The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Substantial discrepancy in the conditions for attainment of different closure modes of a ventilated supercavity has existed widely in the published literature. In this study, supercavity closure is investigated with an objective to understand the physical mechanisms determining closure formation and transition between different closure modes and to reconcile the observations from prior studies under various flow settings. The experiments are conducted in a closed-wall recirculating water tunnel to image ventilated supercavity closure using high speed and high-resolution photography and simultaneously measure pressure inside the cavity. The flow conditions are varied systematically to cover a broad range of water velocity, ventilation flow rate and cavitator size, which correspond to different Froude numbers, air entrainment coefficients and blockage ratios, respectively. In addition to the classical closure modes reported in the literature (e.g. re-entrant jet, twin vortex, quad vortex, etc.), the study has revealed a number of new closure modes that occur during the transition between classical modes, or under very specific flow conditions. Closure maps are constructed to depict the flow regimes, i.e. the range of Froude number and air entrainment coefficient, for various closure modes at different blockage ratios. From the closure map at each blockage ratio, a critical ventilation flow rate, below which the supercavity collapses into foamy cavity upon reduction of Froude number, is identified. The air entrainment coefficients corresponding to such critical ventilation rate are found to be independent of blockage ratio. It has been observed that in the process of generating a supercavity by increasing ventilation flow rate, the cavitation number gradually reduces to a minimum value and stays fixed upon further increments in the ventilation rate. Once a supercavity is formed, the ventilation rate can be decreased to a much lower value with no change in cavitation number while still maintaining a supercavity. This process is accompanied by a change in closure modes, which generally goes from twin vortex, to quad vortex, and then to re-entrant jet. In addition, the blockage effect is shown to play an important role in promoting the occurrence of twin-vortex closure modes. Subsequently, a physical framework governing the variation of different closure modes is proposed, and is used to explain mode transition upon the change of flow conditions, including the blockage effect. This framework is further extended to shed light on the occurrence of closure modes for ventilated supercavitation experiments across different types of flow facilities, the natural supercavity closure and the pulsating supercavity reported in the literature. Finally, in combination with a recent numerical study, our research discusses the role of the internal flow physics on the observed features during supercavity formation and closure-mode transition, paving the way for future investigations in this direction.

An experimental investigation into supercavity closure mechanisms

Large Eddy Simulation (LES) was coupled with a mass transfer cavitation model to predict unsteady 3-D turbulent cavitating flows around a twisted hydrofoil. The wall-adapting local eddy-viscosity (WALE) model was used to give the Sub-Grid Scale (SGS) stress term. The predicted 3-D cavitation evolutions, including the cavity growth, break-off and collapse downstream, and the shedding cycle as well as its frequency agree fairly well with experimental results. The mechanism for the interactions between the cavitation and the vortices was discussed based on the analysis of the vorticity transport equation related to the vortex stretching, volumetric expansion/contraction and baroclinic torque terms along the hydrofoil mid-plane. The vortical flow analysis demonstrates that cavitation promotes the vortex production and the flow unsteadiness. In non-cavitation conditions, the streamline smoothly passes along the upper wall of the hydrofoil with no boundary layer separation and the boundary layer is thin and attached to the foil except at the trailing edge. With decreasing cavitation number, the present case has σ = 1.07, and the attached sheet cavitation becomes highly unsteady, with periodic growth and break-off to form the cavitation cloud. The expansion due to cavitation induces boundary layer separation and significantly increases the vorticity magnitude at the cavity interface. A detailed analysis using the vorticity transport equation shows that the cavitation accelerates the vortex stretching and dilatation and increases the baroclinic torque as the major source of vorticity generation. Examination of the flow field shows that the vortex dilatation and baroclinic torque terms increase in the cavitating case to the same magnitude as the vortex stretching term, while for the non-cavitating case these two terms are zero.

Three-dimensional large eddy simulation and vorticity analysis of unsteady cavitating flow around a twisted hydrofoil

It is well known that shark skin surface can effectively inhabit the occurrence of turbulence and reduce the wall friction, but in order to understand the mechanism of drag reduction, one has to solve the problem of the turbulent flow on grooved-scale surface, and in that respect, the direct numerical simulation is an important tool. In this article, based on the real biological shark skin, the model of real shark skin is built through high-accurate scanning and data processing. The turbulent flow on a real shark skin is comprehensively simulated, and based on the simulation, the drag reduction mechanism is discussed. In addition, in order to validate the drag-reducing effect of shark skin surface, actual experiments were carried out in water tunnel, and the experimental results are approximately consistent with the numerical simulation.

Numerical Simulation and Experimental Study of Drag-Reducing Surface of a Real Shark Skin

http://profdoc.um.ac.ir/articles/a/1053123.pdf

Investigation of cavitation around 3D hemispherical head-form body and conical cavitators using different turbulence and cavitation models

Large eddy simulation and investigation on the laminar-turbulent transition and turbulence-cavitation interaction in the cavitating flow around hydrofoil

Unsteady turbulent cavitation flows in a Venturi-type section and around a NACA0012 hydrofoil were simulated by two-dimensional computations of viscous compressible turbulent flow model. The Venturi-type section flow proved numerical precision and reliability of the physical model and the code, and further the cavitation around NACA0012 foil was investigated. These flows were calculated with a code of SIMPLE-type finite volume scheme, associated with a barotropic vapor/liquid state law which strongly links density and pressure variation. To simulate turbulent flows, modified RNG k − ɛ model was used. Numerical results obtained in the Venturi-type flow simulated periodic shedding of sheet cavity and was compared with experiment data, and the results of the NACA0012 foil show quasi-periodic vortex cavitation phenomenon. Results obtained concerning cavity shape and unsteady behavior, void ratio, and velocity field were found in good agreement with experiment ones.

Modelling and Computation of Unsteady Turbulent Cavitation Flows

The finite volume method based on a multiphase model is adopted to solve the Reynolds-Averaged Navier-Stokes (RANS) equations, which takes into account the effects of fluid compressibility, viscosity, gravity, medium mixture and energy transfer of water and combustion gas. The program Fluent User Define Function (UDF) module combined with the dynamic mesh method is employed to simulate the coupling flow field of combustion gas, water field and trajectory of projectile. The results show that the volume of gas cavity at the bottom of projectile and tail pressure will fluctuate after bottom of the projectile leaving the launch tube. The cause of the fluctuation is analyzed and its effects on the trajectory of projectile are presented. The numerical and experimental results agree well with each other.

Research on the base cavity of a sub-launched projectile

LES investigation on cavitating flow structures and loads of water-exiting submerged vehicles using a uniform filter of octree-based grids

When a body navigates with cavity in shallow water, both flexible free surface and rigid bottom wall will produce great influences on the cavity shape and hydrodynamic performances, and further affect the motion attitude and stability of the body. In the present work, characteristics of the natural cavitating flow around a 2-D symmetrical wedge in shallow water were investigated and the influences of two type boundaries on the flow pattern were analyzed. The Volume Of Fluid (VOF) multiphase flow method which is suitable for free surface problems was utilized, coupled with a natural cavitation model to deal with the mass-transfer process between liquid and vapor phases. Within the range of the cavitation number for computation (0.07–1.81), the cavity configurations would be divided into three types, viz., stable type, transition type and wake-vortex type. In this article, the shapes of the free surface and the cavity surface, and the hydrodynamic performance of the wedge were discussed under the conditions of relatively small cavitation number (

Chuan-jing Lu

Papers

Large eddy simulation and investigation on the laminar-turbulent transition and turbulence-cavitation interaction in the cavitating flow around hydrofoil

Modelling and Computation of Unsteady Turbulent Cavitation Flows

Research on the base cavity of a sub-launched projectile

LES investigation on cavitating flow structures and loads of water-exiting submerged vehicles using a uniform filter of octree-based grids

Properties of natural cavitation flows around a 2-d wedge in shallow water