脈衝爆震發動機的基礎是由於爆震波在爆震管中點燃後向下游傳播，通過噴嘴後會產生超音速的流體並且提供推力。為了清楚的模擬出爆震管中物理現象，在本研究中，我們提出了HMSTH type AUSMD方法來解決爆震波中的問題，同時這個方法是也可以擴展到二維問題的。而為了驗證上述數值方法捕獲震波，擴散波和爆炸波的能力，我們比較了另外兩種數值方法，分別是MUSCL type AUSMD和THINC-EM type AUSMD方法。最後，我們觀察並且比較哪種方法可以在相同條件下得到爆震管中最明顯的物理現象，並提出HMSTH type AUSMD方法確認有向未來發展的可能性。

A pulse detonation engine (PDE) is replied on the detonation wave propagates downstream after being ignited in the detonation tube, then producing supersonic wave through the nozzle to achieve the thrust. In order to simulate the physical phenomenon in the detonation tube, in this study, we propose the Hybrid MUSCL with THINEM (HMSTH) type AUSMD flux scheme to solve the stiffened Euler equations to achieve accurate simulation of the detonation waves, shock waves, and the expansion fans. To verify the proposed numerical methods, one and two-dimensional shock tube and the detonation tube and nozzles are chosen as benchmark test cases. The numerical results show that the proposed HMSTH type AUSMD scheme has great potential for further complicated detonation and PDE flow problems.

Table of Contents
Nomenclature ......................................................................................... VI
List of Figure........................................................................................ VIII
1. Introduction ...................................................................................... 1
1.1. Background......................................................................................... 1
1.2. Review of Numerical Algorithm Literature ......................................... 5
1.3. Review of Pulse Detonation Engine .................................................... 7
1.4. Detonation Physics ........................................................................... 15
1.4.1 CJ Theory ................................................................................... 16
1.4.2 ZND Detonation Wave Structure ............................................... 23
1.5. Pulse detonation engines .................................................................. 25
1.5.1 The simple introduction of the PDE ........................................... 25
1.5.2 The Structure and Mesh Contour of the PDE ............................. 28
1.5.3 The Cycle Operation Process of the PDE ................................... 29
2. Numerical Models ........................................................................... 32
2.1. Governing Equations ........................................................................ 32
2.2. AUSMD ........................................................................................... 35
2.3. Strang Splitting ................................................................................. 41
2.4. MUSCL ............................................................................................ 43
2.5. THINC-EM....................................................................................... 45
2.6. HMSTH ............................................................................................ 47
3. Numerical Results ........................................................................... 49
3.1. One-Dimension Simulation Result .................................................... 49
3.1.1 Detonation Wave Case 1 ............................................................ 49
3.1.4 Detonation Wave Case 2 ............................................................ 55
3.1.5 Grid Independence Case ............................................................. 64

3.2 Two-Dimensional Simulation Results ............................................... 66
3.2.1 Two-Dimensional Benchmark Case ........................................... 66
3.2.1.1 Grid independent test........................................................... 68
3.2.1.2 Calculations by THINC-EM ................................................ 71
3.2.1.3Calculations by HMSTH ...................................................... 74
3.2.2 Single-nozzle tube case .............................................................. 96
3.2.1 The Propagation Process of the Detonation Wave .................. 98
3.2.2 Enlarged Views of Pressure Contours .................................. 107
4. Conclusions ................................................................................... 121
5. References ..................................................................................... 123

Appendix .............................................................................................. 130

Figure 1 Schematic diagram of a stationary 1D combustion wave ..... 16
Figure 2 Rayleigh lines and Hugoniot curve for p versus 1/density (from Kuo, 1986) ................................ ................................ ................... 20
Figure 3 Properties of a ZND detonation wave (adapted from Kuo, 1986)................................ ................................ ................................ ...... 25
Figure 4 Single detonation tube with CD nozzle ................................ . 29
Figure 5 The PDE’s operation process ................................ ................ 30
Figure 6 the Structure of the One-dimensional Riemann problem ..... 37
Figure 7 .Numerical results for One-Dimensional Detonation Wave Case at time = 1×10−7 The HMSTH method and the THINC-EM method is using the beta=1.2. ( A) density ;(B) pressure ;(C) temperature ;(D)Z. ................................ ................. 51
Figure 8 Numerical results for One-Dimensional Detonation Wave
Case at time = 1×10−7 s . The HMSTH method and the THINC-EM method is using the  beta =1.5. ( A) density ;(B) pressure ;(C) temperature ;(D)Z. ................................ ................. 52
Figure 9. Numerical results for One-Dimensional Detonation Wave Case at time = 1×10−7 s . The HMSTH method and the THINC-EM method is using the  beta =1.7. ( A) density ;(B) pressure ;(C) temperature ;(D)Z. ................................ ................. 54
Figure 10. Numerical results for One-Dimensional Detonation Wave Case at time = 1×10−7 s . The HMSTH method and the THINC-EM method is using the  beta =2.3. ( A) density ;(B) pressure ;(C) temperature ;(D)Z. ................................ ................. 55
Figure 11 Numerical results for One-Dimensional Detonation Wave Case at time = 1×10−7 s . The HMSTH method and the THINC-EM method is using the  beta =1.2. (A) density ;(B) pressure ;(C) temperature ;(D)Z. ................................ ................. 57
Figure 12 Numerical results for One-Dimensional Detonation Wave Case at time = 1×10−7 s . The HMSTH method and the
THINC-EM method is using the  beta =1.5. (A) density ;(B) pressure ;(C) temperature ;(D)Z. ................................ ................. 58
Figure 13 Numerical results for One-Dimensional Detonation Wave Case at time = 1×10−7 s . The HMSTH method and the THINC-EM method is using the  beta =1.7. (A) density ;(B) pressure ;(C) temperature ;(D)Z. ................................ ................. 59
Figure 14 Numerical results for One-Dimensional Detonation Wave Case at time = 1×10−7 s . The HMSTH method and the THINC-EM method is using the  beta =1.8. (A) density ;(B) pressure ;(C) temperature ;(D)Z. ................................ ................. 61
Figure 15 . Numerical results for One-Dimensional Detonation Wave Case at time = 1×10−7 s . The HMSTH method and the THINC-EM method is using the  beta =1.9. (A) density ;(B) pressure ;(C) temperature ;(D)Z. ................................ ................. 62
Figure 16 . Numerical results for One-Dimensional Detonation Wave Case at time = 1×10−7 s . The HMSTH method and the THINC-EM method is using the  beta =2.3. (A) density ;(B)
pressure ;(C) temperature ;(D)Z. ................................ ................. 63
Figure 17 Grid Independence test. ................................ ....................... 65
Figure 18. The initial state. As can be seen, z on the left side of the interface is 0, and z on the right side of the interface is 1. .......... 67
Figure 19. The density contour by the MUSCL’s grid independent test................................. ................................ ................................ ...... 69
Figure 20. The density contour by the HMSTH method,  beta=1.2,alpha =100, grid-independent test chart. ................................ ......................... 71
Figure 21. The density contour by the THINC-EM method,  beta =1.2–1.8................................. ................................ ................................ ...... 73
Figure 22 .The density contour by the HMSTH method,alpha =50, beta =1.2–2.1 ................................ ................................ ....................... 77
Figure 23. The density contour by the HMSTH methodalpha =100,  beta =1.2-1.9 ................................ ................................ ........................ 80
Figure 24 The density contour by the HMSTHalpha =500,  beta =1.2–1.8................................. ................................ ................................ ...... 83
Figure 25 The density contour by the HMSTHalpha =1000,  beta =1.2–1.6................................ ................................ ................................ ...... 85
Figure 26 The density contour by HMSTH methodalpha = 2000,  beta =1.2-1.5 ................................ ................................ ................................ 87
Figure 27 The density contour by the HMSTH,  beta =1.5,alpha =1000, 2000, 10000, 100000, comparison chart ................................ ................ 89
Figure 28 Good results. ................................ ................................ ........ 92
Figure 29 Density field along the central line of x-direction by the HMSTH method at t =1.7×10−7. ................................ ............ 95
Figure 30 Density field along the central line of x-direction by the MUSCL and the THINC-EM at t =1.7×10−7. ........................ 96
Figure 31 170,000 grid cells contour. ................................ .................. 97
Figure 32 Multiblock divided into six parts. ................................ ....... 98
Figure 33 Time evolution of Mach number (left-side) and the density-gradient (right-side). ................................ ................................ .. 101
Figure 34 Time evolution of Mach number (left-side) and pressure (right-side) along the central line. ................................ .............. 106
Figure 35 The enlarged views of pressure contours with the result at time=10.4μs and time=6.5μs ................................ .............. 108
Figure 36 The enlarged views of pressure contours with the result at time=10.4μs and time=6.5μs ................................ .............. 109
Figure 37 The enlarged views of pressure contours with the results by HMSTH,alpha=100, beta=1.2~1.5, 1.7~1.8. ................................ 111
Figure 38 The enlarged views of pressure contours with the result by HMSTH,alpha=100,1000,10000 , beta=1.7. .............................. 112
Figure 39 The enlarged views of pressure contours by THINC-EM method,  beta=1.2~1.3 ................................ ................................ ... 115
Figure 40 The Mach number field along the central line of x-direction in the single nozzle tube. ................................ ........................... 119

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