B.C. Khoo and J.G. Zheng K.M. Lim and S.S. Ramesh Department of Mechanical Engineering

1. Dynamics of Unsteady Supercavitation Impacted by Pressure Wave and Acoustic Wave Propagation in Supercavitating Flow B.C. Khoo and J.G. Zheng K.M. Lim and S.S. Ramesh Department of Mechanical Engineering National University of Singapore

2. Outline • Part 1: to simulate supercavitating flow based on compressible Euler flow solver • Part 2: to simulate acoustic wave propagation due to various hydrodynamic sources present in the vicinity of subsonically moving supercavitating vehicle using boundary element method

3. Part 1: numerical simulation of supercavitation • Background • Physical model and numerical method • Numerical results and discussion • Summary

4. Background on cavitation/supercavitation Cavitation types Sheet cavitation [1] Bubble cavitation [1] Super-cavitation [1] Vortex and sheet cavitation [2] Cloud cavitation [1] [1] J.P. Franc, J.M. Michel. Attached cavitation and the boundary layer: Experimental and numerical treatment. Journal of Fluid Mechanics. (1985) Vol. 154, pp. 63-90. [2] G. Kuiper. Cavitation research and ship propeller design. Applied Scientific Research. (1998) Vol. 58, pp. 33-50.

5. What is supercavitation? • Supercavitation is formation of gas bubble in a liquid flow arising from vaporization of fluid. • The flow pressure locally drops below the saturated vapour pressure. • The gas bubble is large enough to encompass whole object. Supercavitation image [3] [3]J.D.Hrubes. High-speed imaging of supercavitating underwater projectiles. Exp. Fluids. (2001) Vol. 30, pp. 57–64.

6. Cavitationdamages: • erosion of devices • noise • vibration • loss of efficiency eroded propeller drag reduction • Benefits of supercavitation: • drag reduction • (The viscosity is much larger in • liquid water than in vapour.) • Stability mechanism by tail slapping stability effect [3]

7. Objective of part 1 • Our interest is focused on numerical resolution of supercavitation bubble over an underwater object subjected to pressure wave. • The supercavitating flow is quite complex due to its two-phase and highly unsteady nature. • Few works on this topic are found in the literature except for say [4, 5]. Interaction of pressure wave and supercavitation [4] J.G. Zheng, B.C. Khoo and Z.M. Hu. Simulation of Wave-Flow-Cavitation Interaction Using a Compressible Homogenous Flow Method. Commun. Comput. Phys. (2013) Vol. 14, No. 2, pp. 328-354. [5] Z.M. Hu, B.C. Khoo and J.G. Zheng. The simulation of unsteady cavitating flows with external perturbations. Computer and Fluids. (2013) Vol. 77, pp. 112-124.

8. Physical models and numerical methods Available physical models • Two-phase model: • Both phases coexist at every point in flow field and one has to solve separate governing equations for each phase. • Model is complex and difficult to implement. • One-fluid model with finite-rate phase transition [6] • The finite-rate phase change can be taken into account. • It is difficult to determine the parameters associated with phase transition a priori. • One-fluid model with instantaneous phase change [4, 5] • There are no empirical parameters in governing equations. • It is easier to implement this kind of model. Continuity and momentum equations for mixture Continuity equation for each phase [6]L.X. Zhang, B.C. Khoo. Computations of partial and super cavitating flows using implicit pressure-based algorithm (IPA). Computer and Fluids. (2013) Vol. 73, pp. 1-9.

9. Physical model employed here Axisymmetric compressible Euler equation where with • A homogeneous model is employed and liquid and vapour • phases are assumed to be in the kinematic and thermodynamic • equilibrium. • The mixture density and momentum are conserved. • Phase transition is assumed to occur instantaneously.

10. Two cavitation models (equations of state (EOS)) (a) Isentropic cavitation model[7] Tait EOS Isentropic model with • This model is mathematically sound • and physically reasonable. • The pressure is the implicit function of density. • The energy equation can be neglected. Sound speed model of Schmidt: [7] T.G. Liu, B.C. Khoo, W.F. Xie. Isentropic one-fluid modeling of unsteady cavitating flow. J. Comput. Phy. (2004) Vol. 201, pp. 80–108.

11. Equations of state and sound speed model: (a) Tait EOS and isentropic cavitation model; (b) speed of sound versus void fraction. • In liquid phase modeled by Tait EOS, the pressure is highly sensitive to small change in the density. This poses challenges to numerical simulation. • The sound speed varies dramatically between liquid phase and cavitation region.

12. (b) Model of Saurel based on modified Tait EOS[8] • Temperature-dependent Tait EOS for liquid water • Ideal gas EOS for vapour • For mixture of liquid and vapour in cavitation region, the pressure is set to be saturated pressure, Tv = Tl = Tsatand pv = pl = psat. with where [8] R. Saurel, J.P. Cocchi and P.B. Butler, Numerical study of cavitation in the wake of a hypervelocity underwater projectile. J. Propul. Power. (1999) Vol. 15, No. 4, pp. 513.

13. Numerical method Time marching: Semi-discrete form: • The inviscidfluxes are numerically discretized using the cell- centered finite volume MUSCL scheme. • The time-marching is handled with the two-stage • Runge-Kutta scheme. • The geometric source terms are dealt with separately. Ghost cell on a wall boundary. Schematicof mesh.

14. Boundary conditions Implementation of boundary conditions is important to the simulation of cavitation/supercavitation. • Supersonic inlet: all flow variables on boundary are determined by freestream values. • Supersonic outlet: all variables are extrapolated from solution inside the computational domain. • Subsonic inlet: velocity is specified whereas other quantities are extrapolated from interior of the domain. • Subsonic outlet: background pressure is given and remaining variables have to be extrapolated from interior of physical domain.

15. Numerical results and discussion Case1: 1D single-phase (liquid) shock tube problem Initial condition: t=0.2ms • The results from Saurel’s and isentropic models are in good agreement. • The shock and rarefaction are well captured.

16. Case2: 1D cavitation bubble Initial condition: t=0.2ms

17. (a) (b) (c) Case 3: cavitating flow over a high-speed underwater projectile (isentropicmodel) Axisymmetricsubsonic flow at U∞=970m/s and P∞=105Pa. Results for the subsonic projectile: (a) experimental image of Hrubes [3]; (b) density map with the isentropic cavitation model; (c) comparison between axisymmetric (upper half) and planar (lower half) supercavitation.

18. The comparison of supercavity profiles between the theoretical prediction, experimental measurements and numerical simulation. • The numerical results concur well with experimental data. • The supercavity size is larger in the planar flow than in the axisymmetric flow.

19. Transonic projectile travelling at speed of Mach 1.03. (a) (b) The comparison of the experimental shadowgraph (a) and computed density contour map (b) for the transonic projectile. • The detached bow shock in front of cavitator, supercavity and wake are all well resolved numerically. • The calculated shock and cavity wake agree well with their counterparts in the experimental shadowgraph.

20. Case 4: 2D supersonic supercavitation (Isentropic model) • Here, • The underwater body consists of three parts: a nose cone with half-angle of 45o and base radius of 1cm, a cylinder of length 1cm and a rear cone with semi-vertex angle of 45o.

21. Comparison of calculated cavity half widths, L=1cm. • For the supersonic flow simulation, Saurel’s model failed. Isentropic model is more stable and robust. • The resolved flow features including detached shock shape and standoff distance and cavity half width are quantitatively consistent with those reported in [9]. [9] D.M. Causon and C.G. Mingham. Finite volume simulation of unsteady shock-cavitation in compressible water. Int. J. Numer. Meth. Fluids (2013) Vol. 72, pp. 632–649.

22. Case 5: 2D axisymmetric supercavitation (Isentropic model) • Here, • The cylinder has radius of 10mm and length of 150mm. Re-entrant jet Transient density field and its close-up view near cylinder with streamlines.

23. Interaction between pressure wave and supercavitation • The pressure wave is introduced by increasing freestream velocity suddenly, i.e. Pressure wave Density and pressure fields at 0.2ms after the abrupt freestream velocity increase.

24. Density and pressure fields at 0.4ms.

25. Density field showing supercavity collapse. • Cavitation bubble is large enough to envelop the whole cylinder, forming a supercavity. • Re-entrant jet is formed behind trailing edge of cylinder. • When impacted by pressure wave, the supercavity locally shrinks from its leading edge and eventually collapses.

26. Impingement of pressure wave on supercavitation • Here, Density field images. • The higher the freestream velocity, the longer the supercavity. • The supercavitation is unstable with respect to perturbations.

27. The schematic of simulation setup. Case 6: supercavitation subjected to sudden freestream velocity increase (Isentropic model) • The initial freestream flow state is U∞=100m/s, P∞=105Pa. • After a steady supercavity is formed, the freestream velocity is suddenly increased to U∞=120m/s. • The radius of cylinder is 10mm. • The flow is assumed to be axisymmetric. • To save computational cost, only part of supercavitation is • resolved.

28. The density field evolution with ∆U=20m/s. Here, =0.1ms.

29. The pressure distribution along the cylinder surface at three different times. • The supercavity is completely destroyed by the pressure wave due to sudden freestream velocity increase. • The pressure wave is relatively weak and not visible in the density field. • The collapse of cavity is followed by huge pressure pulse. • It takes a relatively long time for the cavity to appear again and eventually envelop the cylinder.

30. Case 7: smooth freestream velocity increase (Isentropic model) • The initial freestream flow speed is U∞=100m/s. • The freestream pressure is set to P∞=105Pa. • After a steady supercavity is formed, the freestream velocity • is changed. • Three scenarios are considered: • Scenario 1: the upstream velocity is suddenly increased by 10% (∆U=10m/s). • Scenario 2: the upstream speed is linearly increased to 110m/s via The acceleration is a=10/(nT) with T=Rc/aw where Rc and aw denote radius of cylinder and sound speed in water, respectively. Here, n is set to 50. • Scenario 3: the acceleration is reduced by setting n=100.

31. The supercavity evolution process. Column 1: sudden freestream velocity increase of ∆U=10m/s; column 2: linear velocity increase with n=50; column 3: linear velocity increase with n=100. Here, =0.1ms.

32. Animation for density field evolution • sudden freestream velocity increase, ∆U=10m/s. • constant acceleration, n=50. • constant acceleration, n=100.

33. Case 8: supercavitation subjected to freestream velocity perturbation (isentropic model) • The freestream flow speed is U∞=100m/s. • The freestream pressure is set to P∞=105Pa. The value of n is taken to be 5, 10 and 30, respectively. The larger value of n results in a perturbation with longer period.

34. The supercavity evolution subjected to the freestream velocity perturbation. The three columns (from left to right) correspond to n=5, 10 and 30, respectively. Here, =0.1ms.

35. Animation for density field evolution • sinusoidal perturbation in freestream velocity, n=5. • sinusoidal perturbation in freestream velocity, n=10. • sinusoidal perturbation in freestream velocity, n=30.

36. Case 9: supersonic supercavitation impacted by Mach 3.1 shock wave (Isentropic model) • Here, The time evolution of supercavitation impacted by a Mach 3.1 shock wave. • The supercavity experiences deformation but quickly recoveries to • its original profile. It is relatively stable at high freestream speed.

37. Animation for density field evolution

38. Case 10: 2D partial cavitation (Saurel’s model) • Cavitation number: • Here, Streamline Density Void fraction

39. Cavitation shedding Flow recirculation • The trailing edge of cavity is characterized by an unsteady re-entrant jet. • The re-entrant jet pinches off bubble and leads to cavitation shedding.

40. Case 11: 2D unsteady supercavitation (Saurel’s model) • Steady cavity: • The pressure wave is generated by suddenly increasing freestream velocity, Numerical setup Density Pressure along cylinder surface

41. Flow recirculation due to adverse pressure gradient • Local collapse of supercavity is accompanied by large pressure surge. • The pressure increase associated with left cavity collapse is high enough to create an adverse pressure gradient at trailing edge. This leads to flow recirculation and re-entrant jet, which cause cavitation shedding and full collapse.

42. Animation for density field evolution

43. Case 12: 2D unsteady supercavity impacted by a weaker pressure wave (Saurel’s model) • Smaller velocity increase: Density Pressure along cylinder surface

44. The left partial cavity breakup is accompanied by a weaker pressure surge. • There is no re-entrant jet formed and the left cavity expands downstream, developing into a new supercavity.

45. Animation for density field evolution

46. Summary • The isentropic model is proved to be more stable and robust than Saurel’s model. • It is found that the re-entrant jet is responsible for complete collapse of upstream cavity.However, if the introduced pressure wave is not relatively strong, the partial cavity can grow into a new supercavity. • When impacted by a weak shock, the supercavitation at high freestream speed undergoes deformation. • The higher the freestream flow speed is, the more stable the supercavitation is.

47. Part 2: Acoustic wave propagation in supercavitating flows • Supercavity inception/development by means of ‘natural cavitation’ and its sustainment through ventilated cavitation (caused by injection of gases into the cavity) result in turbulence and fluctuations at the water-vapour interface • Consequently, three main sources of hydrodynamic noise are • (1) Flow generated noise  turbulent pressure fluctuations around the supercavity • (2) Flow generated small scale pressure fluctuations at the vapor-water interface • (3) Pressure fluctuations due to direct impingement of ventilated gas-jets on the supercavity wall • These sound sources interfere with high frequency acoustic sensors (mounted within the nose region) that are crucial for the underwater object’s guidance system

48. Objective • To simulate acoustic wave propagation due to various hydrodynamic sources present in the vicinity of subsonically moving supercavitating vehicle • By using flow data from an unsteady CFD solver developed in Part 1 of the present research, BEM based acoustic solver has been developed for computing flow generated sound.

49. Numerical model and method Axisymmetric Boundary Integral Equation (BIE) for Subsonically Moving Surface • To study flow generated sound caused by turbulent pressure fluctuations (quadrupole/volumetric sources) present in the cavity’s vicinity, the convective Helmholtz equation is modified to include double divergence of Lighthill’s stress tensor Tij • Assuming linear acoustic source region and neglecting viscosity effects of water, the Lighthill’s stress tensor Tij is expressed in terms of Reynold’s stress tensor • M = VS /c denotes Mach number of moving surface • By adopting Prandtl-Glauret transformation, the convective Helmholtz equation is transformed to the standard form (corresponding to the stationary problem)

50. Axisymmetric BIE (contd.) • The axisymmetric BIE for transformed Helmholtz equation is given by denotes source point, denotes field point where - a constant whose value depends on location of source point denotes free space Green’s function  involves gradients of Lighthill’s stress tensor • Discontinuous Constant boundary elements are employed for approximating acoustic variables p (sound pressure) and dp/dn (normal derivative of sound pressure). Quadratic boundary elements are used to model the geometry