本文利用數值模擬分析超音速氣流與側向噴流的交互關係，研究目的在於建立一套模型能夠預測並捕捉當一道噴流噴入超音速流場中兩相交互的物理現象。我們利用尤拉-拉格朗日法來描述，並用KH-RT模型描述噴流，我們將液滴視為粒子並將粒子資訊儲存於網格之中，當粒子移動時，網格之間會交換粒子資訊並更新粒子在不同時間點的位置、速度等。流場是使用MUSCL+Roe通量分離法並由三階Runge-Kutta計算提高時間離散準確度。模擬結果有明顯的壓縮波，噴流前方的壓縮波下的迴流區也非常明顯，而噴流高度和實驗值相比也有不錯的結果。

This paper describes a numerical investigation of the interaction between supersonic crossflow and a side liquid jet. The goal is to develop an in-house code that is capable of predicting the detailed evolution of a side jet injected to a supersonic crossflow. In this study, we present a two-way coupling Eulerian-Lagrangian method to simulate the multiphase flow. The atomization of liquid droplets are simulated using the KH-RT breakup model, in which droplet breakup is induced by liquid instability. This breakup model considers both the surface instability and the sudden deformation of a high-speed droplet. The transport of liquid droplets considers the interaction of the liquid and gas phases. The gas phase equations are solved using the Roe-type flux-difference splitting with a MUSCL spatial differencing scheme and a third-order of Runge-Kutta time-splitting method for better accuracy. Predicted droplet distributions are compared with experimental data in various conditions. Good levels of agreement between simulations and measurements are achieved.

Table of Contents
Nomenclature	viii
List of Figure	xiii
1.	Introduction	1
1.1.	Background	1
1.2.	Literature Review	3
1.3.	Purpose and Motivation	11
2.	Numerical Models	13
2.1. Navier-Stokes Equations	13
2.2.	Turbulence Model	15
2.3.	Particle Motion Equations	17
2.4.	Droplet Momentum Coupling	20
2.5.	Breakup Model	21
2.5.1.	KH-RT Breakup Model	21
2.5.1.1.	Kelvin-Helmholtz Model	22
2.5.1.2.	Rayleigh-Taylor Model	25
2.6.	Droplet Evaporation	27
2.7.	Numerical Scheme	29
3.	Numerical Results	32
3.1.	Part I	33
3.1.1.	Influence of Momentum Flux Ratio	34
3.1.2.	Influence of Droplet Size	35
3.1.3.	Influence of Injector Diameter	35
3.1.4.	Penetration Height	36
3.1.5.	Contour of Test Domain	38
3.2.	Part II	40
3.2.1.	Effects of Breakup Model	41
3.2.2.	Effects of KH and RT Breakup Model	43
3.2.3.	Sauter Mean Diameter	44
3.2.4.	Effects of Breakup Time Constant B_1	46
3.2.5.	Influence of Momentum Ratio	49
3.2.6.	Influence of Injector Diameter	52
3.2.7.	Influence of droplet number	55
3.2.8.	Liquid Trailing Phenomenon	58
3.2.9.	Penetration Height	59
3.3.	Part III	63
4.	Conclusions	66
5.	References	68
Appendix	75

List of Figure
Figure 1 1: Schematic of the liquid jet into cross flow( Liu et al. 2014)	3
Figure 2 1: Spray breakup region and the model	22
Figure 2 2: Illustrated the KH-Breakup process	22
Figure 3 1: Computational domain of the flow field	32
Figure 3 2 Schematic of the spray droplet velocity profile	33
Figure 3 3: Influence of momentum ratio on penetration height: d_p=50 μm, d_j=0.5 mm, and q= (a): 1, (b): 5, (c): 20.	34
Figure 3 4: Influence of droplet size on spray profile: q=10, d_j=0.5 mm, and d_p = (a): 10 μm, (b): 30 μm, (c): 50 μm.	35
Figure 3 5: Influence of injector diameter: d_p=50 μm, q=10, and d_j= (a): 1mm, (b): 2mm, (c): 5mm.	36
Figure 3 6: Penetration height and correlation functions: d_p=50 μm, q=10, and d_j=0.5mm.	37
Figure 3 7: Contour of computational domain: (a): Numerical Schlieren with droplets, (b): Numerical Schlieren without droplets, (c): Mach number, (d): pressure contour.	39
Figure 3 8: Distributions of droplets: q=7, d_j=0.5 mm(a): No breakup, (b) KH breakup(B_1=10)	42
Figure 3 9: Droplet quantity vs. time.	42
Figure 3 10: Droplet trajectory: q=7, d_j=0.5 mm, B_1=10 (a):KH breakup (b):KH-RT breakup	44
Figure 3 11: SMD: q=7, d_j=0.5 mm, B_1=10 (a):Global (b):By location(KH-RT)	45
Figure 3 12: Influence of constant B_1 on penetration height: q=7, d_j=0.5 mm, B_1= (a) 5, (b) 10, (c) 20	47
Figure 3 13: Influence of constant B_1 on droplet size: q=7,  d_j=0.5 mm, B_1= (a) 5, (b) 10, (c) 20	47
Figure 3 14: Contour of computational domain: q=7, d_j=0.5 mm, B_1= (a) 5, (b) 10, (c) 20	48
Figure 3 15: Influence of momentum ratio on penetration height: d_j=0.5 mm, B_1=10, and q= (a): 5, (b): 10, (c): 15, (d): compare of (a),(b) and (c).	50
Figure 3 16: Influence of momentum ratio on droplet size: d_j=0.5 mm, B_1=10, and q= (a): 5, (b): 10, (c): 15.	50
Figure 3 17: Contour of computational domain: d_j=0.5 mm, B_1=10, and q= (a): 5, (b): 10, (c): 15.	51
Figure 3 18: Influence of injector diameter on penetration height: q=7, B_1=10, and d_j= (a): 0.1 mm, (b): 0.3mm, (c): 0.5mm.	53
Figure 3 19: Influence of injector diameter on droplets size: q=7, B_1=10, and d_j= (a): 0.1 mm, (b): 0.3mm, (c): 0.5mm.	53
Figure 3 20: Contour of computational domain: q=7, B_1=10, and d_j= (a): 0.1 mm, (b): 0.3mm, (c): 0.5mm.	54
Figure 3 21: Influence of droplet number on penetration height: q=7, d_j=0.5 mm, B_1=10, droplet number is (a) 2, (b)3, (c)4	56
Figure 3 22: Influence of droplet number on droplet size: q=7, d_j=0.5 mm, B_1=10, droplet number is (a) 2, (b)3, (c)4	56
Figure 3 23: Contour of computational domain:q=7, d_j=0.5 mm, B_1=10, droplet number is (a) 2, (b)3, (c)4	57
Figure 3 24: (a)The liquid-trailing phenomenon of jet spray (Li et al. 2017) (b) The liquid-trailing phenomenon predicted by our simulations: q=7, d_j=0.5 mm, B_1=10.	58
Figure 3 25: Penetration height and correlation functions: q=7, B_1=10, d_j=0.5 mm, refill droplet in every (a)100, (b)300, (c)500 computational time step	61
Figure 3 26: Penetration height and droplet size: q=7, B_1=10, d_j=0.5 mm, refill droplet in every (a)100, (b)300, (c)500 computational time step	61
Figure 3 27: Contour of computational domain: q=7, B_1=10, d_j=0.5 mm, refill droplet in every (a)100, (b)300, (c)500 computational time step.	62
Figure 3 28: Influence of constant B_1 on penetration height: q=7, d_j=0.5 mm, liquid jet temperature =50℃, B_1= (a) 20, (b) 15, (c) 10	64
Figure 3 29: Influence of constant B_1 on droplet size: q=7, d_j=0.5 mm, liquid jet temperature =50℃, B_1= (a)20, (b) 15, (c) 10	64
Figure 3 30: Influence of injector diameter on penetration height: q=7, B_1=10, liquid jet temperature =50℃, and d_j= (a): 0.1 mm, (b): 0.3mm, (c): 0.5mm.	64
Figure 3 31: Influence of injector diameter on droplet size: q=7, B_1=10, liquid jet temperature =50℃, and d_j= (a): 0.1 mm, (b): 0.3mm, (c): 0.5mm.	65
Figure 3 32 Numerical Schlieren of computational domain: q=7, B_1=10, d_j=0.5 mm, liquid jet temperature =50℃.	65

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