A Dissertation
entitled
Towards Improvement of Numerical Accuracy for Unstructured Grid Flow Solver
by
Qiuying Zhao
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Doctor of Philosophy Degree in Mechanical Engineering
______Dr. Chunhua Sheng, Committee Chair
______Dr. Abdollah A. Afjeh, Committee Member
______Dr. Glenn Lipscomb, Committee Member
______Dr. Ray Hixon, Committee Member
______Dr. Terry Ng, Committee Member
______Dr. Patricia R. Komuniecki, Dean College of Graduate Studies
The University of Toledo
December 2012
Copyright 2012, Qiuying Zhao
This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of
Towards Improvement of Numerical Accuracy for Unstructured Grid Flow Solver
by
Qiuying Zhao
Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Doctor of Philosophy Degree in Mechanical Engineering
The University of Toledo
December 2012
An effort to improve the numerical accuracy of a three dimensional unstructured grid finite volume scheme is pursued in the present work. Unstructured grid methods have been widely used in computational fluid dynamics for the convenience of modeling complex geometries in realistic applications. In the present work, improvements towards high order unstructured grid schemes are proposed using high order flux formula for spatial discretizations. The Riemann variables on the left and right sides of the interface are reconstructed using quadratic and quartic polynomials composed of both flow variables and their gradients. The high order flux is then calculated using the concept of
MUSCL (Monotone Upstream-centered Schemes for Conservation Laws) approach. In order to maintain the accuracy for the finite volume scheme, an innovative method based on Radial Basis Function interpolation is introduced as a substitution to Gaussian quadrature to achieve the higher order surface integration on mixed element unstructured grids.
iii The proposed high order improvements for unstructured grid schemes have been tested for a wide range of flows from very low Mach number to supersonic speeds. The observed accuracy for the improved schemes is verified using a benchmark case about an inviscid vortex transporting in a free stream flow. In addition, the ability to capture the tip trailing vortex, which is a major challenge in computational fluid dynamics today, is extensively verified on two vortex dominated viscous flows, a fixed NACA0015 wing at a subsonic Mach number and a rotating NACA0012 hovering rotor at a transonic tip speed. The numerical validations are also performed on two realistic industrial applications including a marine propeller P5168 and a Bell Helicopter aircraft 427 main rotor. Computational results indicate that the methods proposed in the present work can significantly improve numerical accuracy in predicting the strong vortical flows in smooth regions, while maintaining the stability of the schemes in discontinuous regions such as shockwaves.
iv
To my parents
Acknowledgements
First, I sincerely thank my advisor, Dr. Chunhua Sheng, for his support and guidance involved in this thesis. His patient supervision, encouragement, and inspiration have helped me as a PhD student over the years at the University of Toledo. His challenging and rigorous research approach has impressed and trained me as a research assistant, as well as, shown me the importance and responsibility of being an engineer.
I would like to express my appreciation to Dr. Abdollah A. Afjeh, Chair of
Mechanical, Industrial, and Manufacturing Engineering, Dr. Glenn Lipscomb, Chair of
Chemical and Environmental Engineering, Dr. Ray Hixon, and Dr. Terry Ng. for serving on my thesis committee. Their advice and suggestions are very valuable to my dissertation work. I thank Dr. Hixon’s systematic tutoring on the CFD course, from which the knowledge laid the foundation of my research work afterwards.
I would like to especially thank Dr. Christian Allen from the Department of
Aerospace Engineering, University of Bristol, UK, for his patient explanation of the
Radial Basic Function method and longtime discussion on my research work. His practical suggestions have gone a long way in shaping my ideas and helped me conquer the difficulties of developing a robust and efficient CFD tool.
I am very grateful to Dr. Li Wang from the Graduate School of Computational
Engineering, the University of Tennessee at Chattanooga, for discussions about the higher order methods and suggestions on code verifications.
vi I would like to thank my colleagues and friends, Kenneth Miller, Ahmed Magdy, and Jingyu Wang in Dr. Sheng’s CFD group, for the good atmosphere of research discussion and friendship. I express my gratitude to my great colleagues and friends from
MIME, Adrian Sescu, Daniel Ingraham, Nima Mansouri, Chuanbo Yang, Steven Koester,
Boya Zhang, Hao Jiang, Yue Hou, Ting Wen, Yaoying Wang, and Xiaotong Li for the friendships and memories we shared. I truly thank my friend Qi Zhao for her continuous friendship since high school. I would like to express my appreciation to Aaron Kirgesner for his valuable help in correcting this dissertation. Their encouragement and support cheered me up to face challenges from both daily life and research work.
I am deeply indebted to my parents for their endless love, encouragement, and understanding throughout my life. Their selfless affection is my great source of power and gives me the confidence to face anything in my life fearlessly. I express my true love and gratitude and dedicate this dissertation to them.
vii
Table of Contents
Abstract ...... iii
Acknowledgements ...... vi
Table of Contents ...... viii
List of Tables ...... xii
List of Figures ...... xiii
List of Abbreviations ...... xix
List of Symbols ...... xxi
1 Introduction ...... 1
1.1 Motivation ...... 1
1.2 Literature Review about Higher Order Schemes ...... 3
1.2.1 Finite Difference Schemes ...... 3
1.2.2 Compact Schemes ...... 4
1.2.3 Spectral Difference Schemes ...... 5
1.2.4 Finite Volume Schemes ...... 5
1.2.5 Spectral Volume Schemes ...... 7
1.2.6 END and WEND Schemes ...... 7
1.2.7 Finite Element Schemes ...... 8
1.2.8 Discontinuous Galerkin Method ...... 9
1.3 Current State of CFD ...... 10
viii 1.4 Research Objectives and Scope ...... 10
1.5 Organization of Dissertation ...... 12
2 Computational Fluid Dynamics Overview ...... 14
2.1 Historical Perspective of CFD...... 15
2.2 The Importance of CFD in Virtual Prototyping ...... 20
2.3 Procedures of Applying CFD ...... 22
3 Computational Methodology ...... 24
3.1 The CFD Solver ...... 24
3.2 Mathematical Formulation ...... 25
3.2.1 Governing Equations in an Absolute Frame ...... 25
3.2.2 Governing Equations in an Rotating Relative Frame ...... 28
3.2.3 Non-Dimensionalization of Governing Equations ...... 29
3.2.4 Preconditioned Governing Equations ...... 31
3.2.5 Preconditioning Parameter ...... 35
3.3 Numerical Formulation ...... 37
3.3.1 Discretized Governing Equations ...... 37
3.3.2 Convective Flux Evaluation ...... 38
3.3.3 Diffusive Flux Evaluation ...... 39
3.3.4 Time Marching Method ...... 40
3.3.4.1 Newton’s Method ...... 40
3.3.4.2 Gauss-Seidel Relaxation ...... 42
3.3.5 Turbulence Model ...... 43
3.3.6 Boundary Conditions ...... 44
ix 3.3.6.1 Characteristic Far field ...... 45
3.3.6.2 Characteristic Inflow ...... 46
3.3.6.3 Characteristic Outflow ...... 47
3.3.6.4 Impermeable Wall ...... 48
3.3.6.5 Wake Model ...... 48
4 Improvements to CFD Methodology ...... 51
4.1 Inviscid Flux Reconstruction ...... 51
4.1.1 Quadratic Polynomial Construction ...... 51
4.1.2 Quartic Polynomial Construction ...... 53
4.2 Surface Integration ...... 54
4.2.1 General Procedure of Surface Integration ...... 54
4.2.2 Comparisons of Gaussian Quadrature and RBF Integration...... 57
4.2.3 Formulation of RBF Interpolation ...... 60
4.3 Weighted Limitation ...... 65
5 Results and Discussion ...... 67
5.1 2D Vortex Transportation ...... 68
5.2 NACA0015 Wing ...... 81
5.3 NACA 0012 Rotor ...... 97
5.4 Waisted Body ...... 110
5.5 Marine Propeller P5168 ...... 122
5.6 M427 Main Rotor ...... 134
5.6.1 M427 Main Rotor in Hover ...... 136
5.6.2 M427 Main Rotor in Forward Flight ...... 140
x 5.7 Chapter Summery ...... 145
6 Conclusions and Recommendations ...... 147
6.1 Conclusions ...... 147
6.2 Recommendations ...... 149
References ...... 150
Appendix A Coeffients of the Spalart-Allmaras Turbulence Model ...... 166
xi
List of Tables
4.1 Properties of RBF and Gaussian Quadrature methods ...... 57
4.2 RBF functions with global and local piecewise smoothness ...... 62
5.1 Maximum value of voticities at the 10000th step ...... 77
5.2 Maximum value of voticities at the 20000th step ...... 77
5.3 Maximum dissipation ratio of voticities between the 10000th step and the 20000th
step ...... 78
5.4 Observed order of accuracy on the various grids ...... 80
5.5 Computational costs for NACA 0015 wing using different schemes ...... 96
5.6 Computational costs for NACA 0012 rotor ...... 110
5.7 Computational costs for waisted body using different schemes ...... 121
5.8 Flow parameters associated with different advance ratios ...... 123
5.9 Computational costs and comparisons of thrust and torque coefficients ...... 124
5.10 Geometric and Air Property Definitions for M427 Main Rotor ...... 135
xii
List of Figures
2-1 The application of CFD in the design of Boeing 777 ...... 15
4-1 Control volumes of the unstructured grids (a) cell-centered control volume (b) node-
centered control volume ...... 56
4-2 Interpolation on control surfaces of the two-dimensional unstructured grid using RBF
method (a) 3D physical grid projected to 2D, plane (b) one of the control surfaces
...... 59
4-3 Error estimation for Wendland's type RBF function in quadratic polynomials ...... 64
4-4 Error estimation for Wendland's type RBF function in quartic polynomials ...... 65
5-1 Comparison of meshes for vortex transportation case: (a) coarse grid, (b) medium
grid, (c) fine grid ...... 71
5-2 Convergent histories of different schemes on a series of meshes ...... 73
5-3 Comparison of vorticities on the coarse grid at 10000 steps on the coarse grid: (a)
original, (b) improved, (c) exact ...... 74
5-4 Comparison of vorticities on the middle grid at 10000 steps on the medium grid: (a)
original, (b) improved, (c) exact ...... 75
5-5 Comparison of vorticities on the fine grid at 10000 steps on the medium grid: (a)
original, (b) improved, (c) exact ...... 76
5-6 Comparison of vortical velocities on different grids at the 2000th step ...... 79
5-7 Comparison of vortical velocities on different grids at the 1000th step ...... 79
xiii 5-8 Trailing tip vortex generated from the NACA 0015 wing ...... 82
5-9 Three mesh resolutions along the path of trailing tip vortex: (a) coarse mesh, (b)
medium mesh, (c) fine mesh ...... 83
5-10 Computed vortical velocities at 2 chord downstream using the 2nd scheme on
different meshes: (a) no RBF, (b) with RBF...... 85
5-11 Computed vortical velocities at 2 chord downstream using the 3rd scheme on
different meshes: (a) no RBF, (b) with RBF...... 86
5-12 Computed vortical velocities at 2 chord downstream using the 5th order scheme on
different meshes: (a) no RBF, (b) with RBF...... 87
5-13 Computed cross section velocity contours at 6 chords downstream using the 2nd
order scheme on the medium mesh: (a) without RBF, (b) with RBF ...... 88
5-14 Computed cross section velocity contours at 6 chords downstream using the 3rd
order scheme on the medium mesh: (a) without RBF, (b) with RBF ...... 89
5-15 Computed cross section velocity contours at 6 chords downstream using the 5th
order scheme on medium mesh: (a) without RBF, (b) with RBF ...... 90
5-16 Comparison of the vortical velocities using different schemes at one chord
downstream ...... 92
5-17 Comparison of the vortical velocities using different schemes at 2 chords
downstream ...... 92
5-18 Comparison of the vortical velocities using different schemes at 4 chords
downstream ...... 93
5-19 Comparison of the vortical velocities using different schemes at 6 chords
downstream ...... 93
xiv 5-20 Predicted pressure coefficients at 23.8% wing span ...... 94
5-21 Predicted pressure coefficients at 49% wing span ...... 94
5-22 Predicted pressure coefficients at 89.9% wing span ...... 95
5-23 Predicted pressure coefficients at 97.4% wing span ...... 95
5-24 NACA 0012 rotor coarse grid ...... 98
5-25 NACA 0012 rotor fine grid ...... 98
5-26 Predicted 2nd order vorticity contours (left) and iso-surface swirl parameter (right)
on the vertical cutting plane for the coarse grid ...... 100
5-27 Predicted 3rd order vorticity contours (left) and iso-surface swirl parameter (right)
on the vertical cutting plane for the coarse grid ...... 100
5-28 Predicted 5th order vorticity contours (left) and iso-surface swirl parameter (right)
on the vertical cutting plane for the coarse grid ...... 101
5-29 Predicted 2nd order vorticity contours (left) and iso-surface swirl parameter (right)
on the vertical cutting plane for the fine grid ...... 102
5-30 Predicted 3rd order vorticity contours (left) and iso-surface swirl parameter (right)
on the vertical cutting plane for the fine grid ...... 102
5-31 Predicted 5th order vorticity contours (left) and iso-surface swirl parameter (right)
on the vertical cutting plane for the fine grid ...... 103
5-32 Predicted helicity contours by 2nd order scheme for the fine grid ...... 104
5-33 Predicted helicity contours by 3rd order scheme for the fine grid ...... 104
5-34 Predicted helicity contours by 5th order scheme for the fine grid ...... 105
5-35 Predicted tip vortex strengths (peak vorticity) by schemes ...... 106
5-36 Predicted trajectories of trialing tip vortices by various schemes ...... 107
xv 5-37 Predicted pressure coefficients along the 68% of the blade span on the fine grid . 108
5-38 Predicted pressure coefficients along the 80% of the blade span on the fine grid . 108
5-39 Predicted pressure coefficients along the 89% of the blade span on the fine grid . 109
5-40 Predicted pressure coefficients along the 96% of the blade span on the fine grid . 109
5-41 The coarse grid for the waisted body ...... 111
5-42 The medium grid for the waisted body ...... 112
5-43 The fine grid for the waisted body...... 112
5-44 Predicted pressure (left) and Mach number (right) by the 2nd order scheme on the
medium grid ...... 114
5-45 Predicted pressure (left) and Mach number (right) by the 3rd order scheme on the
medium grid ...... 114
5-46 Predicted pressure (left) and Mach number (right) by the 5th order scheme on the
medium grid ...... 115
5-47 Computed Cp distributions on waisted body surface, medium ...... 116
5-48 Computed Cf distributions on waisted body surface, medium mesh ...... 116
5-49 Computed pressure profiles at 0.5 body length above the axis using different
schemes on the coarse meshes ...... 118
5-50 Computed pressure profiles at 0.5 body length above the axis using different
schemes on the medium meshes ...... 119
5-51 Computed pressure profiles at 0.5 body length above the axis using different
schemes on the medium meshes ...... 120
5-52 Surface and volume grids for propeller 5168 ...... 122
5-53 Predicted surface pressure and trailing tip vortices by the 2nd order scheme ...... 125
xvi 5-54 Predicted surface pressure and trailing tip vortices by the 3rd order scheme ...... 125
5-55 Predicted surface pressure and trailing tip vortices by the 5th order scheme ...... 126
5-56 Predicted trajectories of trailing tip vortices by various schemes ...... 126
5-57 Predicted axial velocity components at section x/R=0.2386 and advance ratio J=1.1
...... 127
5-58 Predicted radial velocity component at section x/R=0.2386 and advance ratio J=1.1
...... 128
5-59 Predicted tangential velocity component at section x/R=0.2386 and advance ratio
J=1.1 ...... 129
5-60 Circumstantially averaged axial velocities at x/R=0.2386 ...... 131
5-61 Circumstantially averaged axial velocities at x/R= 0.8378 ...... 131
5-62 Circumstantially averaged tangential velocities at x/R=0.2386 ...... 132
5-63 Circumstantially averaged tangential velocities at x/R= 0.8378 ...... 132
5-64 Circumstantially averaged radial velocities at x/R=0.2386 ...... 133
5-65 Circumstantially averaged radial velocities at x/R= 0.8378 ...... 133
5-66 M427 main rotor and surface mesh resolution ...... 135
5-67 Vorticity distributions on x-y cutting plane in hover for the 2nd order ...... 136
5-68 Vorticity distributions on x-y cutting plane in hover for the improved 3rd order
scheme...... 137
5-69 Comparison of normalized thrust coefficients ...... 139
5-70 Comparison of normalized torque coefficients ...... 139
5-71 Vorticity distributions on x-y cutting plane in forward flight by original scheme . 142
xvii 5-72 Vorticity distributions on x-y cutting plane in forward flight by improved scheme
...... 142
5-73 Comparisons of normalized pitching moment coefficients ...... 143
5-74 Comparisons of normalized thrust coefficients ...... 143
5-75 Comparisons of normalized roll moment coefficient ...... 144
5-76 Comparisons of normalized torque coefficient ...... 144
xviii
List of Abbreviations
2D ...... Two Dimensional 3D ...... Three Dimensional
ADI ...... Alternating Direction Implicit AIAA ...... American Institute of Aeronautics and Astronautics
CFD ...... Computational Fluid Dynamics CFL ...... Courant, Friedrichs, and Lewy CPU ...... Central Processing Unit
DES ...... Detached Eddy Simulation DG ...... Discontinuous Galerkin DRP ...... Dispersion Relation Preserving
ENO ...... Essentially Non-Oscillatory
FD ...... Finite Difference FE ...... Finite Element FV ...... Finite Volume
MUSCL ...... Monotone Upstream-centered Schemes for Conservation Laws
NACA ...... the National Advisory Committee for Aeronautics
PIC ...... Particle in Cell PNS ...... Parabolized Navier-Stokes
RBF ...... Radius Basis Function
SP ...... Spectral Method SOR ...... Successive Over-Relaxation SV ...... Spectral Volume
TVD ...... Total Variation Diminishing TURNS ...... Transonic Unsteady Rotor Navier-Stokes
xix U2NCLE ...... Unsteady Unstructured Computational Field Equations
WENO...... Weighted Essentially Non-Oscillatory
xx
List of Symbols
English letters
A ...... Jacobian matrix, area of control surface c ...... Speed of sound ...... Coefficient of skin friction ...... Specific heat at constant pressure ...... Pressure Coefficient ...... Specific heat at constant volume d...... Distance to the closest wall, first derivative operator D ...... Diagonal matrix of Jacobian matrix, detector function
...... Eckardt number e ...... Internal energy per unit mass E ...... Total energy per unit volume f ...... Function in the inviscid flux construction equation F ...... Flux vector in direction ...... Viscous term G ...... Flux vector in direction ht ...... Specific enthalpy H ...... Total enthalpy, flux vector in direction J ...... Advance coefficient k ...... Coefficient of thermal conductivity, system matrix ...... Coefficient of torque ...... Coefficient of thrust ...... One of the vectors parallel to face L ...... Length, lower triangle matrix of the Jacobian matrix ...... Mass flow rate ...... One of the vectors parallel to face M ...... Mach number, transfer matrix from conservative variables to primitive variables ...... Normal vector to face N ...... Newton’s function p...... Pressure P ...... Inviscid and viscous fluxes Pr...... Prandtl number ...... Vector of conservative variables
xxi q...... Vector of primitive variables, heat stress term ...... Distance vector ...... Eigenvector matrix of the system matrix ...... Distance vector normal to the rotating axis Re ...... Reynolds number S...... Surface, source term t ...... Time T ...... Temperature u...... Absolute velocity components in x direction U ...... Free stream velocity, upper triangle matrix of the Jacobian matrix ...... Linear rotating velocity vector v ...... Absolute velocity components in y direction ...... Control volume ...... Absolute velocity vector ...... Characteristic variable ...... Relative velocity vector x, y, z ...... Cartesian coordinates
Greek letters
β...... Preconditioning parameter ...... Ratio of specific heat ...... Diagonal matrix of the eigenvalues of the system of matrix ...... Preconditioning matrix ...... Preconditioning matrix Laplace operator, incremental indicator of time or space ...... Time step , , ...... Spatial mesh size in , ,and , directions ...... Velocity vector normal to face with the consideration of the grid speed, temporal accuracy control parameter ...... Azimuthal angle ...... Eighenvalue of the preconditioned system of equations ...... Dynamic viscosity ...... Density ...... Reynolds stress tensor ...... Kinematic viscosity ...... Flow angle, one half of square of the magnitude of the total velocity, Barth slope limiter, RBF basis function ...... Rotational speed
Subscripts b ...... Boundary
xxii ...... Mesh point locations in directions, index of nodes, inner point within a control volume ...... Mesh point locations in directions, index of surfaces ...... Mesh point locations in directions ...... Laminar flow ...... Left o ...... Reference value for characteristic variables, outer point of a control volume r ...... Reference value ...... Right t ...... Total condition, turbulence flow ...... Viscous term ...... direction ...... direction ...... direction
Supperscripts
...... Time level m ...... Gauss-Seidel’s iterative level n ...... Newton’s iterative level r ...... Relative frame ...... First order of derivative ...... Second order of derivative
Overheads
...... Nondimensional value ...... Vector ...... Numerical value ...... Time ratio
xxiii Chapter 1
Introduction
1.1 Motivation
Computational Fluid Dynamics, known as CFD, is being widely used in the design, analysis and optimization of many engineering systems such as turbo machinery, rotorcraft, and wind turbines, etc. involving aerodynamic, acoustic, and/or turbulent phenomena. In these engineering problems, the convection of flow characteristics is important for understanding the underlying physical phenomena, which require a robust, efficient, and accurate CFD solver to capture the convectional flows in order to assist the traditional experiment testing, or even as the only practical solution at the current level of developments. For these convection dominated flow problems, accurate numerical methods are required to reduce the numerical dissipation as much as possible to help capture the true flow physics. A large number of researchers have been dedicated to developing high order numerical methods, and significant progresses have been made in the last decade.
For example, the prediction of helicopter rotor aerodynamics requires accurate predictions of rotor tip vortices, which is recognized as one of the major challenges in
1 rotorcraft CFD today. These problems are highly complicated because the rotor wake/vortex is unsteady and exhibits three-dimensional phenomena. For many flight conditions the rotor trailing tip vortices may persist in the vicinity of the rotor blades for several revolutions, causing blade-vortex or airframe-vortex interactions. In order to predict the impact of rotor tip vortices on the aeromechanic and aeroacoustic performance, each trailing vortex system must be accurately captured in a relatively large domain of the flow field. However, conventional Euler/Navier-Stokes solution methods that employed a second-order spatial discretization often generate excessive artificial dissipations, which smear the high gradients in the rotor vortex region before a wake age of even one revolution is attained. Therefore, there is an acute need for development of accurate methods that are able to capture the rotor wake system at an affordable computational cost.
The goal of the present study is to develop a high order unstructured grid scheme that is suitable for solving realistic aerodynamic problems involving complex geometries.
Although several higher order methods have been developed in the past, they are primarily focused on structured grids, which may encounter difficulties in dealing with complex configurations in grid generations. On the other hand, unstructured grid methods have demonstrated a promising ability in handling complex geometries with relative ease comparing with the structured grid counterparts. Meshes can be generated in days instead of months, which has been the main driving force for the development of unstructured grid technologies for the last decade.
In regards to the higher order schemes using both structured and unstructured grids, the robustness and the efficiency of schemes are the two major issues in realistic
2 engineering applications. This is especially true in the unstructured grid category, where the larger stencil, the preservation of the accuracy, and the numerical stability are the three major challenges due to irregular shapes of unstructured grids. The use of node- centered schemes further increases the difficulty to achieve a high order surface integration in finite volume schemes. In the present work, the concept of MUSCL scheme in structured grids is extended to the unstructured grids in developing the higher order schemes, and an innovative method based on the radial basis functions (RBF) interpolation is employed to increase the high order accuracy for the finite-volume integration. All proposed works are developed based on an existing unstructured three- dimensional unsteady Reynolds-averaged Navier-Stokes CFD solver U2NCLE [1].
1.2 Literature Review about Higher Order Schemes
As decades of evolutions, the computational fluid dynamics has been evolved into several categories for practical applications, such as the finite difference (FD) method, the finite volume (FV) method, the finite element (FE) method, the spectral volume (SV) methods, and so on. These are the prevailing methods using different higher order spatial discretizations for the governing equations, and structured and unstructured [2] grids are still the two most popular ones used in the computational fluid dynamics today.
1.2.1 Finite Difference Schemes
Finite difference [3] types of high-order methods are most popular in structured grids, which are derived from Taylor series expansions. The definition of derivatives is very straightforward, particularly on uniform grids. However the finite difference method cannot be applied to unstructured grids due to their irregular shapes. In the traditional
3 finite difference method, the value of the first order derivative at a point, for example, depends only on the function values of the neighbor point. In order to obtain higher order of accuracy, more points must be involved in the stencils.
Direct numerical simulations require all the relevant scales to be properly represented in the numerical model, and these requirements result in the development of spectral method (SP) [4]. This is the other extreme case compared with the traditional finite difference method. In the spectral methods, the value of first order derivative at a point depends on all the nodes. The application of spectral method is limited to flows in simple domains and simple boundary conditions.
1.2.2 Compact Schemes
The compact scheme is a type of method which is in between the traditional finite difference method and the spectral method. Lele [5] developed a compact finite difference scheme in 1992 with a range of spatial scales instead of all. It was originated from the implicit Pade scheme [6] using both function and derivatives at grid points.
Sescu and Hixon [7] developed an optimized multidimensional high-order finite difference scheme. Tam and Webb [8] developed a Dispersion-relation-preserving (DRP) scheme for predictions of computational aeroacoustics. The Pade or compact finite difference schemes mimic the global dependence. Compared to the standard finite difference approximations, the compact scheme can achieve a higher order of accuracy without increasing the stencil width, and make the scheme closer to the spectral method.
Finite difference types of high-order methods are easy to understand and to evaluate its formal order of accuracy. However, the freedom in choosing the mesh geometry and the
4 boundary condition is still maintained, and the application to unstructured grids is difficult.
1.2.3 Spectral Difference Schemes
Z.J. Wang and Yen Liu [9] developed a high-order, conservative spectral difference method based on unstructured grids. This method applies the concept of discontinuous and high-order local representations to achieve the conservation and high accuracy based on a finite difference formula for the sake of simplicity. Two different sets of grid points, i.e., the solution points inside cell and flux points on nodes or midpoints of edges are used. The solutions can be constructed using Lagrange-type polynomial basis to an element-wise degree p polynomial. Fluxes at cell corner points do not have the unique values for all cells sharing the corner. In spite of that, local conservation is guaranteed because neighboring cells do share a common normal flux at all the flux points. Once the fluxes at all the flux points are re-computed, they are used to form a degree polynomial. For two-dimensional triangular elements, first order accuracy needs one solution point in the cell and three midpoints on the edges. Second order accuracy needs three solution points, three node points, and three midpoints on the edges within one cell. The third order needs six solution points, three node points, and six midpoints on edges.
1.2.4 Finite Volume Schemes
The finite volume method was first introduced into the CFD field by McDonald
[10] in 1971 and Mac-Cormack and Paullay [11] in 1972 independently for the solution of two-dimensional, time dependent Euler equations. It then was extended to three- dimensional flows by Rizzi and Inouye [12] in 1973. This technique gives a way to
5 discretize directly the integral formulation of the conservation laws in the physical space.
The finite volume method takes full advantage of using arbitrary types of grids, both structured and unstructured, and gives a lot of options to define the control volumes to satisfy the conservation laws. For example, changing the shape and location of the control volumes associated with a specific grid point, or varying the ways and accuracy of the numerical fluxes through the control surfaces, provides tremendous flexibility to develop different finite volume schemes. In addition, the conservation property of the mass, momentum and energy will be maintained at the discrete level in control volume methods by directly discretizing the integral form of the governing equations. This is a most fundamental property in finite volume schemes.
The first modern finite volume scheme is the Godunov's scheme [13], which is a first order method suggested by S. K. Godunov in 1959. In this method, the conservative variables are considered as piecewise constant over the finite volume cells. The basic idea of Godunov method is to solve a set of Riemann problems exactly at each interface, and then integrally average the Riemann problem over the cell (Hirsch [14], 1990). Because the exact solution is averaged over a cell, many details obtained by solving Riemann problems are lost. In 1979, a second order MUSCL scheme was publish by Bram van
Leer [15], which stands for Monotone Upstream-centered Schemes for Conservation
Laws. The idea is to replace the piecewise constant approximation of Godunov's scheme by a reconstructed variable using linear extrapolations.
A high order, actually second order, finite volume scheme using unstructured grids was first reported by Barth [16] in 1989 for solving the two dimensional Euler equations. A piecewise linear distribution of cell averaged variables replaced the
6 piecewise constant cell distribution in a control volume through the multi-dimensional monotone linear reconstruction procedure. The second order of accuracy then was achieved for both cell-centered and node-centered unstructured grids.
1.2.5 Spectral Volume Schemes
The Spectral volume method [18] to [20] is ultimately a finite volume method on unstructured grids. Each spectral volume is composed of the several control volumes.
Mean state-variables for the control volumes inside a spectral volume are employed to construct a high-order polynomial, which is then utilized to update the mean variables for the control volumes. The reconstruction problem can be solved analytically, and is identical for all simplexes. Therefore a high-order spectral volume method is much more efficient than a high-order k-exact finite volume method, in which a reconstruction problem must be solved for each control volume. The spectral volume method is fully conservative at the sub-cell control volume level, and the flux calculation is based on the upwind concept. However, the spectral volume method is limited to specific element types, such as tetrahedrons, and always needs a large stencil. Therefore, the spectral volume method has practical difficulties in applying to complex and realistic problems.
1.2.6 END and WEND Schemes
Essentially non-oscillatory (ENO) [21], [22] and Weighted ENO (WENO)
[23],[24] and [25] can be classified into finite difference or finite volume schemes. The first ENO scheme was constructed by Harten et. al. in 1987 [21]. The first WENO [23] scheme was constructed in 1994 by Liu, Osher and Chan for a third order finite volume version. In 1996, the third and fifth order finite difference WENO schemes in multi-space dimensions were constructed by Jiang and Shu [24], with a general framework for the
7 design of smoothness indicators and nonlinear weights. A key idea in WENO schemes is using a linear combination of lower order reconstructed fluxes to obtain a higher order approximation. Both ENO and WENO schemes use the idea of adaptive stencils to automatically choose the locally smoothest stencil. This minimizes the possibility of interpolation crossing the discontinuities, and thus achieves the high order accuracy and non-oscillatory property at discontinuities. For the system case, WENO schemes use local characteristic decompositions and flux splitting to avoid spurious oscillatory. ENO and WENO schemes are designed for problems with piecewise smooth solutions containing discontinuities, and have showed promising results for three-dimensional, high-resolution simulations of compressible flows in canonical configurations using mildly stretched meshes. Nevertheless, computations using both ENO and WENO in complex geometries are very costly even for a moderate order of accuracy as a result of high-order reconstructions.
1.2.7 Finite Element Schemes
The finite element method was originated from 20 years of research in the structural analysis field between 1940 and 1960. In the structural analysis, structure is divided into small substructures (elements) of different shapes, and then these substructures are re-assembled after each “element” has been analyzed for stress calculations. This is how the term “element” came from. Turner et al. [26] first introduced the formal elaboration of “finite element method” in a paper in 1956, dealing with the properties of a triangular element in plane stress problems. The expression
“finite elements” was introduced by Clough [27] in 1960. Zienkiewicz and Cheung [28] in 1965 applied this method to solve continuous field problems. Since then, the finite
8 element method has been used as a general approximation method for physical problems described by field equations in any continuous media.
1.2.8 Discontinuous Galerkin Method
In 1973, Reed and Hill introduced a Discontinuous Galerkin (DG) [29] method to solve the hyperbolic neutron transport equation. The discontinuous Galerkin method was used for CFD simulations at the early 1990's when it was first used to solve the Euler equations by Cockburn and Shu [30], [31]. The discontinuous Galerkin (DG) method is often referred to as a hybrid, or mixed, method since it combines features of both finite element (FE) and Godunov-type finite volume (FV) methods. It can be used for unstructured grids and suitable for any gird types. The solution is represented within each element as a polynomial approximation (as in FE), while the inter-element convection terms are resolved with up-winded numerical flux formulas and discontinuous solution representation (as in FV). By using the DG method, the governing equation is multiplied by a test function, and integrated over the domain of interest. This results in a so-called weak form. Theoretically, solutions may be obtained to an arbitrarily higher order of accuracy and has a compact stencil compared to the finite volume method. In addition to being arbitrarily high order of accuracy, the discontinuous Galerkin also allows the formulation of very compact numerical schemes. That is due to the fact that the solution represented in each element are purposefully kept independent of the solutions in other cells, with the inter-element communication occurring only with adjacent cells (elements sharing a common face). Bram van Leer and Shohei Nomura [32] reformulated the DG method for the diffusion equation by applying a smooth locally recovered solution that is indistinguishable from the DG in the weak sense. However, the DG method is not very
9 intuitive to understand the oscillating behavior associated with discontinuous interfaces.
In addition, the DG method is relatively costly to be implemented in 3-D CFD applications.
1.3 Current State of CFD
The literature research indicates that, although many higher order unstructured schemes have been proposed and developed, they revealed some restrictions and disadvantages due to the nature of unstructured grids. The larger stencil, the preservation of accuracy, and the instability are the three major obstacles for many high order unstructured schemes being used as a practical tool for aerodynamic design problems. As mentioned in the previous sections, the spectral volume method is limited to specific element types, such as tetrahedrons, and always generates larger stencils. ENO and
WENO methods have the ability to choose the smoothest stencil in structured grids only, and computations using both ENO and WENO in complex geometries can be costly even for a moderate order of accuracy. Finally, discontinuous Galerkin method is not very intuitive to understanding the oscillating behavior associated with discontinuous interfaces, and is costly in multi-dimensional spaces as well. In summary, most high order unstructured grid schemes are still in their research stage, and are not mature enough to be used as practical tools for engineering designs involving complex geometries and/or complex physics.
1.4 Research Objectives and Scope
The current effort is aimed at developing an efficient and accurate computational tool for the analysis and design of realistic engineering problems. The baseline CFD code
10 is U2NCLE [1], a three-dimensional unsteady Reynolds-averaged Navier-Stokes solver based on multi-element unstructured grids. The primary objective here is to address the three major challenges faced by many unstructured high order schemes in practical applications: the larger stencil, the preservation of accuracy, and the instability of the schemes.
In the present work, a new flux reconstruction method is introduced in an unstructured grid finite volume scheme using high order polynomial reconstructions. As an extension to the MUSCL approach, flow variables on the interface are reconstructed using 2nd and 4th degree of polynomials to provide higher order spatial discretizations in finite volume schemes. The high order polynomials are formulated with both flow variables and their derivatives to achieve compact stencils on unstructured mixed element grids, which is a major distinction from other unstructured grid higher order schemes.
In preserving the numerical accuracy, a novel concept based on radial basis function (RBF) interpolation [33][34][35][36] is introduced in the present study. In traditional high order unstructured schemes, the Gaussian quadrature rule [37] is implemented to ensure the order of accuracy for numerical integration corresponding to the highest polynomial order. However, for most practical unstructured grid solvers, which adopted a node-based finite volume schemes on mixed element grids [38][39][41], the face of the control volume is usually multilateral shapes. The exact implementation of
Gaussian quadrature on multilateral shapes can be very costly, in both memory and computational time. The radial basis function method is introduced as a substitute to the
Gaussian quadrature in the present study.
11 To improve the stability of the high order unstructured grid scheme, a weighted smart limiter is investigated in the current framework. The introduction of a weighted smart limiter notably improves the stability and robustness of the high order schemes in practical applications, which maintains the high order accuracy of the solutions in smooth region while maintaining the stability of the scheme in discontinuous regions such as shocks.
1.5 Organization of Dissertation
This dissertation work is focused on improvements of the numerical accuracy of the current unstructured CFD solver. The motivation of this work is introduced first in
Chapter 1. A brief literature review is followed on current popular higher order schemes.
The current state of CFD method is analyzed through the literature review. The goal and scope of this dissertation are proposed after a short introduction.
Chapter 2 gives an overview of computational fluid dynamics. The historical development of CFD is introduced. The importance and position of CFD in the current engineering area are emphasized after the introduction of the CFD history. A general application procedure of CFD method is also presented.
In Chapter 3, a general methodology of computational fluid dynamics is first introduced. Both mathematical and numerical formulations are presented in details. The normalization of the governing equations is presented after the numerical formulations. A brief description of relative boundary conditions as well as the turbulence model is presented at the end of this chapter.
In Chapter 4, the methodology of improving the current CFD method is discussed in details. The reconstruction of higher order variables for inviscid fluxes is presented.
12 The surface integration is accompanied with the Radius Basic Function (RBF) interpolation as a substitution of the Gaussian quadrature method. A smart limiter technique for discontinuous interfaces is presented at the end of the chapter.
The validation and verification of the newly developed method are presented in
Chapter 5. A vortex dominated subsonic flow over a fixed wing NACA 0015 [42] is computed in the absolutely frame to investigate the numerical dissipation of the new scheme. A rotating NACA 0012 [43] rotor in hover is computed in a rotating frame to demonstrate its ability to capture the tip trailing vortex. A supersonic viscous flow over a waisted body with oblique shocks [44] is used to examine the new method working on a flow with strong discontinuities. Finally, two realistic applications are presented about a marine propeller P5168 [45] and a Bell Helicopter main rotor blade M427 [46].
In Chapter 6, conclusions of the current high order scheme are provided including recommendations for further improvement and development.
13 Chapter 2
Computational Fluid Dynamics Overview
Computational Fluid Dynamics, known as CFD today, is defined as the set of methodologies that enable the computer to provide a numerical simulation of fluid flowing around an object, such as an airplane, ship, car, launch vehicle, etc. The following Figure 2-1 shows an application using CFD tools for the aerodynamic design of a modern commercial jet. The results from the numerical simulation can be used to analyze the performance of the wing lift and drag, wind-fuselage interaction, as well as engine-airframe integration studies.
14
Figure 2-1 The application of CFD in the design of Boeing 777 (Johnson et al. [47])
2.1 Historical Perspective of CFD
The development of CFD is closely bound to the evolution of the digital computer, especially the evolution of the computer hardware. Before the end of the
World War II, almost all the problems were resolved using analytical or empirical methods. Although a very limited number of persons at that time used numerical method to solve engineering problems, all the computations were executed by hands instead of machines. A tremendous number of calculations could be included just for a single result.
Only after the digital computer was developed, the numerical method gradually forms a routine calculation procedure.
The important work done by Richardson [48] in 1910 could be viewed as the real beginning of the CFD. He introduced the point iterative schemes for numerically solving
15 Laplace’s equation and the bi-harmonic equation. Besides, he distinguished the problems needed solved using a relaxation method and a marching method. In 1918, Liebmann
[49] improved Richardson’s relaxation method to solve the Laplace’s equation. Unlike the Richardson method which updated the unknowns only from the previous iteration,
Liebmann updated the unknown variables using both of the variables at current and previous iterative steps. This simple modification immediately shortened the convergence time.
The acronym “CFL” stands for three names, Courant, Friedrichs, and Lewy [50].
Their publication [50] in 1928 is treated as the beginning of modern numerical analysis.
Uniqueness and existence questions were explained for the numerical solutions of partial differential equations in this paper, which is the original source for the CFL stability requirement for the numerical solution of hyperbolic partial differential equations.
In 1940, Southwell [51] developed a relaxation scheme designed for hand calculations. This relaxation was widely applied to computations of both structural and fluid dynamic problems. In his method, point residuals were computed, and the point with the largest residual was always relaxed at the next step. In 1955, Allen and
Southwell [52] used Southwell’s scheme to calculate the incompressible, viscous flow over a cylinder by hand.
During time of World War II, a lot of researcher focused their study on solving fluid dynamics problems with the numerical method. John von Neumann developed marching scheme, and a detail explanation of his method was published by O’Brien,
Hyman, and Kaplan [53] later in 1950. John von Neumann’s work provides a practical way to evaluating stability of the numerical method on solving the fluid dynamics
16 problems, and is the most extensively used technique in CFD field. Almost at the same time, Peter Lax [54] (1954) developed a method for computing flows with shock waves by using the conservative form of the governing equations, and no other treatment was included.
In the meantime, computational methods for elliptic and parabolic types of equations were developed. The first version of the successive over-relaxation scheme,
SOR, is developed by Frankel [55] in 1950 for solving Laplace’s equation. A considerable improvement in the convergence rate was achieved by using this SOR technique. Peaceman and Rachford [56] (1955) and Douglas and Rachford [57] (1956) developed an alternating direction implicit (ADI) schemes for parabolic and elliptic equations. This method alternates sweep directions and removes the restriction of step sizes. The ADI method was applied to computations of fluid mechanics by Briley and
McDonald [58] (1974) and Beam and Warming [59][60] (1976, 1978), and thus fast solutions of Euler and Navier-Stokes equations were gained.
During the decade of sixties, a lot of significant work in CFD was achieved on dealing with shock wave problems. Lax’s method [54] or an aritificial viscosity scheme by von Neumann and Richtmyer [61] (1950) was used for solving the shock wave problems. Also, Los Alamos employed the dissipative nature of the finite-difference method to smear the shock over several mesh intervals (Evans and Harlow [62], 1957), which was named as particle-in-cell (PIC) method. In 1960, Lax and Wendroff [63] developed a second-order accurate method for computing shock flows, and avoided the excessive smearing comparing to the earlier approaches. The MacCormack [64] (1969) version of this technique is one of the most popular schemes. Gary [65] in 1962
17 introduced a technique for fitting moving shocks. Moretti and Abbett [66] (1966) and
Moretti and Bleich [67] (1968) applied shock-fitting technique to multidimensional supersonic flow over different configurations.
In 1959, Godunov [68] proposed a method for solving multidimensional compressible flows, in which a flux on the interface was calculated based on a solution to a Riemann problem. This approach became really popular after van Leer [69] [70] (1974,
1979) published a higher-order reconstruction scheme based on Godunov’s concept. In
1980, Roe [71] suggested used an approximate solution to the Riemann problem related with Godunov’s approach, famous as flux-difference splitting method, in order to improve the efficiency. This flux-difference splitting method substantially reduced the heavy work load on computations of multidimensional problems. Roe scheme resulted in the current trend of practical approach on conservation-dominated flows, as well as on convection-dominated flows. Steger and Warming [72] in 1979 introduced an upwind flux splitting method. van Leer [73] in 1982 also presented an improvement on the flux splitting technique. Many modern production programs are still using this technique today.
The limiter technique as a partner of approaches dealing with Riemann problem flows was also developed in the meantime. The work of Boris and Book [74] in 1973 was the basis for the nonlinear limiting technique. The famous TVD scheme, which represents total variation diminishing, was developed by Harten [75] in 1983. The concept of TVD scheme led to an important direction in which the nonlinear limiting technique on fluxes was followed. Limiter technique has become one of the important factors in state-of-the- art CFD field today. Many other researchers also made significant contributions to the
18 computations of the shock flows, such as Enquist and Osher [76][77] (1980, 1981), Osher and Chakravarthy [78] (1983), Yee [79][80] (1985a, 1985b), and Yee and Harten [81]
(1985) and so on.
Beside of the stability issue, techniques to improve the efficiency of computations were also developed. Multi-grid method and preconditioning method are two typical approaches to improve the convergence rate of iterative computations. Fedorenko
[82][83] (1962, 1964) first applied the multi-grid approach to the computation of elliptic equations. Later, Brandt [84][85] (1972, 1977) used the multi-grid method to the equations of fluid mechanics. McDonald and Shamroth [86] (1983) first introduced the precondition technique to the computations of low Mach number flows. They added a simple constant preconditioning matrix to a nondimensional form of the isoenergetic equations, and the convergence of the test case with Mach number of 0.005 was improved. Choi and Merkle [87] (1985) applied this preconditioning method to the inviscid compressible Euler and the conservative artificial compressibility equations with an approximate factorization. The application of this method to viscous flows was performed [88] in 1993. Turkel [89] (1987) introduced a general preconditioning matrix form for the artificial compressibility method working in compressible equations based on primitive variables and the isoengergetic variables. A good review on preconditioning methods was given by Turkel in [90] (1993).
Meanwhile, the simplification of the Euler and Navier-Stokes equations were also being explored by researchers. The thin-layer equations can be obtained from the Navier-
Stokes equations by neglecting terms of the order of or smaller. The representative applications can be found in the publication [91] by Pulliam and Steger in
19 1978. The usage of the parabolized Navier-Stokes (PNS) equations is predicting the complex three-dimensional supersonic viscous flow field in an efficient way. Rudman and Rubin [92] introduced the earliest form of PNS in 1968.
With the development of the CFD, as alternative approaches of the finite difference method, more and more computational methods, such as finite volume method
[10], finite element method [26], discontinuous Galerkin method [29] and so on were developed and widely used in many different application fields. Moreover, unstructured grid [2] technology was the developed and more suitable than the structured grids for the complex geometry problems. Researchers are making effort to improve the current CFD technology and increase the computation efficiency and accuracy to meet the requirement of the modern practical analysis and designs. With the occurrence of large parallel computers, it is believed more and more solutions of fluid problems may mainly depend on the CFD simulations in the future.
2.2 The Importance of CFD in Virtual Prototyping
In modern design and analysis of a problem in fluid mechanics, experimental, theoretical, and computational (CFD) approaches are the three mostly used methods.
Although they use different ways to resolve problems, they are actually closely connected to each other. Cross-references always happen among these approaches as evidences to themselves. These three methods have their own advantages and limitations in dealing with flow problems. The theoretical method is often referred to as an analytical method which normally can provide a general formulaic form of solution. However, the theoretical method can only solve a limited number of problems which are always simple
20 geometries or physics. This is far less to meet the engineering requirements. On the other hand, the experimental method has the ability to provide the most realistic results for the most of flow problems. However, for the experimental method, a lot of factors have to be taken considerations. First, the experimental equipment has to be available and suitable to the objective flow problem. New devices need to be redesigned or purchased for a particular experiment. Besides of the facility and operating cost, the period of an experiment is relatively long. Scaling the prototype of the real model, mimic the realistic flow conditions, severe operating conditions of turbo-engines for example, also bring difficulties to the experimental method.
In comparing the above two methods, CFD method has removed the restrictions of the analytical method and the experimental method. Not like the analytical method,
CFD method can solve complex geometries and complicated flow conditions. With the better understanding of physics, CFD method can simulate very severe flow conditions and there are no restrictions to any flow region or any type of flows, which may occur in the experimental method. The computational period and the computing cost are also much less than the experimental method. These major advantages of the computational method make wind tunnels test a secondary role to the computer for many aerodynamic problems. With development of the computer hard ware and the correct understanding to the physics, CFD has played a more and more important role in modern industry.
However, many flow problems involving complex physical process that still require experimental facilities for their solution.
21 2.3 Procedures of Applying CFD
The application of CFD metods to engineering designs and analysis basically has five distinct steps. The brief explanation of each step is summarized here.
The first step is the mathematical model selection. In this step, the physical problem being studied has to be fully examined and understood by the researcher, so that the level of the approximation to reality can be defined. As a consequence, a suitable simplified or modified mathematical model to the realistic physical problem can be chosen.
As long as the mathematical model is selected, the discretization has to be executed in the second step as a preparation to the numerical simulation. The discretization step includes two parts. The first part is the discritization to the computational domain, which contains the geometry of the object studied, defined by the grid generation. The second part is the discretization to selected mathematical model equations, which defines the numerical schemes going to be used.
The third step is the analysis of the numerical schemes. The stability of selected numerical schemes has to be evaluated first to make sure a solution to this physical problem can be obtained at least. Then the property of the accuracy of the numerical schemes has to be examined to make sure the solution to the model equations through this method is correct or close enough to the physical problems.
Computations of the objective problem are performed in the fourth step. In order to obtained solutions, the most appropriate time integration methods, the relaxation method of the subsequent algebraic systems, boundary conditions, the turbulence model and the convergence acceleration techniques have to be decided in this step.
22 The fifth step is the post-process of numerical solutions. Graphically post- processing the numerical data is helpful to understand and interpret the physical properties of the obtained simulation results. This is made possible by the existence of powerful visualization software.
After all these steps are finished, the whole procedure of CFD simulation is performed. Luckily, the solutions obtained meet the expectation, and therefore this procedure is only executed for once. However, CFD is always type of reformation technology, and thus repetition most likely happens at each of the step for further verification, validation, and improvement.
23 Chapter 3
Computational Methodology
3.1 The CFD Solver
The baseline CFD code used for the current work is U2NCLE [1], which solves the three-dimensional unsteady Reynolds-averaged Navier-Stokes equations using multi- element unstructured grids in both fixed and rotating frames. A global preconditioning technique [94][95] is used to compute low and high speed rotor flows for rotorcraft aerodynamic analysis. The system of governing equations is formulated in a conservative flux formula using primitive variables. The inviscid flux is evaluated using the second order spatial discretization with various improvements [96]. The viscous flux is calculated by a second-order directional or normal derivative method on mixed elements.
The nonlinear system of equations is solved by Newton’s method, with symmetric Gauss-
Seidel relaxations implemented at each Newton sub-iteration. The Newton implicit scheme implemented in the U2NCLE solver allows a relatively large time step to be used, up to one or two degrees of blade azimuth angle per time step. This provides significant computational efficiency compared to other explicit or semi-implicit CFD solvers.
The turbulence model used here is the one-equation Spalart-Allmaras turbulence model [97]. Its variant, the Detached Eddy Simulation (DES) model [98] is also
24 implemented in the solver. The turbulence model is solved separately from the flow equations at each time step. Although several two-equation turbulence models such as k-
ε, k-ω, and q-ω models [98] are available in the solver, the Spalart-Allmaras model was found to be very reliable with good accuracy in predicting rotor hover and forward flight performance, and therefore was used for all computations performed in the present study.
The resulting linear system is solved using the same Newton iterative scheme employed for the flow equations. The turbulence model is integrated all the way through the boundary layer to the wall without using a wall function.
3.2 Mathematical Formulation
In this section, the three-dimensional Navier-Stokes system of governing equations without external forces is presented. The section will start with the governing equations in a fixed absolute frame, followed by the governing equations in a rotating relative frame. The normalization for both the flow variables and the governing equations is then presented. A key feature of the current computational algorithm is the preconditioning method for the compressible governing equations, which is introduced in this section. The general procedure of the Newton’s implicit algorithm for the preconditioned system of equations is described, followed by the common boundary conditions for internal and external viscous flow computations.
3.2.1 Governing Equations in an Absolute Frame
The unsteady Reynolds averaged Navier-Stokes equations represent the physical conservation laws of mass, momentum and energy. There are many forms of governing
25 equations. The integral forms of the compressible flow Navier-Stokes equations can be expressed in the fixed absolute frame as
Conservation of mass:
Conservation of momentum:
Conservation of energy:
Where is density, is the velocity vector, is pressure, T is temperature, is the stagnation enthalpy, is identity matrix, is the total energy, is the coefficient of thermal conductivity, is shear stress tensor, is time, is control surface,
is the normal vector of the surface of control volume .
26
here, is internal energy per unit mass, is enthalpy per unit mass, is specific heat at constant pressure, is specific heat at constant volume, is ratio of specific heats, is gas constant per unit mass, and is coefficient of dynamic viscosity, , , and are the
Cartesian coordinates in three directions, and , , and are the components of velocity vector in , , and direction, respectively.
27 3.2.2 Governing Equations in an Rotating Relative Frame
Most unsteady problems in isolated rotating machineries can be solved in rotating reference frame, in which the governing equations are casted and solved with a more efficient steady flow approach. Assume the system is rotating steadily with an angular rotating speed of around an axis which is parallel to the axis. The governing equations in a rotating reference frame can be written as the following forms:
Conservation of mass:
Conservation of momentum:
Conservation of energy:
where, is the relative velocity in the rotating reference frame. Term is the
Coriolis force, and is the centrifugal force per unit mass. is the position vector perpendicular to the rotation axis and is the position vector from the relative reference rotation center to the position considered. is the total energy, and
is the total enthalpy in the rotating frame (rothalpy), respectively. The definitions of these variables are presented in the following formulas:
28
3.2.3 Non-Dimensionalization of Governing Equations
It is helpful to present the governing equations in a non-dimensional form for the
CFD algorithms. Therefore, the following reference quantities are chosen to normalize
29 the flow variables and governing equations, where the subscript represents the state of reference values, and the over-bar represents the non-dimensional values: density, ;
velocity, ; temperature, ; pressure, ; length, ; time, ; energy and enthalpy, . The non-dimensional variables are defined as:
In the following formulations, the non-dimensional values are presented without over-bar for the sake of simplicity. The following non-dimensional relationships are true for the ideal gas:
30
where is the constant specific heat, is an Eckert number. and are the total energy and total enthalpy. is the total velocity, and is the speed of sound.
3.2.4 Preconditioned Governing Equations
The general form of governing equations is presented in the relative rotational frame using the absolute velocities components ([99] Shoe & Knight, 1989, [100] Chen et al. 1997). This formulation in the rotating frame can be easily switched to the one in the absolute frame by simply setting the rotational speed to zero. This provides a unified solution algorithm for solving rotating flows in both fixed and relative frames. It also provides a good framework for the preconditioning algorithm for solving arbitrary Mach number flows.
The non-dimensional and integral form of the governing equations without preconditioning technique can be written as:
The nondimensional integral form of the governing equations with preconditioning technique to solve arbitrary Mach number flows can be written as
where is the vector of conservative variables, is the vector of primitive variables, is the combination of the inviscid and viscous flux, and is the source term due to rotation of the system. is the transformation matrix which transform conservative variables to
31 the primitive variables. is the preconditioning matrix. The conservative and primitive variable can be presented as.
The flux is:
Where the convective fluxes at three directions are:
32
The diffusive fluxes at three directions are:
33
The Reynolds shear stress and heat flux are given as:
where is the viscosity, is the Prandt number, and is the
Reynolds number based on the reference length and velocity.The source term can be expressed as:
34
This source term is related to the local flow variables and the rotational speed
, which is zero in the fixed frame. This formulation provides a flexible way to express conservation laws for both rotating and non-rotating flows, where the same numerical algorithm may be applied.
3.2.5 Preconditioning Parameter
The preconditioning matrix can be expressed as [1]:
The transformation matrix from conservative to primitive variables is
35
Where
In order to improve the convergence and dissipation properties of discretized schemes for low Mach number flows involving rotating speeds, the choice of preconditioning parameter need to take consideration on both free stream and rotating speed. In the present study, a general form of preconditioning is applied, which combines both reference Mach number, , and rotating Mach number, ,
where is the global reference Mach number based on the free stream velocity and is the rotating Mach number based on the characteristic rotating speed of the rotating flow. An alternative way to calculate is to use the local rotating speed, so that the modified preconditioning parameter would be depend on both global free stream Mach number and local rotating speed. In both options the method restores to the original preconditioning if there is no rotating effect involved ( ).
36 3.3 Numerical Formulation
In the context of Computational Fluid Dynamics, the mathematical formulations of the governing equations are discretized on computational grids to obtain the numerical solutions. In this work, the finite volume method is employed, and the numerical formulations of governing equations are solved iteratively to meet the convergence criterion of the finite volume method. In the following sections, the numerical procedures are detailed based on the present CFD solver U2NCLE. 3.3.1 Discretized Governing Equations
The preconditioned governing equations are discretized using a finite volume formulation. The basic concept of the finite volume method is to divide the computational domain into a finite number of sub-volumes through the gridding technique. Then, the control volumes, which the discretized governing equations work on, are formed basically based on those sub-volumes. There are the two different algorithms to construct control volmes, cell-center scheme and node-center scheme. For the cell-center scheme, these sub-volumes are control volumes. The flow variables are stored at the center of the control volume. On the other hand, for the node-center scheme, control volumes are formed by the new constructed surfaces surround at each vertex in the mesh. The flow variables are stored at the vertices of the mesh, which is also the center of the control volume. In this work, the node-center scheme is employed.
The preconditioned mathematical governing Equation (3.42) is rewritten in to following discretized form over a control volume with the convective flux and diffusive flux distinguishing from each other.
37
where and are the numerical convective and diffusive fluxes in the governing
equations, respectively. It is seen that the preconditioning matrix is applied to the convective (inviscid) term only through modifying the system matrix. The diffusive
(viscous) term of the governing equations remains unchanged. The subscript denotes the vertex, and denotes the surface of the control volume surrounding the vertex .
3.3.2 Convective Flux Evaluation
The convective numerical flux of preconditioned governing equations is calculated using in a formula similar to Roe’s flux, as reported by Briley, et al. [Briley et al., 2003]. Consider the surface of a control volume whose left and right states are denoted by and , the numerical flux projected onto the surface can be written as:
where is preconditioned system of matrix. The appendix A described detailed
formulation of .
Since the eigensystem and the flux are evaluated using the above averaged variables instead of Roe variables, the current approach is considered to be an extension of the Roe flux approximation.
In the above Equation (3.67) quantities and are the values of the primitive variables on the left and right side of the face of the control volume. For a first-order
38 accurate differencing, quantities and are set equal to the value at the vertices lying on either side of the face. For the second-order scheme, these face values ( ) are computed with a Taylor series expansion about the central node ( ) of the control volume
where is the vector that extends from the central node to the midpoint of each edge, and
is the first derivatives of the primitive variables at node and is evaluated with an un- weighted least-squares procedure ([101] Anderson et al., 1995).
3.3.3 Diffusive Flux Evaluation
The diffusive term of the Navier-Stokes equations can be described as:
where is the general form of viscous term. Consider an incompressible viscous flow with constant laminar viscosity, the viscous diffusive terms in the momentum equations are identically Laplace operators acting upon the velocity field. One important property of the Laplace equation is to satisfy the maximum principle, which requires that all of the stencil weights of a discrete scheme be positive. Positivity is a key property for numerical stability. Another key property is the accuracy of gradients, in which a linearity- preserving reconstruction is required for a second order accurate discretization.
Therefore, the optimal reconstruction scheme for the diffusive fluxes should maintain high accuracy as well as positivity of the operator. Nevertheless, these two properties often cannot be achieved simultaneously on arbitrary grids, as investigated by Coirier
([102] Coirer, 1994). Another consideration is that the current finite volume discretization is formulated on an unstructured mesh with mixed elements (tetrahedron,
39 quadrahedron, prism, and hexahedron), where an edge-based data structured is adopted.
This requires the calculation of diffusive (viscous) fluxes to be compatible with the edge- based data structure, where information is stored on the two vertices of an edge.
3.3.4 Time Marching Method
The temporal discretization is performed using an implicit Euler method for the system of governing equations. A general expression can be described for the different order of accuracy of the temporal discretization, which is given as:
where , . is the time marching step increment between steps and . represents both inviscid and viscous fluxes. Subscript denotes the index of the control volume or the node, and is the index the dual faces that compose of the control volume. A first order of temporal accuracy of the Euler implicit scheme is given by the choice and a second order time accurate Euler implicit scheme is given by .
3.3.4.1 Newton’s Method
The nonlinear system of Equation (3.70) is solved by Newton’s method [37], which solves a linear system of equations at each time step instead. The Newton’s iterative method allows relatively large time step to be used in numerical computations, a considerable savings compared to the explicit method, in particular for the unsteady time- accurate problems.
Define a function as:
40
The Newton’s method solves the following equation:
where . are the Newton’s iteration steps,
with the initial guess of . is the Jacobian matrix of Equation
(3.72) and can be written as:
In the Equation (3.73), the first term on the right hand side is the contribution from the unsteady time derivative of , the second term is the contribution from the steady state residual (both inviscid and viscous) of the governing equations, and the third comes from the source term in the rotating reference frame. The flux Jacobian of the steady state residual can be evaluated using an approximate method. Consider the matrix
in Equation (3.73) as a local constant matrix, the approximate flux Jacobian for the inviscid flux can be written as
where
41
A similar method is used to evaluate the viscous flux Jacobian by taking the derivatives of discretized viscous flux formulation.
3.3.4.2 Gauss-Seidel Relaxation
The nonlinear system of equations is linearized by Newton’s method, which results in a sparse system of equations at each time. The solution of the sparse system of equations is obtained by a relaxation scheme in which is obtained through a sequence of iterations, , which converge to . Several variations of classic relaxation procedures have been used in the past for solving this kind of linear system of equations ([102] Coirier, 1994; Anderson, 1992, [103] Batina, 1990). In this work, a symmetric implicit Gauss-Seidel procedure as described in [103] (Batina, 1990) is used. To clarify the scheme, the system matrix is first written as a linear combination of matrices representing the diagonal , upper triangular , and lower triangular parts at each time step.
where
42 Based on the above definitions, the symmetric Gauss-Seidel relaxation can be described as the following two-step process:
where, is the vector of unsteady residuals, and represents the change in the dependent variables. In first step, the forward pass, are obtained through the previously updated , which were set to zero at the initial stage. In the second step, backward pass, are obtained with the most recent value of from the previous step. Normally six to eight symmetric Gauss-Seidel sub-iterations are adequate at each time step.
3.3.5 Turbulence Model
Several turbulence models are popular for high-Reynolds number flows, such as the one-equation Spalart-Allmaras turbulence model [97] (Spalart and Allmaras, 1991), and two-equation k-, k-, and q- turbulence models [98]. In this work Spalart-
Allmaras turbulence model is used for all simulations for the easy comparison purpose. It formulates a transport equation for the turbulence Reynolds number. Based on the original Spalart-Allmaras formulation, a transport equation for the turbulence viscosity- like , which is the working variable, can be expressed as:
43 where the first and second terms on the right hand side of Equation (3.83) are the source production term, and the third term is the turbulence dissipation term. The coefficients in above equation can be found in Appendix A.
3.3.6 Boundary Conditions
In order to compute both internal and external flows at arbitrary Mach numbers, four characteristic-based boundary conditions with the consideration of the preconditioning method were developed by Sheng and Wang (Sheng and Wang [104]
2003). They are far field boundary, inflow boundary based on total conditions or mass flow, outflow boundary based on back pressure, and impermeable wall boundary conditions. The characteristic variables were derived based on preconditioned system matrix in order to be consistent with interior point treatments. In order to apply boundary conditions in a rotating frame, a source term is considered and is implicitly integrated into the characteristic boundary conditions. In the following, characteristic-based boundary conditions are briefly outlined.
where the characteristic variable vector is defined as:
In the Equation (3.85), is the matrix whose rows are the left eigenvectors of the preconditioned system matrix , and the subscript 0 represents a reference value.
, , and are the diagonal matrix in three directions, and their diagonal values are the eigenvalues of the in three directions, respectively.
44 The five components of characteristic variables for the three-dimensional preconditioned system are
The values of above five characteristic variables at a boundary point are decided by the value at either the interior point or the outer point, which depends on the propagation directions of five characteristic waves.
3.3.6.1 Characteristic Far field
At far field, the direction of wave propagation is determined by the sign of each associated eigenvalue. Since all boundary faces have a positive outward pointing normal with the current unstructured grid topology, a negative sign of the eigenvalue means that the related characteristic wave propagates from the free stream to the boundary nodes, while a positive sign of the eigenvalue means that the characteristic wave propagates from the interior of the computational domain to the boundary. The following relation is valid along the characteristic line:
45 where is the distance incremental value at a specific direction, and is the engenvalue, known as the velocity of the characteristic wave, at the corresponding direction. and can be express as:
For governing equations containing a source term, the Equation (3.93) can be rewritten as the following form.
The above equation can be integrated and the discretized form is
where subscript denotes a point on the boundary, and subscript denotes either a interior point or a outer point, depending on the direction of the characteristic wave propagation on the boundary. If the characteristic wave propagates from the interior domain to the outer domain, and thus the sign of the eigenvalue is positive. Therefore, the reference value is chosen to be the value at inner point, and vice versa. The primitive variables at the far field boundary can then be computed by solving the following system of equations:
3.3.6.2 Characteristic Inflow
For internal flows in rotating machineries with total conditions, normally the three conditions, total pressure , total temperature , and the flow angle are given at the inlet. The following nondimensional relations exist.
46
There are six unknowns in above five equations. For the subsonic inflow, the fourth characteristic relation Equation (3.89) where the characteristic wave propagates from the interior field to the boundary is used to close the above equations.
If a mass flow is specified at the inflow boundary, the total temperature , the mass flow rate , and the inflow angle are known. The following relations are valid.
Again, for above three equations, there are four unknowns , , , and .
These equations combining with the fourth characteristic relation Equation (3.89) can solve the total velocity at the boundary point.
3.3.6.3 Characteristic Outflow
Characteristic variable boundary conditions are applied to outflows as well. For the subsonic outflow, the first four eigenvalues are positive and the fifth eigenvalue is
47 negative. Flow variables at the boundary are connected with the interior values through the first four characteristic relations. The fifth relation is replaced by the static pressure specified at the exit:
For a supersonic outflow case, the fifth eigenvalue is positive, and all characteristic waves propagate from downstream to the upstream.
3.3.6.4 Impermeable Wall
For impermeable surfaces, the specified condition is that of no flow past througt the surface, i.e. .
The remaining boundary conditions are found from the first four characteristic relations. It is worth to note that in all above boundary conditions, the source term contributions are added to the equations implicitly.
3.3.6.5 Wake Model
A rigid/free wake model [105] developed at Georgia Tech has been implemented into the U2NCLE solver [1]. For flows around the rotor blade, the wake shed from a rotor system extends up to 4-6 rotor diameters. A CFD solver used for rotor simulations must be able to capture this far wake. In the hybrid methodology based approach, a single blade is resolved through the Navier-Stokes solution, and the influence of the other blades and of the trailing vorticity in the far field wake is accounted for by modeling them as a collection of piece-wise linear bound and trailing tip vortex elements. The near wake is captured inherently in the Navier-Stokes analysis. This hybrid Navier
Stokes/wake modeling method allows for an accurate and economical modeling of
48 viscous features near the blades, and an accurate “non-diffusive” modeling of the wake in the far field.
The wake model is based on a Lagrangian approach where a collection of vortex elements are shed from the rotor blade trailing edge and are convected downstream by a combination of the free-stream velocity and the bound and trailing vortex self-induced velocities. The far field wake can be modeled using a single tip vortex or a multiple trailer representation. The strength of the vortex elements in the single tip-vortex model is set to be equal to the peak bound circulation on the rotor blade at the instant that the element is shed. The shedding point of the vortex element is based on the centroid of trailed circulation between the tip and location of peak bound circulation. The single tip vortex is replaced by user specified multiple vortex segments in the multiple trailers model. The trailers are equally distributed along the blade span. The strength of the vortex elements is based on the radial gradient of bound circulation and the number of wake trailers chosen by the user.
The convection of the tip vortex elements is determined based on either rigid wake or free wake model. In the rigid wake model, the wake elements are non-distorting and hence maintain their initial helical structure, which are convected at a speed determined by linear superposition of free stream velocity components and a uniform inflow velocity. This inflow velocity is determined using Prandtl-Glauert’s formula
[106]. The convective velocity components in the free/distorting wake model include velocity components induced by wake elements on each other in addition to the above mentioned velocity components. The effect of the bound vortex is also included for the computation of self-induced velocity components. These self-induced velocity
49 components are determined using Biot-Savart’s law [107]. Free wake methods provide more generality with a minimum dependence on experimental data but they are also computationally more expensive than rigid wake modeling.
50 Chapter 4
Improvements to CFD Methodology
4.1 Inviscid Flux Reconstruction
Based on the concept of upwind schemes, high order spatial accuracy can be achieved by introducing more spatial points to represent flow variables within the control volume, which is an extension of Godunov’s approach [68]. If the solution at the center of the control volume is directly projected onto the interface on a piecewise constant base, it is equivalent to the first order spatial accuracy of the Godunov scheme. However the Riemann problems are solved at the interface of the control volumes, in which the projected approximation is decoupled from the physical stage. Therefore in order to produce a higher order spatial accuracy, the piecewise constant approximation within each control volume should be replaced by a higher order representation of the flow variables without modifying the Riemann solver. The second order upwind scheme can be formulated using a linear extrapolation. Similarly, a quadratic representation leads to a 3rd order scheme; and a quartic representation leads to a 5th order scheme; and so on.
4.1.1 Quadratic Polynomial Construction
In the previous work, a flux reconstruction attempting to achieve higher-order spatial discretizations on unstructured grids was introduced [96], which employed a high-
51 order polynomial evaluated with combined solution variables and their first and the second gradients. In a general finite-volume scheme, a solution variable with a linear variation over the control volume yields a second-order spatial discretization scheme. A third-order discretization scheme requires a quadratic variation, and a fifth-order scheme requires a quartic polynomial extrapolation within a control volume. Using the solution variable and its first order gradients stored at the vertices of a control volume, the reconstructed solution variables based on a quadratic polynomial can be expressed at the interface of the control volume as
where is the first order gradient of the state variable. The extra term in Equation
(4.1) is to account for the high-order polynomial, which is constructed based on an upwind-biased formula using both primitive variables and their gradients. In this work, is constructed as:
Here, subscripts and denote the node state inside and outside of the control volume at a given boundary interface. The reconstruction of a piecewise quadratic polynomial within each control volume is essential for developing a third-order spatial discretization scheme. If the above reconstructed piecewise-discontinuous polynomial is accurate to the degree of two, a third order flux reconstruction scheme can be achieved at the interface of the control volume.
52 4.1.2 Quartic Polynomial Construction
Following the previous section, an even higher order discretization can be achieved by using the higher degree of polynomials. Using both flow variables and their
1st and 2nd gradients that are stored on the left and right vertex of the interface of control volumes, a 4th degree of quartic polynomial for the reconstructed flow variables within the control volume can be expressed as
and
where is the second order gradient of the state variable the extra term , , and in Equation (4.3) is coefficient of the higher degree terms of the polynomial, and their forms are described in Equation (4.4) to (4.5) where and are functions of the node solution value, and are the first derivatives of the node solution value, and and
are the second derivatives of the node solution value.
53
4.2 Surface Integration
With the reconstructed left and right variables ( at the interface using the higher degree of polynomial (4.3), the high order discretizations can be obtained based on the numerical flux evaluated using Roe’s [108] formula:
In preserving the numerical accuracy for high order finite volume schemes, a novel concept based on radial basis function (RBF) interpolation [33][34][35][36] is introduced in the present study as an alternative way to the Guassian quadrature
[37][108][110][111]. Instead of calculating the Guassian quadrature points on irregular surfaces using a costly decomposition method or iterative method [110][111], all sub-cell faces that compose of the interface of control volumes are used for the high order surface integration.
4.2.1 General Procedure of Surface Integration
The surface integration is processed within each individual control volume. For the finite volume method, fluxes are integrated on control surfaces around the control volume. The net integrated fluxes balance the volume change rate of the conservative variables, so that the conservation laws [112] can be satisfied. Flow variables are stored at the control volume center (node for node-center scheme, cell center for the cell- centered scheme). The fluxes on control surfaces are computed based on the flow variables on both sides of the control surface. Those flow variables on the control surfaces are interpolated or extrapolated from the flow variables stored at the center of
54 the control volume. Therefore, the flux integration can be divided into two steps. The first step is interpolation/extrapolation, and the second step is integration. In the first step, the variable values stored at the control volume center are interpolated or extrapolated to the control surfaces surrounding the control volume. The second step is integration of fluxes on the control surfaces. To maintain the high-order accuracy of a discretized numerical scheme, the high-order interpolation/extrapolation of the flow variables is necessary, and the higher order integration is also indispensable.
In this work, node-centered unstructured scheme is used. Normally the cell- centered unstructured scheme basically has regular shapes of control surfaces, such as triangles or quadrilaterals. On the other hand, the node-centered scheme has more complicated irregular shapes of control surfaces, which adds difficulties to the surface integration to maintain the higher order accuracy. In Figure 4-1, example control volumes as well as control surfaces are presented for both cell-centered and node-centered unstructured grids.
55 O: Control volume center N1-N4: Physical nodes
(a) Control volume of the cell-centered scheme with triangle
control surfaces
N0: Control volume center N0-N8: Physical nodes
(b) Control volume of the node-centered scheme with irregular
control surfaces
Figure 4-1 Control volumes of the unstructured grids (a) cell-centered
control volume (b) node-centered control volume
56 4.2.2 Comparisons of Gaussian Quadrature and RBF
Integration
After the general surface integration procedure was discussed in above section, the two surface integration methods Gaussian quadrature and RBF methods are compared, and RBF method was selected as the surface integration method in this work in this section. In Table 4-1, properties of these two methods are recorded.
Table 4.1 Properties of RBF and Gaussian Quadrature methods
Features RBF Gaussian Quadrature Data points and weights can Basic element types No restriction be precalculated Quadrature points need to Irregular element types No restriction be calculated iteratively or by decomposition Fixed data points and Basic points All Scattered data weights Accuracy Approximate Exact order of accuracy
The traditional integration method is usually Gaussian quadrature method
[37][108][110][111]. Guassian quadrature chooses the integration points in an optimal way, and produces the exact integration result for the largest class of polynomials. For one-dimensional polynomial integration problem, for example, Gaussian quadrature method can guarantee an exact integral solution for a polynomial of order of less than or equal to , with quadrature points. The positions and the weights for the quadrature points are fixed, and they need to be calculated in advance of the integration.
The calculation of the positions and the weights of quadrature points is easy for one
57 dimensional problem, however, for a multi-dimensional problem, it becomes a complicated issue. The flux integration on the control surfaces is a two dimensional integration problem. The Gaussian quadrature needs to deal with the calculation of the positions and weights of the quadrature points on the surfaces instead of lines. For regular shapes like triangles or quadrilaterals, researchers have found a way to find out the optimal quadrature points. However, for the node-center unstructured scheme, meshes of mixed element type are common and this makes the control surfaces (Figure 4-1 (b) for example), irregular multi-laterals.
The RBF interpolation is another concept to extrapolation the variables. RBF method using the Radius basis functions as interpolation functions instead of polynomials, and thus oscillations occur during the interpolation due to the property of the RBF function. The real interpolation values cannot be obtained like the Gaussian quadrature. On the bright side, RBF method is good at interpolating the scattered data sets. There is no restriction on the data type, and therefore RBF method can treat any shape of surface, which is very suitable for the node based unstructured schemes.
Moreover, there is no need to calculate the quadrature weights and points in advanced of the integration. Because of the flexibility, RBF method is employed in this work to calculate the engineering problems. Comparing with the Gaussian quadrature, memories of restoring the locations and weights of quadraure points are saved.
In this work, RBF is used to interpolate reconstructed variables on control surfaces comprising the control volumes. All of the reconstructed variables on the surfaces within a control volume are used as bases of the interpolation equation. Figure
4-2 describes the interpolation-integration procedures for RBF method.
58 3rd Order Scheme
(a) 3D physical grid projected to 2D plan
(b) Control Surface
Figure 4-2 Interpolation on control surfaces of the two-dimensional unstructured grid using RBF method (a) 3D physical grid projected to 2D
(Zhao, Sheng [40] ), plane (b) one of the control surfaces
59
The values on these sub-cells are obtained through the RBF interpolation. Figure
4-2 shows a diagram of the sub-cell surfaces in a two-dimensional mesh. For node-based unstructured schemes, flow variables are stored at the node (vertex) of the control volume, which is represented by dash lines. The lines connecting nodes are edges. Values at the center of all edges of a control volume, which are called basis points, are evaluated by the high order polynomial reconstruction using Equation (4.3). Values at the center of each sub-cell faces are evaluated by the RBF interpolation Thus all sub-cell points are used to perform the high order surface integration to improve the overall accuracy of the high order finite volume schemes.
4.2.3 Formulation of RBF Interpolation
To maintain the high order accuracy of a finite volume scheme, the numerical integration of surface fluxes must be implemented in a high order manner without degrading the overall accuracy of the high order schemes constructed using polynomial
Equation (4.1) or Equation (4.3). Due to the difficulty of multi-dimensional integration and complexity associated with irregular surface shapes for traditional Gaussian quadrature, a new but economic way for surface interpolation based on Radial Basis
Functions (RBF) was proposed at [113], which can be viewed as an alternative way to the
Gaussian quadrature integration over the surface of control volumes for high order schemes.
RBF method is a weighted sum of translations of a radially symmetric basic function augmented by a polynomial term. The general theory of RBFs was presented in details by Buhmann [33] and Wendland [34]. RBFs are popular for interpolating
60 scattered data as the associated system of linear equations is guaranteed to be invertible under very mild conditions on the locations of the data points. Rendall and Allen [35] developed a unified mesh deformation method using the RBF method and applied it to fluid-structure interaction calculations. In particular, RBFs do not require that the data lie on regular grid points (connectivity information), making the selection of integration point much easier than the Gaussian or multiquadric method. Since the RFB interpolation can be used to construct a high-order smooth curve based on any scattered points, the evaluation of quadrature points within each control volume for general high-order schemes can be replaced by a more flexible and lower cost RFB interpolation. Another advantage of the RBF interpolation is its global nature, meaning that all high-order variables reconstructed within each control volume can be used as the basis points to build an interpolation polynomial, while other methods can only used a selected number of points.
The solution of an interpolation problem using RBFs begins with the form of the required interpolation
Here, is the function value to be evaluated at location , is the form of a basis function. The index denotes the basis points of the RBFs and is the coordinates on the basis points. The last term is a polynomial up to a linear degree [35] to ensure that the translational and rotational motions are recovered by the RBF interpolation. The polynomial in the direction (equivalent results hold in and directions) is defined as
61 The coefficients in Equation (4.8) are found by requiring exact recovery of the original function, in this case the basis function values. When the polynomial term is included, the system is completed by an additional requirement
It was found that the selection of RBF functions and the number of basis points both play an important role in preserving the interpolation accuracy of the high order reconstructed variables within the control volume. Numerical tests indicated that the RBF functions with a locally piecewise smoothness property (such as Wendland’s) performed better comparing to the ones with the global smoothness (such as Gaussian’s, multiquadratic, and inverse multiquadratic) [35]. Furthermore, there are also requirements on the number of RBF basis points within each control volume [113].
Several types of RBF functions are tabulated in Table 4-1. The Wendland order of 2 basis function was used in the current work.
Table 4.2 RBF functions with global and local piecewise smoothness (Zhao and Sheng
[40])
Global Multiquadratic
Smoothness Gaussian Wendland order 2 Piecewise Wendland Order 4 Smoothness Wendland Order 8
It was found that the selection of different RBF functions and the number of basis points both play an important role in preserving the interpolation accuracy of the high-
62 order polynomial within the control volume. In the current study, Wendland’s C2 RBF function [35] is employed, which has the following form
To evaluate the impact of the Wendland’s C2 function in Equation (4.11) on the accuracy of polynomial interpolations, the interpolation errors are analyzed by choosing different number of basis points for interpolating a quadric and a quartic polynomial, which represents the third and fifth order schemes being investigated in this study. The emphasis here is to find out the resolution characteristics of the Wendland’s C2 function
(4.11) rather than the formal order of the accuracy (i.e., truncation error). The resolution characteristics means the error bound by which different approximations represent the exact value of a polynomial over the full range of length scales that is realized in a given mesh. One dimensional formula is used for the purpose of simplicity. From the interpolation point of view, an ideal interpolation scheme should have the property that the values evaluated at the basis points (known values) must satisfy the exact solutions of the polynomial, and the values evaluated at other interpolated points are close enough to the exact solutions within the specific error tolerance. The first condition is automatically satisfied with the Wendland’s C2 RBF function [35], and the second condition is the issue that needs to be evaluated and addressed here. Figure 4-3 and Figure 4-4 illustrate the relative errors by choosing different number of basis points on a uniform grid of 21 discretized grid points within a control volume. Obviously, the interpolated values by
RBF exhibited an oscillatory behavior around the basis points. The magnitude of the oscillation, which is also the error bound, has a direct impact on the interpolation accuracy of the RBF method, and thus affects the overall stability and accuracy of the
63 finite-volume schemes. The error bound is significantly reduced with the increase of number of basis points. Numerical tests indicated that at least 14 basis points for the quadratic polynomial, or 16 basis points for the quartic polynomial, should be adopted to maintain an error bound below 0.2% of the exact solution.
3rd Order Scheme
Figure 4-3 Error estimation for Wendland's type RBF function in quadratic
polynomials (Zhao and Sheng [113])
64 Figure 4-4 Error estimation for Wendland's type RBF function in quartic
polynomials(Zhao and Sheng [113])
4.3 Weighted Limitation
If the flow field contains strong gradients in vortical regions or discontinuities such as shock waves, limiters are normally required by imposing monotonicity principles on the reconstructed high order variables. However, applying a monotonic limiter designed for the second order scheme often reduces the spatial accuracy of the scheme to second order, and thus losses all the benefits associated with the high order flux computations. A proper limiter therefore plays a key role in maintaining both stability and accuracy of the high order scheme. A good limiter should work in such a way that it has no or very little effect in the smooth region, but acts firmly in the discontinuous region to avoid the oscillation of the high order solutions. A thorough discussion on this type of limiters can be found in reference [114]. Two widely used slope limiters for
65 unstructured grid schemes are that proposed by Barth, et al. [116] and Venkatadrishnan, et al. [117]. The Barth’s limiter works well for the second order schemes to maintain the monotonicity conditions, however, it diffuses too much in the smooth region. The
Venkatadrishnan’s limiter works well with weak shocks by alleviating the effect of the limiter in the smooth region, but solutions are sometimes hard to converge for complex flow fields with strong gradients. In the present study, a limiter suitable for the high order scheme is investigated similar to the Barth’s limiter, but with a high order term. A discontinuity detector, similar to the work by Delanaye [118], is introduced to adjust limiters between lower and higher order terms of the reconstructed solution variables, i.e..
where is the reconstructed high order variables on the interface and is the lower
order flow variables of the scheme. For example, if is the fifth order reconstructed
variables, will be the third order reconstructed variables. Or if is the third order
reconstructed variables, will be the second order reconstructed variables. is the
Barth’s slop limiter similar to the second order scheme. is the detector function depending on , which adjusts the weight of the limiters in the smooth and discontinuous regions based the constant values and suggested by [119].
66 Chapter 5
Results and Discussion
In order to examine the newly developed higher order schemes, six validation cases with different flow conditions are computed and compared with the original and/or the experimental data. Since the current development is aiming at solving realistic flows, two important flow phenomena, the trailing vortices (continuous flows) and shockwaves
(discontinuous flows), are extensively verified in this work.
First in Section 5.1, a single two-dimensional transporting vortex is computed to provide quantitative validations of numerical accuracy of the new method. Then in
Section 5.2 and 5.3, two viscous vortex-dominated flows are presented to validate the high order methods with/without the RBF interpolation. One is a stationary wing NACA
0015 [42] mounted on a flat wall in a subsonic flow field, in which the new method is validated on the subsonic flow condition and for capabilities to capture vortices generated by the wing tip. The second case is a rotary wing NACA 0012 rotor [43] in hover. The solutions were obtained by putting the whole system in a relative rotating reference frame to examine the ability of the method working on the viscous flow with vortices and shocks in the rotating frame. In Section 5.4, a supersonic flow past a waisted body [44] is computed with the high order method. This computation is to demonstrate the newly developed method to capture strong shock waves with the assistance of the weighted
67 smart limiter technique, which probably is the bottleneck of all higher order schemes. A realistic viscous flow about a marine propeller P5168 [45] at nearly incompressible Mach number is presented in Section 5.5. This case provides validations not only to vortex- dominate flows but also to low Mach number incompressible flows. Finally in Section
5.6, a Bell helicopter M427 main rotor blade [46] is employed to validate the method for predicting the aerodynamic performance of a realistic configuration. All viscous meshes were generated with mixed unstructured elements types including tetrahedron, prism, pyramid, and hexahedron. The y+ value is about one on all solid surfaces to resolve the viscous sublayers. One-equation Spalart-Allmaras turbulence model [97] is used for all computations.
5.1 2D Vortex Transportation
The first case is an inviscid vortex transported by a uniform flow. This case is referred from one of the test problems from the first international workshop on High- order CFD Method [120] at the 50th AIAA Aerospace Science Meeting holding at
Nashville, Tennessee. This case is designed to test the ability of the numerical scheme to accurately transport the vortex and maintain the strength of the vortex in an inviscid flow.
Vortices are common physical phenomena in the real world, such as rotorcraft rotors, wind turbines, marine propellers, and etc. Accurate and efficient prediction of vortex flows is very important for understanding the flow physics and improving the aerodynamic designs. Therefore, the capability of numerical schemes to accurately capture the vortex flows is an essential requirement in any rotorcraft CFD codes. The biggest issue in computing vortical flows is the numerical dissipation of the CFD solvers.
In this work, the 3rd order spatial discretization is evaluated quantitatively on numerical
68 dissipations for the vortical flow by comparing with the solution obtained with the original 2nd order numerical scheme.
The vortex flow being investigated can be treated as a superposition of a two dimensional vortex into a free stream flow. The initial vortex rotates around the vortex center and travels along with the free stream. The vortical moment is generated based on the free stream speed so that the vortex disappears when the free stream is stationary. The nondimensional formulas are listed below [115].
where, and are the coordinates of the vortex rotating center. The Mach number of free stream is chosen as , and the strength of the vortex is , and the characteristic radius of this vortex is . In this work, the reference value is based on the free stream. Thus the nondimensional free stream velocity is , and free stream temperature is with the ratio of specific heat . The above
Equation (5.1) through Equation (5.3) represent the isolated vortex flow variables, and
Equation (5.4) through Equation (5.6) are the combined flow variables. The whole flow region is initialized with the exact values based on the above equations.
69 This case is an unsteady two-dimensional problem calculated with the three- dimensional unsteady Euler equations. Thus, the grid generation and boundary conditions need to be considered to meet requirements for both the flow solver and flow conditions.
A series of three-dimensional isentropic tetrahedrons meshes of thin squares are generated, which are shown in Figure 5-1. There are no boundary layers generated for this inviscid flow problem. The nondimensional length at direction is 0.1, 0.1, and 0.003. The very thin thickness on the direction is for the purpose of saving the mesh size and the computational time.
In order to exam the order of accuracy of the numerical scheme, the computational mesh has to be within the asymptotic range. This means that the grid size has to be small enough so that the truncation error can be ignored compared to the numerical solutions, but also large enough to overcome the runoff error and let the truncation error dominate other types of errors. Several tests were performed to determine the appropriate grid size. In the present study, the grid spacing is 0.000834 for the coarse grid, 0.000417 for the medium grid, and 0.0003475 for the fine grid, with the corresponding number of grid points of 72749, 463538, and 707214, respectively.
70
Computational Domain
(a) Coarse grid, N=72749
(b) Medium grid, N=463538 (c) Fine grid, N=707214
Figure 5-1 Comparison of meshes for vortex transportation case: (a)
coarse grid, (b) medium grid, (c) fine grid (Zhao and Sheng [115])
71 The purpose of calculating this vortex flow is to exam the spatial order of accuracy of the improved scheme, and thus the temporal effect should be minimized.
Several calculations are performed to test the minimum time step on the fine grid, so that the residual of the equation is not affected too much by decreasing the current time step.
The minimum nondimensional time step is chosen as 0.0001 for all the cases. The vortex center of the initial condition is set to the center of the computational domain.
Maintaining the unsteady moving vortex stationary at the center of the computational domain can save significant amount of computational grid points. This is achieved by setting the grid speed in an opposite direction to the free stream velocity. The Dirichlet type of the boundary condition is set to all far field boundaries. A symmetric boundary condition is set to the -direction surfaces to eliminate the three-dimensional effect in this two-dimensional flow.
All the cases are computed using the implicit iterative scheme. Four Newton’s nonlinear iterations and six Gauss-Seidel linear relaxations are applied at each time step.
Shown in Figure 5-2 is the convergence history of the residual for different cases calculated on three sets of meshes using the 2nd order scheme and improved one. The residual obtained on the finer grid is generally smaller than the one obtained on the coarse grid, and the the residual obtained by the improved scheme is also smaller than that obtained by the original scheme. The residuals are reduced to the machine accuracy in all cases. This is reasonable because the unsteady moving vortex case is treated as a steady problem by fixing the vortex center at the initial location.
72 Figure 5-2 Convergent histories of different schemes on a series of meshes
(Zhao and Sheng [115])
The calculated vorticities at the 10000th time step are compared with the exact solution, which are shown in Figure 5-3 through Figure 5-4. The predicted vorcitiy by the improved scheme reveals less dissipation comparing with the one obtained by the 2nd order scheme. The shape of improved results is more condense and smooth than the 2nd order one. The results of the finer grid solutions present better values close to the exact solutions. Although only the vorticity contour plots at the 10000th step are shown here, and the results obtained at the 20000th step show a similar but more dissipated pattern comparing with the results at the 10000th step.
73 10.0 10.0
5.0 5.0
0.0 0.0
(a) Original Scheme (b) Improved Scheme
10.0
5.0
0.0
(c) Exact Solution
Figure 5-3 Comparison of vorticities on the coarse grid at 10000 steps on the
coarse grid (Zhao and Sheng [115]): (a) original, (b) improved, (c) exact
74 10.0 10.0
5.0 5.0
0.0 0.0
(a) Original Scheme (b) Improved Scheme
10.0
5.0
0.0
(c) Exact Solution
Figure 5-4 Comparison of vorticities on the middle grid at 10000 steps on the
medium grid(Zhao and Sheng [115]): (a) original, (b) improved, (c) exact
75 10.0 10.0
5.0 5.0
0.0 0.0
(a) Original Scheme (b) Improved Scheme
10.0
5.0
0.0
(c) Exact Solution
Figure 5-5 Comparison of vorticities on the fine grid at 10000 steps on the
medium grid(Zhao and Sheng [115]): (a) original, (b) improved, (c) exact
76 The same conclusion is true for the maximum values of the voritcity at the
10000th time step in the Table 5.1 and the 20000th time step Table 5.2. Although discrepancies are observed between the exact values and the numerical solutions at the same grid level, stronger peak values are obtained by the improved scheme and on the finer grids. The dissipation ratios of the vorticities for all the cases are examined and given in Table 5.3. Increasing the order of accuracy of schemes has the similar effect as increasing the number of computational grid points to the numerical solutions. Smaller dissipation ratio for the improved scheme or finer grid is expected. The percentage of the dissipation ratios for different schemes and levels of grids presented in Table 5.3 meet this expectation. Solutions of the finer grid have smaller dissipation ratio, and the average dissipation ratio of the improved scheme is 40% less than the 2nd order scheme for all the grid levels.
Table 5.1 Maximum value of voticities at the 10000th step (Zhao and Sheng [115])
Exact maximum 2nd order maximum Improved, RBF maximum
Coarse grid 9.84801 5.60538 6.97256, 7.6503
Medium Grid 9.96724 8.90235 9.38907, 10.5306
Fine Grid 9.97425 9.14941 9.52515, 10.604
Table 5.2 Maximum value of voticities at the 20000th step (Zhao and Sheng [115])
Exact maximum 2nd order maximum Improved, RBF maximum
Coarse grid 9.84801 4.14693 5.59333, 6.54535
Medium Grid 9.96724 8.10449 8.8997, NAN
Fine Grid 9.97425 8.52187 9.14571, NAN
77
Table 5.3 Maximum dissipation ratio of voticities between the 10000th step and the
20000th step (Zhao and Sheng [115])
Exact maximum 2nd order maximum Improved, RBF maximum
Coarse grid 9.84801 26.018753% 19.780827%, 14.443224%
Medium Grid 9.96724 8.962353% 5.212124%, NAN
Fine Grid 9.97425 6.858803% 3.983559%, NAN
Shown in Figure 5-6 and Figure 5-7 are the velocity (in direction) distributions along the centerline of the computational domain across the center of the vortex, computed by two schemes on different levels of grids. Again, the improved scheme provides better results with less dissipation rate of the vortical flow than the 2nd order scheme. The RBF surface integration method was also evaluated. It maintained the better peak velocity and vorticity strength among all solutions. However, the oscillation property of the RBF interpolation has caused the vortex contour in the non-smooth and skewed shapes. Therefore these results are not included in this dissertation. Moreover, unlike the Gaussian quadrature, the RBF collects all the point from all the direction as the base points for the interpolation, and there is no control on the points selection for this two dimensional problem which is computed in the three dimensional domain. This issue needs to be worked out in the future work.
78 Figure 5-6 Comparison of vortical velocities on different grids at the 2000th step
(Zhao and Sheng [115])
Figure 5-7 Comparison of vortical velocities on different grids at the 1000th step
(Zhao and Sheng [115])
79 The quantitative analysis on the observed order of accuracy is performed in the present study. Velocity in the free stream direction is used for the analysis of the accuracy of the schemes, and direction velocity indicated a very similar result. The density and the pressure show a little bit lower, but the total average of four variables excluding the velocity approaches the same observed order of accuracy. The volume weighted norm of the difference of calculated value and the exact solution is used as the parameter to calculate the observed orders. Because of the restriction on grid generation tool, -refined meshes cannot be generated. Therefore the grid spacing is a kind of global averaged spacing calculated based the number of grid points in the computational domain. Shown in Table 5.4 are the observed orders of accuracy calculated based on the coarse grid (G1), the medium grid (G2), and the fine grid (G3).
The results for the observed order of accuracy are consistent among two sets of grids. The observed order of accuracy for the 2nd order scheme is around the 2.2, and the observed order of accuracy for the improved scheme is close to 2.6. The reason of the observed order of accuracy for the improved order scheme less than 3 is because the surface integration is not taken into the consideration.
Table 5.4 Observed order of accuracy on the various grids (Zhao and Sheng [115])
2nd Order Scheme Improved Scheme
Observed Observed
Order Order
G1 1.2638E-003 8.5924E-004
G2 0.54023869 3.2091E-004 2.22 1.7550E-004 2.57
G3 0.468553 2.2514E-004 2.28 1.2089E-004 2.59
80
From this simple 2-D vortex transportation case it shows that the improved scheme does perform better than the original 2nd order scheme in preserving the vortex strength. As exampled results, the observed order of accuracy of the improved 3rd order scheme is approaching to 3 even without the consideration of the surface integration, which demonstrates quantitatively the order of accuracy is raised. The main purpose of the work is to find a way to improve the accuracy of the simulations on the practical engineering problems instead of the order of accuracy of the scheme. Therefore, it remains interesting to see how this improvement affects simulations of the 3-D problems.
5.2 NACA0015 Wing
NACA0015 wing is a three-dimensional untwisted wing tested by McAlister, et al. [42]. This case is used to examine the capability of new developed method on vortex- dominate flow in the absolutely frame in a subsonic flow condition. The wing was positioned at 12 degrees angle of attack and mounted on a flat wall. The incoming free stream Mach number is 0.1235, and Reynolds number is 1.5 million based on the chord length. A typical feature of this flow is the trailing vortex generated from the wing tip that may lasts to a very long distance the downstream to. Figure 5-8 displays the illustration of the simulation grid and the trailing vortex of the wing. In order to better capture the trailing tip vortex, an interior cylindrical interior surface was built along the vortex trajectory to provide refined grid resolutions. Three unstructured meshes were generated with the coarse, medium, and fine grid resolutions along the path of the vortex, as shown in Figure 5-8. The only difference among three meshes is the mesh resolution along the path of the tip vortex. That is, the medium mesh has a half point spacing of the
81 coarse mesh, and the fine mesh has an half point spacing of the medium mesh. The mesh sizes are 2.77, 4.39, and 14.48 million nodes, respectively. Numerical computations were performed on all three meshes using the 2nd, 3rd, and 5th order discretization schemes with and without the RBF interpolations. Since the flow is steady state, a local time step is used with CFL numbers from 5 to 10. One Newton’s iteration and 8 symmetric Gauss-
Seidel relaxations are applied in all computations.
NACA 0015 Trailing tip vortex wing 90 80 70 60 50 East 40 West 30 North 20 10 Figure 5-8 Trailing tip vortex generated from the NACA 0015 0wing (Zhao, 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr Sheng [40]) vortex
82 (a) Coarse mesh, 2.77M (b) Medium mesh, 4.39M
(c) Fine mesh, 14.48M
Figure 5-9 Three mesh resolutions along the path of trailing tip vortex (Zhao,
Sheng [40]): (a) coarse mesh, (b) medium mesh, (c) fine mesh
In order to look at the improvement of the accuracy of the schemes, one of the good indicators is the trailing vortex strength. The grid resolution study was executed on these three grids. Shown in Figure 5-10 to Figure 5-12 are comparisons of computed vortical velocity component within the vortex core at 2 chords downstream obtained by various schemes on different meshes. In general, higher order schemes captured
83 stronger vortical velocity or strength than the lower order schemes. The RBF interpolation only shows benefits on the 3rd and 5th order solutions, and does not show improvements on the 2nd order solutions. This behavior is totally expected because the accuracy of the 2nd order discretization cannot be further improved by either RBF interpolation or Gaussian quadrature.
The impact of the RBF interpolation is rather significant in the 3rd and 5th order schemes. As a demonstration, predicted vortical velocity contours plots at 6 chords downstream are presented in Figure 5-13 to Figure 5-15. The captured vortex by the 3rd order scheme using the RBF interpolation is even better than that captured by the 5th order scheme without using the RBF interpolation.
84 (a) 2nd order, no RBF
(b) 2nd order, with RBF
nd Figure 5-10 Computed vortical velocities at 2 chord downstream using the 2 scheme on different meshes (Zhao, Sheng [40]): (a) no RBF, (b) with RBF
85 (a) 3rd order, no RBF
(b) 3rd order, with RBF
Figure 5-11 Computed vortical velocities at 2 chord downstream using the 3rd
scheme on different meshes (Zhao, Sheng [40]): (a) no RBF, (b) with RBF
86 (a) 5th order, no RBF
(b) 5th order, with RBF
Figure 5-12 Computed vortical velocities at 2 chord downstream using the 5th order scheme on different meshes (Zhao, Sheng [40]): (a) no RBF, (b) with RBF
87 0.50
0.25
0.00
(a) 2nd order no RBF
0.50
0.25
0.00
(b) 2nd order with RBF
Figure 5-13 Computed cross section velocity contours at 6 chords downstream using the 2nd order scheme on the medium mesh (Zhao, Sheng [40]) : (a) without RBF, (b)
with RBF
88 0.50
0.25
0.00
(a) 3rd order no RBF
0.50
0.25
0.00
(b) 3rd order with RBF Figure 5-14 Computed cross section velocity contours at 6 chords downstream using the 3rd order scheme on the medium mesh(Zhao, Sheng [40]): (a) without
RBF, (b) with RBF
89 0.50
0.25
0.00
(a) 5th order without RBF
0.50
0.25
0.00
(b) 5th order with RBF
Figure 5-15 Computed cross section velocity contours at 6 chords downstream using the 5th order scheme on medium mesh (Zhao, Sheng [40]) : (a) without RBF,
(b) with RBF
90 In order to further examine the performance of the different schemes, the computed results by different schemes are compared. Although the high order discretizations and RBF interpolation developed in this work have improved the numerical accuracy in capturing vortical flows, there are still discrepancies between the computational results and experimental data, as shown in Figure 5-16 to Figure 5-19.
However, the computational results obtained in this work are the closest ones to the experimental data so far using the unstructured grid schemes [113].
To check the correctness of new developed schemes, the predicted pressure coefficients, , on the viscous surface of the wing with different schemes are compared with the experimental data in Figure 5-20 to Figure 5-23, since normally the pressure distribution is not sensitive to the different numerical method. Pressure coefficients along the wing span are compared with the experimental data. No significant differences are found among various schemes, including the use of the RBF interpolations. This is because that the trailing tip vortex, which receives the most impact by the RBF method, only affects the flow field downstream of the wing.
91 Figure 5-16 Comparison of the vortical velocities using different schemes at one
chord downstream (Zhao, Sheng [40])
Figure 5-17 Comparison of the vortical velocities using different schemes at 2
chords downstream (Zhao, Sheng [40])
92 3rd Order Scheme
Figure 5-18 Comparison of the vortical velocities using different schemes at 4
chords downstream (Zhao, Sheng [40])
Figure 5-19 Comparison of the vortical velocities using different schemes at 6 chords
downstream (Zhao, Sheng [40])
93 Figure 5-20 Predicted pressure coefficients at 23.8% wing span (Zhao and Sheng
[113])
Figure 5-21 Predicted pressure coefficients at 49% wing span (Zhao and Sheng
[113])
94 Figure 5-22 Predicted pressure coefficients at 89.9% wing span (Zhao and Sheng
[113])
Figure 5-23 Predicted pressure coefficients at 97.4% wing span (Zhao
and Sheng [113])
95
Table 5.5 Computational costs for NACA 0015 wing using different schemes (Zhao,
Sheng [40])
Size No. CPU Time (s) / RAM (GB) Grid of nd nd rd rd th 5th RBF (M) CPUs 2 2 RBF 3 3 RBF 5 2.84/ 4.85/ 2.84/ 4.85/ 4.06/ 5.55/ Coarse 2.77 32 0.56 0.95 0.56 0.95 0.56 0.95 5.15/ 11.31/ 5.54/ 11.31/ 7.72/ 11.61/ Medium 4.39 32 0.72 1.2 0.72 1.2 0.72 1.2 5.23/ 11.43/ 5.53 11.43/ 7.35/ 11.71/ Fine 14.48 128 0.75 1.2 /0.75 1.2 0.75 1.2 Mean Unit Cost 40.08/ 91.74/ 42.38/ 91.74/ 56.33/ 93.99/ (x10-6) 6.12 9.68 6.12 9.68 6.12 9.68
Computational cost is a very important element to estimate a scheme, which decides the practicality of the scheme used for the engineering problems. Computational costs in terms of memory requirements and CPU times for all computations on NACA
0015 wing are given in Table 5.5, where the mean unit cost is total CPU time, and memory per mesh point per time step based on medium and fine meshes. In general, computations obtained by all schemes without the RBF interpolation had almost the same memory requirements. This is because all quantities needed to build the 3rd order discretization have been computed and stored in the 2nd order scheme. The CPU costs for the 3rd order scheme is only about 10% more than the 2nd order scheme, and the
CPU costs for the 5th order scheme is about 33% more than the 3rd order scheme.
However if the RBF interpolation procedure is implemented, the memory requirements will increase by 60%, and CPU times will increase by up to 130% over the methods
96 without using the RBF interpolation, due to the extra storage requirements and computational costs associated with the sub-cell surface integrations.
A sectional summery for this fixed wing case is made here. The new developed schemes are fully validated and verified on this 3-D viscous vortex dominate NACA0015 wing. This simple but still challenging case is a good example to verify the ability of the scheme to predict the viscous vortex-dominate flow, since other difficulties such as the rotational issue and discontinuities are not introduced. The grid resolution study and the comparison of predicted results by all the schemes with the experiments demonstrate the new higher order method do increase the accuracy of the original scheme without the loss of the correctness. Although a limited computing cost is included, but is is still acceptable for the computations of viscous problems.
5.3 NACA 0012 Rotor
The third case is about a viscous rotary wing, a NACA 0012 rotor in hover.
Compared with the previous fixed single wing case, this rotor case draws complexities of the rotation action and multi-blades computations into the numerical simulations. The experiment test was conducted by Caradonna, et al. [43]. The rotor has two untwisted, untapered blades constructed with the NACA 0012 airfoil. The diameter of the rotor is
2.286 meter and the aspect ratio of the rotor is 6. Collective pitch angle for the rotor blades is 8 degrees. The rotating speed of the rotor is 2500 RPM, and the blade tip Mach number is 0.877. The Reynolds number here is 4.6x107 based on the blade tip speed and the diameter of the rotor.
97 Figure 5-24 NACA 0012 rotor coarse grid (Sheng and Zhao [96])
NACA 0012 rotor 8 degrees of collective angle Tip Mach number 0.899 Reynolds number 47 million 3.0 million nodes
Figure 5-25 NACA 0012 rotor fine grid (Sheng and Zhao [96])
98 Two unstructured viscous mesh for the NACA 0012 rotor was generated. Shown in Figure 5-24 and Figure 5-25 are the resolutions of two grids on vertical cutting planes of the rotor. The first one is a coarse grid that has 2.96 million nodes in total size. The grid points were clustered only near the rotor disk plane, which caused a rather coarse grid resolution in the rotor wake downwash region. The second grid has an enhanced grid resolution in the rotor wake region with total grid points of 6.64 million to help capture the trailing tip vortices. Computations were carried out in the rotating relative frame, where the flow field can be viewed as a steady state. Therefore, a local time step was used at a CFL number of 2-10. No-slip boundary condition was applied to viscous surfaces, and characteristic variable far field boundary condition was applied to the outer and far wake boundaries.
Numerical solutions obtained on the coarse grid by various discretization schemes were first compared. Figure 5-26 to Figure 5-28 show predicted vorticity contours and the predicted swirl parameters obtained by the 2nd, 3rd, and 5th schemes for the coarse grid.
The predicted strength of trailing tip vortices was stronger in the 5th solution than in the
2nd and 3rd solutions. However, because there was no enough grid resolution along the rotor wake downwash direction, all solutions showed limited wake ages due to numerical diffusions. This clearly indicates that a proper grid resolution in the rotor wake region is required to capture the rotor wake system. Therefore, the following analysis will be focused on the results obtained on the refined grid.
99 3rd Order Scheme
2nd order on the coarse grid Figure 5-26 Predicted 2nd order vorticity contours (left) and iso-surface swirl parameter (right) on the vertical cutting plane for the coarse grid (Sheng and
Zhao [96])
3rd order on the coarse grid Figure 5-27 Predicted 3rd order vorticity contours (left) and iso-surface
swirl parameter (right) on the vertical cutting plane for the coarse
grid(Sheng and Zhao [96])
100 5th order on the coarse grid Figure 5-28 Predicted 5th order vorticity contours (left) and iso-surface
swirl parameter (right) on the vertical cutting plane for the coarse grid
(Sheng and Zhao [96])
Figure 5-29 to Figure 5-31 present the predicted vorticity contours for the fine grid and wake ages obtained by different schemes on the second refined grid. Improved predictions of the rotor tip vortex system were seen in all solutions because of increased grid resolutions in the wake region. The vortex sheets generated by the rotating blades were clearly visible in all solutions. The higher-order solutions obtained by the 3rd and the 5th order schemes clearly captured strong vortex strengths than the original the 2nd order solution. It is interesting to notice that, although the predicted tip vortex strength by the 5th scheme was the strongest among all solutions, the third-order solution captured the longest unbroken wake age, as represented by the iso-surface of the swirl parameter. This structural change observed in the rotor wake system may be due the significant
101 circulations predicted by the 5th solution in the far wake region, while these circulations were rather weak in the second and third-order solutions.
3rd Order Scheme
2nd order on the fine grid Figure 5-29 Predicted 2nd order vorticity contours (left) and iso-surface swirl
parameter (right) on the vertical cutting plane for the fine grid (Sheng and Zhao [96])
3rd order on the fine grid Figure 5-30 Predicted 3rd order vorticity contours (left) and iso-surface swirl
parameter (right) on the vertical cutting plane for the fine grid (Sheng and Zhao [96])
102
5th order on the fine grid
Figure 5-31 Predicted 5th order vorticity contours (left) and iso-surface swirl
parameter (right) on the vertical cutting plane for the fine grid (Sheng and Zhao [96])
Figure 5-32 to Figure 5-34 shows the evolution of tip vortex sheets predicted by various schemes at different azimuthal angles. The helicity contour maps on these cutting planes were illustrated for every 30 degrees of blade rotation. These comparisons provided a general assessment of the current high-order discretization method in capturing the rotor wake system for helicopter rotors.
103
-0.2 -0.1 0.0 0.1 0.2
Figure 5-32 Predicted helicity contours by 2nd order scheme for the fine grid
(Sheng and Zhao [96])
-0.2 -0.1 0.0 0.1 0.2
Figure 5-33 Predicted helicity contours by 3rd order scheme for the fine grid
(Sheng and Zhao [96])
104
-0.2 -0.1 0.0 0.1 0.2
Figure 5-34 Predicted helicity contours by 5th order scheme for the fine grid
(Sheng and Zhao [96])
To quantify the difference in captured tip vortex strengths by different discretization schemes, the predicted peak vorticity values along the rotor wake age are plotted in Figure 5-35. The tip vortex strength increases in the initial stage after generated by the blade, and then gradually decreases as the wake ages. The selected starting location in the wake age was at 30 degrees azimuth, where the maximum vortex strengths were observed for all solutions and then started to decrease rapidly. However, predicted peak vorticity values by a higher-order method were generally higher than that predicted by a lower-order method at all wake ages. In general, the vortex strength predicted by the
5th order method was 36-50% higher than that obtained by the 3rd order solution, and 60-
80% higher than that obtained by the 2nd solution. It clearly shows that the current high- order method has less numerical diffusions compared to the original 2nd order method, resulting in improved accuracy in predicting the trailing tip vortex system for helicopter rotors.
105 Figure 5-35 Predicted tip vortex strengths (peak vorticity) by schemes (Sheng,
and Zhao [96])
The predicted tip vortex trajectories are compared with the measured data in
Figure 5-36. The trajectory of each trailing vortex was identified based on the location of the peak vorticity value predicted at different vortex ages. It seems that the stronger the predicted tip vortex, the closer the distance to the rotating axis. Overall, predicted vortex trajectory matched reasonably well with the measured data.
106 Figure 5-36 Predicted trajectories of trialing tip vortices by various
schemes (Sheng, and Zhao [96])
Unlike the previous low Mach number flow, a shock is generated at around 80% blade span location in this transonic flow. Shown in Figure 5-37 to Figure 5-38 are the predicted pressure coefficients ( ) at different blade span locations comparing with the measured data. No significant differences were observed in the predicted distributions among different discretization schemes. All solutions were in a reasonably good agreement with the experimental data. A similar pattern is obtained in all schemes with and without the RBF interpolation.
107 Figure 5-37 Predicted pressure coefficients along the 68% of the blade span on the
fine grid (Sheng, and Zhao [96])
Figure 5-38 Predicted pressure coefficients along the 80% of the blade span on the
fine grid (Sheng, and Zhao [96])
108 Figure 5-39 Predicted pressure coefficients along the 89% of the blade span
on the fine grid (Sheng, and Zhao [96])
Figure 5-40 Predicted pressure coefficients along the 96% of the blade span
on the fine grid (Sheng, and Zhao [96])
109 Table 5.6 lists the grid sizes and CPU times used by various schemes on the two grids. Not too much cost loading is added to the higher order computations comparing to the original scheme, which is consistent with the conlusion of the NACA0015 case.
Table 5.6 Computational costs for NACA 0012 rotor (Sheng, and Zhao [96])
Nodes CPU Time (s) Grid (x106) 2nd 3rd 5th
Coarse 2.96 0.555 0.596 0.647
Fine 6.64 1.353 1.418 1.589
CPU time per node 1.96E-07 2.07E-07 2.29E-07
From the computation results of this viscous and rotating NACA0012 rotor case, the new method worked well and obtained the higher accuracy than the original scheme, which is consistent with conclusions from the NACA0015 fixed wing case This rotary rotor case demonstrates the new method has the ability to predict the rotor craft problems in the relative frame system, which gives the confidence to further investigate the real rotor craft models. Beside, for the transonic with problem, a shock wave appeared in the flow field. The new method captured this discontinuity reasonably. This observation provides a clue that the new method can working on the flow problems with strong shock waves. This was verified at the next waisted body case.
5.4 Waisted Body
The previous NACA0012 rotor case was about a rotational flow with vortex shedding and a minor shock wave in the flow field. In order to further evaluate the new
110 method in predicting discontinuous flows such as shocks, a supersonic viscous flow about a waisted body [44] at Mach umber 1.4 is investigated in the present study.
The waisted body is a simplified missile type of configuration with an approaching flow Mach number of 1.4 at zero degrees angle of incidence. Three unstructured meshes, with the size of 1.23M, 3.84M, and 5.33M grid points on the coarse, medium, and fine meshes, were generated using different point spacing to control the grid resolution around the waisted body, see Figure 5-41 to Figure 5-43.
Computations were performed on all three meshes with the 2nd, 3rd, and 5th order discretization methods with and without the RBF interpolations. The same CFL number of 2-5, and 1 Newton’s iteration with 8 Gauss Seidel relaxations were used in all computation performed in this work.
Coarse mesh, 1.23M
Figure 5-41 The coarse grid for the waisted body(Zhao and Sheng [40])
111 Medium mesh, 3.84M
Figure 5-42 The medium grid for the waisted body (Zhao amd Sheng [40])
Fine mesh, 5.33M
Figure 5-43 The fine grid for the waisted body (Zhao and Sheng [40])
112 Shown in Figure 5-44 to Figure 5-45 are predicted pressure and Mach number contours on a vertical cutting plane through the body rotational axis. The results were obtained with the 2nd, 3rd, and 5th order discretizations on the medium sized mesh, where the RBF interpolation procedure was implemented. Although not shown, the results obtained without using the RBF interpolation were very close to the ones using the
RBF interpolation. The basic flow feature is that a strong oblique shock is formed from the leading edge of the body, followed by a secondary oblique shock near the middle part of the body. In the downstream of the trailing edge, a series of expansion and shock waves are also formed, making a complicated flow structure around the waisted body. In the pressure contours, the 3rd and 5th order solutions showed sharper shock waves than the 2nd order solution. In addition, a slightly longer wake region behind the waisted body was predicted by the higher order methods than that predicted by the lower order solution for the Mach number contours.
113 1.8 1.0
0.9 0.6
0.0 0.2
2nd order pressure 2nd order Mach number Figure 5-44 Predicted pressure (left) and Mach number (right) by the 2nd order
scheme on the medium grid (Zhao and Sheng [40])
1.8 1.0
0.9 0.6
0.0 0.2
3rd order pressure 3rd order Mach number
Figure 5-45 Predicted pressure (left) and Mach number (right) by the 3rd order
scheme on the medium grid (Zhao and Sheng [40])
114 1.8 1.0
0.9 0.6
0.0 0.2
5th order pressure 5th order Mach number Figure 5-46 Predicted pressure (left) and Mach number (right) by the 5th order
scheme on the medium grid (Zhao and Sheng [40])
The computed surface pressure distribution in Figure 5-47 and skin friction in
Figure 5-48 coefficients were compared with the experimental data measured by Winter et. al. [44]. It was found that all numerical solutions obtained by various schemes (with and without using the RBF interpolation) showed very similar results on the surface and distributions. Nevertheless the results shown in Figure 5-7 and Figure 5-48 are obtained based on the medium mesh solution, the same conclusion holds for the coarse and fine meshes as well. This indicates that the high order methods proposed have less impact on the surface flow variables than the volume flow variables.
115 Figure 5-47 Computed Cp distributions on waisted body surface, medium
(Zhao, Sheng [40])
Figure 5-48 Computed Cf distributions on waisted body surface, medium mesh
(Zhao, Sheng [40])
116 This can be further verified by examining predicted pressure distributions along the free stream direction across multiple shocks at a location off the waisted body, such as 0.5 body length above the body as shown in Figure 5-49 to Figure 5-51. A limiter has been used to maintain the stability of numerical solutions with strong discontinuous solutions (shocks). In all solutions obtained on the coarse, medium, and fine meshes, sharper shock interfaces have been predicted by the 3rd and 5th order solutions than the 2nd order results. In particular, refined details of the peak pressure pattern for the secondary oblique shock have been captured by the 3rd and 5th order schemes, which have been completely omitted by the 2nd order solution. However, the 3rd and 5th order solutions do show the overshot behaviour at the first shock interface, which is typical for the high order discretization schemes.
117 Coarse mesh, no RBF
Coarse mesh, with RBF
Figure 5-49 Computed pressure profiles at 0.5 body length above the axis using
different schemes on the coarse meshes (Zhao and Sheng [40])
118
Medium mesh, no RBF
Medium mesh, with RBF
Figure 5-50 Computed pressure profiles at 0.5 body length above the axis using
different schemes on the medium meshes (Zhao and Sheng [40])
119 Fine mesh, no RBF
Fine mesh, with RBF
Figure 5-51 Computed pressure profiles at 0.5 body length above the axis using
different schemes on the medium meshes (Zhao and Sheng [40])
120 Computational costs for memory requirements and CPU times on waisted body are given in Table 5.8. Similar to the previous case of the NACA 0015 wing, computations obtained by the 2nd, 3rd, and 5th order schemes without the RBF interpolation have almost the same memory requirements. The CPU cost per time step of computations is 15% more on the 3rd order scheme over the 2nd order scheme, and 55% more on the 5th order scheme over the 3rd order scheme. If the RBF interpolation procedure is implemented, the memory requirements will increase by 140% and CPU times will increase by up to 130% over the ones without using the RBF interpolation.
Certainly there is a penalty associated with the implementation of high order discretizations, in particular to the RBF interpolation for surface integration over the sub- cells of all control volumes.
Table 5.7 Computational costs for waisted body using different schemes (Zhao, Sheng
[40])
Grid Size No. CPU Time (s) / RAM (GB) of CPUs (M) 2nd 2ndRBF 3rd 3rdRBF 5th 5thRBF Coarse 1.23 16 2.31/ 4.26/9. 2.32/6.7 4.26/9.82 3.34/6.7 4.72/9.82 6.71 82 1 1 Medium 3.84 32 4.28/ 11.82/1 4.28/0.9 11.82/1.5 6.55/0.9 12.3/1.5 0.90 .5 0 0 Fine 5.33 64 3.33/ 8.83/1. 3.91/0.5 8.83/1.30 6.93/0.5 11.2/1.30 0.54 30 4 4 Mean Unit Cost (x10-6) 37.8 100.29/ 44.41/9. 100.29/22 69.24/9. 118.24/22 2/9.4 22.85 49 .85 49 .85 9
The waisted body case provides an evidence of the reliability of the new developed method on computing the supersonic flow with strong discontinuities. In the
121 mean time, it also verified the weighted limiter is working well in the flows with shock waves by keeping the computations stable without sacrificing too much accuracy.
5.5 Marine Propeller P5168
For the previous cases, the new method was concerned with the vortex-dominate subsonic flow, transonic rotating flow with minor shocks, and supersonic flow with series of strong shock waves. The numerical results obtained from those computations meet the expectation, in which the higher order scheme provides the more accuracy solutions. In this section, a realistic incompressible flow of a David Taylor P5168 marine propeller
[45] (Chesnakas and Jessup, 1998) is presented for this investigation. Figure 5-52 shows the geometric form and the computational mesh illustration of propeller P5168.
Figure 5-52 Surface and volume grids for propeller 5168 (Sheng and Zhao [96])
122 P5168 is a five bladed, controllable-pitch propeller with a design advance coefficient of ( , where is the reference velocity, is the propeller rotational speed, and is the diameter of the propeller). The diameter of the propeller is
15.856 inches (402.7mm) with the test condition at a rotational speed of 1,450 RPM, or the tip speed of 30.57 m/s. The free stream velocity is of 10.70 m/s. The detailed experimental results were reported by Chesnakas and Jessup [45] for various advance ratios. Four advance ratios with respect to the flow parameters were presented in Table
5.8. The current study chose an advance ratio of 1 for code validations. The nondimensional rotational speed of the propeller was 5.71 based on the diameter of the propeller. The Reynolds number was 4.26 million based on the free stream speed and the diameter of the propeller.
Table 5.8 Flow parameters associated with different advance ratios (Sheng and Zhao
[96])
Advance Free Rotational Tip Tip Mach Reynolds Coefficient Stream Speed Velocity Number Number 6 J=V/nD V(m/s) RPM Vtip (m/s) Mtip Re (10 ) 0.98 7.893 1200 25.302 0.0744 3.14 1.1 10.705 1450 30.574 0.0899 4.26 1.27 11.081 1300 27.411 0.0806 4.41 1.51 11.730 1150 24.248 0.0713 4.64 Only one computational mesh was generated for this geometry which contains approximately 3.8 million unstructured nodes for the whole propeller. It is worth to mention that a slightly increased grid resolution was implemented in the downstream of the propeller for about one blade radius distance. It may be necessary to extend the distance to two blade diameters to better capture the trailing tip vortex. For the current external hydrodynamic viscous flow, the preconditioned algorithm was turned on for
123 computing the low Mach number flow using a reference Mach number of 0.1. The tip trailing vortex characteristics and hydrodynamic performance of the propeller were predicted and solved by various discretization schemes. Table 5.9 shows the CPU time costs and predicted thrust (Kt) and torque (Kq) coefficients by various schemes, and the comparison with the measured data.
Table 5.9 Computational costs and comparisons of thrust and torque coefficients (Sheng
and Zhao [96])
Schemes Kt Kq CPU Time (s)x106
2nd 0.2978 0.0877 0.686
3rd 0.3108 0.0875 0.739
5th 0.3160 0.0870 0.858
Exp 0.313 0.0783
Figure 5-53 to Figure 5-55 show the comparison of predicted tip vortices at a swirl parameter value of 0.1 by the second, third, and fifth order-schemes. It is seen that higher-order scheme has the ability to preserve stronger and longer trailing tip vortex.
Figure 5-56 shows predicted tip vortex trajectories along axial direction using various schemes, which agreed well with the experimental data. However, due to a limited local grid refinement in the downstream region (about one blade radius distance), the trailing tip vortex trajectory beyond cannot be effectively captured by all solutions, although the experiment has detected the trailing tip vortex up to two radius distance from the blade trailing edge. It is clear that future investigations should consider grid refinement studies with longer refined region in the downstream of the propeller.
124 2nd order
Figure 5-53 Predicted surface pressure and trailing tip vortices by the 2nd order
scheme (Sheng and Zhao [96])
3rd order
Figure 5-54 Predicted surface pressure and trailing tip vortices by the 3rd order
scheme (Sheng and Zhao [96])
125 5th order
Figure 5-55 Predicted surface pressure and trailing tip vortices by the 5th order
scheme (Sheng and Zhao [96])
Figure 5-56 Predicted trajectories of trailing tip vortices by various schemes
(Sheng and Zhao [96])
126
(a) 2nd order (b) 3rd order
(c) 5th order (d) Experiment
0.42 0.82 1.22 1.62
Figure 5-57 Predicted axial velocity components at section x/R=0.2386 and
advance ratio J=1.1 (Sheng and Zhao [96])
127 (a) 2nd order (b) 3rd order
(c) 5th order (d) Experiment
-0.8 -0.4 0.07 0.49
Figure 5-58 Predicted radial velocity component at section x/R=0.2386
and advance ratio J=1.1 (Sheng and Zhao [96])
128 (a) 2nd order (b) 3rd order
(c) 5th order (d) Experiment
0.32 1.19 2.05 2.91
Figure 5-59 Predicted tangential velocity component at section x/R=0.2386
and advance ratio J=1.1 (Sheng and Zhao [96])
129 The most important flow feature affecting the marine propeller performance is the near blade tip vortex in the vicinity of the propeller blades, which may cause cavitation inception or increased noises under certain conditions. To investigate the near flow details, predicted axial, radial, and tangential velocity components in rotating frame at a section that is just downstream of the propeller ( ) were plotted in Figure 5-
57 to Figure 5-59.
The blade tip vortices generated by the propeller blade were clearly visible in both computation and experiment, where the vortex core was represented by the blue “spokes” in the velocity field. The computed results, in particular the fifth-order solutions, showed a favorable agreement with the experimental data. There was a flow circulation around the tip vortex, as well as the radial velocity generated by the blade wake. The strong asymmetry of the tip vortex can be observed. It clearly shows that the current high-order spatial method helps capture the fine flow detail in the near blade tip region. This was most obviously seen in the axial and radial velocity components predicted by the fifth- order method, where the dark spots in the plots indicated the strength of the tip vortex that looked very the measured data.
The comparisons of circumferentially averaged velocities at two downstream distances and are shown in Figure 5-60 to Figure 5-65.
Computational results of axial and tangential velocities matched very well with the experimental data. A slight deviation occurred for radial velocities between the prediction and with the measurement. Since these velocities were circumferentially averaged quantities, no significant differences were observed among different spatial order solutions.
130 Figure 5-60 Circumstantially averaged axial velocities at x/R=0.2386 (Sheng and
Zhao [96])
Figure 5-61 Circumstantially averaged axial velocities at x/R= 0.8378 (Sheng and
Zhao [96])
131 Figure 5-62 Circumstantially averaged tangential velocities at x/R=0.2386(Sheng
and Zhao [96])
Figure 5-63 Circumstantially averaged tangential velocities at x/R= 0.8378 (Sheng
and Zhao [96])
132
Figure 5-64 Circumstantially averaged radial velocities at x/R=0.2386 (Sheng
and Zhao [96])
Figure 5-65 Circumstantially averaged radial velocities at x/R= 0.8378 (Sheng and
Zhao [96])
133 The validation of the marine propeller P5168 manifested that the newly developed higher order scheme can be applied to the incompressible rotational flow problems. The computational results do not deviate from the experiment, and the details of the tip velocities are captured by the higher schemes.
5.6 M427 Main Rotor
After all the computations about subsonic, supersonic, vortex dominate, strong discontinuities, and incompressible flows, a realistic, compressible, rotary rotor mouted on the bell helicopter is used for verification and validation of the new methods. The following test is performed based on a realistic Bell aircraft M427 main rotor [46]. M427 main rotor is a four-blade rotor system with a blade radius of 5.625 meter. The blade rotational speed is 395 RPM, and the blade tip Mach number is 0.691. The geometric form and the surface mesh exhibition are shown in Table 5.10. The M427 main rotor geometric and air flow conditions for the hovering and forward flight are summarized in
Table 5.11.
134 Blade Hub Blade Middle Blade Tip Section Figure 5-66 M427 main rotor and surface mesh resolution (Zhao and Sheng [115])
Table 5.10 Geometric and Air Property Definitions for M427 Main Rotor (Zhao and
Sheng [115])
Geometric and Air Conditions Hover Forward Flight
Air speed (m/s) 0 72
Shaft angle or angle of attack (deg) 0 -5.9
Sideslip angle (deg) 0 4
Main rotor radius (m) 5.625 5.625
Main rotor speed (RPM) 395 395
Main rotor collective (+up, deg) 3, 7.46, 10.46, 13.96, 17.46 10.04
Main rotor F/A cyclic (+fwd, deg) 0 7.65
Main rotor LAT cyclic (+right, deg) 0 0.96
Air temperature (k) 288.15 288.15
Rotor tip speed (m/s) 299.24 299.24
advanced ratio=Vinf/Vtip 0 0.306
135 5.6.1 M427 Main Rotor in Hover
M427 rotor in hover has been predicted using the 2nd order scheme at five collective pitch angles in the previous work [121]. In the present study, the rotor with a collective angle of 13.96 degrees is selected and predicted by both the 2nd order scheme and the improved the 3rd scheme. The computational mesh with a single blade model was generated in a 90-degree azimuthal space. Since the flow in hover is axisymmetric, a periodic boundary condition is applied to the axisymmetric surfaces and no free wake mode is needed. The number of grid points for the single blade rotor in hover is about 7.9 million.
2.0
1.0
0.0
Original scheme Figure 5-67 Vorticity distributions on x-y cutting plane in hover for the 2nd
order (Zhao and Sheng [115])
136
2.0
1.0
0.0
Improved scheme
Figure 5-68 Vorticity distributions on x-y cutting plane in hover for the improved
3rd order scheme (Zhao and Sheng [115])
The unsteady simulations of the hovering rotor are started by rotating the rotor blade at a rate of one degree of azimuth per time step, with a local time step during the first 3 blade revolutions. After this initial phase, the simulation is switched to the minimum time step and carried out in a time-accurate unsteady mode. Three Newton iterations with 8 Gauss-Seidel relaxations are used at each time step during the unsteady simulation. A converged periodic solution is obtained after 20 rotor revolutions. The computations are carried out on a Linux cluster using 64 parallel processors.
137 Shown in Figure5-67 and Figure5-68 are the vorticity contours on a vertical cutting plane parallel to the rotating direction and across the center of rotation. For the viewing purpose, the contours generated on the cutting plane by the single blade are rotated and duplicated to give a 180-degree view of the flow field. The vorticity contour values are limited to a range of 0 to 2. Both 2nd –order scheme and the improved scheme have generated a strong downwash flow which is about 20 times of the radius length. The blade tip vortices are also generated by the rotating blades, and are pushed downward.
The strengths of the tip vortex are also displayed in Figure5-67 and Figure5-68, where the 2nd order scheme shows quick dissipation of the trailing vortices lasting only about one fourth way of the downwash path. The strength of the tip vortex predicted by the improved scheme, on the other hand, maintains strong at a consistent level almost to the end of the downwash path. It is interesting to see that two sets of vortex structures appear in the middle of the downwash path with the improved scheme. However, the results obtained by the 2nd order scheme smear out the details.
138 Figure 5-69 Comparison of normalized thrust coefficients (Zhao and Sheng [115])
Figure 5-70 Comparison of normalized torque coefficients (Zhao and Sheng [115])
139 Predicted rotor normalized thrust and torque coefficients at various blade collective angles are shown in Figure 5-69 and Figure 5-70. Results obtained by the
TURNS code [122] are also plotted in the figure for comparison purpose. The predicted thrust and torque coefficients by the improved scheme at 13.96 degrees collective angle are displayed in green color. It seems that no significant changes are observed on the integrated rotor force and moment predicted, as they both are matched well with the flight test data (not shown here). However, the high-order improvement scheme does show less numerical dissipation, and provide better preservation on the rotor trailing vortices as shown in Figure 5-69 and Figure 5-70.
5.6.2 M427 Main Rotor in Forward Flight
The following validation case for the high order improvement scheme is the
M427 main rotor in a level forward flight at 140 knots and an advance ratio of 0.306. The inputs to the blade control and cyclic motions are obtained based on the COPTER [123] analytical trimmed solution listed in Table 5.11. For the forward flight case, a viscous unstructured mesh with a single rotor blade is generated in combination with the free wake model [124]. The total number of grid points here is about 8.5 million. The free wake model rather than full rotor simulation is used to save the computational time. The computation of the forward flight condition is performed in the absolute coordinate system. The boundary conditions are set the similar with the hovering case, except for the axisymmetric surfaces which are replaced with the free wake model to calculate the far field velocities. The free wake model requires the blade normal force distributions from
CFD solutions in order to calculate and update the trailing vortex structure and strength.
Therefore enough number of airstations (64) on the blade surface is necessary to provide
140 a reasonable load representation for the wake model. In the current hybrid computations, the number of trailing vortex filaments is selected as one, with 15 wake revolutions. The free wake structure is updated every 5 degrees azimuthal angle in the blade rotation to minimize the computational costs. Viscous turbulence effect is modeled with Spalart-
Allmaras one equation turbulence model.
Predicted trailing tip vortices are again examined to investigate the impact of the high-order improvement scheme over the original 2nd order scheme. Shown in Figure 5-
71 and 5-72 are the vorticity contours on the cutting plane perpendicular to the blade surface predicted by the 2nd order scheme and the improved scheme. The range of the vorticity value is selected from 0 to 2. An obvious vortex sheet extending to the downstream is observed in both the 2nd –order and the improved results. The vortex is initiated from the one fourth of the blade span location, and then shed to the downstream.
Although the free wake model also has an impact on the vortex sheet predicted, the strength of the trailing vortex is stronger in the improved solution than in the 2nd order solution. The vortices tend to smear out quickly in the 2nd order scheme even if with the use of the free wake model. This is consistent with the conclusions observed in the hovering rotor case. This validation shows that the high-order improvement scheme does increase the strength of predicted vortices, and decrease the dissipation of the numerical solution in vortex-dominated viscous flows. It also shows that even though the high-order improvement is only applied to the inviscid flux, noticeable impacts are observed in viscous flow computations as well.
141 Original scheme
Figure 5-71 Vorticity distributions on x-y cutting plane in forward flight by
original scheme (Zhao and Sheng [115])
Improved scheme
Figure 5-72 Vorticity distributions on x-y cutting plane in forward flight by
improved scheme (Zhao and Sheng [115])
142 Figure 5-73 Comparisons of normalized pitching moment coefficients (Zhao
and Sheng [115])
Figure 5-74 Comparisons of normalized thrust coefficients (Zhao and Sheng [115])
143 Figure 5-75 Comparisons of normalized roll moment coefficient(Zhao and Sheng [115])
Figure 5-76 Comparisons of normalized torque coefficient (Zhao and Sheng [115])
144 Comparisons of normalized rotor thrust (CT), hub pitching moment (CMX), hub roll moment (CMY), and hub torque coefficients (CMZ) computed with the 2nd order scheme and the improved scheme are shown in Figure 5-73 to Figre 5-76. The forces and moments predicted by these two schemes generally capture the similar trends. However, a noticeable difference in predicted hub torques is observed between the the high order improvement scheme and the 2nd order scheme, especially in the third quarter of azimuth
(180-270 degrees). This is the retreating side of the rotor blade in forward flight, where the flow most likely experiences separation or even stall. This means that the impact of the high order improvement scheme is most likely evident in flow regions with adverse conditions. The peak torque value in the 2nd order solution is 25% more than the improved scheme. Because there is no experimental data available for those integrated quantities, a sound judgment cannot be made at this point. Nevertheless, this discovery indicates the improved scheme does affect the flow field more in adverse flow conditions than in the normal and smooth flow regions.
5.7 Chapter Summery
Numerical validations were performed on the improved high order unstructured grid schemes on six different cases. The observed numerical accuracy was verified on an inviscid vortex flow transporting in a free stream. The ability to capture the trailing tip vortex was validated on two vortical flows, a stationary wing NACA0015 and a rotating rotor NACA0012 in hover. To verify the stability and robustness of the high order improvement, a supersonic oblique shock flow over a waisted body was computed using a weighted smart limiter. Finally, the new high order method was verified on two
145 complex and realistic applications about a low Mach number incompressible flow for a marine propeller P5168 and a realistic Bell aircraft M427 main rotor.
All validations performed in the present work showed an improved observed order of accuracy of the new high order unstructured grid schemes. It also showed better ability to capture strong trailing tip vortices in both fixed wing and rotating rotor, and the stability and robustness in discontinuous flows such as shockwaves for the newly developed high order methods. It also demonstrated the applicability of the current method for complex and realistic engineering applications, which is the ultimate goal of the present investigation.
146 Chapter 6
Conclusions and Recommendations
6.1 Conclusions
An attempt to improve the numerical accuracy of an unstructured finite volume
CFD solver was presented in this work. Three key issues for the unstructured grid high order schemes, the large stencil, the preservation of numerical accuracy on mixed element meshes, and the ability to maintain stability for discontinuities, have been addresses in details in the present investigation. To reduce the stencil for unstructured high order schemes, high order polynomials based on both the flow variables and their first and second order gradients were employed to reconstruct high order flow variables for flux calculations. To maintain the high order accuracy of the finite volume discretization schemes, an innovative concept based on Radial basis Function method was adopted as a substitute for the Gaussian quaderdrature rule for high order flux integration. Finally, to maintain the stability of the scheme in discontinuous flows
(shocks), a weighted smart limiter was introduced to provide a stable solution in the shock region while maintaining the high order accuracy in the smooth region. Six
147 different test cases were presented to validate the new attributes of the improved high order unstructured grid scheme. The major conclusions for the present investigation are:
1. The improved high order scheme based on the quadratic polynomial flux reconstruction provides the most stable solutions and better accuracy on all calculations, including two realistic complex flows about the marine propeller P5168 and Bell aircraft
M427 main rotor. However, the method based on quartic polynomial representation sometimes shows instability in complicated flows, in particular for flows with discontinuities.
2. The Radial Basis Function method is a cost effective way to substitute the traditional Gaussian quadrature in high order surface integration on general mixed element unstructured meshes, which showed significant improvement of the overall accuracy for the high order method comparing to the one without using the RBF interpolations. However, the effect of the RBF method on the very fine meshes are not as good as on the coarse meshes due to the oscillatory behavior of the RBF interpolation method.
3. The key for the high order unstructured grid schemes to work in general and complex applications is to develop a smart limiter technique, which will impose more weights on the discontinuous flow region to maintain the stability, but less weights on the smooth flow region without degrading the numerical accuracy. The weighted smart limiter technique introduced in the present work seemed to work well in general cases.
4. The present high order unstructured grid schemes have demonstrated improved observed accuracy on a benchmark flow case, and showed improved numerical accuracy in capturing trailing tip vortices generated by fixed or rotary wing geometry.
148 Furthermore, the new high order methods have been applied to realistic viscous flows with complex geometries, which showed applicability of the current method as a practical
CFD tool for engineering applications.
6.2 Recommendations
Although significant improvement of the numerical accuracy has been demonstrated in the present method, further improvement and development of the current high order unstructured grid methods are being pursued. In particular, improvements for the Radial Basis Function interpolation and smart limiter technique are extremely important to make the method more practical and accurate for general applications.
1. Radial Basis Faction interpolation should be further improved to eliminate the oscillatory behavior especially in the fine meshes. This may be pursued by evaluating different RBF functions or introducing the error correction method.
2. A better limiter technique should be developed to work seamlessly in both low order and high order unstructured grid schemes.
149
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165 Appendix A
Coefficients of the Spalart-Allmaras Turbulence Model
Spalart-Allmaras model is a one equation model which solves a transport equation for a viscosity-like variable . This can be referred to as the Spalart-Allmaras variable.
The coefficients used in the Spalart-Allmaras turbulence model in this work are:
166
where, d is the distance to the closest surface. The constants in the above formulas are
167