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

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