ti. GXY DJPL I M11111111111111MI 3IBRARIES 111111Il 3 c1080 0.601118 0 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 ........ ............. ...