Numerical simulation of supercavitation

NUMERICAL SIMULATION OF SUPERCAVITATION ZENG HUIMING (B.S., M.E., HUST) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledgements I would like to express my deepest and most sincere gratitude to Professor Khoo Boo Cheong for all the continuous support, invaluable encouragement, and academic guidance provided as the thesis supervisor in the last two years, as well as his comprehensive instruction and assistance during my thesis writing My gratitude also goes to the co-supervisor Dr Liu Tiegang for his support, instruction and expertise which greatly help me during my studies in numerical techniques for supercavitating flow Furthermore, I would give my greatest appreciation to him for providing the basic programme codes and valuable advice throughout the whole study Also, I would sincerely appreciate the help of Dr Xie Wenfeng, my friends in the Fluid Lab and my two roommates My entire family deserves a special gratitude for their unlimited support during my oversea study It is their constant love and encouragement that make the completion of this work possible At last, I would like to thank the National University of Singapore for their Research Scholarship Zeng Huiming Jul, 2007 i Table of Contents Acknowledgements i Table of Contents ii Summary v List of Tables vii List of Figures viii Nomenclature xiv Chapter Introduction……………………………………… ……1 1.1 Background……………………………………………………… …… 1.2 Literature Review of Difficulties in Numerical Simulations of Cavitating Flow………………………………………………………… …………2 1.3 Literature Review of Experiments and Numerical Simulations of Supercavitation…………………………………………………………5 1.4 Motivation of the Research on the Numerical Simulation of Supercavitation………………………………………………………… 1.5 Outline of Thesis…………………………… ………………………… Chapter Methodologies…………………………………… ……10 2.1 Governing Equations………………………………………… ………10 2.2 Equation of State for Gas and Water………………………………… 12 ii 2.3 The Eulerian Solver…………………………………………………… 13 2.4 The Level Set Technique and its application to Structure Interface…….17 2.5 The Modified Ghost Fluid Method……………………………….…… 21 Chapter Cavitation Models……………………………… …….22 3.1 Assumptions for the simulation… ………………………………….….23 3.2 Relationship across the Cavitation Boundary……………………….….24 3.3 Cavitation Models………………………………………………….… 25 3.3.1 Cut-off Model…………………………………………………….…… 25 3.3.2 Isentropic Model……………………………………………….……… 26 3.4 The One-fluid Cavitation Models for Multi-dimensions……………… 28 3.5 Verification of Isentropic One-fluid Cavitation Model………… …… 29 3.5.1 Verification of Cavitation Models by 2D flow passing over a forward facing step………………………………………… ………………… 30 Chapter Special Treatment for the Corner Point at a Forward Facing Step………………………………………………… ……33 4.1 the Mesh Refinement for Multiphase Flow……… ……………….……33 4.2 The Special Treatments of Corner………………………………….……36 4.2.1 The Entropy and Enthalpy Corrections for Single Phase (gas) Flow… 38 4.2.2 Restriction of Pressure at the corner for water flow……………… … 41 4.3 Concluding Remarks of the Chapter………………………………… 44 iii Chapter Numerical Simulations of Supercavitation inception and evolution over a high-speed Projectile… …… 46 5.1 Numerical Simulation of Supercavitation Flow over a Forward-facing Step………………………………………………………………….… 47 5.2 Numerical Simulation of Flow over a Hemisphere head Projectile…… 50 5.3 Concluding Remarks of the Chapter…………………………………….53 Chapter Conclusion and Future Work…………………… … 55 References……………………………………… …………………57 Tables…………………………………………………… ……… 62 Figures…………………………………………………… ……… 63 iv Summary In this work, the research is focused on simulating the cavitating flow and supercavitating flow Based on the previous theoretical analysis on existing two one-fluid cavitation models and numerical tests, the isentropic model is proved to be more consistent and physically realistic than the cut-off cavitation model In addition, to further verify the isentropic model, cavitating flow occurring in two-dimensional cavitating flow is simulated To study the supercavitating flow over high speed projectiles, two different underwater projectile models are simulated, one is a forward-facing step, and the other is a hemisphere head projectile In the simulation of flow over a forward-facing step, the corner point is a singular point Based on the fixed Eulerian grids, if no special treatment is applied on the singular point, an unphysical expansion shock will emanate from the corner In the work, two special corrections are employed to treat the corner, which are entropy and enthalpy corrections and the restriction of pressure at the corner point It is found that the entropy and enthalpy correction, which works well for single phase flow, does not work for water flow with phase transition (cavitation) From the experimental results, the flow always starts to cavitate at the corner point: by restricting the pressure of the corner point to the saturated pressure whenever there is phase change at the corner, the unphysical expansion shock can be efficiently removed for the multi-phase flow In the simulation of flow over a hemisphere head projectile, the reflective boundary v condition is employed to treat the rigid boundaries However, the coding for treatment of a rigid boundary of complex geometry is usually quite complex To develop a consistent way of rigid boundary treatment, the level set technique is extended to treat the rigid boundary together with the reflective boundary condition Numerical simulation of supercavitaton has been carried out for these two underwater projectiles by employing MUSCL scheme and the isentropic cavitation model Furthermore, the process of cavitation inception, evolution over a high speed underwater projectile is investigated The captured cavitation pocket and pressure distribution along the projectile surface are also investigated It is shown that the simulation results are reasonable and physical comparing with the experimental data (Rouse & McNown 1948) vi List of Tables Table 4-1 Error pertaining to grid refinement 62 vii List of Figures Fig 2-1 Construction of Point A and B in extrapolation 63 Fig 3-1 Convergence contour with computing loops using the cut-off cavitation model 64 Fig 3-2a Contour plots of numerical approximation of the pressure with MUSCL scheme with cut-off cavitation model at t= 4.13E-04s 64 Fig 3-2b Contour plots of numerical approximation of the density with MUSCL scheme with cut-off cavitation model at t= 4.13E-04s 65 Fig 3-2c Contour plots of numerical approximation of the pressure with MUSCL scheme with cut-off cavitation model at t= 8.24E-04s 65 Fig 3-2d Contour plots of numerical approximation of the density with MUSCL scheme with cut-off cavitation model at t= 8.24E-04s 66 Fig 3-3 Convergence contour with computing loops using the Isentropic cavitation model 66 Fig 3-4a Contour plots of numerical approximation of the pressure with MUSCL scheme with meshes [301, 101] at t= 8.20E-04s 67 Fig 3-4b Contour plots of numerical approximation of the density with MUSCL scheme with meshes [301, 101] at t= 8.20E-04s 67 Fig 4-1 The mesh distribution for Case 4.1.1, Case 4.2.1.1, Case 4.2.2.1 and Case 4.2.2.2 in the vicinity of the cylinder surface 68 Fig 4-2a Contour plots of Experimental result of the density by Rouse and McNown(1948) at the velocity of 25.3m/s and pressure of 1atm 68 Fig 4-2b Contour plots of Experimental result of pressure distribution along the projectile surface at different cavitation number by Rouse and McNown (1948) 69 Fig 4-3a Contour plots of numerical approximation of the pressure with MUSCL scheme with meshes [301, 101] at t= 8.20E-04s 69 viii Fig 4-3b Contour plots of enlarged section on the pressure distribution near the corner with MUSCL scheme with meshes [301, 101] at t= 8.20E-04s 70 Fig 4-3c Contour plots of numerical approximation of the density with MUSCL scheme with meshes [301, 101] at t= 8.20E-04s 70 Fig 4-3d Contour plots of pressure distribution along the projectile surface with MUSCL scheme with meshes [301, 101] at t= 8.20E-04s 71 Fig 4-4a Contour plots of numerical approximation of the pressure with MUSCL scheme with meshes [601, 201] at t= 1.78E-03s 71 Fig 4-4b Contour plots of enlarged section on the pressure distribution near the corner with MUSCL scheme with meshes [601, 201] at t= 1.78E-03s 72 Fig 4-4c Contour plots of numerical approximation of the density with MUSCL scheme with meshes [601, 201] at t= 1.78E-03s 72 Fig 4-4d Contour plots of pressure distribution along the projectile surface with MUSCL scheme with meshes [601, 201] at t= 1.78E-03s 73 Fig 4-5a Cell locations where Entropy and Enthalpy corrections are applied for the forward facing step test at Mach flow 73 Fig 4-5b Initial Diagram of Mach flow 74 Fig 4-6a Contour plots of numerical approximation of the pressure without fixing at t= 1.61E-04s 74 Fig 4-6b Contour plots of numerical approximation of the density without fixing at t= 1.61E-04s 75 Fig 4-6c Contour plots of numerical approximation of the pressure with MUSCL scheme using entropy and enthalpy corrections at t= 1.88E-04s 75 Fig 4-6d Contour plots of numerical approximation of the density with MUSCL scheme using entropy and enthalpy corrections at t= 1.88E-04s 76 Fig 4-6e Contour plots of numerical approximation of the pressure with 76 ix cavitation 5.5E+06 5E+06 4.5E+06 4E+06 3.5E+06 3E+06 2.5E+06 2E+06 1.5E+06 1E+06 500000 76573.4 8511.79 1.50918E-07 Fig 4-8c Contour plots of numerical approximation of the pressure with MUSCL scheme with the restriction of pressure fix at the corner point at t= 6.20E-04s cavitation 1001.52 1000.81 1000.18 999.992 900 800 700 600 500 400 300 200 100 Fig 4-8d Contour plots of numerical approximation of the density with MUSCL scheme with the restriction of pressure fix at the corner point at t= 6.20E-04s 80 0.8 k 0.6 0.4 0.2 0 s/d Fig 4-8e Pressure Distribution along the projectile surface with MUSCL scheme with the restriction of pressure fix at the corner point at t= 6.20E-04s 81 Cavitation 600000 550000 500000 450000 337296 250000 150000 100000 50000 17873.7 5377.41 0.972388 0.00203202 Fig 5-1a Contour plots of numerical approximation of the pressure with MUSCL scheme with the restriction of pressure fix at the corner point at t= 1.24E-03s Cavitation 1000.15 1000.06 1000.01 999.986 999.96 900 800 700 600 500 400 300 200 100 Fig 5-1b Contour plots of numerical approximation of the density with MUSCL scheme with the restriction of pressure fix at the corner point at t= 1.24E-03s 82 0.8 0.6 k 0.4 0.2 -0.2 0.5 1.5 2.5 s/d Fig 5-1c The Pressure distribution along the projectile surface for Case 5.1.1 at t= 1.24E-03s Fig 5-2a Mean Cavitaion Pockets for blunt head projectile at different cavitation number by Rouse & McNown (1948) 83 Fig 5-2b Contour plots of Experimental result of pressure distribution along the projectile surface at different cavitation number by Rouse and McNown (1948) cavitation 4.2E+07 3.6E+07 3E+07 2.4E+07 1.74739E+07 1.05675E+07 1E+07 8E+06 6E+06 890764 300545 84720 0.000178339 Fig 5-3a Contour plots of numerical approximation of the pressure with MUSCL scheme with the restriction of pressure fix at the corner point at t= 2.98E-04s 84 1014.05 1009.64 1005.18 1001.71 1000.44 1000 900 800 700 600 500 400 300 200 100 cavitation Fig 5-3b Contour plots of numerical approximation of the density with MUSCL scheme with the restriction of pressure fix at the corner point at t= 2.98E-04s cavitation 4.2E+07 3.4E+07 3E+07 2.4E+07 2.2E+07 1.8E+07 1.4E+07 1E+07 6E+06 2.38286E+06 2E+06 877903 194971 3.24636E-05 Fig 5-3c Contour plots of numerical approximation of the pressure with MUSCL scheme with the restriction of pressure fix at the corner point at t= 5.98E-04s 85 1012.94 1012.78 1008.09 1004.52 1001.18 1000 900 800 700 600 500 400 300 200 100 cavitation Fig 5-3d Contour plots of numerical approximation of the density with MUSCL scheme with the restriction of pressure fix at the corner point at t= 5.98E-04s 4.2E+07 3.8E+07 3.4E+07 2.8E+07 2E+07 1.6E+07 1.4E+07 1E+07 6E+06 2E+06 961078 200084 0.0012463 6.24172E-09 cavitation Fig 5-3e Contour plots of numerical approximation of the pressure with MUSCL scheme with the restriction of pressure fix at the corner point at t= 1.19E-03s 86 cavitation 1013.96 1008.29 1005.75 1002.4 1000.6 1000 900 800 700 600 500 400 300 200 100 Fig 5-3f Contour plots of numerical approximation of the density with MUSCL scheme with the restriction of pressure fix at the corner point at t= 1.19E-03s 30d 5.83d d d 14d Fig 5-4 Schematic of the computational domain of a hemisphere head projectile 87 3.4 3.2 2.8 2.6 2.4 2.2 1.8 1.6 1.4 1.2 0.8 0.6 Fig 5-5a Contour plots of numerical approximation of the density of gas at t= 3.06E-04s 3.4 3.2 2.8 2.6 2.4 2.2 1.8 1.6 1.4 1.2 0.8 Fig 5-5b Contour plots of numerical approximation of the density of gas at t= 5.09E-04s 3.4 3.2 2.8 2.6 2.4 2.2 1.8 1.6 1.4 1.2 Fig 5-5c Contour plots of numerical approximation of the density of gas at t= 1.02E-03s 88 3.4 3.2 2.8 2.6 2.4 2.2 1.8 1.6 1.4 1.2 Fig 5-5d Contour plots of numerical approximation of the density of gas at t= 2.03E-03s 700 650 600 550 500 450 400 350 300 250 200 150 100 50 Fig 5-6a Contour plots of numerical approximation of the x velocity of gas at t= 2.04E-02s 180 160 140 120 100 80 60 40 20 Fig 5-6b Contour plots of numerical approximation of the y velocity of gas at t= 2.04E-02s 89 550000 500000 450000 400000 350000 300000 250000 200000 150000 Fig 5-6c Contour plots of numerical approximation of the pressure of gas at t= 2.04E-02s 2.8 2.6 2.4 2.2 1.8 1.6 1.4 1.2 Fig 5-6d Contour plots of numerical approximation of the density of gas at t= 2.04E-02s cavitation 3.4E+07 3.2E+07 3E+07 2.8E+07 2.6E+07 2.2E+07 2E+07 1.4E+07 1E+07 8E+06 6E+06 4E+06 2E+06 Fig 5-7a Contour plots of numerical approximation of the pressure of water at t= 2.13E-04s 90 1008.3 1004.86 1002.96 1000 850 800 600 550 500 400 350 250 200 100 50 cavitation Fig 5-7b Contour plots of numerical approximation of the density of water at t= 2.13E-04s cavitation 3.7E+07 3.4E+07 3.2E+07 3E+07 2.8E+07 2.5E+07 2.3E+07 1.8E+07 1.6E+07 1E+07 6.01624E+06 4.16152E+06 3E+06 2E+06 1E+06 Fig 5-7c Contour plots of numerical approximation of the pressure of water at t= 3.36E-04s cavitation 1008.09 1003.18 1002.07 1001.95 1001.51 1000 900 800 700 600 500 400 300 200 100 Fig 5-7d Contour plots of numerical approximation of the density of water at t= 3.36E-04s 91 cavitation 3.4E+07 3.3E+07 3E+07 2.9E+07 2.1E+07 1.9E+07 1.8E+07 1.7E+07 9.12628E+06 4.14958E+06 2.3655E+06 1.62042E+06 1.26808E+06 0.0258536 3.719E-05 3.30052E-05 Fig 5-7e Contour plots of numerical approximation of the pressure of water at t=6.73E-04s cavitation 1009.91 1005.29 1000.89 1000 950 900 800 700 600 500 450 350 300 200 100 50 Fig 5-7f Contour plots of numerical approximation of the density of water at t=6.73E-04s cavitation 3.3E+07 3.2E+07 3.1E+07 3E+07 2.9E+07 2.8E+07 2.7E+07 2.6E+07 2.5E+07 1.4E+07 1.08489E+07 1E+07 8E+06 4E+06 155915 24.2409 Fig 5-7g Contour plots of numerical approximation of the pressure of water at t= 1.42E-03s 92 cavitation 1006.83 1003.42 1001.03 1000.12 1000 999.009 950 900 850 800 750 700 650 600 550 Fig 5-7h Contour plots of numerical approximation of the density of water at t= 1.42E-03s 262.302 257.401 250 249.496 246.898 243.088 229.706 200 150 100 50 Fig 5-8a Contour plots of numerical approximation of the x velocity of water at t= 2.85E-03s 130 120 110 100 90 80 70 60 50 40 30 20 10 Fig 5-8b Contour plots of numerical approximation of the y velocity of water at t= 2.85E-03s 93 3.2E+07 3E+07 2.8E+07 2.6E+07 2.4E+07 2.2E+07 2E+07 1.8E+07 1.29658E+07 7.88082E+06 6E+06 4E+06 2E+06 25274.7 0.297568 2.43824E-11 cavitation Fig 5-8c Contour plots of numerical approximation of the pressure of water at t= 2.85E-03s cavitation 1012.06 1008.87 1004.79 1003.36 1001.11 999.99 900 800 700 600 500 400 300 200 100 Fig 5-8d Contour plots of numerical approximation of the density of water at t= 2.85E-03s 94 [...]... plots of numerical approximation of the density of gas at t= 2.04E-02s 90 Fig 5-7a Contour plots of numerical approximation of the pressure of water at t= 2.13E-04s 90 Fig 5-7b Contour plots of numerical approximation of the density of water at t= 2.13E-04s 91 Fig 5-7c Contour plots of numerical approximation of the pressure of water at t= 3.36E-04s 91 Fig 5-7d Contour plots of numerical approximation of. .. approximation of the density of gas at 88 xi t= 1.02E-03s Fig 5-5d Contour plots of numerical approximation of the density of gas at t= 2.03E-03s 89 Fig 5-6a Contour plots of numerical approximation of the x velocity of gas at t= 2.04E-02s 89 Fig 5-6b Contour plots of numerical approximation of the y velocity of gas at t= 2.04E-02s 89 Fig 5-6c Contour plots of numerical approximation of the pressure of gas... difficulties in numerical simulations of cavitating flow The literature review of experiments and numerical simulations of supercavitation is presented in section 1.3 The motivation of this work is presented in section 1.4 Lastly, the outline of this thesis is shown in section 1.5 1 1.2 Literature Review of Difficulties in Numerical Simulations of Cavitating Flow From the viewpoint of numerical simulation of cavitating... of the density of water at t= 3.36E-04s 91 Fig 5-7e Contour plots of numerical approximation of the pressure of water at t=6.73E-04s 92 Fig 5-7f Contour plots of numerical approximation of the density of water at t=6.73E-04s 92 Fig 5-7g Contour plots of numerical approximation of the pressure of water at t= 1.42E-03s 92 Fig 5-7h Contour plots of numerical approximation of the density of water at t=... 1.42E-03s 93 Fig 5-8a Contour plots of numerical approximation of the x velocity of water at t= 2.85E-03s 93 xii Fig 5-8b Contour plots of numerical approximation of the y velocity of water at t= 2.85E-03s 93 Fig 5-8c Contour plots of numerical approximation of the pressure of water at t= 2.85E-03s 94 Fig 5-8d Contour plots of numerical approximation of the density of water at t= 2.85E-03s 94 xiii Nomenclature... approximation of the density with MUSCL scheme with the restriction of pressure fix at the corner point at t= 1.19E-03s 87 Fig 5-4 Schematic of the computational domain of a hemisphere head projectile 87 Fig 5-5a Contour plots of numerical approximation of the density of gas at t= 3.06E-04s 88 Fig 5-5b Contour plots of numerical approximation of the density of gas at t= 5.09E-04s 88 Fig 5-5c Contour plots of numerical. .. the homogeneous mixture of fluids via a cell-centered finite-volume method to obtain the supercavitation Both Owis and Nayfeh 6 (2003) and Neaves and Edwards (2006) employ the preconditioned algorithm for the simulation which is quite complex to implement 1.4 Motivation of the Research on the Numerical Simulation of Supercavitation On the past works on numerical simulations of supercavitation, none employs... index of Gas i Index of a grid in x direction j Index of a grid in y direction l Index of the fluid flow in the left side of the interface; Index of liquid L Left side m Index of mixture medium n Temporal index N Index of normal direction R Right side sat Index of saturated status t Time x x coordinate y y coordinate xvii xviii Chapter 1 Introduction 1.1 Background Cavitation is formation of pockets of. .. of pressure fix at the corner point at t= 5.98E-04s 85 Fig 5-3d Contour plots of numerical approximation of the density with MUSCL scheme with the restriction of pressure fix at the corner point at t= 5.98E-04s 86 Fig 5-3e Contour plots of numerical approximation of the pressure with MUSCL scheme with the restriction of pressure fix at the corner point at t= 1.19E-03s 86 Fig 5-3f Contour plots of numerical. .. (1948) 84 Fig 5-3a Contour plots of numerical approximation of the pressure with MUSCL scheme with the restriction of pressure fix at the corner point at t= 2.98E-04s 84 Fig 5-3b Contour plots of numerical approximation of the density with MUSCL scheme with the restriction of pressure fix at the corner point at t= 2.98E-04s 85 Fig 5-3c Contour plots of numerical approximation of the pressure with MUSCL scheme ... plots of numerical approximation of the density of gas at t= 3.06E-04s 88 Fig 5-5b Contour plots of numerical approximation of the density of gas at t= 5.09E-04s 88 Fig 5-5c Contour plots of numerical. .. plots of numerical approximation of the density of gas at t= 2.04E-02s 90 Fig 5-7a Contour plots of numerical approximation of the pressure of water at t= 2.13E-04s 90 Fig 5-7b Contour plots of numerical. .. of numerical approximation of the pressure of water at t= 1.42E-03s 92 Fig 5-7h Contour plots of numerical approximation of the density of water at t= 1.42E-03s 93 Fig 5-8a Contour plots of numerical