臺灣博碩士論文加值系統

我們採用預處理純量無矩陣隱式法和AUSMD(R)來解多相流的問題，因為多尺度的聲速所造成的數值穩定性問題，我們是透過加入預處理的尤拉系統所得到的特徵值，來改善收斂性的問題。本文所介紹的預處理隱式法所模擬出來的結果與完整的隱式法和Runge-Kutta 是非常接近的。由模擬解果得知純量無矩陣隱式法式能夠提高計算效率，並且在許多例子中都得到很好的驗證。

Here, a scalar matrix-free implicit type preconditioning hybrid AUSMD(R) solver for multi-phase flows is developed. The numerical stability problem caused by the multi-scale speed of sound due to uncertain dissipation terms in the current schemes which can be resolved by rescaling the eigenvalues of the Euler type system equations to enhance computational convergence. This paper presents implicit pre-conditioning approaches which indicate similarly accurate results obtained with the fully implicit and Runge-Kutta explicit schemes. The current used homogeneous two-phase mixture model with the assumption of kinematics and thermodynamics equilibriums. The thermodynamics behaviors of liquid phase, vapor phase and their phase transitional process are described by a temperature dependent hybrid equation of state which includes a mass-fraction averaged formula of water-vapor saturation process. The current work shows that the scalar matrix free implicit schemes are capable of improving the computational efficiency over its explicit counterpart. Several benchmark tests are used for numerical validations.

Brief Contents

1. Introduction 1
1.1 General Background Information 1
1.2 Literature Review 3
1.3 Motivation and Purpose 11

2. Numerical Methods 12
2.1 Governing Equations 12
2.2 Numerical methodology 13
2.3 Equation of state 16
2.4 Cavitation model 17
2.5 Time discretization 19
2.6 Numerical flux 25
2.7 Turbulence model (RNG ) 28

3. Results and Discussions 29
3.1 Steady-state 1D cavitating nozzle flow 30
3.2 Two-dimensional cavitating nozzle flow 39
3.3 Two-dimensional Blunt Body Flow 43
4. Concluding Remarks 51

5. References 53

Appendix A: Characteristic Boundary Conditions 60

Appendix B: Paper submission 63

List of Figure
Figure 3.1 One-dimensional nozzle boundary conditions and mesh........30
Figure 3.2 1-D caviated nozzle flow: (A) pressure distributions(B) density distributions (C) total velocity distributions (D) convergence history on different grids.......................................................................................33
Figure 3.3 1D caviated nozzle flow: (A) pressure distributions (B) density distributions (C) Vapor volume fractions (D) convergence history on different time discretization......................................................................36
Figure 3.4 The pressure distributions of 1D caviated nozzle flow...........37
Figure 3.5 The density distributions of 1D caviated nozzle flow.............38
Figure 3.6 convergence history on different Numerical flux method ......38
Figure 3.7 Two-dimensional nozzle boundary conditions and mesh .....39
Figure 3.8 The contours of vapor volume fraction at different cavitation number ......................................................................................................41
Figure 3.9 the pressure distributions at different cavitation number........42
Figure 3.10 The density distributions at different cavitation number ......42
Figure 3.11 Two-dimensional blunt body mesh .....................................43
Figure 3.12 Surface pressure coefficient distributions .............................47
Figure 3.13 The volume of fraction..........................................................48
Figure 3.14 The streamlines over the 2D blunt body ...............................50

[1]C. L. Merkle “Preconditioning methods for viscous flow calculations.” Computational luid Dynamics Review 1995, pages 419–436J.
[2]J. M. Weiss and W. A. Smith. “Preconditioning applied to variable and constant density flows”, AIAA Journal, Vol. 33, No. 11 (1995), pp. 2050-2057.
[3]J.R. Edwards, R.K. Franklin and M.S. Liou, “Low-Diffusion Flux-Splitting Methods for Real Fluid Flows with Phase Transitions”, AIAA Journal, pp1624–1633, 38, 2000
[4]M. D. Neaves and J.R. Edwareds, “Time-Accurate All-Speed Multiphase Calculations Using A Low-Diffusion Flux Splitting Scheme”, J. Fluids Eng. 128(2), 284-296, 2005
[5]Handbook of ‘IAPWS-IF97, Bernhard Spang, Hamburg, Germany,1997
[6]B. R. Shin, Y. Iwata, T. Ikohagi, “Numerical simulation of unsteady cavitating flows using a homogenous equilibrium model”, Computational Mechanics, Volume 30, Issue 5, April, 2003,cpp 388-395
[7]Yang-Yao Niu, “A Simple and Robust Advection Upwind Flux Splitting to Simulate Transient Cavitated Water-Vapor Flows”, Numerical Heat Transfer, Part B, Volume, Issue 7 , January 2007 , pages 679 – 696.
[8]C. H. Chang and M.S. Liou, “A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM+-up scheme”, Journal of Computational Physics, Vol. 225, July 2007, pp 840–873.
[9]M. S. Liou and Y. Wada “An accurate and robust flux splitting scheme for shock and contact discontinuities”, SIAM Journal on Scientific Computing 18 (3), 633-657,1995
[10]Rouse H. and McNown, J. S., “Cavitation and Pressure Distribution: Head Forms at Zero Angle of Yaw,” State Univ. of Iowa Engineering Bulletin 32, Ames, IA, 1948.
[11]Cole, R. H., “Underwater Explosions” , Princeton Univ.Press , Princeton , NJ,1948
[12]Kubota A , Kato H. and Yamaguchi H., “A New Modelling of Cavitating Flows: A Numerical Study of Unsteady Cavitation on A Hydrofoil Section”, Journal of Fluid Mechanics, Vol.240, pp59-96,1992
[13]Sankaran V. Jules W.L. , R.F. Kunz and Charles L. Merkle “Preconditioning Algorithms for the Computation of Multi-Phase Mixture Flows ” AIAA 2001-0279
[14]H. Kazem and F.H. Kasra , “Assessment of A Central Difference Finite Volume Scheme for Modeling of Cavitating Flows Using Preconditioned Multiphase Euler Equations” Journal of Hydrodynamics, Ser. B Volume 23, Issue 3, June 2011, Pages 302–313
[15]Y. Rong and Y. Wei , “A flux vector splitting scheme for low Mach number flows in preconditioning method ” Applied Mathematics and Computation 242 (2014) 296–308
[16]J.C. Boniface “ Rescaling of the Roe Scheme in Low Mach-Number Flow Regions ” Journal of Computational Physics, April 1, 2016, hal-01286182
[17]H. Guillard and A. Murone “On the Behaviour of Upwind Schemes in the Low Mach Number Limit” II. Godunov Type Schemes INRIA report, No 4189, May 2001
[18]X.S. Li, C.W. Gu “ Mechanism of Roe-type Schemes for All-Speed Flows and its Application” Computers & Fluids, 86 (2013), 56-70
[19]H. Paillere, C. Corre , J.R. Garcıa Cascales “On the extension of the AUSM+ scheme to compressible two-fluid models ” Computers & Fluids 32 (2003) 891–916
[20]Clint N. D. , Hector K. , Mary F. W. and Carol S. W. “A parallel, implicit, cell-centered method for two-phase flow with a preconditioned Newton–Krylov solver ” Computational Geosciences (1997) 215–249
[21]K. N. Volkov and A. G. Karpenko “Preconditioning of Gas Dynamics Equations in Compressible GasFlow Computations at Low Mach Numbers” Computational Mathematics and Mathematical Physics, 2015, Vol. 55, No. 6, pp. 1051–1067
[22]Toumi, A. Kumbaro and H. Paillère “Approximate Riemann Solvers and Flux Vector Splitting Schemes for Two-Phase Flow” VKI LS 1999-03 CFD, 8-12 March 1999.
[23]Richard S., Pierre B. and Olivier L. M. “A general formulation for cavitating, boiling and evaporating flows.”Computers and Fluids 128 (2016) 53–64.
[24]A. J. Chorin. A numerical method for solving incompressible viscous flow problems. Journal of Computational Physics, 2:12–26, 1967.
[25]E. Turkel. Preconditioned methods for solving the incompressible and low speed compressible equations. Journal of computational physics, 72:277–298, 1987.
[26]G. Volpe. Performance of compressible flow codes at low mach numbers. AIAAJournal, 31:49–56, 1993.
[27]Chen, Y., and Heister, S.D., “Two-Phase Modeling of Cavitated Flows,” Computers & Fluids, Vol.24, (1995), pp.799-809
[28]Thakur S, Wright J, Shy W. STREAM “ A computational fluid dynamics and heat transfer Navier–Stokes solver: theory and applications ” Streamline Numerics, Inc. and Computational Thermo-Fluids Laboratory, Department of Mechanical and Aerospace Engineering Technical Report, Gainesville, FL, 2002.
[29]Klaus A. H. and Steve T. C.(2000) “ Computational fluid dynamics (4th) ” Engineering Education System , USA
[30]M.S. Liou “A sequel to AUSM, Part II: AUSM+-up for all speeds” Journal of Computational Physics 214 (2006) 137–170
[31]Kunz R.F., Boger D.A., Stinebring D.R., Chyczewski T.S., Lindau J.W., Gibeling H.J. “A preconditioned Navier–Stokes method for two-phase flows with application to cavitation.” Computers and Fluids, 2000 ,849–875.
[32]Y. Utturkar., J. Wu , G Wang and W. Shyy “ Recent progress in modeling of cryogenic cavitation for liquid rocket propulsion Yogen” Progress in Aerospace Sciences 41 (2005) 558–608
[33]L. Rayleigh and Strutt, J.W. “On the pressure developed in a liquid during the collapse of a spherical cavity”, Phil. Mag., 34, pp. 94-98, 1917.
[34]Plesset, M.S. and A. Prosperetti “Bubble dynamics and cavitation” Annu. Rev. Fluid Mech. Vol. 9, pp. 145-185, 1977.
[35]Hosangadi, A. and Ahuja, V., Arunajatesan S. “Simulation of Cavitation Flows Using Hybrid Unstructured Meshes” Journal of Fluids Engineering, Vol. 123, pp. 331, 2001.
[36]Hosangadi, A., Ahuja, V. and Ungewitter, R.J. “Simulation of Cavitation Cryogenic Inducers” AIAA 2004-4023.
[37]Guillard H, Viozat C. “On the behaviour of upwind schemes in the low Mach number limit”, Comput Fluids 1999;28:63–86.
[38]C.Viozat “Implicit Upwind Schemes for Low Mach Number Compressible Flows” INRIA Report No 3084, January 1997
[39]Can F. D. , Zafer B., Steffen J. S. and Günter H. S. “ Unsteady Bubbly Cavitating Nozzle Flows ” 7th International Symposium on Cavitation, August 17-22, 2009
[40]Z.M. Hu, B.C. Khoo , J.G. Zheng “The simulation of unsteady cavitating flows with external perturbations” Computers & Fluids 77 (2013) 112–124
[41]W. Briley, L. Taylor, D. Whitfield, “High-resolution viscous flow simulations at arbitrary Mach number” Journal of Computational Physics 184 (1) (2003) 79 – 105.
[42]Margarida M. G., Mariano V., Guillaume H. “Local preconditioning and variational multiscale stabilization for Euler compressible steady flow” Comput. Methods Appl. Mech. Engrg. ,2015
[43]P.J. Zwart , A.G. Gerber and T. Belamri “A Two-Phase Flow Model for Predicting Cavitation Dynamics” ICMF 2004 International Conference on Multiphase Flow , Yokohama, Japan, May 30-June 3, 2004 , Paper No.152
[44]Merkle C.L., Feng J., Buelow P.E.O. “Computational modeling of the dynamics of sheet cavitation” In: Third International Symposium on Cavitation, Grenoble, France. 1998.
[45]Roe P.L. “ Approximate Riemann Solvers, Parameter Vectors, and Difference Scheme ”, Journal of Computational Phusics,Vol.43, (1981),pp.357-372,