Quadrimestre de primavera 2015. Defensa Abril 2015. Enginyeria Aeronàutica Miguel Alejandro ...

162 Pages·2015·13.03 MB·English

Convocat(cid:242)ria de lliurament del PFC
Quadrimestre de primavera 2015. Defensa Abril 2015.
Titulaci(cid:243)
Enginyeria Aeron(cid:224)utica
Alumne
Miguel Alejandro del Moral Cejudo
T(cid:237)tol PFC
StudyofNumericalMethodsfortheSolutionofCompressibleExternalFlows
Director del PFC
Roberto Maurice Flores Le Roux
Contingut d’aquest volum -MEM(cid:210)RIA-

Study of Numerical Methods for the Solution of
Compressible External Flows
A Final Year Project submitted to Universitat PolitŁcnica de Catalunya for the title of
Aeronautical Engineer at the Escola TŁcnica Superior d’Enginyeries Industrial i
Aeron(cid:224)utica de Terrassa
Miguel Alejandro del Moral Cejudo
March, 2015
Director: Roberto Maurice Flores Le Roux

Acknowledgments
The development of this study would not have been possible without the support of
some people. These lines are to express them my gratitude.
First of all I want to thank the project director, Roberto Flores, whose knowledge and
support during all these months help me out in the crossroads of this study.
Special mention is required to my colleagues and friends Daniel SÆnchez and Joel Tem-
prano. Thanks, not only for helping me in the validation of results but, even more
important, for sharing this wonderful and hazardous university adventure.
The long road that bring me here has been full of di(cid:30)culties which would have not
been overcome without the unconditional support of my family. Alberto, Mercedes
and Timoteo, thanks for standing there even when I could not. My future family also
deserves mention. Ms. Zapater and Mr. Mart(cid:237)nez, thanks for your a(cid:27)ection and great
generosity.
Finally, but not less important, I want to thank my lawyer, Lourdes. Thanks for your
inspiring bravery, for your patience, for bringing light in the dark moments and for
deciding to go out that summer night.

Contents
Abstract xi
Nomenclature xiii
1 Aim 1
2 Scope 3
2.1 In Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Out of Project Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Justi(cid:28)cation 5
4 Introduction 9
5 Environmental Impact 11
6 Budget 13
7 State of the art 17
7.1 History and background . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.2 Methods for Computational Fluid Dynamics . . . . . . . . . . . . . . . 18
7.2.1 LinearizedPotentialCompressibleModelandPrandtl-Glauertanal-
ogy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
7.2.2 Numerical approaches . . . . . . . . . . . . . . . . . . . . . . . . 20
7.2.2.1 Finite di(cid:27)erences method . . . . . . . . . . . . . . . . . 20
7.2.2.2 Finite volumes method . . . . . . . . . . . . . . . . . . 21
7.2.2.3 Finite elements method . . . . . . . . . . . . . . . . . . 21
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7.2.3 Solver algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7.2.3.1 Iterative methods . . . . . . . . . . . . . . . . . . . . . 22
7.2.3.2 Time-marching methods . . . . . . . . . . . . . . . . . . 22
7.2.3.3 Other methods. . . . . . . . . . . . . . . . . . . . . . . 23
7.2.4 Contour methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
7.3 Note on commercial software . . . . . . . . . . . . . . . . . . . . . . . . 25
8 Fundamentals 27
8.1 Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8.1.1 Boundary conditions for airfoils . . . . . . . . . . . . . . . . . . . 29
8.2 Fundamentals of (cid:28)nite-volume method . . . . . . . . . . . . . . . . . . . 29
8.2.1 Second order centered scheme . . . . . . . . . . . . . . . . . . . . 31
8.2.2 Centered scheme with average value in cells vertex . . . . . . . . 33
8.3 Fundamentals of explicit time marching method . . . . . . . . . . . . . . 35
8.3.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
8.3.2 Runge-Kutta method . . . . . . . . . . . . . . . . . . . . . . . . 37
8.4 Adimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
8.5 Numerical special recipes . . . . . . . . . . . . . . . . . . . . . . . . . . 40
8.5.1 Recipe 1: Initial conditions close to the stationary solution. . . . 40
8.5.2 Recipe 2: Internal boundary conditions . . . . . . . . . . . . . . 41
8.5.3 Recipe 3: External boundary conditions . . . . . . . . . . . . . . 42
8.5.4 Recipe 4: Avoiding the odd-even decoupling problem . . . . . . . 43
8.5.4.1 Jameson’s arti(cid:28)cial dissipation model . . . . . . . . . . 44
8.5.5 Recipe 5: Local time step . . . . . . . . . . . . . . . . . . . . . . 46
8.6 Fundamentals of meshing . . . . . . . . . . . . . . . . . . . . . . . . . . 47
8.6.1 Note on unstructured grids . . . . . . . . . . . . . . . . . . . . . 48
8.6.2 Structured grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
8.6.2.1 The C-type grid . . . . . . . . . . . . . . . . . . . . . . 49
9 Software description 53
9.1 Programming language election . . . . . . . . . . . . . . . . . . . . . . . 53
9.2 Software construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Miguel del Moral Cejudo ii
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9.2.1 Software structure . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.2.2 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
9.2.3 Meshing subroutine. . . . . . . . . . . . . . . . . . . . . . . . . . 57
9.2.3.1 Grid design parameters: airfoil points zp(2,:) . . . . . . 59
9.2.3.2 Grid design parameters: nodes distance delta_wake(i) . 60
9.2.3.3 Grid design parameters: Separation between levels k(j) 62
9.2.3.4 Grid design parameters: inclination angle theta_vector(i) 62
9.2.3.5 Grid creation . . . . . . . . . . . . . . . . . . . . . . . . 64
9.2.3.6 Finite-volume grid . . . . . . . . . . . . . . . . . . . . . 65
9.2.3.7 Function for area calculation . . . . . . . . . . . . . . . 66
9.2.4 Solver subroutine . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
9.2.4.1 Resolution algorithm. . . . . . . . . . . . . . . . . . . . 69
9.2.4.2 Final computations . . . . . . . . . . . . . . . . . . . . 70
9.2.4.3 Function for integral calculation . . . . . . . . . . . . . 71
9.2.5 Printing results subroutine. . . . . . . . . . . . . . . . . . . . . . 73
9.3 Input (cid:28)les . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.4 Output (cid:28)les . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.5 Notes about software limitations . . . . . . . . . . . . . . . . . . . . . . 76
10 Results 81
10.1 Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
10.1.1 Cylinder with second order centered scheme . . . . . . . . . . . 84
10.1.2 Cylinder with average values in cell vertex . . . . . . . . . . . . . 85
10.1.2.1 Cylinder with average values in cell vertex with rougher
grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
10.2 Drop airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
10.2.1 Drop airfoil with second order centered scheme . . . . . . . . . . 97
10.2.2 Drop airfoil with average values in cell vertex . . . . . . . . . . . 99
10.3 Generic airfoil: NACA 0012 . . . . . . . . . . . . . . . . . . . . . . . . . 105
10.3.1 Low velocity results . . . . . . . . . . . . . . . . . . . . . . . . . 106
10.3.1.1 Results with (cid:28)ner mesh . . . . . . . . . . . . . . . . . . 107
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10.3.2 High velocity results . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.3.3 Medium velocity results . . . . . . . . . . . . . . . . . . . . . . . 113
11 Conclusions 123
12 Future work 127
Appendix A 129
Appendix B 131
Appendix C 133
Appendix D 135
Appendix E 137
Appendix F 139
Bibliography 141
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