NEWTON SOLUTION OF STEADY TWO-DIMENSIONAL TRANSONIC FLOW by Michael B. Giles GTL Report No. 186 October 1985 NEWTON SOLUTION OF STEADY TWO-DIMENSIONAL TRANSONIC FLOW by Michael B. Giles GTL Report No, 186 October 1985 Research sponsored by the Air Force Office of Scientific Research (AFSC), under Contract F49620-78-C-0084, with technical monitor Dr. James D. Wilson. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon. Abstract A new method is developed for the solution of the steady, two- dimensional Euler equations for transonic flows. The discrete steady- state equations are derived in conservative finite-volume form on an intrinsic streamline grid, and are solved using Newton's method. Direct solution of the linear system of Newton equations is shown to be more efficient than iterative solution. Test cases include duct, cascade, and isolated airfoil flows, and demonstrate the speed and robustness of the method. The accuracy of the solutions is verified by comparison against values obtained analytically, experimentally and by other numerical methods. 2 Acknowledgements Many friends and colleagues have helped in this research effort. Firstly, I wish to thank Mark Drela for sharing two years of hard work and fun, with all of the long hours, frustrations and successes, brain- storming and tedious programming. Many thanks also to my thesis super- visor Professor Tilt Thompkins for sharing our enthusiasm and letting us have fun; he knew when to let us charge ahead boldly (or blindly?), and when to be a critical observer, and I greatly appreciate his help during the writing of this thesis. I also wish to thank the other members of my committee, Professors Jack Kerrebrock and Earll Murman, for their helpful discussions, and the other students in the group for their technical comments and for their company during the late nights and the long weekends. Finally, I wish to thank my family who has supported me throughout my studies, and John Dannenhoffer who has helped me so much as a friend and a colleague, with his encouragement, advice and thought- ful criticisms. 3 Table of Contents Page Abstract 2 Acknowledgements 3 Table of Contents 4 List of Figures 6 List of Tables 10 List of Symbols 11 1. Introduction 12 2. Steady State Equations 20 2.1 Euler Equations 20 2.2 Duct Boundary Conditions 28 2.3 Auxiliary Pressure Relation 30 2.4 Possible Solution Methods 34 3. Artificial Compressibility 36 3.1 Introductory Discussion 36 3.2 One-Dimensional Analysis 40 3.3 Second Order Corrections 49 4. Newton Linearization 55 4.1 Euler Equations 57 4.2 Duct Boundary Equations 66 4.3 Artificial Compressibility 67 4.4 Initialization of Solution 69 4.5 Updating of Solution 71 5. Direct Method of Solving Newton Equations 74 5.1 Assembling the Equations 74 4 5.2 Block Tridiagonal Solution Method 79 6. Modified Direct Method for Choked Flow 80 6.1 Boundary Conditions 80 6.2 Solution Procedure 83 7. Iterative Method for Subsonic Flow 86 7.1 Pre-conditioning 88 7.2 Solution Procedure 92 8. Global Variables and Equations 94 8.1 Concept and Numerical Procedure 94 8.2 Cascade Boundary Conditions 98 9. Results 101 9.1 Duct with Sin2(l(x) Bump 104 9.2 Duct with Elliptic Bump 108 9.3 Incompressible Gostelow Cascade 116 9.4 T7 Turbine Cascade 126 9.5 Garabedian Compressor Cascade 134 9.6 NACA 0012 Airfoil 147 9.7 Two-Dimensional Laval Nozzle 154 10. Conclusions 159 10.1 Discretization of Euler Equations 159 10.2 Newton Solution Method 161 10.3 Versatility of Approach 164 References 167 Appendix: Program Listing 170 5 List of Figures Page Figure 2.1: Location of grid nodes and cell geometry. 22 Figure 2.2: Definition of vectors A , A , B~, B+. 22 1 2 Figure 2.3: Location of flow variables. 23 Figure 2.4: Indexing for duct geometry. 27 Figure 2.5: Unconstrained "sawtooth" mode. 32 Figure 2.6: Cross-sectional areas for pressure correction. 32 Figure 3.1: Location of grid nodes and variables for Potential 37 equation. Figure 3.2: Mach number distributions illustrating analytic 41 "boundary layer" behavior. Figure 3.3: Results for 1-D streamtube, with ucon=1.5 46 Figure 3.4: Results for 1-D streamtube, with 4con=1.0 47 Figure 3.5: Results for 1-D streamtube, with ucon=0 .5 48 Figure 3.6: Results for 1-D streamtube, with Uc =1.5, and second 52 order density corrections. Figure 3.7: Results for 1-D streamtube, with uco =1.0, and second 53 order density corrections. Figure 3.8: Results for 1-D streamtube, with Uc =0.5, and second 54 order density corrections. Figure 4.1: Pressures on shared streamline face. 60 Figure 5.1: Global indexing system for unknown variables. 73 Figure 5.2: Indexing system for a particular pair of cells. 73 Figure 5.3: Structure of matrices Z., B., A., C . 76 Figure 5.4: Structure of matrices A C 1,. , 78 16 Figure 5.5: Structure of matrices B, A . 78 Figure 6.1: Shift of rows in choked-flow equations. 84 Figure 7.1: Regular sheared grid for perturbation analysis. 89 6 Figure 9.la: Duct and grid geometry for test case 1: 107 duct with sin2(i(x) bump. Figure 9. 1b: Mach number contours with increments of 0.02. 107 Figure 9. 1c: Stagnation density contours with increments 107 of 0.00004. Figure 9.2a: Duct and grid geometry for test case 2: 109 duct with ellptic bump. Figure 9. 2b: Mach number contours with increments of 0.05. 109 Figure 9.2c: Stagnation density contours with increments 109 of 0.001. Figure 9.3: Close-up of grid near stagnation point on 110 elliptic bump. Figure 9.4: Streamlines in a stagnation point flow. 110 Figure 9.5a: Duct and grid geometry for test case 2: 113 duct with elliptic bump using modified grid. Figure 9.5b: Mach number contours with increments of 0.05. 113 Figure 9.5c: Stagnation density contours with increments 113 of 0.001. Figure 9.6: Close-up of grid near. stagnation point on elliptic 114 bump with modified grid. Figure 9.7a: Airfoil and grid geometry for Gostelow cascade. 117 Figure 9.7b: Mach number contours for Gostelow cascade 118 with increments of 0.005. Figure 9.7c: Stagnation density contours for Gostelow cascade 119 with increments of 0.00001. Figure 9.8: Comparison of calculated and theoretical surface 120 pressure coefficients for Gostelow casade. Figure 9.9a: Close-up of the grid near the leading edge 121 stagnation point of the Gostelow cascade. Figure 9.9b: Close-up of the grid near the trailing edge 121 of the Gostelow cascade. 7 Figure 9.10a: Airfoil and grid geometry for T7 turbine cascade. 127 Figure 9.10b: Mach number contours for T7 turbine cascade 128 with increments of 0.05. Figure 9.10c: Stagnation density contours for T7 turbine cascade 129 with increments of 0.001. Figure 9.11: Close-up of the grid near the leading edge 130 stagnation point of the T7 turbine cascade. Figure 9.12: Comparison of calculated and experimental surface 131 Mach numbers for T7 turbine cascade. Figure 9.13: Variation of lift with inflow angle for T7 133 turbine cascade. Figure 9.14a: Airfoil and grid geometry for Garabedian cascade, 136 using first order artificial compressibility. Figure 9.14b: Mach number contours for Garabedian cascade, 137 using first order artificial compressibility with increments of 0.1. Figure 9.14c: Stagnation density contours for Garabedian cascade, 138 using first order artificial compressibility with increments of 0.005. Figure 9.15: Comparison of calculated and hodograph surface Mach 139 numbers for Garabedian cascade, using first order artificial compressibility. Figure 9.16a: Airfoil and grid geometry for Garabedian cascade, 140 using second order artificial compressibility. Figure 9.16b: Mach number contours for Garabedian cascade, 141 using second order artificial compressibility with increments of 0.1. Figure 9.16c: Stagnation density contours for Garabedian cascade, 142 using second order artificial compressibility with increments of 0.005. Figure 9.17: Comparison of calculated and hodograph surface Mach 143 numbers for Garabedian cascade, using second order artificial compressibility. Figure 9.18: Close-up of the grid near the leading edge stagnation 144 point of the Garabedian cascade. 8 Figure 9.19a: Airfoil and grid geometry for NACA 0012 airfoil. 149 Figure 9.19b: Mach number contours for NACA 0012 airfoil, with 150 increments of 0.1. Figure 9.19c: Stagnation density contours for NACA 0012 airfoil, 151 with increments of 0.005. Figure 9.20: Surface pressure coefficients for NACA 0012 airfoil. 152 Figure 9.21a: Duct and grid geometry for 2-D Laval nozzle flow. 155 Figure 9.2 1b: Mach number contours for 2-D Laval nozzle flow, 155 with increments of 0.1. Figure 9.21c: Stagnation density contours for 2-D Laval nozzle 155 flow, with increments of 0.01. Figure 9.22a: Stagnation density changes on center streamtubes 156 of 2-D Laval nozzle flow. Figure 9.22b: Mach number distribution on center streamtubes 157 of 2-D Laval nozzle flow. 9