Force analysis of underwater object with supercavitation evolution

Indian Journal of Geo-Marine Sciences Vol. 42(8), December 2013, pp. 957-963 Force analysis of underwater object with supercavitation evolution B C Khoo 1,2,3* & J G Zheng 1,3 1 Department of Mechanical Engineering, National University of Singapore, Singapore 119260 2 Singapore-MIT Alliance, National University of Singapore, Singapore 117576 3 Temasek Laboratories, National University of Singapore, Singapore 117411 * [E-mail: mpekbc@nus.edu.sg] Reeived 5 December 2012; revised 9 September 2013 Supercavitation generally occurs as a result of flow acceleration along underwater body surface and is numerically investigated in this study using a compressible Navier-Stokes equations solver. Here, the supercavitating flow is assumed to be the homogeneous mixture of pure liquid water and vapour which are in kinematic and thermodynamic equilibrium. Liquid phase and cavitation are modeled by Tait equation of state (EOS) and isentropic one-fluid formulation, respectively. Convective terms of the governing equations are numerically integrated using Godunov-type, cell-centered finite volume MUSCL scheme on unstructured triangular mesh, whereas time integration is handled with the second-order accurate Runge-Kutta approach. Our interest is focused on the force analysis of traveling object with the formation, growth, evolution and even collapse of supercavity enveloping the object. It is found that skin friction drag exerted on the object can be reduced significantly by the formation of supercavity where viscosity of vapour is much smaller than that of liquid water. It is also observed that form drag acting on the object is influenced by the supercavitation. Collapse of supercavity over the body due to external perturbation not only damages underwater object but also alters form drag on it. [Keywords: Supercavitating flow, Isentropic cavitation model, Underwater object, Navier-Stokes equations] Introduction The numerical simulation and prediction of cavitation/supercavitation are of great importance for efficient design of engineering devices such as hydrofoil ship, hydraulic turbine and propeller which may suffer from the damage due to cavitation bubble collapse. However, a lot of numerical and physical modeling difficulties are encountered in the computational fluid dynamics simulation and analysis. The discontinuities of fluid properties are present across cavitation bubble interface, which include large density ratio of liquid to vapour, drastic sound speed variation and completely different equations of state for liquid and vapour. On the other hand, the cavitation phenomenon is related to a number of complex physics such as significant mass transfer rate, viscous effect, interfacial dynamics and so on. All the issues mentioned above constitute challenges for any numerical methods which should be able to model the cavitation problem accurately and be robust as well as efficient. Various numerical methods have been developed to resolve the cavitation. Among them, multi-phase method is implemented by solving the separate system of governing equations for each phase. This type of approach is able to model fluid dynamics and thermodynamics of liquid and vapour, respectively. Although the multi-phase model is physically reasonable, it is usually quite complex, involving various transfer terms, and therefore not easy to implement. On the contrary, one-fluid model treats the cavitating flow as the homogeneous mixture of liquid and vapour and is used widely 1,2,3,4. In most cavitation simulations, the liquid flow is assumed to be incompressible as the flow speed is relatively low. This assumption is consistent with the experimental observation that the fluid density almost does not change when the flow remains in low or moderate Mach number regime. A consequence of the incompressible flow assumption is that the method cannot model the perturbation propagation through the flow at finite speed. However, under a number of situations such perturbations like shock wave and other pressure wave may have significant effect on the cavitation evolution and should be modeled. In this paper, a homogeneous cavitating flow model based on compressible Navier-Stokes equations is employed. The liquid is described as

958 INDIAN J. MAR. SCI., VOL. 42, NO. 8, DECEMBER 2013 being compressible and modeled by Tait equation of state (EOS). The cavitation is treated as the homogeneous mixture of isentropic liquid as well as vapour and its inception and evolution is modeled by isentropic cavitation model. The phase transition is assumed to occur instantaneously and other physics associated with the phase change are neglected. The present method allows us to model the shock wave propagation through liquid media and its interaction with cavitation bubble. This study is concentrated on the simulation of pressure wave induced supercavitation deformation and collapse and on their influence on the underwater object. Materials and Methods In the present study, the cavitating flow is assumed to be the homogeneous mixture of isentropic liquid and vapour which remain in kinematic and thermodynamic equilibrium, that is, liquid and vapour share the same velocity, pressure and temperature. Baseline partial differential equation system describing the cavitating flow is the compressible Navier-Stokes equations, U F G i Fv G v i + + + S = + + S t x y y x y y v (1) where i takes value of 0 and 1 for planar and axisymmetric flows, respectively. Here, U, F, G and S are the vector of conserved variables, inviscid flux vectors in x and y direction, geometric source term vector associated with axisymmetric flow, respectively, and are given by, ρ ρu ρv ρv ρ ρ ρ ρ 2 U= u F= u + p G= uv S= uv 2 2 ρv ρuv ρv + p ρv (2) In (2), ρ is the averaged density, u is the averaged x velocity component, v is the averaged y velocity component and p is the averaged pressure. Energy equation is neglected in model system (1) as the flow is assumed to be isentropic and the pressure depends on the density only. The viscous flux vectors F v, G v and S v are not presented here to save space. The liquid phase is modeled by Tait EOS, 1/N p + B A ρ = ρ0, p psat. B (3) When the flow pressure drops below the saturated vapour pressure p sat, the cavitation occurs and is modeled by isentropic cavitation model 3, kρ + ρ =, p < p p + B A p + k p + B A p cav cav g l ρ 1/ N 1/ γ cav cav sat (4) where k=α 0 /(1-α 0 ) and α 0 is the known void fraction of the mixture density at p cav. Here, ρ and ρ are cav g cav l the associated vapour and liquid densities at the cavitation pressure p cav. For liquid water, values of parameters in (3) are set to be N=7.15, A=10 5 Pa, B = 3.31 10 9 Pa, p o =1000 Kg/m 3, respectively. The sound speed is calculated through Wallis s formula 5, α ( 1 α ) ρ ρ ρ a = + 2 2 sv. av sw. aw 1 2 (5) where void fraction α is determined from the mixture density definition, sv ( 1 ) ρ = αρ + α ρ (6) sw In (6), p sw and p sv are the saturated liquid and vapour densities, respectively. Viscosity coefficient for the two-phase mixture in (1) is given by, v ( 1 ) µ = α µ + α µ (7) l where µ v and µ 1 are viscosity coefficients of vapour and liquid, respectively. The convective and viscous terms of model system (1) closed by (3) and (4) are discretized using secondorder MUSCL 6,7 scheme and an algorithm based on calculation of average of gradients 8, respectively, while the time-marching is dealt with using two-stage Runge-Kutta method. Results and Discussion Supercavitating flow around a hemispherical head cylinder To validate our method, a supercavitating flow around an underwater cylinder with hemispherical head is resolved. The computational domain is shown in Fig. 1 where the inflow boundary condition is imposed on the left hand side of the domain. The

KHOO & ZHENG: FORCE ANALYSIS OF UNDERWATER OBJECT 959 Fig. 1 The supercavitating flow around a hemispherical head cylinder. Upper left: the triangular mesh; upper right: close-up view of the mesh around the cylinder head; lower left: the intermediate stage of supercavitation evolution; lower right: the steady state supercavitation. In all the coordinate systems shown in the figures of this paper, unless stated otherwise, unit of length is meter non-slip boundary is implemented along the cylinder surface while the remaining sides are treated as open boundary. The freestream pressure and velocity are set to be P =10 5 Pa, U =30 m/s, respectively. The flow is assumed to be symmetric. The triangular mesh employed in the simulation and its close-up view around cylinder head are illustrated in the first row of Fig. 1. The unstructured mesh is better able to conform to the geometric features. The density contour maps showing the supercavitation evolution process is demonstrated in the second row of Fig. 1. In the lower left frame of Fig. 1, the cavitation bubble has appeared. The cavitation bubble continues growing and finally develops into a supercavity enveloping the whole cylinder as illustrated in the lower right frame of Fig. 1 where the flow has reached steady state. The density in the cavitation region is several orders of magnitude smaller than that of the liquid outside the supercavitation bubble. In all the simulations presented in the paper, Courant-Friedrichs-Lewy (CFL) number is set to be 0.8. The code is seen to accurately resolve the supercavitating flow and is stable and robust. Supercavitating flow past a high-speed underwater projectile The second validation case is on the computation of a high-speed conical-shaped projectile travelling at velocity of 970 m/s in water. A disk cavitator of radius 0.71 mm is mounted on the head of the projectile which has length of 157.4 mm and base radius of 6.56 mm. The freestream pressure is 1.4 10 5 Pa. The experimental 9 and numerical images for supercavity encompassing the entire object are compared visually in Fig. 2(a,b). The supercavitation

960 INDIAN J. MAR. SCI., VOL. 42, NO. 8, DECEMBER 2013 Fig. 2a,b Experimental (a) and numerical (b) images for a high-subsonic underwater projectile at speed of 970 m/s and comparison between experimental and calculated supercavity profiles (c). Fig. 2c profile and wake resolved with isentropic model are qualitatively comparable to their counterparts in experimental image, as shown in Fig. 2. In Fig. 2(c), the computed supercavity thickness concurs quantitatively with Hrubes supercavity size measurements 9, which demonstrates the validity of our method. Supercavitating flow interacting with a pressure wave In this case, the supercavitation bubble impacted by a pressure wave is numerically investigated. Initially, a blunt head cylinder with radius of 10 mm is immersed in a uniform water flow at pressure of 10 5 Pa and velocity of 100 m/s. The cylinder is at the angle of attack of 0. The boundary conditions are the same as those in the first case except that the freestream pressure and velocity at inlet are set to be P =10 5 Pa, U =100 m/s, respectively. Due to the symmetry of the flow, only the upper half of the flow field is calculated. The above initial conditions will give rise to a supercavitation bubble enveloping the entire cylinder. Fig. 3 depicts part of the density field where time t=0 corresponds to the time instant when the numerical solution is converged. The supercavity interface is smooth and there are no any pressure oscillations. The supercavitating flow is well resolved as illustrated in Fig. 3.

KHOO & ZHENG: FORCE ANALYSIS OF UNDERWATER OBJECT 961 In the steady state calculation, the pressure and velocity at inlet are fixed as constants. For unsteady simulation, after the supercavity is formed and the solution is converged, the freestream pressure is increased suddenly to a higher value and then returns to its initial value after a short time period of 66 µs. The pressure pulse will generate complex pressure wave which is to collide with the supercavity. The situation is schematized in Fig. 4. Four sets of density evolution maps corresponding to different pressure pulses are illustrated in Fig. 5. In Fig. 5 (a1-e1), the freestream pressure is abruptly increased by 100 times, from 10 5 Pa to 10 7 Pa. It is observed that the supercavity is deformed by the pressure wave. The perturbation travels through water quite fast and the locally deformed boundary gradually recovers to its original profile, see Fig. 5 (a1-e1). In Fig. 5 (a2-e2), the pressure is increased to P=1.2 10 7 Pa. The supercavitation is seen to undergo more severe deformations. The qualitative behaviors of supercavitation evolution associated with the two pressure pulses are similar. With increasing peak pressure to P=1.4 10 7 Pa, the supercavity locally collapses and is divided into two parts as shown in Fig. 5 (a3-e3). The right cavity fully collapses with time advancing while the left one expands downstream and finally develops into a new supercavity. The evolution process for P=1.6 10 7 Pa is similar to that with the peak pressure of 1.4 10 7 Pa. The skin friction drag on the cylinder is reduced significantly by the supercavitation formation as the viscosity in vapour is much smaller than that in water. However, the form drag exerted on the body increases due to impact of generated pressure wave on cylinder head. Supercavitating flow around a 2D plate Next, we consider the supercavitating flow past two dimensional (2D) plate inclined relative to the flow Fig. 3 The density field map of a supercavitation around a blunt head cylinder. Here, P =10 5 Pa and U =100 m/s. Fig. 4 The schematic of simulation setup for the supercavitation interacting with a shock wave. Fig. 5 The density field evolution for the supercavitation bubble impacted by a pressure wave. Frames (a1-e1): P=10 7 Pa; frames (a2-e2): P=1.2 10 7 Pa; frames (a3-e3): P=1.4 10 7 Pa; frames (a4-e4): P=1.6 10 7 Pa.

962 INDIAN J. MAR. SCI., VOL. 42, NO. 8, DECEMBER 2013 Fig. 6 The supercavitating 2D plate at different angles of attack (the first row) and shock induced supercavity collapse (the second row) at angle of attack α=5. direction. The flow is not symmetric and therefore assumed to be planar. The plate is 150 mm long and 20 mm high. The freestream pressure and velocity are 10 5 Pa and 100 m/s, respectively. The density contour images associated with angles of attack of 5, 10 and 15 at steady state are presented in Fig. 6. The angles of attack are selected in such a way that supercavitating flow exhibits different patterns. It is obvious that cavity shape depends on the angle of attack. The upper cavity over the upper surface of the plate becomes thicker with increasing attack angle. The cavity below lower surface appears to be more affected by inclined angle. At α=10, the lower cavity is narrower than that at α=5. For the larger angle of 15, only a thin layer of cavitation is formed near plate head. It is known that the drag acting on an object may be influenced by cavity over it. The form drag coefficients for α=5 and α=10 are 0.064 and 0.063, respectively. Here, the drag coefficient is 2 defined as C = F / (0.5P U A ) where F d is the d d ref form drag, P and U are the freestream density and velocity, respectively, and A ref is the reference area. For the two angles of attack of 5 and 10, the force on the left side of the plate is the major source of drag and hence the two coefficients are close to each other. However, at α=15, the pressure contribution to the drag on the lower surface outside the cavity has to be accounted for and therefore larger coefficient of 0.076 results. The calculation indicates that in engineering practice, devices may need to operate within a certain range of angle of attack to avoid the possible situation like that at α=15, which may lead to larger drag and cause a degradation of devices efficiency. For the case with α=5, a Mach 1.1 incident shock is introduced after a steady state solution is obtained and the numerical results are presented in Fig. 6. As illustrated in Fig. 6(d), at t=2 τ, the cavities originally generated and anchored by plate corners are shrinking from their leading edges. Here, τ represents a time period of 0.1 ms. Simultaneously, a relatively small cavity appears near the upper left corner of plate and grows. The lower cavity appears to shrink more rapidly than the upper one as time advances, see Fig. 6(e) at t=4 τ. On the other hand, two water jets are formed along top and bottom of the plate and the lower jet penetrates deeper into cavity than the upper one. By t=12 τ, the cavities have completely collapsed as shown Fig. 6(f). After the cavities vanish, the form drag coefficient for the flow configuration with 5 angle of attack is around 0.13 which is about 2 times the value with the cavities.

KHOO & ZHENG: FORCE ANALYSIS OF UNDERWATER OBJECT 963 Conclusion The supercavitation interacting with the pressure wave is numerically simulated using a homogeneous cavitation fluid method. It is found that the supercavity may be locally distorted or destroyed by the imposed pressure wave. The skin friction drag can be reduced significantly by the supercavity and the form drag can also be affected by the cavitation evolution. References 1 Kunz R F, Boger D A & Stinebring D R, A preconditioned Navier-Stokes method for two-phase flows with application to cavitation prediction, Comput. Fluid, 29 (2000) 849-875. 2 Saurel R, Petitpas F & Berry R A, Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures, J Comput Phys, 228 (2009) 1678-1712. 3 Liu T G, Khoo B C & Xie W F, Isentropic one-fluid modeling of unsteady cavitating flow, J Comput Phys, 201 (2004) 80-108. 4 Zheng J G, Khoo B C & Hu Z M, Simulation of Wave-Flow- Cavitation Interaction Using a Compressible Homogenous Flow Method, Commun Comput Phys, 14(2) (2013) 328-354. 5 Wallis G B, One-dimensional two-phase flow, (McGill-Hill, New York) 1969. 6 Van Leer B, Upwind and High-Resolution Methods for Compressible Flow: From Donor Cell to Residual- Distribution Schemes, Commun Comput Phys, 1(2) (2006) 192-206. 7 Toro E F, Riemann solvers and numerical methods for fluid Dynamics: A practical introduction, (Springer-Verlag, Berlin Heidelberg) 1999. 8 Blazek J, Computational fluid dynamics: principles and applications, (ELSEVIER) 2001. 9 Hrubes J D, High-speed imaging of supercavitating underwater projectiles, Exp. Fluids, 30 (2001) 57-64.