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A Dual-Potential Formulation of the Navier-Stokes Equations Steven Gerard Gegg Iowa State University

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A Dual-Potential Formulation of the Navier-Stokes Equations Steven Gerard Gegg Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1989 A dual-potential formulation of the Navier-Stokes equations Steven Gerard Gegg Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Aerospace Engineering Commons, and the Mechanical Engineering Commons Recommended Citation Gegg, Steven Gerard, "A dual-potential formulation of the Navier-Stokes equations " (1989). Retrospective Theses and Dissertations. 9040. https://lib.dr.iastate.edu/rtd/9040 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS The most advanced technology has been used to photo­ graph and reproduce this manuscript from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. 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University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 9003619 A dual potential formulation of the Navier-Stokes equations Gegg, Steven Gerard, Ph.D. Iowa State University, 1989 UMI 300N.ZcebRd. Ann Arbor, MI 48106 A dual potential formulation of the Navier-Stokes equations by Steven Gerard Gegg A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major: Mechanical Engineering Signature was redacted for privacy. In Charge of Major Work Signature was redacted for privacy. FOI the Major Dwartment Signature was redacted for privacy. For the Graduate College Iowa State University Ames, Iowa 1989 ii TABLE OF CONTENTS ABSTRACT xiii ACKNOWLEDGMENTS xv NOMENCLATURE xvii 1. INTRODUCTION 1 1.1 Background 1 1.2 Primitive Variable Methods 3 1.3 Non-primitive Variable Methods 4 1.3.1 Vorticity/velocity approach 6 1.3.2 Dual potential approach 7 1.4 Scope of the Present Study 12 1.5 Organization 15 2. DUAL POTENTIAL FORMULATION 17 2.1 Introduction 17 2.2 Governing Fluid Dynamics Equations 17 2.3 Derivations 21 2.3.1 Velocity decomposition 21 iii 2.3.2 New dependent variables 24 2.4 Derivation of Non-Primitive Variable Equations 26 2.4.1 Two dimensions 26 2.4.2 Three dimensions 32 3. PRELIMINARY ANALYSIS 35 3.1 Introduction 35 3.2 Boundary Conditions 35 3.2.1 Scalar potential boundary conditions 36 3.2.2 Vector potential boundary conditions 38 3.2.3 Vorticity boundary conditions 38 3.2.4 Dilatation boundary conditions 46 3.2.5 Density boundary conditions 48 3.2.6 Temperature boundary conditions 49 3.2.7 Velocity boundary conditions 49 3.3 Cartesian Grid Clustering 50 3.3.1 Clustering near boundaries 53 3.3.2 Clustering near an interior point 54 3.3.3 Arbitrary user defined clustering 56 3.4 Poisson Equation Solvers 57 3.4.1 Vectorized point Gauss-Seidel with SOR 58 3.4.2 AF scheme 60 3.4.3 ADI scheme 62 3.5 Poisson Solver Comparisons 62 iv 3.5.1 Scalar potential test problem 64 3.5.2 Vector potential test problem 70 3.5.3 Summary of Poisson solver experience 71 3.6 Time Marching ADI Solver for the Transport Equations 75 4. NUMERICAL RESULTS 79 4.1 Introduction 79 4.2 Solution Strategy 79 4.3 Two-dimensional Cases 82 4.3.1 Incompressible flow 83 4.3.2 Compressible flow 121 4.4 Three-dimensional Cases 172 4.4.1 Incompressible flow 172 4.4.2 Compressible flow 180 5. DISCUSSION AND CONCLUSIONS 181 6. RECOMMENDATIONS FOR FUTURE WORK 184 7. REFERENCES 187 8. APPENDIX A: THREE-DIMENSIONAL VORTICITY AND DILATATION TRANSPORT EQUATIONS 194 9. APPENDIX B: ALTERNATE NON-PRIMITIVE VARIABLE EQUATIONS 200 V LIST OF TABLES Table 3.1: Poisson solver comparison on a 21 x 21 uniform grid 66 Table 3.2: Poisson solver comparison on a 41 x 41 uniform grid 67 Table 3.3: Poisson solver comparison on the scalar potential test problem 67 Table 3.4: Poisson solver comparison on the vector potential test problem 70 Table 4.1: Speed-up affect of limiting the vorticity solution domain . 115 Table 4.2: Fully-developed centerline velocity and skin friction 174 vi LIST OF FIGURES Figure 3.1: 3-D Cartesian coordinate system with example boundary con­ ditions for the potentials on a solid impermeable surface in the x-z plane 39 Figure 3.2: An initial uniform velocity profile along a viscous boundary for an impulsive start 40 Figure 3.3: 2-D channel boundary conditions 51 Figure 3.4: Typical 2-D channel grid, (a) Physical plane using cr = 1.05, a = 0.5,(3 = 1.2; (b) Computational plane 55 Figure 3.5: Typical 2-D grid for an isolated airfoil a,t y = h, (a) Physical plane using r = 5.0, a = 0, « = 0,/3 = 1.5; (b) Computational plane 56 Figure 3.6: Scalar potential test problem 64 Figure 3.7: Convergence behavior of the Poisson solvers on the scalar po­ tential test problem 68 Figure 3.8: Convergence history for the scalar potential test problem . 69 Figure 3.9: Vector potential test problem 71 Figure 3.10: Convergence behavior of the Poisson solvers on the vector po­ tential test problem 72 vii Figure 3.11: Convergence history for the vector potential test problem . 73 Figure 4.1: Boundary conditions for laminar incompressible irrotational flow over a bump 83 Figure 4.2: Parabolic arc and sine wave bumps 87 Figure 4.3: Pressure coefficient for Moo = 0.6 flow over a parabolic arc airfoil 88 Figure 4.4: Pressure coefficient for Moo = 0.6 flow over a sine wave arc airfoil 89 Figure 4.5: Centerline velocity development for a 2-D channel inlet ... 95 Figure 4.6: Skin-friction development along a 2-D channel 96 Figure 4.7: Basic wall heat transfer boundary conditions 97 Figure 4.8: Local Nusselt number for flow between parallel plates at con­ stant temperature. Pe = 105. Asymptotic Nu = 7.541 .... 100 Figure 4.9: Local Nusselt number for flow between parallel plates at con­ stant temperature. Pe = 210. Asymptotic Nu = 7.541 .... 101 Figure 4.10: Local Nusselt number for flow between parallel plates at con­ stant temperature. Dual potential code with boundary-layer assumptions 104 Figure 4.11: Proflles of the temperature parameter at various x* for the constant wall temperature case 105 Figure 4.12: Local Nusselt number for flow between parallel plates with constant wall heat flux. Pe = 210. Asymptotic Nu = 8.235 . 106 viii Figure 4.13: Profiles of the temperature parameter at various x* for the constant wall heat flux case 107 Figure 4.14: Local Nusselt number for flow between parallel plates with mixed wall conditions. Pe = 210. Asymptotic Nu = 5.385 for one wall insulated and the other at constant heat flux. Asymptotic Nu = 4.861 for one wall insulated and the other at constant temperature 108 Figure 4.15: Profiles of the temperature parameter at various z* for the case of one wall insulated and the other at constant heat flux 109 Figure 4.16: Profiles of the temperature parameter at various x* for the case of one wall insulated and the other at constant temperature110 Figure 4.17: Boundary conditions for laminar incompressible flow over a flat plate 112 Figure 4.18: Laminar incompressible boundary-layer velocity profile at Re^ = 312,500 117 Figure 4.19: Skin friction along a flat plate 118 Figure 4.20: Velocity vectors for laminar flow over a flat plate with an ir­ rotational inlet condition 119 Figure 4.21: Velocity vectors for laminar flow over a flat plate with a rota­ tional inlet condition 120 Figure 4.22: Boundary conditions for laminar compressible irrotational flow over a bump 122 Figure 4.23: Pressure coefficient for Moo = 0.6 flow over a sine wave arc airfoil with maximum thickness = 4% of chord Figure 4.24: Dilatation contours for Moo — 0.6 flow over a sine wave arc airfoil Figure 4.25: Density profiles from inlet to exit for the sine wave arc airfoil Figure 4.26: Dilatation profiles from inlet to exit for the sine wave arc airfc Figure 4.27: V velocity profiles from inlet to exit for the sine wave arc airfc Figure 4.28: Comparison of the dilatation profiles for the stream function and Euler mode of computation by the DP code.
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