements
in y were calculated similarly.
154 Figure 6.10 displays the solution on the finest of the randomized grids in Fig. 6.9
obtained with the second order version of the present scheme. Compared with Fig 6.7, this gives an indication of the lack of solution degradation with grid distortion. A further
measure is presented in Fig. 6.11. This figure shows total pressure errors computed
on the randomized grids of Fig. 6.9 compared to the smooth grids of Fig. 6.6. The
results demonstrate that the scheme remains second order accurate even for these very nonsmooth grids. In fact, there is only a minor increase in total pressure errors compared to the smooth grids.
It should be clarified what is responsible for the second order accuracy of the present
algorithm. The lowest curve in Fig. 6.8 was obtained with the spatial discretization of Section 3.4.3, which uses two-point Gauss quadrature of the fluxes on each face. How-
ever, this Gauss quadrature is not the reason for the second order accuracy; its main
purpose is to constrain decoupled modes in the gradients. The element that is respon- sible for the accuracy is storing both averages and gradients, where the gradients are
known to at least first order accuracy. As a result, all interpolations to faces (which
are used in the evaluation of fluxes) are second order accurate regardless of cell shape. Addition evidence for this statement is given by Allmaras and Giles [5], where a sim-
ilar order of accuracy study was performed. In that study, fluxes on each face where evaluated by the midpoint rule, similar to that described in Section 3.4.2. Like the present study, second order accuracy was obtained in that paper. Note also that the flux-splitting does not degrade the second order accuracy.
6.1.5 1-D Shock Tube
The first unsteady test of the Euler algorithm is a 1-D shock tube problem. Results are compared to the analytic solution. The particular case presented here has been used by Sod [74] for the comparison of several finite differencing schemes.
The domain extends from x = 0 to x = 1 and is divided into 100 uniformly spaced cells. The air (-y = 1.4) in the tube is initially at rest and a diaphragm is placed at x = 0.5. Across the diaphragm, there is a pressure jump of 10 and a density jump of
155 8; to the left p = 1 and p = 1, and to the right p = 0.1 and p = 0.125. When the diaphragm is burst (at t = 0+), a shock wave propagates to the right followed by a contact discontinuity. To the left an isentropic expansion fan propagates. Across the shock, which moves at a constant speed, all flow quantities are discontinuous and satisfy the unsteady Rankine-Hugoniot relations. The pressure and velocity are continuous across the contact discontinuity, but the density has a jump discontinuity. The speed of the contact discontinuity is constant and equal to the local fluid velocity. Within the expansion fan, the fluid velocity increases linearly. The front and back of the expansion fan propagate at different constant speeds; the front at the speed of sound in the stagnant fluid to the left. Between the shock and the contact discontinuity, the flow properties are uniform. They are also uniform between the contact discontinuity and the back of the expansion fan.
This flowfield is an example of Riemann's initial value problem, briefly discussed in Section 3.2, and has an analytic solution given by Liepmann and Roshko [52]. The flowfield is self-similar with constant properties along rays of constant x/t emanating from the diaphragm at t = 0.
Starting from the specified initial conditions, the flow was integrated at a CFL of
A = 0.55 for 57 time steps to t = 0.14492; this corresponds to a shock position of x = 0.754, which is approximately the position Sod used for his comparisons. The instantaneous density, pressure, velocity and energy in the tube are plotted in Fig. 6.12.
These distributions are compared to the analytic solution at the same time (t = 0.14492). Agreement is very good in general. The positions of the shock, contact discontinuity, and expansion fan are all correctly predicted. The shock is captured in essentially one cell. These results are superior to all schemes presented by Sod and compare closely the results of van Leer's second order Godunov scheme [83], shown in Fig. 6.13.
Few noticeable errors are seen in the numerical solution. Small oscillations appear on both sides of the shock, contact discontinuity, and expansion fan. These are mainly the result of dispersion errors. Part of these errors may also be attributed to van Leer's flux-splitting. To the right of the shock and to the left of the expansion fan, the flow is
156 stagnant, where the split fluxes are farthest from the physical fluxes. Another error is that the contact discontinuity in the density and energy plots is smeared over 3 or 4 cells.
Again, the flux-splitting may be partly responsible, since it has not been developed to maintain contact discontinuities without smearing. Since the flux-splitting formulation is treated as a "black box" in the present algorithm, these errors may be reduced by other flux-splitting techniques.
6.1.6 Unsteady Quasi-1-D Laval Nozzle
Self-consistency of the algorithm for unsteady shock tracking is studied using a quasi-
1-D channel flow with forced oscillation of the exit pressure. Results are compared for
a coarse grid of 33 cells and a very fine grid of 257 cells. For purposes of comparison, the 257-cell grid is essentially grid resolved to plotting accuracy and may be considered the "exact" solution, based on the results of the previous test case. In addition, results have also been computed using a quasi-1-D version of Jameson's cell-centered central-
differencing scheme [43] on a grid of 65 cells. The smoothing formulation and coefficients for Jameson's scheme are identical to that used by Allmaras and Baron [3,4] for steady- state calculations.
The test problem is a converging/diverging nozzle with area ratio given by,
h(x)_ _ h(x.5) = 1.25 - x(1 - X), 0 < X 1, (6.4)
where the throat is located at x = 0.5. Actual computations for the present algorithm
were performed on a symmetric 2-D nozzle with throat height of h(O.5) = 0.01 using a single row of cells. The exit pressure ratio (exit static to inlet stagnation) is equated
to 0.787 to produce a shock at x = 0.75 in the steady-state. This same steady test case
was studied by Anderson et al [7] for comparing different flux-splitting methods.
Figure 6.14a shows the steady-state pressure distribution along the nozzle for the 33
and 257 cell grids, both using uniform Ax spacing. The shock is captured on both grids in one cell, and the two solutions are identical to plotting accuracy everywhere except
the shock. As in the 2-D transonic circular bump case, there are no oscillations in the
157 vicinity of the shock on either grid. Indeed, the difference in the shocks is due solely to the fact that the shock cannot be better resolved than this on a 33-cell grid.
As a comparison, the solution for Jameson's scheme on a uniformly spaced grid of 65 cells is also shown in Fig. 6.14b. The symbols represent Jameson's solution at cell- centers (no plotting interpolation is done), and the dashed line represents the present solution on the very fine grid. As stated in Chapter 1, Jameson's scheme was originally developed and tuned for steady calculations; hence, the shock is captured with only minor oscillations and the agreement with the very fine grid is quite good. The odd/even oscillations in the solution could be greatly reduced by post-processing (e.g., averaging to nodes).
Using the steady solution as a starting condition, the exit pressure was sinusoidally oscillated with a magnitude of 15% (of the exit pressure) and a reduced frequency of 2 (based on channel length and sonic speed of sound). Both the coarse and fine grid calculations were run at a CFL of A = 0.55, corresponding to 816 iterations per cycle for the coarse grid and 6368 for the fine grid. Figure 6.15 displays a perspective plot of the coarse grid pressure variation along the channel during the initial three cycles of the exit pressure oscillation. The pressure at every 22.50 of phase is shown. The plot shows that compression waves issuing from the exit steepen into shocks as they move upstream and that the nozzle becomes unchoked during part of the cycle. It also shows that the entire flowfield is nearly periodic after three cycles.
Figures 6.16a-6.19a show a comparison of the solutions for the present algorithm on the coarse and fine grids at four phase angles of the third cycle. The plots show excellent agreement in the position and structure of the moving shock, even though the grids differ by a factor of eight in resolution. This demonstrates the accuracy of the scheme in predicting shock motion.
The results for the coarse grid are even more impressive when compared to those of Jameson's scheme at twice the resolution, shown in Figs. 6.16b-6.19b. The numerical oscillations at the shock that are minor in the steady-state become very pronounced for the moving shock. In particular, the solution at 90* phase (Fig. 6.17b) is quite revealing.
158 The strong shock near the exit is predicted in the correct position with approximately the correct pressure jump, but significant odd/even modes are present. Even more disturbing is the weaker shock moving upstream of the throat. Not only is it smeared over several cells by numerical errors (making it more a compression wave than a shock), but its predicted position is simply wrong.
It may be argued that the smoothing coefficients in Jameson's scheme could be tuned to give better results for this unsteady case. But the results of Fig. 6.17b alone disprove this. Decreasing the smoothing would result in worse numerical oscillations near the strong shock. Increasing the smoothing would further smear the weak shock and increase the error in its predicted position.
Only Jameson's central-differencing scheme has been presented here, but the re- sults should be representative of an entire class of Euler algorithms. The same type of problems should occur for any scheme which uses explicitly added smoothing to cap- ture shocks, especially when the smoothing is constructed and tuned for steady-state calculations.
159 ------1111 T------
a) Computational Grid (64 x 16)
| .55
b) Mach Number Contours (A = 0.01)
0.86S
c) Density Contours (A =0.005)
:zA 0.0
d) Total Pressure Loss Contours (A =0.1%)
Figure 6.1: Subsonic Circular Bump: M = 0.5, T = 10%, 64 x 16 grid, Characteristics Procedure
160 0.8 -
M
0.6- upper
lower 0.4 -
0.2 e) Mach Number Distribution on Walls
1.0-
p
0.9- lower
upper
0.8
f) Density Distribution on Walls
1 -N lower loss '/ 0
upper (sero)
-1
g) Total Pressure Loss Distribution on Walls
Figure 6.1: Subsonic Circular Bump: M = 0.5, r = 10%, 64 x 16 grid, Characteristics Procedure
161 o5~5
7 (
a) Mach Number Contours (A = 0.01)
b) Total Pressure Loss Contours (A = 0.1%)
3.0 loss lower
2.0
1.0
upper 0.0
-1.0 -I
c) Total Pressure Loss Distribution on Walls
Figure 6.2: Subsonic Circular Bump: M = 0.5, r = 10%, 64 x 16 grid, Velocity Reflection Procedure
162 a) Computational Grid (64 x 16)
o0.70
b) Mach Number Contours (A = 0.05)
a0.80
c) Density Contours (A = 0.01)
- I I
d) Total Pressure Loss Contours (A = 0.5%) Figure 6.3: Transonic Circular Bump: M = 0.675, r = 10%, 64 x 16 grid
163 1.5-
M
1.0-
upper
0.5 Vv ower
01 e) Mach Number Distribution on Walls
0.9-
upper
0.7- lower
0.5-
0.31 1 1 f) Density Distribution on Walls
10- loss lower
0-
5 II I I II g) Total Pressure Loss Distribution on Walls Figure 6.3: Transonic Circular Bump: M = 0.675, r = 10%, 64 x 16 grid
164 0.70
a) Mach Number Contours (A = 0.01)
1.5-
M
1.0
0.5- lower
0- b) Mach Number Wall Distribution
Figure 6.4: Transonic Circular Bump: M = 0.675, r = 10%, 64 x 16 grid, Solution Interpolated to Nodes for Plotting
165 a) Computational Grid (64 x 16) .220
IT
b) Mach Number Contours (A = 0.05)
-.+8
c) Density Contours (A = 0.01)
2.0 %
Tai I(. =0.5% d) Total Pressure Loss Contours (A = 0.5%) Figure 6.5: Supersonic Circular Bump: M = 1.4, r = 4%, 64 x 16 grid
166 1.8 -
M
1.4- A"
V--Iower 1.0 - I- lower upper
0.6 e) Mach Number Distribution on Walls
0.9 -
p
0.7 - lower upper
,
0.5-
0.3 f) Density Distribution on Walls
10 -
%loss
5 - lower
lower upper
0- i
-5- g) Total Pressure Loss Distribution on Walls Figure 6.5: Supersonic Circular Bump: M = 1.4, r = 4%, 64 x 16 grid
167 8 x 2
16 x 4
32 x 8
64 x 16 Figure 6.6: Order of Accuracy Study: Computational Grids
168 a) Mach Number Contours (A = 0.01)
0.9-
M
-lower 0.7- 0.7
0.5-
0.31 b) Mach Number Wall Distribution Figure 6.7: Order of Accuracy Study: Mach Number Distribution for 64 x 16 Grid
169 -1.0, log Ill - P/Piietlrms
-1.5-
-2.0- N First order (averages only)
-2.5 -
-3.0- Central-differencing [431
-3.5-
Second order (averages and gradients) -4.01 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 log(I)
Figure 6.8: Order of Accuracy Study: Effect of Grid Resolution on Total Pressure Error
170 -F
8 x2
16 x 4
32 x 8
I
64 x 16
Figure 6.9: Order of Accuracy Study: Randomized Computational Grids
171 O.S4
a) Mach Number Contours (A = 0.01)
0.9-
M -
0.7 lo\* r
0.5
0.3 b) Mach Number Wall Distribution
Figure 6.10: Order of Accuracy Study: Mach Number Distribution for 64 x 16 Ran- domized Grid
172 -1.0 . log |11 - P/PinietIIr=s
-1.5 Second order (averages and gradients) to Smooth grids -2.0- + + Randomized grids
-2.5 -
-3.5 -
-4.0 *1 C .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 log(I)
Figure 6.11: Order of Accuracy Study: Effect of Grid Resolution on Total Pressure Error for Smooth and Randomized Grids
173 1.2- Present Algorithm (100 cells)
------Analytic Solution 1.0-
0.8-
0.6-
0.4-
0.2-
0.0- 0.8 1.0 0.0 0.2 0.4 0.6 x a) Density 1.2-
1.0-
0.8-
0.6
0.4
0.2-
0.0- 0.0 0.2 0.4 0.6 x 0.8 1.0 b) Pressure
Figure 6.12: 1-D Shock Tube Problem: Solution at t 0.14492
174 1.0- -A
0.8.
U
0.6-
0.4-
0.2-
0.0
-0.2 - 0. 0 0.2 0.4 0.6 0.8 1.0 x c) Velocity 3.0-
e
2.5 --
2.0-
1.5 0.0 0.2 0.4 0.6 0.8 1.0 z d) Static Internal Energy
Figure 6.12: 1-D Shock Tube Problem: Solution at t = 0.14492
175 1.0 1.0 U
C
0.5 05
I ... ~I.,.. I 0.25 aI 0.50 0.75 X 0.. 0.50 0
I U p e
3' 0-
0.51 2. 5 --
2.
'0 ' ' ' '
I I 025 050 075 025 0.50 075 x
Figure 6.13: 1-D Shock Tube Problem: Solution for Second Order Godunov Scheme of Van Leer [83] (t = 0.14154), Exact solution (lines) and cell averages (circles)
176 0.9-
0.8 - P Pinlet 0.7 -
0.6-
0.5 -
0.5 -
------Present Algorithm (33 cells) 0.4- Present Algorithm (257 cells)
0.3 e 0. 0 0.2 0.4 0.6 0.8 1.0 A a) Present Euler Algorithm 0.9,
0.8- P Pinlet 0.7-
S
0.6-
0.5-
3 Jameson Algorithm (65 cells) 0.4- Present Algorithm (257 cells)
0.3 +- 0.0 0.2 0.4 0.6 0.8 1.0 x b) Jameson Cell-Centered Algorithm
Figure 6.14: Laval Nozzle: Grid Comparison of Steady Pressure Distribution
177 P P,Miet
1.0 I t
0.5
0. .0 0.2 0.4 0.6 0.8 1.0 z
Figure 6.15: Laval Nozzle: Unsteady Distribution of Pressure Due to a Sinusoidal Exit Pressure Oscillation of 15% Amplitude and Reduced Frequency of 2
178 0.9-
0.8- P Pmlet 0.7 -
0.6-
0.5.
------Present Algorithm (33 cells) Present Algorithm (257 cells)
0.3 0.0 0.2 0.4 0.6 X 0.8 1.0 a) Present Euler Algorithm 0.9-
0.8- P Pinlet 0.7-
0.6- 0n 0
0.5-
.o Jameson Algorithm (65 cells) 0.4- Present Algorithm (257 cells)
0.3 0.0 0.2 0.4 0.6 z 0.8 1.0 b) Jameson Cell-Centered Algorithm
Figure 6.16: Laval Nozzle: Grid Comparison of Unsteady Pressure Distribution Due to a Sinusoidal Exit Pressure Oscillation: Phase = 0*
179 0.9
0.8 P Pinlet 0.7-
0.6-
0.5-
- Present Algorithm (33 cells) 0.4- Present Algorithm (257 cells)
0.3 I ) 0.2 0.4 0.6 z 0.8 1.0 a) Present Euler Algorithm 0.9- U U U
U 0.8. P r. Pinlet 0.7- mno
0.6-
0.5-
I 13 Jameson Algorithm (65 cells) 0.4- Present Algorithm (257 cells)
0.3 I ______C .0 0.2 0.4 0.6 0.8 1.0 z b) Jameson Cell-Centered Algorithm
Figure 6.17: Laval Nozzle: Grid Comparison of Unsteady Pressure Distribution Due to a Sinusoidal Exit Pressure Oscillation: Phase = 90*
180 0.9-
0.8- p Pinlet 0.7-
0.6-
0.5-
------Present Algorithm (33 cells) 0.4- Present Algorithm (257 cells)
0.3 i I ______0. 0 0.2 0.4 0.6 0.8 1.0 X a) Present Euler Algorithm 0.9-
*u U a U * * 0.8- * * - U U P U Pinlet 0.7-
0.6-
2
0.5-
3 Jameson Algorithm (65 cells) 0.4- Present Algorithm (257 cells)
0.3 4 0... 0. 0 0.2 0.4 0.6 0.8 1.0
b) Jameson Cell-Centered Algorithm
Figure 6.18: Laval Nozzle: Grid Comparison of Unsteady Pressure Distribution Due to a Sinusoidal Exit Pressure Oscillation: Phase = 180*
181 0.9-
0.8 P PinIet 0.7
0.6.
0.5-
------Present Algorithm (33 cells) 0.4- Present Algorithm (257 cells)
0.3 1 0.2 0.4 0.6 0.8 1.0 z a) Present Euler Algorithm 0.9-
0.8 - P Pinlet 0.7-
U Un 0 0 0.6-
0.5-
o Jameson Algorithm (65 cel 0.4- Present Algorithm (257 ce]
0.3 'I. 0.0 0.2 0.4 0.6 0.8 1.0 X b) Jameson Cell-Centered Algorithm
Figure 6.19: Laval Nozzle: Grid Comparison of Unsteady Pressure Distribution Due to a Sinusoidal Exit Pressure Oscillation: Phase = 2700
182 6.2 Results for TSL Navier-Stokes Algorithm
In this section, two numerical test cases are presented to demonstrate the TSL
Navier-Stokes algorithm of Chapter 4. Both test cases presented in this section have a uniform outer inviscid flow and were run without coupling to an outer inviscid solver.
The first case is a Blasius similarity solution to verify the second order accuracy of the i7-discretization. The second test case is an incompressible turbulent flat plate flow to verify that the Cebeci-Smith turbulence model has been correctly implemented.
6.2.1 Blasius Similarity Solution
The first test case is a Blasius similarity solution for incompressible laminar flow on a flat plate. The purpose of this case is to verify the second order accuracy of the rj-discretization developed in Section 4.4.1. The numerical results presented here were obtained with the similarity profile generator discussed in Section 4.11. Its s7- discretization is equivalent to that of the full TSL algorithm but contains no temporal or streamwise discretization. To simulate incompressible flow a freestream Mach number of M = 0.05 was specified. Figure 6.20a shows u/U velocity profiles (where U is the edge velocity) obtained on grids containing 4, 8, 16 and 32 nodes. These are compared with the "exact" solution taken as that obtained on a very fine grid of 256 nodes. Each grid was generated by (4.121) with t7 = 14 and P = 1.5. Figure 6.20b shows the corresponding profiles of skin friction coefficient cfp/e, where
cf = , (6.5) IpU2 Re' and the Reynolds number based on z is obtained from the reference Reynolds number (2.20) by, Uzi Ux Re,, -- = -- Re. (6.6)
The velocity and skin friction profiles are plotted against the Blasius similarity param- 1 eter 7B given by,
17 = = 0.469600 t, (6.7)
183 since q = y/O and 0 (664115 z/.b~.
Figure 6.21 presents the effect of grid resolution on the error in the skin friction coefficient at the wall and the root-mean-square and maximum errors in the velocity profiles. The error in the friction coefficient at the wall is the difference from the generally accepted value of cfJjez = 0.664115 [87, p. 264] (that obtained on the 256- node grid was 0.66408). The two velocity error norms are defined by,
r m s "I ( U j - tex ( j)i)2 ] 1/2 (6 .8 a ) Um [1 =1 11/
max = max Uj - U x (6.8b) where U is the edge velocity and uex(i1j) is the velocity on the 256-node grid interpolated to t5. The lines in Fig. 6.21 have slope -2.12, verifying second order accuracy.
6.2.2 Turbulent Flat Plate
The second test case for the TSL solver is an incompressible turbulent flat plate flow to confirm the correct implementation of the Cebeci-Smith turbulence model. Numerical results are compared with both theory and the experimental data of Weighardt and Tillmann [86]. The case is T1400 from the 1968 Stanford Conference [21]; conditions are a freestream velocity of U = 33 m/s and kinematic velocity of V = 0.151 cm 2 /8 over a plate of length 5 m. This corresponds to a Reynolds number per unit length of Re, = U/ = 2.19 x 106 /m. Assuming sea level conditions and reference length of
Tref = 1 m, this corresponds to a freestream Mach number of M = 0.1 and a reference
Reynolds number of Re = creflref/7ref = 2.20 X 107 .
Figure 6.22 shows the converged viscous grid in physical space; it is composed of 30 streamwise stations with constant Ax and 48 nodes per station. A plate length of 6 was chosen so that downstream boundary conditions would not contaminate the section of plate where data is known. The skin friction coefficient along the plate is presented in Fig. 6.23 and compared to experiment and experimental correlation (4.117). Agreement is very good. As indicated in Section 4.12, correct prediction of wall shear is dependent
184 on the location of the first cell being well within the laminar sublayer (y+ < 35). This is confirmed in the plot of Ayi in Fig. 6.24.
In the experiment of Wieghardt and Tillmann, velocity profiles were measured at stations along the plate. Figure 6.25 shows a comparison of the computed velocity profile at x = 4.083, the experimental velocity profile at x = 4.087, and the analytic law-of-the-wall (u+ = y+) and log law (u+ ' log(y+) + 5.0). Agreement is again very good.
The numerical results for this case were obtained using a shear velocity u' based on the local shear stress (4.45) in the Cebeci-Smith model. For this type of flow, differences are negligible if the wall shear stress is used to define the shear velocity (4.39).
185 8-
6- o 4 Nodes -> 8 Nodes + 16 Nodes 4 - 4G X 32 Nodes 256 Nodes
2-
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 u/U a) Axial Velocity Profile
8-
6
4-I
2 -
0* -0.4 -0.2 0 0.2 0.4 0.6 0.8 Cf /Rez b) Skin Friction Coefficient Profile
Figure 6.20: Blasius Similarity Solution: Effect of Grid Refinement on Profiles
186 log(Error)
-1.0-
0 error in skin friction 0 RMS error in velcity profile -1.5 - MAX error in velocity profile
-2.0-
-2.5-
-3.0-
-3.5-
I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 log(J)
Figure 6.21: Blasius Similarity Solution: Effect of Grid Refinement on Solution Error
187 0.125-
0.100-
y/ L
0.075-
0.050-
0.025-
0.000- 0 1 2 3 4 6 x1 L
Figure 6.22: Turbulent Flat Plate: Computational Grid, 30 x 48 (vertical scale mag- nified 40 times)
188 (xO.001) 7.0-
6.0-
. Present Algorithm + Experiment [86] Expt. Correlation (4.117)
4.0-
ae , G 3.0-
C G+ G++ 0 +
2.0 0 1 2 3 4 5 6 z/L Figure 6.23: Turbulent Flat Plate: Skin Friction Distribution
189 6-
5- un
4-
3-
2-
1-
0- 0 1 2 3 4 5 6 x/L Figure 6.24: Turbulent Flat Plate: Minimum Grid Spacing Along Plate
190 30 -
eeiO
25
00 U+
20-
15-
o Present Algorithm 10 - + Experiment [86] Law-of-the-Wall, Log-Law
5-
0 i 0. i 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 log(y+) Figure 6.25: Turbulent Flat Plate: Velocity Profile at z = 4.08
191 6.3 Results for Coupled Euler/Navier-Stokes Algorithm
In this section, three test cases are presented for the fully coupled Euler/Navier-
Stokes algorithm. The first is a steady subsonic compression duct with a laminar boundary layer on the lower wall. The second is a standard shock/boundary layer interaction test case, where an oblique shock impinges on a flat plate laminar boundary layer. The final case is an unsteady transonic diffuser with turbulent boundary layers on the upper and lower walls. Results for the last two test cases are compared with experiment.
6.3.1 Subsonic Compression Duct
The first case presented is a steady subsonic compression duct with a laminar bound- ary layer on the lower wall. Its purpose is to demonstrate the improvement in numerical results obtained with the solution of the Defect equations (2.44b) rather than the TSL Navier-Stokes equations (2.39) within viscous regions.
Figure 6.28a shows the geometry and composite grid for this case. The duct has an exit to inlet area ratio of 0.8, with a cos(x) variation in the upper wall from x = 0 to z = 2. The Reynolds number is Re = 106 and the exit Mach number is Me, = 0.5.
Inviscid slip boundary conditions are imposed on the upper wall, and no-slip conditions on the lower. At the inlet, a flat plate similarity boundary layer profile is specified with a thickness of 3.4% of the inlet height. The fixed Euler grid is 48 x 8, and the lower wall viscous grid is 15 cells across.
Mach number contours and wall pressure distribution are also shown in Fig. 6.28. These show the flow acceleration through the constriction.
This case was first run with the discretized TSL Navier-Stokes equations (4.20) solved within the boundary layer. Velocity profiles at three streamwise stations are shown in Fig. 6.29. The stations are located in the constriction where the flow is accelerating as indicated in Fig. 6.28. The plots in Fig. 6.29 show both the viscous
192 profile U and the pseudo-inviscid profile U through the boundary layer; labels on the plots correspond to the discrete operator analysis of Section 2.3. The plots show that the viscous and inviscid solutions do not quite match at the the edge of the viscous grid. Although the viscous solution asymptotes to an inviscid solution towards the outer edge of the viscous grid, it is not the same as that predicted by the outer Euler solver. Also worth noting is the difference in 8v/ay between the viscous and inviscid profiles; this indicates a difference in the streamwise velocity gradient, au/ax.
As stated in Section 2.3, the cause of the mismatch is the difference in truncation er- rors between the Euler and TSL Navier-Stokes algorithms. Hence, solution of the Defect equations should cancel the difference in truncation errors, allowing the the solutions to match. Because the Defect equations have been approximately discretized, assum- ing a linear pseudo-inviscid profile, we cannot expect to match the solutions precisely. However, they should be close since the viscous grid is completely contained within the first Euler cell. Figure 6.30 shows the velocity profiles obtained from the solution of the discrete Defect equations (2.53) within the boundary layer. The plots show a dramatic improvement over those in Fig. 6.29. The u velocities match to within plotting accuracy, and the v velocities are very close. In addition, av/ay matches between the viscous and inviscid solutions. The v velocity is the most difficult solution component to match because it is typically quite small, and matching is almost completely dependent on the mass transpiration flux rather than edge boundary conditions on the viscous solution.
6.3.2 Oblique Shock Impinging on a Laminar Boundary Layer
The second test case is an oblique shock wave impinging on a laminar flat plate boundary layer. This flowfield was the subject of two series of experimental and the- oretical investigations of shock/boundary layer interaction conducted at M.I.T. in the
1950's ([8] and [39],[1],[35]). The experiments were conducted in a supersonic wind- tunnel where a wedge was positioned above a flat plate as shown in Fig. 6.26. An oblique shock forming at the leading edge of the wedge impinged on the plate boundary layer, where pressure and skin friction measurements were taken [39].
193 FLOW2 DIRECTION
t 4 t3 Lti 6
't ' LtA X t
-Xi
Figure 6.26: Oblique Shock/Boundary Layer: Experimental Geometry (from [8])
The particular case presented in this section has also become a standard shock/boundary layer interaction test case for 2-D Navier-Stokes solvers [58,10,2,79]. In this case inci- dent shock is strong enough to induce a small separation bubble in the boundary layer at the base shock.
Comparison with experiment has always been fair, but not great, even with the most accurate algorithms on fine grids. In part this may be due to incorrect interpretation of the experimental test conditions, further discussed in Appendix E. Experimental test conditions reported in Ref. [39] were as follows: nomirral freestream Mach number of
M = 2, static pressure rise through the interaction zone of p6/pl = 1.40 (see Fig. 6.26), and Reynolds number Re, = 2.96 x 105, based on a shock impingment point on the plate of z = 1.95. Using the value of M = 2.0, MacCormack [58] deduced an incident shock angle of 8 = 32.58* and a shock generator angle of 6 = 3.09* (i.e., the angle of the flow downstream of the incident shock). Reexamination of the experimental data suggests the test conditions were closer to M = 2.03, 8 = 32.040, and 6 = 3.06*; these
194 conditions are used in the present computations.
This case is solved on a coarse and fine grid to show the effects of grid refinement.
The coarse grid, shown in Fig. 6.31, consists of a 64 x 16 cell Euler grid and a viscous grid with 15 cells across the boundary layer. The Euler grid is algebraically stretched away from the lower wall. The viscous grid is completely contained in the first Euler cell adjacent to the wall upstream of the shock, and is two Euler cells thick downstream of the shock. For the fine case, the grid density is doubled in each dimension as shown in Fig. 6.32. It consists of an Euler grid with 128 x 32 cells and a viscous grid with 31 cells across the boundary layer. There are two to three Euler cells overlapping the viscous grid. Figures 6.33 and 6.34 show density contours for the steady solutions obtained on the coarse and fine grids, respectively. The separation bubbles obtained for the two grids are compared in Fig. 6.35, which are plots of the streamlines within the boundary layer.
The adverse pressure gradient induced by the incident shock causes the boundary layer to thicken ahead of the shock. To the freestream flow this appears as a compression ramp, which results in compression waves (physically these are shocklets but they are not resolved here). These cause the primary oblique shock to steepen slightly before it impinges the boundary layer. Physically, the reflection of the shock involves coalescence of complex compression shocks and expansion fans off the boundary layer. As a result the reflection is much more spread out than an inviscid regular reflection off the wall.
The fine grid shows three reflected shocks in Fig. 6.34c. The boundary layer separates at the shock and is slightly thicker downstream of the interaction. In addition, a weak shock forms at the leading edge of the plate due to the presence of the boundary layer.
In general, all of these flow features, particularly the shocks, are better resolved on the fine grid. An exception is the boundary layer; it seems to be resolved as well on the coarse as the fine, except for the outer region. This is consistent with the profile plots for the Blasius similarity test case (Fig. 6.20). The only difference in the boundary layers appears to be a slightly thicker separation bubble caused by the better resolved shock on the fine grid. Both the coarse and fine grids have mismatches in the density at the
195 interface. The solutions are close at the incident shock but are noticeably different for the reflected shock; this is more evident in the fine grid solution. Because the solutions mismatch, the interface is not conservative.
Figure 6.36 shows a comparison of the surface pressure and skin friction coefficient of the coarse and fine grid solutions. Also plotted are the experimentally measured values. Within the separation bubble there is a pressure plateau as indicated by the experiment. The fine grid predicts a pressure plateau, but it is different from that of the experiment. The coarse grid does not have enough resolution in the interaction region to predict a pressure plateau. The two grid solutions agree closely with each other in the skin friction coefficient, except in the separation bubble. The fine grid also has a slightly more rapid decrease in cf just prior to separation. They agree well with experiment in both pre-shock distribution and length of the separation bubble, but fail to predict the same skin friction rise after separation. This may be due to the solution mismatch for the reflected shock.
These solutions were not computed with the complete coupling algorithm as devel- oped in the previous two chapters. In fact, the algorithm only converges on this test case for coarse grids, such as the 64 x 16 grid used in this study. The reason is an instability in the coupling, specifically the pressure boundary condition at the outer edge of the viscous grid (4.28) and the approximation to the pseudo-inviscid profile made in the mass transpiration integral (5.7). The grids and solutions presented in Figs. 6.31-6.36 were obtained by the following modifications to the coupling algorithm in decreasing order of importance: First, the pressure boundary condition at the outer edge of the viscous grid has been replaced by specifying the Riemann invariant normal to the edge of the grid. Second, in the mass transpiration flux integral, the fluxes based on the in- viscid solution (Fi 1/2,J-1/2) have been replaced by the viscous fluxes at the outer edge
(Fi1/2 ,J-1/ 2). Third, the discretized TSL Navier-Stokes equations are solved rather than the discretized Defect equations. This last modification is not necessary for stabil- ity; it produces somewhat better results. The need for the second modification makes sense in light of the discussion in Section 5.2 on the possible unstable feedback mech- anism between the wall transpiration fluxes and the boundary conditions at the outer
196 edge of the viscous grid. The first modification is crucial for convergence on fine grids. It was discovered by chance and is completely non-intuitive.
Figures 6.37-6.38 show the solution obtained on the coarse grid using the original unmodified coupling (i.e., without the three changes just listed). Downstream of the shock the solution oscillates; it is actually exponentially growing downstream, but this grid is too coarse to show it. The wavelength of the oscillation is 6.0 times the viscous grid height and remains constant as the Euler grid is refined. The fact that the oscilla- tions correlate with the viscous grid height (i.e., the boundary layer thickness), rather than the grid spacing in z or y, lends further evidence that they are a result of the coupling.
An additional point to note is that this streamwise divergence does not occur when the Prandtl Boundary Layer equations (Section 2.2.3) are solved instead of the TSL Navier-Stokes equations (the coding change is minor). As discussed in Section 2.2.3, the removal of the unsteady and convective terms in the y-momentum equation de- stroys acoustic wave propagation within the boundary layer. Therefore, the feedback mechanism between the wall transpiration fluxes and the viscous edge boundary condi- tions is eliminated. This streamwise divergence has not been observed before, because all previous work on coupling through transpiration fluxes has involved solution of the Boundary Layer equations. Of course, alteration of acoustics waves in the Boundary Layer equations also eliminates much of the shock/boundary layer interaction for this case.
6.3.3 Transonic Diffuser
The second test case for the coupled Euler/Navier-Stokes algorithm is a 2-D tran- sonic diffuser with turbulent boundary layers. The geometry, depicted in Fig. 6.27, has been the subject of extensive experimental investigations at McDonnell-Douglas [12,13,70,71] into the nature of 2-D unsteady shock/boundary layer interaction. The experimental studies have included both self-induced and forced oscillations of the diffuser flowfield.
197 Side view Trip
a/3 8 6 63 4
Inlet Throat Exit'' 381 Exciter reference .178 572 position 305 635
Top view 5.9% 2.4% 12.7% Tr t 191 180 178 165 -
Dimensions in mm Vertical dimensions doubled Slot sizes exaggerated
Figure 6.27: Transonic Diffuser: Experimental Geometry (from [71])
For all cases containing terminal shocks in the diffuser section, low amplitude natural oscillations were observed in the experimental investigations. The physical mechanisms responsible for the unsteadiness were found to depend on the strength of the terminal shock. For weak shocks (i.e., a pre-shock Mach number of M < 1.27), the boundary layers on the upper and lower walls remain attached throughout the diffuser and do not merge together. The natural unsteadiness was identified as longitudinal acoustic modes resulting from the interaction of the shock and the diffuser exit. The frequency and number of observed modes changed with the diffuser length [12]. For strong shocks, the upper wall boundary layer separates at the shock and the boundary layers merge prior to the exit of the diffuser. Only one natural frequency was observed; its mechanism is more complex, involving acoustic modes in the post-shock inviscid core and waves within the boundary layer.
The flowfield has also been the subject of Navier-Stokes simulations [41,42,561 using MacCormack's algorithm [591. These simulations also investigated both forced and self-induced oscillations. Reference [42] included self-induced oscillations for a strong shock case. Unfortunately, this case, with its merging boundary layers, is beyond the capabilities of the present algorithm. Furthermore, the applicability of viscous/inviscid coupling becomes questionable when the channel becomes completely viscous.
198 The case presented in this thesis has a weak terminal shock. The flow has a pre- scribed exit pressure ratio of R, = 0.826 (exit static to inlet stagnation), producing a weak shock with a pre-shock Mach number of M = 1.235. The Reynolds number is 1.1 x 106 based on the throat height h. and stagnation conditions at the inlet. As stated above, the turbulent boundary layers remain completely attached throughout the diffuser for this case. It is also naturally unsteady due to longitudinal acoustic modes downstream of the shock. To this author's knowledge, computation results for the nat- ural oscillations occurring in this weak shock case have not been reported previously.
This case is computationally difficult for three reasons. The first is that the un- steadiness is naturally occurring rather than forced. Hence, the physics causing the unsteadiness must be correctly modeled. The second reason is is that the upper and lower wall boundary layers grow by an order of magnitude downstream of the shock.
This presents difficulties for any viscous solver, and is an excellent test of the coordinate rescaling in the present algorithm. The third reason is that the boundary layers nearly merge at the exit, pushing the limit of applicability of viscous/inviscid coupling in gen- eral. It also presents a test of the present coupling technique, and the approximations made herein, since several inviscid cells are located within each boundary layer near the exit.
Quasi-Steady Results
The experimentally observed unsteadiness in this flow has also been detected by the present algorithm. It was initially run using the steady-state acceleration techniques of
Appendix D until the residuals leveled off. An indication of its unsteadiness is the fact that the residuals leveled off two orders above machine zero, whereas most other cases attempted have converged to machine zero. The quasi-steady results are presented in Figs. 6.39-6.41.
Figures 6.39a and 6.40a show the composite grid (with inviscid cells inside the bound- ary layers hidden). The computational domain extends from the experiment's nomi- nal inlet station at z/h. = -4 upstream of the throat to the nominal exit station at
199 x/h. = 8. The Euler grid is constructed from a uniformly spaced 48 x 8 cell background grid with streamwise clustering in the vicinity of the shock increasing the resolution to
71 x 8. The upper and lower wall viscous grids, each 31 cells across, are exponentially stretched away from the wall (the values of a in Eq. (4.122) are 5.47 and 5.21 for the upper and lower walls, respectively). The minimum grid spacing for the upper wall grid is Ayi = 1.5 from the inlet to the shock and increases to 5 at the exit. Likewise, the minimum lower wall grid spacing is 1.5 from the inlet to the shock and increases to 7.5 at the exit. This places the first few cells well within the laminar sublayer which extends to approximately y+ _ 30. The viscous grids are thin upstream of the shock and grow dramatically downstream of the shock. The initial guesses for the upper and lower wall viscous grids were flat plate turbulent boundary layers beginning at the specified inlet profiles.
The Mach number distribution in the entire flowfield is shown in Figs. 6.39b-c. Contour plots in Fig. 6.40 and wall distribution plots in Fig. 6.41 show a blowup of the flowfield in the diffuser section. These figures show that the flow accelerates through sonic conditions at the throat with a terminal shock located at x/h* = 1.45. The exper- imentally determined mean shock position was located at x/h* = 1.47. The grid plot and some of the wall plots indicate that the presence of the shock is felt upstream within the boundary layers, beginning at about x/h. = 1.1. In accordance with experimental observation, the skin friction coefficient on the upper and lower walls (Fig. 6.41c) show that the boundary layers remain attached throughout the diffuser section. Downstream of the shock, the boundary layers grow by almost a factor of ten as shown in Figs. 6.41d-e. Because the y-coordinate rescaling in the viscous grid is dynamically depen- dent on the local displacement and momentum thicknesses, these growths are accurately duplicated by the viscous grid.
The Mach contours (Fig. 6.40b) indicate an overshoot in the viscous profiles down- stream of the shock on both walls. These are the result of shear layers originating at the base of the shock. Within both boundary layers, the shock becomes a compression fan, and at the shock-fan junction, a shear layer forms. Although the grid is not fine enough to resolve the fan at the base of the shock, the shear layer is captured.
200 Results of the present computation are compared with experimental data in Fig. 6.41.
Unfortunately, static pressures for R, = 0.826 have not been published; Figure 6.41b shows data for R, = 0.80. In general the agreement is quite good. The major reason for any discrepancies is the lack of experimental data on the boundary layers at the inlet. Bogar et al [12] give displacement and momentum thickness data for the upper wall at the inlet; however, the nominal inlet station at x/h. = -4 is in accelerating flow. Hence, it is difficult to specify an inlet profile replicating these values. No data is given for the lower wall. Reference [12] states that a suction slot is located upstream of the throat producing a new laminar boundary layer off its lip. This boundary layer remains laminar until it transitions at the shock. This causes problems for the present viscous algorithm since no transition mechanism is built into it. In previous computa- tions, Hsieh et al [41] specified flat plate turbulent profiles for the inlet upper and lower wall boundary layers; the thicknesses were reported to be from experimental data and specified to be 9% and 4.5% of the throat height, respectively. These same conditions are specified here.
The contour plots of Fig. 6.40 give an indication of the quality of solution matching at the edge of the viscous grid. The pressure is correctly matched throughout the diffuser since it is a prescribed boundary condition at the outer edge of the viscous grid; it also transitions smoothly through the interface. The Mach number, density, and other flow quantities that are not prescribed at the edge, match fairly well ahead and just aft of the shock. Farther downstream, the mismatch becomes more noticeable as the boundary layer grows rapidly. Part of the reason for the mismatch is the rapid growth of the boundary layers and the fact that they becomes several inviscid cells thick at the exit. As a result the linear inviscid profile assumption in the discrete Defect equations
(see Section 4.6) and the mass transpiration flux is beginning to break down. Another reason may be the lack of cross-stream resolution in the outer portion of the viscous grid. With a proper discretization of the Defect formulation, the solution mismatches should disappear completely.
201 Unsteady Results
For the weak shock case (R, = 0.826), unsteady results were reported for two diffuser
lengths by Bogar et al [121. For a diffuser length of ze,it/h. = 14.4, two natural fre-
quencies in the shock motion were observed; these correlated well with measured wall
static pressures. The two frequencies where 60 and 230 Hz, which correspond to re- duced frequencies of 0.046 and 0.178, respectively, based on the throat height h. and inlet stagnation speed of sound Coiniet. The amplitude (rms) of the shock motion was
measured to be 1.4% of the throat height. For a diffuser length of xe, t/h. = 30.5, three natural frequencies were observed.
In the present computations, a constant-pressure exit boundary condition was spec- ified at x/h* = 8, which was the nominal exit station in the experiments. Starting with the quasi-steady solution, the flowfield was marched in time at a CFL of A = 0.55 (and
a At based on the Euler grid) for approximately 90,000 iterations.
Figure 6.42a shows a trace of the unsteady pressure component on the upper wall at x/h* = 5.4. The unsteady component was taken as the difference between the
instantaneous and time-averaged pressures. This trace is typical of streamwise stations downstream of the shock. It shows distinct periodic oscillations that reach a constant
amplitude after 5 or 6 periods. The last 20 periods of the trace was Fourier transformed
and the resulting power-spectral-density plotted in Fig. 6.42b. A peak is present at a
reduced frequency of 0.135, and a second, smaller peak is present at 0.275.
The computed natural frequencies are difficult to compare to the experiments. The
reason is the effect of the differing experimental and computational diffuser lengths.
Since the experimentally observed oscillations were identified as longitudinal acoustic modes, their frequencies would vary inversely with the diffuser length. Thus, the values
of 0.046 and 0.178 experimental observed with the exit at z/h* = 14.4 should increase
for a shorter diffuser. By how much they should increase for a diffuser of length 8 is
clouded by the presence of boundary layer suction and the unused exciter downstream
of z/h. = 8 in Fig. 6.27.
202 An additional uncertainty is the constant-pressure exit boundary condition, chosen for practical reasons, in the computation. It does not correctly model the physical wave reflection and attenuation at the open exit. This should affect both the strength of reflected waves at the exit and the effective length of the diffuser.
Figure 6.43 shows the amplitude (rms) of the unsteady pressure component along the diffuser. The amplitude is greatest in the immediate vicinity of the shock (x/h. = 1.45), producing a sharp peak there. Downstream of this peak, the amplitude decreases slowly to zero at the exit, where constant pressure is prescribed. Essentially no oscillations occur upstream of z/h. = 1, which from the quasi-steady results is approximately the farthest upstream point in the boundary layer that is affected by the shock. Experimen- tal amplitudes are also plotted in Fig. 6.43; the values are approximately an order of magnitude higher than predicted by the computations. This difference may be caused by the fact that the unsteady shock motion is not resolved on the present grid. The experimentally measured shock motion was 0.014h., but the streamwise resolution is only Ax/h = 0.063 at the shock in the present computations. This calculation was also performed on a coarser grid composed of a 48 x 8 uniform Euler grid with a streamwise resolution of Az/h* = 0.25 at the shock; the resulting flow was completely steady. This suggests that as the shock motion is resolved, the amplitude of the pressure fluctuations will increase.
Advantages of the Defect Equations
Numerical advantages of the Defect equations have been demonstrated in the case of
Section 6.3.1. That case showed improvements in the matching between the viscous and inviscid solutions resulting from the solution of the Defect equations. The present test case provides another demonstration of the Defect equations. For coarse grids, better solution matching can also dramatically improve the overall solution accuracy. This finding is particularly useful for industrial applications where economic considerations necessitate the use of coarse grids.
The present weak shock test case was rerun to a steady solution on a coarse grid
203 composed of a 48 x 8 uniform Euler grid (as stated above, no unsteadiness was observed on this coarse grid). The only modification to the grid shown in Fig. 6.39a is removal of the clustering at the shock. Figures 6.44 and 6.45 show the resulting flowfields obtained by solution of the discrete Defect equations (4.23) and the discrete TSL equations (4.20), respectively. The Defect equation flowfield is similar to the results already presented, except the region in the vicinity of the shock is not as well resolved. On the other hand, the TSL equation flowfield on this coarse grid is simply wrong. The flow remains subsonic throughout the diffuser. The reason for this is the level of numerical dissipation generated by the first order streamwise discretization in the boundary layer. It is enough to prevent the inviscid core flow from accelerating through sonic conditions at the throat. Thus, the character of the predicted flowfield is entirely different than what it should be.
Time Step Study
This turbulent test case also examplifies the computational efficiency of a semi-implicit time integration of the viscous equations over that of a fully-explicit scheme.
Both the Euler and Navier-Stokes algorithms have their individual stability restric- tions on the time step, as discussed in Sections 3.5.1 and 4.5.1, respectively. The Euler algorithm has a 2-D time step restriction of,
Ate = A[ + ,] (Euler) (6.9) where the maximum CFL is A, 5 0.55. The Navier-Stokes algorithm uses semi-implicit time integration, which results in a viscous time step restriction depending only on Ax, S|uI+ c1' AtV = AV , (viscous) (6.10) where the maximum CFL is A, < 1. Because of grid assumptions and the CFL num- bers, the global time step for the coupled algorithm is always determined by the Euler restriction. For the present test case, the time step was Atgobal = 1.43 X 10-2.
Since the present test case is turbulent and the Reynolds number fairly high, the viscous grid spacing Ay is very small at the walls (in fact, the maximum cell aspect ratio
204 is 4700 on the upper wall). With the present semi-implicit viscous time integration, this does not matter. The minimum viscous time step from (6.10) was At,, = 3.19 x 10-2. If
a fully-explicit time integration were used for the viscous solver, then the small spacing in y would matter a great deal. The resulting stability restriction on the time step would be (6.9) with A = 0(1). In the present case this would result in a viscous time step of 4.6 x 10-5 (assuming A = 1). This is 700 times more restrictive than the present viscous constraint and 310 times smaller than the time step for the Euler algorithm. Thus, the unsteady computation of this section would have taken 28 million time steps using fully-explicit time integration, compared to 90,000 for the present algorithm.
The computational speed of a fully-explicit time integration would have to be 0(100) times faster than the present semi-implicit technique to make up for the difference in At.
This is not the case; at best a fully-explicit technique might be several times faster (but still 0(1)). Thus, for test cases similar to that discussed here, the present semi-implicit time integration is approximately two orders faster than a fully-explicit scheme.
205 - 1 1ThnTIwrnThTFTT~
0
velocity profiles a) Composite Computational Grid (vertical scale doubled)
1
I 'J
0 b) Mach Number Contours (A = 0.01) (vertical scale doubled)
0.92
P/Pinlet 0.90
0.88
0.86 lower
0.84
upper 0.82 -2 0 2 4 z c) Wall Pressure Distribution
Figure 6.28: Compression Duct: Composite Grid and Flowfield
206 0.05 -
0.04-
0.03-
0.02-
0.01-
0 0 0.1 0.2 0.3 0.4 0.5 a) Axial Velocity
0.05-
1' 0
0.04-
0.03-
0.02-
0.01 V
0 -4 -3 -2 -1 0 v (x0.001) b) Vertical Velocity
Figure 6.29: Compression Duct: Velocity Profiles Using TSL Navier-Stokes Equations (Profiles at Stations Indicated in Fig. 6.28a)
207 0.05-
0.04-
0.03-
0.02-
U
0.01-
S0 .1 0.2 0.3 0.4 0.5 U a) Axial Velocity
0.05-
0.04-
0.03-
0.02-
0.01 -
0 -4 -3 -2 -1 0 v (xo.001) b) Vertical Velocity
Figure 6.30: Compression Duct: Velocity Profiles Using Defect Equations (Profiles at Stations Indicated in Fig. 6.28a)
208 1.5-
Y
1.0-
0.5-
0.0+- -0.5 0.0 0.5 1.0 1.5 2.0 z 2.5 3.0 3.5 a) Entire Domain
0.6-
0.5 -
0.4
0.3-
0.2-
0.1.
0.0 P' 1.2 1.4 1.6 1.8 2.0 2.2 x 2.4 2.6 2.8 b) Close-up of Shock/Boundary Layer Region (vertical scale magnified 2 times)
Figure 6.31: Oblique Shock/Boundary Layer: Coarse Computational Grid (Euler 64 x 16, Viscous 59 x 15)
209 1.5-
1.0-
0.5-
0.0- -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 z 3.5 a) Entire Domain
0.5
0.3
0.2
0.1
0.0 1.2 1.4 1.6 1.8 2.0 2.2 z 2.4 2.6 2.8 b) Close-up of Shock/Boundary Layer Region (vertical scale magnified 2 times)
Figure 6.32: Oblique Shock/Boundary Layer: Fine Computational Grid (Euler 128 x 32, Viscous 123 x 31)
210 1.5
1.0-
0.5- 0.225 ' 0660
0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 -0.5 0.0 0.5 1.0 1.6 2.0 2.5 3.0 3.5 a) Entire Domain 0.6- / / 0.5- /.2 / -A' / 0.4 / / / /
0.3- / /1 / K-2 loll / - / 0.2- // ,/o.zBo / 0.1- /
0.0 I liii 1.4 1.6 1.8 2.0 2.2 x 2.4 2.6 2.8 b) Close-up of Shock/Boundary Layer Region (vertical scale magnified 2 times) 0.30-
0.25-
0.20.1 -0.5 0.0 0.5 1.0 1.5 2.0 x 2.5 3.0 3.5 c) Density Distribution at y = 0.3
Figure 6.33: Oblique Shock/Boundary Layer: Coarse Grid Density Contours (A = 0.005)
211 1.5.
y
0.5- 0.25 'N.4 /o0.220
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 -0. 5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 a) Entire Domain 0.6-
0.5- -N
0.4
0.3- I7/
0.2-
02250 0.1-
0.0 1.:2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
b) Close-up of Shock/Boundary Layer Region (vertical scale magnified 2 times)
0.30-
0.25 - -'
" 2" . -0. 0.0 0,5 1,0 1.5 2.0 2.5 x 3.0 3.5 c) Density Distribution at y = 0.3
Figure 6.34: Oblique Shock/Boundary Layer: Fine Grid Density Contours (A = 0.005)
212 0.06-
0.05-
Y
0.04-
0.03 -
0.02-
0.01-
0.00 1.2 1.4 1.6 1.8 2.0 2.2 x 2.4 2.6 2.8 a) Coarse Grid Streamlines (A = 5.0 x 10-4)
0.06-
0.05-
0.04-
0.03 -
0.02-
.1
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 x b) Fine Grid Streamlines (A = 5.0 x 10-4) Figure 6.35: Oblique Shock/Boundary Layer: Comparison of Separation Bubbles
213 0.18-
0.17- P Pinlet 0.16-
0.15-
0.14- U A 0
0.13- mm ai
0.12 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 X a) Static Pressure (xO.001) 5.-
------Present Algorithm (coarse grid) 4.1 Present Algorithm (fine grid) Cf a Experiment [391 3.
2.
1. - - -
0.
-1~ -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 r b) Skin Friction Coefficient
Figure 6.36: Oblique Shock/Boundary Layer: Comparison of Wall Pressure and Skin Friction Distributions
214 KI
1.5-
0.5-
0.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 x a) Entire Domain 1.0s.. jh
0.6r)
0.54 .-.
0.3-~/
0.2-.//
0.0 1.2 1.4 1.6 1.8 2.0 2.2 x 2.4 2.6 2.8 b) Close-up of Shock/Boundary Layer Region (vertical scale magnified 2 times)
0.30 )
0.25-
0.02.5 0,10 0.5 1.0 1.5 2.0 2.5 3.0 3.5
c) Density Distribution at y =0.3 Figure 6.37: Oblique Shock/Boundary Layer: Coarse Grid Density Contours (A = 0.005) for Unmodified Coupling Procedure
215 0.18-
0.17- ' " '/t:- : inlet 0.16-
0.15-
0.14 -
U I
0.13- .
0.12 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 X 3.5 Static Pressure (x.o1) a) 5.-
4.-
Cf ------Present Algorithm (coarse grid)
3. -1 Experiment [39]
1.-
-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 b) Skin Friction Coefficient
Figure 6.38: Oblique Shock/Boundary Layer: Coarse Grid Wall Pressure and Skin Friction Distributions for Unmodified Coupling Procedure
216 OFF-0 0 0 0U M y/h* 1
0 i F-2 4, z/h. 6 F8 a) Composite Grid (Euler 71 x 8, Viscous 71 x 31 lower and2 upper) y/h* 1 05 0 OSO OSs
0 S -4 -2 0 2 4 6 8 x/h. b) Mach Contours (A = 0.05)
1.5 - M
1.0-
0.5 -
-4 -2 0 2 4 z/h. 56 5 c) Core Mach number
Figure 6.39: Transonic Diffuser: Grid and Mach Number Distribution for Entire Domain
217 1.5 -
y/h*
1.0-
0.51
0 < 2 4 z/h. 6 a) Composite Grid (Euler 71 x 8, Viscous 71 x 31 lower and upper)
1.5 y/h* -ONOW 1.0 W / o-S4 \--- OS,21 0.50
0.5- 0.52 i-00",
U ) 2 4 x/ . 6 8 b) Mach Contours (A = 0.02)
1.5 - y/h.
1.0-
'1 0.5- 0.82
0 2 4 z/h. 6 8 c) Static Pressure Contours (A = 0.01)
Figure 6.40: Transonic Diffuser: Grid and Contour Plots for Diffuser Section
218 1.5. y/h.
1.0
O.5 ?.86
0. - Qp- --- - 0 2 6 8
d) Density Contours (A = 0.01)
1.5. y/h. S. x 10-3
1.0 ammmmmmmmm
0.5
5. x ~ 0 0 x/h. 6 8 e) Friction Coefficient Contours (A = 5.0 x 10-4) Figure 6.40: Transonic Diffuser: Grid and Contour Plots for Diffuser Section
219 -- Present Algorithm 0 Experiment [12] 1.6-
M
1.2\
0.8-
0.4 6 2 4 6 8 z/h* a) Core Mach Number
upper 0.8Over P/Pinlet 0.7-
0.6-
klower 0.5-
upper (Note: Expt. data at R, = 0.80) 0.4 2 4 z/h. 6 8 b) Wall Static Pressure)
(xO.001) 4- Cf
3-
2-
1 - upper lower 0 2 z/h. 8 c) Wall Skin Friction Coefficient
Figure 6.41: Transonic Diffuser: Wall and Core Distributions for Diffuser Section
220 - Present Algorithm Experiment [121 1.5-
1.0
0.5- Ok
z/h* d) Upper Wall Displacement and Momentum Thicknesses (incompressible)
1.5-
1.0. k
0.5-
00 2 4 6 8* x/h. e) Lower Wall Displacement and Momentum Thicknesses (incompressible)
1.0-
U/CO inlet
0.8
0.6 a0 0
0.4 0 2 4 h6 8 f) Core Axial Velocity Figure 6.41: Transonic Diffuser: Wall and Core Distributions for Diffuser Section
221 (xO.0001) 4- 4 P/ in le t
3-
2-
1-
0-
-1
-2 0 0.2 0.4 0.6 0.8 1.0 1.2 t/(h*/coinIet) (xIOOO) (xlO- 8 ) a) Unsteady Trace in Time 1.2-
1.0- -e-- Present Algorithm
PSD Experiment [71]
0.8-
0.6-
0.4-
0.2 -
0 0.1 0.2 0.3 0.4 W/(COinlet/h.) b) Power-Spectral-Density of Trace
Figure 6.42: Transonic Diffuser: Unsteady Static Pressure on Upper Wall at z/h* = 5.4
222 Present Algorithm (xO.OO1) a Experiment [12] 10-
|1Ap/Pinled||rn
8-
6-
4-
2-
0 0 2 4 6 z/h. 8
Figure 6.43: Transonic Diffuser: Variation in Amplitude (rms) of Unsteady Static Pressure Along Upper Wall
223 1.6-
M
1.2-
0.8-
0.4 -4 -2 0 2 4 6 8 x/h. a) Core Mach Number 0.9-
0.8- P/Pinlet 0.7- lower
0.6- upper
0.5-
0.4- -4 -2 0 2 4 6 8 z/h. b) Wall Static Pressure) (xO.001) 4- lower
3- upper Cf 2
1 lower
upper
-4 -2 0 2 46 8 z/h. c) Wall Skin Friction Coefficient
Figure 6.44: Transonic Diffuser: Solution of Defect Equations on Coarse Grid (Euler 48 x 8, Viscous 48 x 32)
224 1.6-
M
1.2-
0.8-
0.4- . -4 -2 0 2 4 6 8 zxh. a) Core Mach Number 0.9.
0.8- P/Pinlet 0.7- lower
0.6- upper
0.5-
0.4- i -2 0 2 4 6 8 z/h. b) Wall Static Pressure) (xO.001) 4-
3- C f 2-
lower 1
upper 0------
-1 A - . - - 1 -v - 0 x/h. c) Wall Skin Friction Coefficient
Figure 6.45: Transonic Diffuser: Solution of TSL Navier-Stokes Equations on Coarse Grid (Euler 48 x 8, Viscous 48 x 32)
225 Chapter 7 Conclusions
This thesis has presented three major contributions all aimed toward the main goal of developing a coupled Euler/Navier-Stokes algorithm for the solution of unsteady 2-D transonic flows. These three major contributions are the Euler algorithm developed in Chapter 3, the Thin-Shear-Layer Navier-Stokes algorithm developed in Chapter 4, and a procedure discussed in Chapter 5 for coupling the two together. Each of these contributions is summarized in the following three sections. The chapter ends with a discussion of possible extensions and recommendations for future research in Section 7.4.
7.1 Euler Algorithm
A novel finite volume time-marching algorithm for the solution of the unsteady Euler equations has been developed in Chapter 3. The objectives of this development were crisp resolution of unsteady shocks combined with high accuracy for nonsmooth grids. Both of these objectives have been met. Moving shocks are captured crisply and with minimal oscillations using upwinding. More specifically, van Leer's flux-splitting [84] is used, but the algorithm is equally suited to other flux-splitting techniques. The second goal has been met in an algorithm formulated from the onset for cells of arbitrary shape and number of sides. Both averages and gradients of the solution are stored for each cell.
Using this linear approximation within each cell, all interpolations to cell faces for flux evaluations are second order accurate. The cell averages are evolved in time by solving the Euler equations, while the gradients are evolved by solving the first moments of the Euler equations about each cell centroid. An explicit 3-stage Runge-Kutta integration is used to march the solution in time. The resulting algorithm has grid independent second order accuracy for both steady and unsteady flows.
226 Further refinement of the algorithm has resulted from more careful discretization of the first moment equations. Two-point Gauss quadrature is used to calculate the flux on each face of the cells. This gives second order accurate gradients for steady flow. It also eliminates all decoupled modes and any need for explicitly added artificial dissipation. In addition, nonlinear consistency in discretized equations has been found to be crucial for convergence near shocks. This nonlinear consistency is obtained using Roe's parametric vector [69].
A von Neumann stability analysis has been performed on the present algorithm using the linearized 1-D Euler equations instead of the more traditional scalar 1-D wave equation. This study has determined a maximum CFL of A = 0.55 and the optimal first stage coefficient of the Runge-Kutta time integration. Also, the effects of the flux-splitting on stability at different Mach numbers has been assessed.
Convergence to steady-state has been accelerated by two techniques. The first is gradient update underrelaxation, which increases the maximum stable CFL number to A = 2.2. The second technique, applicable only to the rectangular grids and duct geometries presented in this thesis, is a simplified implicit approximate factorization.
The Euler algorithm has been demonstrated with a series of quasi-1-D and 2-D channel flow cases using quadrilateral grids. Those flow regimes studied include sub- sonic, transonic and supersonic flow. Both quasi-1-D and 2-D transonic flow cases show that normal shocks are captured in one or two cells without pre- or post-shock numerical oscillations. A numerical accuracy study using a sin2 z bump channel has been undertaken including both smooth and pathological randomized grids. The study demonstrates the algorithms grid independent second order accuracy. In addition, little difference was found in the magnitude of numerical errors and the resulting solutions obtained on the smooth and distorted grids. Thus, the algorithm shows great potential for applications on unstructured meshes.
Unsteady results are presented for a 1-D shock tube and a quasi-1-D transonic
Laval nozzle. In the shock tube problem, the solution was integrated from an initially discontinuous solution with a pressure jump of 10. After a specified time, the numerical
227 solution was compared with the analytic solution. Overall comparison in the shock, contact discontinuity and expansion fan was excellent. Unsteadiness in the Laval nozzle case was forced by a sinusoidally varied exit pressure. Results for the present algorithm
were compared for a coarse grid and a grid eight times finer. Excellent agreement in shock position and structure was found. The coarse grid results were also found to be superior to those of a central-differenced algorithm at twice the grid density.
7.2 Thin-Shear-Layer Navier-Stokes Algorithm
The second major contribution of this thesis has been the development of a new algorithm for the solution of the unsteady 2-D Thin-Shear-Layer Navier-Stokes equa- tions. The algorithm has been developed to be used to solve the viscous regions of a flowfield and to be coupled with an inviscid solver. The three main objectives of this development have been as follows: high accuracy across the boundary layer for both inviscid and viscous terms, a time integration which uses practical time steps, and a means of adapting the grid to the changing boundary layer thickness.
The first objective has been obtained by discretizing the governing equations using two-point differencing across the boundary layer. To facilitate this, the inviscid solu- tion, along with the viscous shear stress and enthalpy flux, is stored at horizontal face
midpoints across the grid. In addition, the governing equations are solved as a system
of six first-order equations. The resulting discretization is similar to the Keller Box scheme [481, used for the solution of the Boundary Layer equations, and is spatially sec- ond order accurate even for highly stretched grids. This accuracy has been numerically
verified using the Blasius similarity solution for a incompressible laminar boundary layer on a flat plate.
First order accurate flux-splitting is used in the streamwise discretization to crisply capture shocks occurring in the outer regions of the boundary layer.
An advantage of the spatial discretization of this scheme is that it admits no decou-
228 pled modes. Thus, no add artificial dissipation is needed.
The solution is integrated in time using a single-stage semi-implicit formulation. The discretization across the boundary layer is integrated implicitly, while the streamwise discretization is integrated explicitly. The implicit system at each streamwise station and time step is iterated by Newton's method; the resulting block tridiagonal system is solved by Gaussian elimination. The semi-implicit time integration avoids impractical time step restrictions imposed on fully explicit techniques by the small grid spacing normal to the wall. This is verified by a von Neumann stability analysis using the 2-D linear advection-diffusion equation, which gives a CFL restriction of A < 1 based only on the streamwise grid spacing.
The problem of specifying the viscous grid without a priori knowledge of the bound- ary layer thickness is solved by a dynamic coordinate transformation. The coordinate across the viscous grid is rescaled by the local unsteady boundary layer thickness, and the transformed equations solved on a fixed grid.
The algorithm also incorporates a Cebeci-Smith algebraic turbulence model, whose correct implementation has been verified by a turbulent flat plate test case.
7.3 Coupling Procedure
The third major contribution of this thesis is a procedure for coupling the Euler and
Navier-Stokes algorithms together. The flowfield is solved on overlapping grids with the Euler grid extending to the body surface. That portion of the Euler solution within the viscous region is computed but is not physically meaningful. Coupling between the Euler and Navier-Stokes solutions is through boundary conditions: specified outer edge values for the viscous solution and wall transpiration fluxes for the Euler solution.
Coupled in this manner, the Euler grid remains fixed while the viscous grid evolves with the changing boundary layer thickness. Thus, a priori knowledge of boundary layer thicknesses is not required to solve the flow accurately.
229 This coupling technique has proven successful with boundary layer schemes coupled with potential or Euler schemes. The present thesis represents the first extension of this coupling technique to the solution of the TSL Navier-Stokes equations. The funda- mental issues addressed have been matching of the inviscid and viscous solutions and performance of the procedure for boundary layers which may have substantial thickness. To better isolate these issues, geometric complexity has been removed by simplifying grid assumptions.
Numerical coupling between the Euler and Navier-Stokes algorithms is through an explicit relaxation procedure. This allows for increased flexibility, since the algorithms are computationally isolated from one another. In the relaxation procedure, the viscous edge boundary conditions and inviscid wall transpiration fluxes are lagged in time, making them first order accurate in time.
An important aspect of the present coupling is the Defect formulation of Le Balleur [503, specifically the solution of the TSL Navier-Stokes equations written as Defect equations within viscous regions. A discrete operator analysis has shown insight into the numer- ical advantages of the solution of the Defect equations. When different algorithms are used to solve the inviscid and viscous equations, the discrete Defect equations subtract off the relative truncation error. This allows the inviscid and viscous solutions to match smoothly at the edge of the viscous grid. It also makes a conservative formulation at the interface possible in principle; this is a potential advantage over other overlapping grid techniques.
As a preliminary investigation of the Defect formulation, a simplified version of the Defect equations has been discretized. Rather than interpolate the known inviscid solution to each point in the viscous grid, a linear inviscid profile has been assumed within the boundary layer based on the solution and gradients at the outer edge of the viscous grid. This subtracts off most of the first order truncation error of the streamwise discretization of the viscous solver. Marked improvement in solution quality and matching over discretizing the TSL Navier-Stokes equations has been demonstrated for steady subsonic and transonic flows.
230 The fully coupled 2-D Euler/Navier-Stokes algorithm has been demonstrated for both steady and unsteady flows using three test cases. Flow regimes include subsonic, transonic and supersonic flow. The first test case is a steady subsonic compression duct, used to demonstrate the numerical improvements resulting from the solution of the Defect equations.
The second case is a steady oblique shock impinging on a laminar flat plate boundary layer. Surface pressure and skin friction coefficients are compared with experiment. Solutions on two grids, one at twice the resolution of the other, are used to show the effects of grid refinement. The case also revealed coupling instabilities for supersonic flows, which do not occur for the solution of the Boundary Layer equations.
The third case is a transonic diffuser with turbulent boundary layers on the upper and lower walls. This case has natural unsteadiness due to self-induced longitudinal acoustic modes. This experimentally observed unsteadiness is also detected by the present algorithm. In addition, this case demonstrates the algorithm's ability to pre- dict shock/boundary layer interaction and boundary layers which grow by an order of magnitude. Quasi-steady results compare well with experimental time-averaged results.
7.4 Recommendations for Future Research
The research leading to this thesis has included new algorithm developments in several areas. It has also pointed to a number of improvements and extensions of the work presented in this thesis. These are discussed below for each of the three major contributions of this thesis.
7.4.1 Euler Recommendations
By far the most successful and refined contribution of this thesis has been the Euler algorithm developed in Chapter 3. Although formulated from the onset for arbitrary control volumes, its use in this thesis has included only structured quadrilateral grids.
231 Hence, the most obvious extension is to unstructured grids, which are finding increased application for complex geometries [44]. The test case using randomized grids in Sec- tion 6.1.4 is a vivid demonstration of the present algorithm's ability to give accurate results on distorted grids, which often occur when unstructured grids are used. The only work required for this extension is development of a suitable data structure (which is easily done), and rewritting the code listed in Appendix F in terms of this data structure.
An additional and very promising extension of the algorithm is to adaptive embed- ded grid techniques, where local grid refinement is by subdivision of cells into smaller cells, as shown in Fig. 7.1. This form of embedding has previously been developed for
Ni's cell-vertex algorithm [81,24,75,47 and Jameson's cell-centered algorithm [11,3,4]. The application of embedding to these algorithms suffers from one drawback-they re- quire special formulations for fluxes, smoothing and time integration at the interface between coarse and fine cells. Great care must be taken in constructing these interface formulations to maintain conservation, accuracy and stability.
coarse cell 0 o
solution location interface fine cell 0 0
0
Gauss point 0 0
Figure 7.1: Embedded Grid
What makes grid embedding for the present Euler algorithm so promising is that no special interface formulations are required. No loss of conservation, accuracy, or stability should occur at the interfaces. As shown in Fig. 7.1, all that is needed is to treat the interface as two fine cell faces. The flux through each face is by two-point
Gauss quadrature (as is done for every other face); interpolation from the coarse cell
232 is straightforward using its average and gradients and the location of the Gauss points.
Again, a data structure must be developed, but the present algorithm fits neatly into that already developed for Jameson's cell-centered algorithm in Ref. [3].
Extension of the algorithm to 3-D should involve no technical difficulties, but may raise questions of excessive storage and CPU costs. In 2-D, 12 solution unknowns are stored for each cell and 12 equations are solved; this number increases to 20 in 3-D.
However, the scheme's accuracy may alleviate this problem by allowing coarser grids to be used. In 3-D the two-point Gauss quadrature of fluxes will be replaced by a four- point Gauss quadrature over the surface of each face. Also, the first moment equations
(3.33) form a coupled system for the integration of the cell gradients; this will require a 3 x 3 matrix inversion for each cell at each time step. This may be avoided and the equations decoupled by solving for gradient components in suitably chosen local natural coordinate directions, rather than the Cartesian components as presently done. More specifically, the linear approximation for U or W on each cell (3.19) should be written as a product of three 1-D Legendre polynomials rather than a single generalized 3-D
Legendre polynomial.
An interesting by-product of the present Euler algorithm is the usefulness of Roe's parametric vector [69]. The nonlinear consistency it provides proved to be the difference between convergence and divergence at shocks. The vector has not been used in previous Euler algorithms because nonlinear inconsistency shows up as a second order error for smooth flows. Thus, the error was probably deemed negligible if it was realized at all.
However, the present work suggests that more careful treatment of nonlinear consistency may improve results near shocks in other algorithms; it certainly cannot hurt. The use of Roe's parametric vector should also increase as higher order algorithms are developed.
7.4.2 Navier-Stokes Recommendations
The TSL Navier-Stokes algorithm developed in Chapter 4, achieved the goal of sec- ond order accuracy across the boundary layer. Its overall accuracy should be improved by better streamwise and temporal discretization. Possible alternatives to the first order
233 interpolation used in the flux-splitting have been discussed in Section 4.4.2. As stated, these alternatives are unstable for the single-stage time integration. A remedy to this problem may be semi-implicit multi-stage techniques, carefully constructed to achieve second order accuracy. Note that the solution of the Defect equations largely subtracts off the first order streamwise truncation error in the present scheme, but a second order discretization would further improve results.
For more complex 2-D geometries, the TSL Navier-Stokes equations should be solved in body-normal coordinates rather than Cartesian coordinates as done in this thesis.
The change is minor and was not done here mainly for reasons of simplified coupling to the outer inviscid flow.
This thesis has dealt only with boundary layer flows. Other types of viscous flows need to be considered, such as free shear layers and wakes. Modification of the present algorithm for wakes involves changes to boundary conditions and tracking the centerline of the wake. Two approaches become apparent. The first is to solve the wake in lower and upper sections with the centerline boundary conditions modified to zero shear
(r = 0) rather than no-slip (u = 0). The second approach is to solve the entire wake as a single system at each streamwise location. This requires inviscid boundary conditions at the lower and upper edges.
As with the Euler algorithm, extension of the present TSL Navier-Stokes algorithm to 3-D flows should involve only one new technical difficulty, grid definition and turbu- lence modeling for corner flows, which all algorithms must deal with. Apart from this, the major modification is the streamwise discretization which will become discretization in the plane of the wall. Discretization across the boundary layer will remain largely unchanged. In addition, the semi-implicit time integration can easily be extended to be implicit across the boundary layer and explicit in the plane of the boundary as is done by Nakahashi [611.
Improvements to the prediction of turbulent flows, particularly those with shocks and separation bubbles, should be provided by better turbulence models. The most promising for this type of algorithm is the Johnson-King model [461, which uses an
234 O.D.E. to determine the maximum eddy viscosity at each streamwise location. Imple- mentation of this model involves defining additional global variables at each streamwise station, which fits easily into the existing Newton solution procedure.
A more fundamental and difficult issue to address is the proper boundary conditions to impose at the outer edge of the viscous grid. This issue is particularly important
for inviscid/viscous coupling, because these boundary conditions have been found to influence solution quality and stability in certain flows. Perhaps 2-D characteristic theory may be helpful in resolving the questions raised by this thesis.
Some final comments pertain to the full Navier-Stokes equations. The present vis- cous algorithm solves the Thin-Shear-Layer Navier-Stokes equations. Extension to the full Navier-Stokes equations raises several technical problems. For example, additional shear stress and heat transfer component must be stored and each horizontal face mid- point. For each of these, an additional defining equation must be solved, and all involve streamwise derivatives. When streamwise viscous stress gradients become important, the streamwise grid spacing must be refined to resolve them. This makes the usefulness of a semi-implicit time integration questionable for full Navier-Stokes because of the streamwise stability constraint. The alternative is a fully implicit technique, such as that used by Beam and Warming [10].
7.4.3 Coupling Recommendations
The coupling procedure presented in this thesis has concentrated on matching the viscous and inviscid solutions for geometrically simple problems and grids. Extensions can be made to improve solution matching and treat more complex geometries.
Solution of the Defect equations within the viscous regions is clearly the direction to take for matching the viscous and inviscid solutions. This direction should be further explored by developing a proper discretization of the Defect equations. This involves interpolating the known inviscid solution to every point in the viscous grid. A proper discretization of the Defect equations is desirable not only because it allows smooth
235 solution matching, but also because it paves the way towards a fully conservative for- mulation at the interface.
Unfortunately, a simple and efficient method for performing the interpolation has not been determined yet. The problem is locating the particular inviscid cell that a given viscous grid point lies within. Since the viscous grid is dynamic, this locating procedure must be repeated at every time step. In addition, preliminary attempts have suggested the possibility of instabilities in a proper discretization if care is not taken.
One approach to the interpolation problem may be to define a viscous grid within each Euler cell. Then interpolation is straightforward since the inviscid solution really is linear, and the Euler cell where a given viscous grid point lies is known a priori. However, this approach raises other questions, such as the variation in the viscous grid from inviscid cell to inviscid cell. Also, it is not clear how to best couple the Euler and Navier-Stokes algorithms. This approach is similar to one first suggested by Dannenhoffer [25].
To be competitive with other Navier-Stokes solvers or other coupled Euler/Navier- Stokes approaches, the present coupling procedure must be extended to more complex geometries and to 3-D. The optimal arrangement may be a completely unstructured Euler grid coupled with a body fitted viscous grid. The two areas that need to be modified are the interpolation of the viscous edge boundary conditions and the mass transpiration flux.
Interpolation of the edge solution involves locating and interpolating to arbitrary points in the inviscid grid. Since the edge of the viscous grid changes, this process must be repeated at each time step. The problem in this context is very closely related to the interpolation problem for the properly discretized Defect equations. The process
can be cumbersome and time consuming, but it is a problem which has been addressed by other overlapping grid approaches, such as the Chimera schemes [27]. Some of the
same techniques may find use here to speed up the interpolation process.
For more complex geometries, the integration of the mass Defect equation (5.4a)
236 across the viscous grid is completely adequate. In 3-D, the mass transpiration flux contains an additional spatial derivative,
(pO)v = (PA dy + ^00= -azfo t0 --p y -71pu) - (Pt - pw)dy + -f(3-p)dy, (7.1) where z and z are the coordinate directions parallel to the wall, and u and w are the corresponding velocity components. The added term is no difficulty if the Defect equations are solved within viscous regions, since (7.1) is just the integral of the discrete mass Defect equation across the viscous grid.
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243 Appendix A Cell Metrics for Euler Algorithm
In this appendix, cell metrics for an n-sided polygon are derived. The metrics include the area, first and second area moments, and the cell centroid location.
The desired cell metrics involve integrals over the area of the control volume. For an arbitrary control volume, these can be converted to line integrals around the bounding curve by the divergence theorem,
//V -rdA = f - A ds, (A.1) where 0 is the control volume and 8af is its boundary with A the outward normal
(see Fig. 2.1). The vector i is chosen to produce the desired integrals over the control volume. The cell area, and first and second area moments about the origin are given by the following choices of f:
A= ffdA -+ 2 (XA + y) A (A.2a) -0 12
- AX = ffz dA 1 (X2- + Xy A), (A.2b) 3 r -= (zS + j, AY = ff ydA (A.2c) -+ 1 (YAs + Y2yA), AXX = ffX2 dA (A.2d) ->y r = (2A + X2/)
Azy = ffzy dA -4 12 (A.2e)
ZYY = ffy2 dA -- > S + y j (A.2f)
The () is used to denote area moment s about the origin. Note that these choices for r are not unique.
The cell centroid (x,, y,) is that point about which the first area moments are zero,
ff(z-z) dA = 0, ff(y -y,) dA = 0. (A.3)
244 The position of the centroid can be obtained from the first area moments about the origin by manipulating the integrals in (A.3). For example,
if(x-xc)dA = 0 = if xdA - xc dA = A.- zA. (A.4)
Thus, the position of the centroid is,
e Ye = . (A.5) A'A
The second area moments about the centroid can be obtained by a similar manipula- tion,
2 Azz = f(xz-C)2 dA = Az - 2zA2 + zC A, (A.6a)
Azy = ff(x -zC)(y - y) dA = Az2 - xAZ, - ycAz + xcycA, (A.6b)
2 AY, = (y _ yC)2 dA = AYY - 2ycAy + y0 A. (A.6c)
For an n-sided polygon control volume, the line integral in (A.1) becomes a sum of the integrals over each straight face,
ffV -dA =f f- Ads, = faces ds. (A.7)
Each of the desired integrals can be evaluated analytically. For example, the calculation of the cell area becomes a sum of the "area" contributions from each face,
A= E 6A, 6A = f(I/b + y) - A ds. (A.8) faces
For a typical face, shown in Fig. A.1, the tangential and outward normal unit vectors are,
8 S + n = -s - -7 (A.9) i8 A8 A8 A8 where Ax and Ay are evaluated consistent with counter-clockwise integration around the control volume. The integral is more easily evaluated if the x and y coordinates are transformed into local coordinates,
() = M + C ,x y() = ym + C Ay ,(A.10)
245 b ((=+1)
m n
a ((=-1) Figure A.1: Notation for Metric Integration Along a Face where (xm, y ..) is the midpoint of the face and C E [-1, +1]. Substitution into the integral (A.8) gives, 6A = (XmdY - YimAZ). (A.11a)
Similarly, the remaining metric contributions from the face are,
6Z = Zm A, (A.11b) 3 6AY = yme5A, (A.I1c) 3
6AX = ( Xm2 + IAZ2) 6A, (A.11d)
6AZy = Zmymmj,+ a zAy} 5A, (A.11e)
6AYY = (1ym2 + 1Aiy2) 6A. (A.11f)
Note that substantial roundoff error reduction results if the nodes of the polygon are translated to a local coordinate frame (e.g., taking one of the polygon nodes to be the local origin) prior to calculation of the metrics.
246 Appendix B Stability Analysis: Euler Algorithm
In this appendix, a von Neumann stability analysis is performed for the Euler al- gorithm developed in Chapter 3. Specifically, the first order spatial discretization of Section 3.3 and the second order spatial discretization of Section 3.4.2. This second or- der version of the scheme uses the midpoint rule for flux evaluations. It is assumed that the stability of this version is the same as that of Section 3.4.3, which uses two-point Gauss integration of fluxes.
The stability is analyzed using the linearized 1-D Euler equations rather than the more traditional scalar wave equation. This choice of equations has been prompted by complications arising from the use of flux-splitting and coupling between the av- erage and gradient equations. In supersonic flow, all characteristics move in the same direction, and the linearized Euler equations completely decouple into the separate char- acteristic equations. As a result, the stability restriction for the linear Euler equations is the same as that for the wave equation. In subsonic flow, the use of van Leer's flux- splitting reduces the maximum CFL number from that obtained for supersonic flow. This reduction is not easily obtained from an analysis of the wave equation.
The 1-D Euler equations in integral form are,
d- Udx + F = 0, (B.1) dt 2, ,,I where the conservation and flux vectors are,
P pu U pu F= pu2+p (B.2)
PE) puH and (zi, Z,) define the limits of the cell. The first moments of the Euler equations about
247 the cell center x, are given by,
dL (-xc)Udx + (x-xe)F = 0. (B.3) dtzi IZI
In the present analysis, the grid is assumed uniform with spacing Ax, and the centroid for the cell in question is located at xz = 0. The governing equations are discretized by approximating the solution within each cell as a linear Taylor series. Here, it is more convenient to use cell differences AU rather than cell gradients Ux,
U(X) e U + -- AU, (B.4) AX where AU = AxUX. Substitution of this solution approximation into the Euler and first moments of the Euler equations gives,
AXd U = - [F, - F], (B.5a) dt I -A2d- AX F2(- F] +AxF(U). (B.5b) 12 dt - K2) The fluxes through the faces are evaluated by van Leer's flux-splitting (3.6), where left and right interpolated solutions are obtained using the linear solution approximation (B.4),
F_ = F+(U+ }iU) + F- (U - !ZU) , (B.6a)
F1 = F+(U + IZU_, + F (U - ZU). (B.6b)
The equations are integrated in time using an explicit 3-stage Runge-Kutta scheme (k = 1, 2, 3),
U,(k) = ~ ~ At -,(k-1) ,k (B.7a) U -a(O)- AxA -, AU=(k) = Ui") - 6pak AR 1) (B.7b) i AX where R and AR are the right-hand-sides of (B.5a). The integration constants are a3 = 1, a 2 = 1/2 and a1 is arbitrary. The parameter p is an gradient update under- relaxation factor used to accelerate convergence to steady-state.
248 To study the stability of this system, it is first linearized about a constant solution.
In this linearization, the fluxes are replaced by their linearized forms,
F(U) = AU, A = OF (B.8)
Fi(U) = A*U, Ai = -U , (B.9)
where the Jacobian matrices A and A are evaluated based on the constant solution. These Jacobian matrices may be obtained by introducing an intermediate solution vector
UM given by, P UM =(C) (B.10)
(M) where c is speed of sound and M is Mach number. With this intermediate solution, the
Jacobian matrices are obtained by change rule differentiation,
F .UMaF aF aUm(B.11) A + au aUM U U A8auau au
The Jacobian of the flux vector F with respect to UM is,
F1 F 1 F 1 P C
2 OUM F 2- 2pc M (B.12)
F3 3 F 3 F3 + PC3M2 P C M+. where F, is the nth component of the flux vector. The subsonic van Leer split-flux
Jacobians are given by,
F F aF2 P C OM aF P 2 !12 (B.13) OUM C OM p aFi 3F3 .P C aM where aF = -PC(1+ M)2 , (B.14a) OM 2 aF2 = - 71)M + 2o] OF +( -)F (B.14b) am 7 a m ,
249 aFs' - y 29 2~ a) F 2 aF,1 a _ y2 - 2 ( ) Fi _ F2 (B.14c) M ~~ 2(' 1) 2 F I 8M F) M where o = +1 for F+ and o = -1 for F-. Finally, the Jacobian of UM with respect to U is given by,
1 0 0
aUM 1(7~1) M2] (B. 15) au 2+ [2jM P'&) (L.25)
[-M (1 + -Y('711)M2)] [1 + (;I-)M2] 1 __(Y1) M
When writing the linearized form of the equations, it is convenient to define the translation operator T,
TU = U +1 , T-'Uj = U- 1 . (B.16)
The linearized average and gradient equations for a single stage of the Runge-Kutta integration can then be written as,
U(k) - '6k' 1A+(1 - T-') + A-(T - 1)] Uk-1)
+ [A+(i - T- 1) - A~(T - 1)] ~U(k1) , (B.17a)
1 (k) = () - 6pak + T- ) + A-(T + 1) - 2A] Uk-1)
S[A+(1 + T- 1 ) - A-(T + 1)] aJ,(k-1) (B.17b)
A von Neumann stability analysis of this system examines the growth of the Fourier mode,
= Qef, -2 <; < +ir (B.18)
where Q is a 6 x 1 constant vector and 1 = V/ . Substitution of this mode into all three stages of the Runge-Kutta integration results in the matrix system,
Qn+l = GQ", (B.19)
250 where G is a 6 x 6 amplification matrix. The maximum eigenvalue of G must have magnitude less than 1 for the system to be stable. The matrix G itself is the product of three amplification matrices for each stage. These are obtained from (B.17) by replacing
T -+* efe and T~ 1 -> e-18. Furthermore, the time step At is determined by defining the CFL number, At A (lul + c) , (B.20) where Jul + c is the largest physical wave speed (and the spectral radius of A).
The matrix G is too complex to study analytically. Thus, it is constructed numeri- cally and its eigenvalues determined by EISPACK. For given values of Mach number M, first-stage integration constant ai, and under-relaxation factor p, the stability bound- ary -is determined by finding the value of A where the maximum eigenvalue of G first exceeds 1. This point is found by a combined bisection and secant iteration. The bi- section method [23, p. 220-221] is used initially to search for and straddle the desired point; once this is accomplished the secant method [23, p. 227-229] is used to speed convergence. At each iteration (i.e., at a given M, ai, p, and A), a search must be per- formed in 9 to find the maximum eigenvalue. This is done by evaluating G at several (typically 9 to 11) values of 9 between 0 and +7r (the eigenvalues for -0 have the same magnitude as those for +0).
Stability boundaries for the unsteady first order (no gradients) and second order versions of the algorithm are shown in Fig. 3.7 (Section 3.5.1). The effect of the under- relaxation factor on the stability boundaries is shown in Fig. D.1 (Appendix D).
251 Appendix C Stability Analysis: TSL Navier-Stokes Algorithm
In this appendix, a von Neumann stability analysis is performed for the TSL Navier- Stokes algorithm developed in Chapter 4. The analysis is first performed for the first order upwind streamwise discretization used in the algorithm. Analysis is also performed for the two suggested (but not used) streamwise discretizations in Section 4.4.2.
The present analysis is conducted for the 2-D advection-diffusion equation, written as a system of two first order equations, au au au ar au -0,-+a-+b at ax y ay r=v--, ay a>0. (C.1) Consistent with the algorithm developed in Chapter 4, this equation is discretized using a single-stage time integration with implicit two-point differencing in the y-direction and explicit first order upwind differencing in the z-direction. The resulting discretized equations are,
(Uj+1 + uj),'1 (ui+ - + +uu++ 1 (C.2a)
- (r+ - r)+1 + 2(Us+1 + u,)' - 2(uj+ 1 + us)_1 =
(rj+1 + r)7+1 - (u,+ 1 - us7+' = 0. (C.2b)
These equations my be written more compactly using the following translation opera- tors:
Zu" =_Un+1, Tui =_ui-1, Kui =_uj+j. (C.3)
With these translation operators the discrete equations may be written as, 1 aAt (K + 1)(Z - 1)u?, + 2ad(K + 1)(1 - T)uZ, (C.4)
252 +b dt ( - )O At 0, + (K - 1)Zt - A(K - 1)Zr=
1(K + 1),rn-' (K - 1) u . (C.5)
The shear definition may be used to eliminate r from the system,
r', = _ ( I-fi. (C.6)
Substituting for rg gives the final result,
(Z - 1) + A( - T) + 2AYZ K - 4Z K= 0, (C.7) (K+1 (K+1 where the CFL numbers are defined,
aot bAt vAt
Equation (C.7) is now examined for the Fourier mode,
9 u =-' g=et( +<<), +r (C.9) where S = 1. For this mode the translation operators become,
Z = g, T = e~", K = e+t, (C.10) and, K - 1 e+to _ 1 K +1 e+#+1 =itan(0/2). (C.11)
Substitution of the Fourier mode (C.9) into (C.7) gives the amplification factor g,
= 1+ 2 1 - Ax(1 - e-f) 1 + 2AY tan(0/2) - 4o tan 2 (0/2)(
For stability, the magnitude of the amplification factor must satisfy IgI 5 1.
Note that the numerator is a function of e only and the denominator a function of 4 only; hence, their amplitudes can be analyzed independently. The denominator has a magnitude greater than 1 for all positive AY and a, except at 0 = 0 where it has a
253 magnitude of 1. Thus, the numerator must have a magnitude that is not greater than
1 for stability,
1 - Az(1 - e) < 1. (C.13)
This is satified for all 0 if the streamwise CFL number is restricted to A, < 1. For the Navier-Stokes equations, the wave speed a is replaced by Jul + c, giving the stability
restriction, A_ = (Jul+c)At 1. (C.14) AX This shows that the numerical stability of the algorithm depends only on the stream- wise spacing Ax. It also shows that the stability limit is determined solely by the explicit streamwise discretization; the implicit discretization across the boundary layer is unconditionally stable.
The streamwise discretization presented in Section 4.4.2 is flux-splitting using first order solution interpolation to faces. Its equivalent discretization for the advection- diffusion equation is first order upwind differencing. Two higher order interpolation formulations are also suggested in Section 4.4.2; these have been found to have stablity problems when integrated in time using single-stage forward Euler.
The first alternate formulation is two-point solution interpolation to faces, as used by Anderson et al [7]. The equivalent discretization for the advection-diffusion equation
is second order upwind differencing,
-1-T ) 12 T) uj. =u- u- u-1) + ('Ui-1 - 1s_ = (C.15) Replacing the streamwise discretization in (C.7) by this formulation leads directly to
the stability requirement,
I - AX(1 - e--)( - < 1. (C.16)
This stability statement has been numerically determined to be satisfied only for A. = 0.
Hence, this higher order formulation is unconditionally unstable.
The second alternate formulation is interpolation assuming a similarity solution,
U(X) = u(0) + X jufd, (C.17) U dx
254 where U is the inviscid edge velocity. For this interpolation, the streamwise discretiza-
2Ud)(1+Ui tion becomes, au AxdU AUd AxdU ,d 1 + s 1 1 - T) (1 2U dx) + 2U dx i (C.18) The inviscid edge velocity has the Fourier mode,
U = #e ) C E [-pr/Az, +7r/A ] (C.19)
where the wave number is related to that in (C.9) by e = CAx. Hence, the interpolation term becomes,
dU ^ AxdU _ dT U dz (C.20) The resulting stability requirement is,
II - A (1 - C~") (1+ /2) I< 1. (C.21)
This has also been numerically determined to result in an unconditionally unstable formulation.
255 Appendix D Steady-State Acceleration
D.1 Euler Algorithm
Unsteady simulations are often periodic excitations about a steady mean flow. The
Euler algorithm of Chapter 3 is not efficient for calculating such steady flows because it has very little temporal or spatial dissipation and a rather restrictive CFL constraint.
To accelerate convergence to steady-state, two modifications to the scheme are used.
The first is gradient update under-relaxation, or marching the average equations at a higher CFL than the gradient equations. A single stage of the Runge-Kutta time integration is,
0 [A;U(k)] = [AU( )] - akAtXRk_'
Axrx(k) + AxyUy(k)] = [A..Ux (0) + AyU-y(0)] - Ck AtRX(k1) (D.1)
[Ay IUx(k) + AyyUy(k)] = [AZyUx (0) + AYYUY (0) - pakAtWRY (k-1). Figure D.1 shows the effect of gradient update under-relaxation on the stability limit of the second order scheme with first-stage coefficient of a = 0.17 (the optimal value for unsteady stability, see Fig. 3.7). Using p = 0.1 the maximum CFL for supersonic
flow increases to A = 2.6 (compared to A = 0.59 for 0 = 1). Table D.1 shows the effect of under-relaxation (p = 1, A = 2.2) on the steady-state convergence for the subsonic circular bump case of Section 6.1.1 on a 32 x 8 grid. This small modification
of the time integration causes it to converge 30 times faster for that case. The effect of under-relaxation is also shown in Table D.2 for the transonic circular bump case (Section 6.1.2); here the speedup is 6.4 times.
The second modification is time integration using implicit approximate factorization
256 2.2-
M =1/2
2.0 -
AM > I UNSTABLE
1.8
1.6-
1.2- STABLE
0.8
0.4 0.0 0.1 0.2 0.3 0.4 1
Figure D.1: Effect of Gradient Update Under-Relaxation on Stability Boundary of Second Order Scheme (a, = 0.17)
(AF). By using implicit AF, two major assumptions are made that are not made in the rest of the formulation of the present Euler algorithm. The first is that the grid is structured and made up of quadrilaterals. Thus, each cell can be assigned a (i,j) index and implicit sweeps in i and j constructed. The second major assumption is that the grid is roughly aligned with the x and y-axes, as it is for the duct flows presented in this thesis. This allows the derivatives 8/8z to be evaluated as differences in i at constant j, and it allows 1-D characteristics in z to be used (likewise for y derivatives).
Consider a scheme of unspecified spatial discretization which is integrated in time using backward Euler. If the fluxes are linearized about the current solution (level n), then the equations for updating the solution (to level n + 1) may be written in delta form as, a9 a aFn aen I+ At-An +,At-BJ bUflAta + = -AtR, 1, x ay Ix( aY (D.2)
257 where 6Un = Un+1- U, A and B are the spatially varying flux Jacobian matricies, and R is the residual or net flux balance on each cell. The multi-dimensional implicit opera- tor used in these equations typically leads to inversion of a large bandwidth matrix. The operator may be approximated to order O(At 2) by the product of two one-dimensional operators as, I +At--A" I +,At-B" Un = -AtRn. (D.3) 5_X ay I Each of the one-dimensional operators leads to matricies of much smaller bandwidth
(typically block tridiagonal). This technique of approximate factorization is used for solution of the Euler and Navier-Stokes equations in the central-difference scheme of Beam and Warming [10]. It has also been used with various other spatial discretiza- tions including the flux-split scheme of Anderson, Thomas and van Leer [7]; in this case, the implicit derivatives become backward and forward differencing using split flux Jacobians. For the present algorithm, the presence of gradients and gradient equations greatly complicates a proper approximate factorization method, if by nothing else than calculation and inversion of 12 x 12 block Jacobian matricies.
The emphasis of the current thesis is unsteady flows, and this AF technique is used only to speed convergence to steady-state. Thus, the goal is to construct a simple but effective technique, rather than to construct or duplicate the most efficient AF method.
As a tool for achieving steady-state faster, (D.3) may be interpreted as updating the solution using a modified residual,
Un+1 = U" + 6U" = U" - dtR** an a n (DA) [I+At-A" 1+ At -B"I R** =R".
With this in mind, several simplifying assumptions can be made. First, the gradient residuals are modified using the same equations as the average residuals. Second, the flux Jacobians are assumed to be locally constant. Thus, at each stage of the Runge- Kutta integration, each of the residuals vectors R, R1 and Ry of (D.1) are modified by the following equations,
I+AtA aR*=R aBz (D.5) R**R* [+,dtB-aay]
258 where the Jacobian matricies are evaluated at the current solution. If the derivative operators are discretized consistent with the explicit spatial discretization, then A and B become the Jacobians of the split flux vectors and the derivatives become upwind differences. These lead to block tridiagonal systems (4 x 4 blocks). Alternatively, each of these systems can be decoupled and solved as scalar tridiagonals if they are transformed to 1-D characteristics in x and y, respectively. The first system, written in terms of 1-D characteristics in the z-direction, becomes,
I+ AtA aR* = R Iax] - I+ AtA a C* = c2 , (D.6) where A. is the diagonal matrix of the eigenvalues (A) of A and C. is the vector of characteristics in the z-direction. There are given by,
SU + C 0 0 0) ( +(p'+pcu' 0 U - c 0 o p'F-pcu' Ax = o C 9-=. (D.7) 0 0 U ) P' - OP) 0 0 0 U,)V1
The transformation from residuals to characteristics is as follows,
p'= R 1 U' = -(R 2 - uR1 ) (D.8) 1 v' = (R - vR1 ) p
2~
The derivatives are evaluated using first order upwind differencing based on the sign of the eigenvalues A,
a i ( ) i-. if |\1 ;> 0 ( f - ,i A (D.9) ( )i,- ( )+1,j if A, AX < 0
The computational procedure is as follows: Convert the residuals into 1-D character- istics; construct the scalar tridiagonal matrix based on the local values of u and c for each characteristic; solve the scalar tridiagonal form the modified characteristics; and
259 transform back to conservation residuals. This final transformation is given by,
R* = pu' + up' (D.10) R* = pt' + Vp'
R* =p + PI + PuU + Pv, 4 - 1 2 once p', u', v' and p' have been recovered from C*.
This process is then repeated for the second system of (D.5), ia * + AtB2 a R**= R* C*, (D.11) cl I I,+ AtAy y C** = where A, and C, are, 0 0' (V +C0 V 0i 0 0 Ay I, (D.12) 0 0 v 0 P' - C2p, 0 0 0 V,
First order upwind differencing in j is used to evaluate the derivatives.
This implicit AF technique does not make the scheme unconditionally stable. Split-
ting errors in the AF, the assumption of locally constant flux Jacobians, and treatment of the gradient equations will all affect the stability and convergence for large CFL. As a result, the optimal CFL must be found experimentally. Tables D.1 and D.2 shows the effect of AF integration on steady-state convergence for both subsonic and transonic flows. For subsonic flow the optimal CFL is A = 8 compared to the unsteady limit of A = 0.55 and the under-relaxed limit of A = 2.2. The speedup is approximately 3.3 over the under-relaxed CPU time, even though an implicit iteration takes 44% longer than an explicit iteration. For transonic flow (Table D.2) the optimal CFL is A = 6, which produces a factor 1.8 speedup over the under-relaxed time.
260 Table D.: Effect of Relaxation and AF on Steady-State Convergence: Subsonic Circular Bump on 32 x 8 Grid
b CPU CPU Scheme y A Iterations to CPU (pusec) Convergence" cell -iter (sec) (normed) explicit 1.0 0.55 M33,200 18.8 154,000 83
explicit 0.1 2.2 1130 " 5450 2.83
AF 0.1 2.0 990 27.1 6865 4.10
AF 0.1 4.0 520 3605 2.15
AF 0.1 6.0 350 2427 1.45
AF 0.1 8.0 240 1664 1
AF 0.1 10.0 340 2358 1.41
AF 0.1 12.0 920 6379 3.83
AF 0.1 14.0 DNCe
a convergence criteria: <_. 5.0 x 10-6 At rms b D.E.C. MicroVax II (scalar machine)
C did not converge
D.2 TSL Navier-Stokes Algorithm
The TSL Navier-Stokes algorithm of Chapter 4 is also modified to accelerate its convergence to steady-state. The modification is implicit AF, similar to that used in the Euler algorithm.
The viscous solution is integrated in time using a semi-implicit technique. Discretiza- tion normal to the boundary is integrated implicitly, while streamwise discretization is integrated explicitly. The stability analysis of Appendix C has shown that the implicit component of the algorithm is unconditionally stable. Hence, only a streamwise implicit sweep is needed to increase the maximum CFL and accelerate convergence.
261 Table D.2: Effect of Relaxation and AF on Steady-State Convergence: Transonic Circular Bump on 32 x 8 Grid
CPU CPU Scheme ~ A Iterations to CPU (' sec) b Convergence' cell - iter (sec) (normed)
explicit 1.0 0.55 ~8700 18.8 41,900 11.6
explicit 0.1 2.2 1370 " 6606 1.83
AF 0.1 2.0 1380 27.1 9569 2.65
AF 0.1 4.0 750 " 5200 1.44
AF 0.1 6.0 520 " 3605 1
AF 0.1 8.0 DNC
a convergence criteria: A < 5.0 x 10-8 1 At b D.E.C. MicroVax II (scalar machine)
C did not converge
For the present modification, the unsteady time integration of Section 4.5 may be simply viewed as a procedure which produces a change 6W in the solution over one time step. This change is modified to 6W* by an implicit streamwise sweep prior to update,
Wn+1 = Wn + 6W*, (D.13a)
[I+AtA]+Atfa- C,(6W*) = Ce(6W"), (D.13b)
where AC and CC are the 1-D characteristics and characteristic variables in the stream- wise direction (D.7). There is only one difference between this streamwise implicit sweep and that used to modify the Euler algorithm. In the viscous solver, the changes in the solution from time level n to n + 1 are determined directly in terms of Roe's paramet- ric vector W, rather than in terms of the conservation vector U as done in the Euler algorithm. Thus, different formulas are required to transform solution changes to and
262 from the characteristic variables:
6wj = -p
2 1 +(D.14)
6w* = v6w* + w 1V',
6w = H6w* + wiH', where H' C +uu'+vv'. (D.15)
Because of 2-D splitting errors in the AF, the modified time integration will not be unconditionally stable, and its effectiveness should not continue to improve as the CFL number is increased. Thus, the optimal CFL for steady-state convergence must be found by experiment, just as in the Euler algorithm. Table D.3 shows the effect of the CFL number on convergence for both the unsteady time integration and this modified integration for the turbulent flat plate case of Section 6.2.1. The maximum speedup from the AF is approximately 3.5 and occurs at a CFL of A = 3.0. This CFL is also used for coupled Euler/Navier-Stokes steady-state calculations.
263 Table D.3: Effect of AF on Steady-State Convergence: Turbulent Flate Plate With a 30 x 48 Grid
CPU CPU Scheme A Iterations to CPU (psec) b Convergencea cell - iter (sec) (normed) S-Id 0.5 45 17.4 1130 3.51
S-I 0.8 29 728 2.26
S-I 1.0 DNCc
S-I w/ AF' 1.0 29 18.8 777 2.41
S-I w/ AF 2.0 15 402 1.25
S-I w/ AF 3.0 12 322 1
S-I w/ AF 4.0 DNC
convergence criteria: -- < 0.31
b D.E.C. MicroVax II (scalar machine)
c did not converge
d semi-implicit time integration
9 semi-implicit with approximate factorisation
264 Appendix E Test Case Descriptions
This appendix contains descriptions of the four viscous test cases presented in Chap- ter 6. The specified computational test conditions refer to the following parameters:
-y = 1.4 specific heat ratio
M inlet Mach number
R, = *eit exit pressure ratio Finlet
Re = PrefCrefLref reference Reynolds number Pref Pr = 0.72 reference Prandtl number Prt = 0.9 turbulent Prandlt number S'= 0.4 Sutherland's temperature constant
Xprof flat plate length for specified inlet profile
Reference condtions ref are taken as inlet stagnation conditions, and all barred quan- tities are dimensional. Those conditions numerically specified above are the same for all the test cases.
265 Turbulent Flat Plate (Section 6.2.1)
Experiment Test Conditions [861:
U = 33 m/s freestream velocity
V = 0.151 cm2 /s kinematic viscosity Lplate = 5 m length of plate
Computational Test Conditions:
Assume sea level conditions (E = 330 m/s) and a reference length of Lrfi = 1 m.
The Mach number is, U M = -= 0.1.
The reference Reynolds number is,
Re = crefLref _ U Lref [ref = 2.195 x 107, Wref VF I F W ref where F/Fref and r/gref are obtained from isentropic relations and Sutherland's law (2.14).
Geometry and Grid:
The geometry is a flate plate of length 6. The turbulent grid is 30 x 48 (i.e., it contains 47 cells across the grid). It is exponentially stretched away from the wall and is generated by Eq. (4.122) with r, = 14 and a = 6.594.
266 Subsonic Compression Duct (Section 6.3.1)
Computational Test Conditions:
M = 0.5, Re = 1.0 x 106, Xprof = 4.5
Geometry and Grid: The geometry is a compression duct with a straight lower wall and cosine constriction on the upper wall,
Yinlet, ; < 0;
Yupper r( (a;-2)1 0 < x < 2; = (yexit + yinlet) + (yexit - yinlet) cos [ 2
yinlet,
The inlet is located at zinlet = -2 and the exit at xetit = 4. The inlet has height Yiniet = 1 and the exit has height yedit = 0.8. The Euler grid is 48 x 8 with uniform spacing in both x and y. The viscous grid is 48 x 16; it has algebraic stretching, Eq. (4.121) with
77j = 14 and a = 1.5.
Oblique Shock/Boundary Layer (Section 6.3.2)
Experimental Test Conditions [39]:
p6/p1 = 1.40 static pressure rise through interaction Ti = 1.95 in shock impingement point on flat plate
Rex = -- = 2.96 x 105 Reynolds number based on Yi
M = 2 nominal freestream Mach number
All subscripts refer to Fig. 6.26.
267 In the first series of investigations using this experimental setup, conducted as part of the M.I.T. Guilded Missiles Program [8], the freestream Mach number was measured by a total head probe. In the second series of experiments, conducted by the M.I.T. Gas
Turbine Laboratory [39,1,35], actual Mach number measurements were only conducted in the initial experiments. For example, Abarbanel [1] states, "The tests were made at an nominal Mach number of M = 2, with the actual ranging between 1.99 and 2.05.
Average Mach was 2.03 and most runs were made at M = 2.03 and 2.04." In later experiments [39,35], the freestream Mach number was not actually measured. This includes Hakkinen et al [39], where the present test case is found.
Computational Test Conditions:
This has become a standard test case for shock/boundary layer interaction [58,10,2,791.
The test conditions reported by MacCormack are,
M=2 nominal freestream Mach number
8 = 32.59* incident shock angle 9 = 3.089* shock generator angle
The shock generator angle is the flow deflection angle through the incident shock. Though not stated in Ref [58], it is believed that these conditions were obtained by an- alyzing an inviscid regular shock reflection off the wall [52, p. 86], where the freestream Mach number is M = 2 and the static pressure rise is pr/pl = 1.40.
These conditions seem to be slightly erroneous. Although the pressure rise through the interaction is correct, the actual pressure levels do not match experiment, which were reported in terms of p/P1 . For a freestream Mach number of M = 2, the pressure is pI/P = 0.1278; this seems high compared to the data. Computational results for this case are normally plotted as a static pressure ratio p/pi, rather than p/P1 . The reason is that these pressure level mismatches do not show up as much if a static pressure ratio is used.
268 To match experimental pressure levels better in terms of p/1I, the experimental data has been reexamined. The measured post interaction pressure is p6 /P 1 = 0.171
(see Fig. 6.36a). Using this value and a static pressure rise of pe/pi = 1.40, an inviscid shock reflection analysis gives the following test conditions:
M = 2.03 nominal freestream Mach number P = 32.04* incident shock angle
9 = 3.059* shock generator angle
These values are used for the computational results presented in Section 6.3.2. With this freestream Mach number, the pre-interaction pressure is pi/P = 0.1220; this is closer to the experimental values.
Geometry and Grid: The experimental geometry is a flat plate with a sharp leading edge. A shock generator is placed above the plate to produce an oblique shock which impinges the flat plate at z = 1.95 (see Fig. 6.26). In the computational geometry, the plate length is
Xexit = 3.5. The inlet is placed at 5Ax upstream of the leading edge (Xiniet = -5.6), where Ax is the uniform grid spacing on the plate (AX = Zexit/(I - 5). The streamwise position of the shock generator (upper wall) leading edge is located at the inlet; the y-position is located so that a shock of angle 8 = 32.04* impinges at xi = 1.95,
XGLE = Zinlet = -5Az, yGLE = (xi + 5Ax) tan(#). (E.1)
The upper wall of the channel is sloped at an angle 0 = 3.059*, equal to the flow deflection angle after the incident shock.
Two grids are used, one is 64 x 16 and the other is 128 x 32. The grid spacing in x is uniform. The spacing in y is algebraically stretched away from the lower wall (similar to Eq. 4.121 with P = 1.15).
269 Transonic Diffuser (Section 6.3.3)
Experimental Test Conditions [12,711:
R, = 0.826 exit pressure ratio (exit static to inlet stagnation)
Re. = * * * = 7.5 x 105 mean Reynolds number based on throat conditions 11* = 0.0441 m throat height
Mahock = 1.235 pre-shock Mach number
Mexd = 0.51 exit Mach number
The exit of the diffuser was vented to constant pressure atmospheric conditions.
Computational Test Conditions:
Use K. as the reference length. The reference Reynolds number is,
Re= Re. [ [ j! 1 = 1.104 x 106, P* J I * JLprefJ where the conversion factors are obtained from the isentropic relations (with M. = 1) and Sutherland's law. No data is given for the inlet boundary layers. They are assumed to be flat plate turbulent boundary layers with dimensions,
lower wall: 6 = 0.045 h + Zprof = 2.0
upper wall: 6 = 0.090h. -+ Zprof = 4.5
These are the same inlet profiles used by Hsieh et al [41].
These conditions are sufficient to specify the flow. The experiments also report natural frequencies in Hz. To convert them to reduced frequencies based on the throat height K. and inlet stagnation speed of sound cref, the stagnation temperature is needed.
Assume the exit is at room temperature ( = 300 K), then using the exit Mach number,
Tref = T, 1 + Me2 = 316 K.
Note the exit Mach number can also be obtained by using R, and the total pressure
loss across the shock, calculated using MAhock. The stagnation speed of sound is given
270 by,
Cref = 7lRTref = 356 m/s, R = 287kg K
Geometry and Grid: The diffuser model (see Fig. 6.27) is a convergent/divergenct duct with a flat lower wall and an upper wall given analytically by Ref. [12),
h(x) = acosh(e) S(a - 1)+ cosh(C)' where C1 (X/l)[1 + C2(X/l)IC'
[1 - (z/l)}C4 All lengths are nondimensionalized by the throat height. The coefficients upstream and downstream of the throat are given as follows:
constant X<0 z<0 z>0
(expt.) (comp.) (expt.)
1.4114 1.4114 1.5
1 -2.598 -3.0 7.216
C1 0.81 0.81 2.25
C2 1.0 3.5 0
C3 0.5 0.5 -
C4 0.6 0.6 0.6
Upstream of the throat, the upper wall in the computations has been slightly altered.
The experimental geometry has a high curvature where the duct first begins to constrict. This requires grid resolution, and it is an area where nothing interesting is happening (i.e., the flow is completely steady there). Hence, this initial constriction is spread out somewhat in the computational geometry, keeping the same inlet to throat area ratio.
The computational inlet and exit stations are Zinlet = -4 and Zexit = +8, respec- tively.
271 The Euler grid is 71 x 8; it is constructed from a uniformly spaced background 48 x 8 grid with clustering at the shock. From x = 1.1 to z = 1.8, the streamwise spacing is
Ax = 0.0625. Outside this region, it is exponetially stretched (Axi+1/Az; = 1.1) to the background spacing of Ax = 0.25. The upper and lower wall viscous grids are each 71 x 32. The grids are exponetially stretched away from the walls, and are generated by Eq. (4.122) with t7, = 14, a = 5.473 for the upper, and a = 5.207 for the lower wall grid.
272 Appendix F Code Listing
This appendix contains the source code for the program UNSCENS (Unsteady Cou- pled Euler/Navier-Stokes). Source codes are not included for the grid generation and graphics packages used in conjuction with UNSCENS.
This program is copyrighted by the Massachusetts Institute of Technology, and may not be used without written license from M.I.T. For additional information, please contact:
M.I.T. Software Center Technology License Office M.I.T. Room E32-300 77 Massachusetts Ave. Cambridge, MA 02139
Code listings available for $15.00 to cover duplication costs.
273