Evaluation of Turbulence Models for the Numerical Prediction of Time-dependent Cavitating Flow During Water Entry of a Semi-closed Cylinder

LU Zhong-lei,SUN Tie-zhi,CUI Lei,WEI Ying-jie

(1.The System Design Insitute of Mechanical-Electrical Engineering,Beijing 100854,China;2.School of Naval Architecture,Dalian University of Technology,Dalian 116024,China;3.Theoretical Research Center of Flying Training Base,China;4.School of Astronautics,Harbin Institute of Technology,Harbin 150001,China)

Abstract:During water entry of the semi-closed cylinder,the air in the cell interacts with the flow field around the cylinder,which generates time-dependent cavitating flow with perturbation.The kε model,k-ω model and detached eddy simulation(DES)model are selected to assess the state-ofthe-art of computational capabilities for the special cavitating flow.The turbulence models mentioned above are evaluated and validated by comparing the numerical time evolution of cavitating flow with the experimental results.The results show that the DES model can better capture the unsteady phenomena including the cavity ripples,shedding,transition flow and multi-scales vortex.Furthermore,the formation mechanism of eddies around the cavity interface is analysed based on the boundarylayer theory.

Key words:DES;cavitating flow;perturbation;water entry

0 Introduction

Water entry is the process that unconstrained mobile object with special speed move across the free surface from air into water or other liquids.This process refers to an unsteady,compressible and turbulent multi-phases flow[1-2].Many researchers have been focusing on applying the CFD method to predict trajectory[3],flow characteristic in near field[4]and the evolution of cavitating flow[5-6].However,the pre-existing researches were almost based on simple body such as sphere,disk,cylinder,and so on.This paper investigates a special cylinder model which has an large hollow cell and one end opened.In the process of water entry,the semi-closed cylinder keeps the posture of the opening end towards moving direction and impacts free surface first.Water flows into and closes the cell after initial impact.An air spring is formed inside the cell and causes a periodic jet form the opening end,which results to the local field and the cavity waved[7]and the hydrodynamics and motion parameters fluctuating[8].The opening hollow cell divides the field into an internal flow field and an external flow field.The two fields are coupling each other to form a complicated flow environment.The flow mechanics problems for the process of semi-closed cylinder water entry involve the unsteady flow,swirling turbulent flow,multiphase flow,and so on,so the complicated flow structure increases the difficulty on the numerical computation.

Turbulence models play very important roles in the simulations of the flow phenomena.The Reynolds-averaged Navier-Stokes(RANS)turbulence models,such as the k-ε model and k-ω model two-equation closures,have been very popular in the numerical simulation of water entry[9-10].However,RANS noticeably over-predicts turbulent production and effective viscosity in stagnation flow region.Large Eddy Simulation(LES)method[11]is an alternative to RANS model that can be employed to simulate the multi-scale vortex field and to provide accurate prediction for cavity ripples and shedding.However,the LES model can not avoid the deficiencies of high resolution requirements for boundary layers and the huge computational cost.In recent years,researchers develop a series of derived models[12-14].Spalart[15]pioneered to put forward the detached eddy simulation(DES)turbulence model which combined RANS modeling with LES.Based on this method,Menter[16]developed a new DES model blended k-ω SST model and LES model.Strelets[17]and Morton[18]have applied DES model respectively to simulate the hydrofoil movement and obtained unsteady vortex shedding flows.

In the present study,three turbulence models including k-ε model,k-ω model and DES model are used to evaluate the application for time-dependent cavitating flow with perturbation during water entry of a semi-closed cylinder.Serving as a reference to evaluate the validity of turbulence model,experimental visualization and data are used for comparisons and errors analysis with numerical results.At last,we based on the calculation results to analyze the typical flow phenomenon of the water entry for a semi-closed cylinder and to evaluate the applicability of DES model for the problem of a semi-closed cylinder water entry.

1 Numerical method

1.1 Governing equations

The governing equations under the homogeneous-fluid modeling consist of the conservative form of the Favre-averaged Navier-Stokes equations and the turbulence closure.In the conditions of low Froude number and isothermal temperature flow,we assume that both fluids are no phase transition and no interpenetrating.The continuity,momentum and energy equations are given below:

where keffis effective thermal conductivity.The thermodynamics relationship between energy and temperature is

where Cpis the specific heat at constant pressure.

The Volume of Fluid(VOF)multiphase flow model is adopted to track the interface of immiscible fluids.The volume-fraction-averaged properties for mixture fluids can be expressed as:

where α is the volume fraction and the subscripts l and g indicate liquid and gas,respectively.

1.2 Turbulence models

The RANS equations have a similar form with the instantaneous Navier-Stokes equations with the solution variables representing ensemble-averaged values.The averaged momentum equations expressed as

Additional terms appear in Eq.(8)and must be modeled in order to close the equations.

The Boussinesq hypothesis[19]is used to relate the Reynolds stresses to the mean velocity gradients,the formulation of hypothesis can be written as

where k is turbulence kinetic energy,k=（u′2+v′2+w′2）/2 andis turbulent viscosity.

(1)k-ε model

The k-ε model is a high Reynolds number turbulence model[20].The turbulent eddy viscosity is defined as

with Cμ=0.09.Turbulence dissipation rate is defined as

The modeled transport equations for k and ε are presented as follows:

where the generation term of turbulence kinetic energy is presented aswithand Pr=0.85.The dissipation term of the turbulent kitnetic energy is presented as Yk=ρk3/2/lrke,where the turbulent length scale is given as lrke=k3/2/ε.The dissipation term of the fluctuating dilatation in compressible turbulence flow is presented as YM=2 ρkε /（γRT）.

(2)SST k-ω model

SST k-ω model is a low Reynolds number turbulence model[21].This model effectively blends the robust and accurate formulation of the k-ω model in the near-wall region with the free-stream independence of the k-ε model in the far field.The turbulent eddy viscosity is defined as

where the specific dissipation rate is defined as ω=ε/（Cμk ）.The coefficient λ damps the turbulent viscosity causing a low-Reynolds number correction.The expression is

The modeled transport equations for k and ω are presented as follows:

where the generation term of turbulence kinetic energy is presented asThe production term of specific dissipation rate is presented asThe turbulent length scale is given by

(3)DES model

As a hybrid LES/RANS models,the DES model is to switch from the RANS model to the LES model in regions where the turbulent length predicted by the RANS model is larger than the local grid spacing.Basing the k-ω SST model,the turbulent length scale is used in the dissipation term of the turbulent kinetic energy,which is modified for the DES turbulence model such as

where lDESis the turbulent length scale instead of lSST,and the expression is

where CDESis a calibration constant used in the DES model and has a value of CDES=0.61.Δ is based on the largest dimension of the grid cell:

According to turbulent length,The LES region is normally associated with the core turbulent region where unsteady turbulence scales play a dominant role.In the small-turbulence scales region,the RANS model is recovered.Fig.1 shows the relation between turbulent length and turbulent models.

Fig.1 Prediction methods of RANS,LES and DES approaches

Fig.2 Schematic of computation domain and boundary conditions

1.3 Solution implementation and discretization method

The semi-closed cylinder studied in this paper is a pipe with one end closed.Geometric parameters of the semi-closed cylinder are as follows:the external diameter is D=0.04 m,the internal diameter is d=0.036 m,the length is l=0.2 m and the depth of cell is lc=0.109 m.The mass of semi-closed cylinder is m=0.345 kg.The opening end is always towards gravity direction and impacts free surface first.

The computation domain,coordinate system and boundary conditions are shown in Fig.2.At the initial time,the semi-closed cylinder is located above the free surface at the position z*with the velocity v*.The time initially impacting water is defined as the time t=0.

The local mesh of the semi-closed cylinder is shown in Fig.3.The structured grid is adopted for whole domain.In order to accurately capture the boundary layer flow,it was necessary to tailor the mesh to satisfy the refinement criterion y+～1,where y+is a dimensionless distance measured from the fluid-wall interface.

The Finite Volume Method(FVM)is used to discretize the governing equations and solve the discretization equations to obtain the values of the flow variable in the field overall process.For time discretization,we choose time-advancement algorithm which the transient term in the transport equation is discretized by the second-order implicit form with second-order accuracy.For spatial discretization,we choose the second-order upwind form to discretize convective term and the least squares cell-based form to discretize the gradients and derivatives in diffusive term.The discrete values of the scalar are stored at the cell centers,but face values are required for the convection terms.The face values must be interpolated from the cell center values.We select Body Force Weighed scheme for pressure interpolation and select Geo-Reconstruct scheme for volume fraction interpolation.

Fig.3 Schematic of local mesh(a)Meshing around the cylinder;(b)Meshing on solid walls;(c)Meshing in the cell

2 Results and discussion

2.1 Kinematic parameters

Fig.4 shows the trajectory of semi-closed cylinder.Water entry is a time-dependent process from cavity formation to cavity collapse.Generally,the time is transitory in water entry.At low Froude numbers,the inertia force magnitude has a same order with gravity,So the position of the center on the opening end presents linear relation with time.The trajectories are well-predicted on the early stage of water entry by three turbulence models.Because the RANS model(such as k-ε and SST)noticeably over-predicts turbulent production and hence effective viscosity in stagnation flow region,the errors created by the overpredicts viscosity are accumulated to cause the predicted value of sinking depth lesser than the actual value.To compare with the experimental data,the DES model performs best after the time t=0.06 s(see the enlarged graph).

Fig.4 Trajectory of the semi-closed cylinder

Fig.5 Velocity of the semi-closed cylinder

Fig.5 shows the velocity of semi-closed cylinder.The velocity keeps linear increase before water impact and begins to decrease at the eve of impact under the effect of the air recoil.The drag force replaces the gravity to play a dominant role after the opening end sinking and leads to the velocity to decrease.A physical phenomenon that leads to an additional force on the closed end prominently is the air compressed and expanding alternately in the cell.The additional force results in a fluctuating velocity.By comparison with experimental data,DES model is more suitable to capture the periodical fluctuation according to the amplitude and frequency of velocity.

Fig.6 shows the acceleration of semi-closed cylinder.The acceleration presents fluctuant variation synchronized with the air compression and expansion.The difference between three models results is more obvious,especially at both time of air compressed completely and air expanding process,the time-dependent acceleration of cylinder appear high frequency wave.For the impact process,water flow into cell and compress the air.In the meantime,an impact wave is produced and spreads toward the closed end.As the properties of impact wave are high frequency and low energy,the effect for the motion of cylinder is very tiny.The force created by the impact wave exerts pressure on the inner surface of closed end.

Fig.7 shows the time history of force on the closed end.It is consistent with the impact wave action from the enlarged graph.Comparing the numerical results for the three turbulence models,DES model can capture the impact wave.

Fig.6 Acceleration of the semi-closed cylinder

Fig.7 Force on the closed end

2.2 Cavity visualization

As the important characteristic for water entry,the cavity visualization is another evaluation parameter for the validity of turbulence model.Comparisons of the cavity profile between experimental measurement and numerical results associated with different turbulence models are listed in Tab.1.The cavity visualizations present a waved profile.The time evolution of cavity goes through in sequence the smooth shape(t=0.01 s),the waved shape(t=0.02 s),pinchoff(t=0.04 s),necking(t=0.06 s)and segmental shedding(t=0.08 s).The air expansion forms the liquid jet on the opening end,and makes the flow structure to be disordered.As a interface of two phases which are water and air,the cavity will be disturbed by the chaos field to present a fluctuating flow pattern.

Tab.1 Comparisons of the cavity profile at typical time

As for the DES model in Tab.1,the features of every stage in experiment can be well captured,including the profile and the size of fluctuating cavity,the appearance time and position of detached cavity in the trailing at the last stage,and the frequency of fluctuating and shedding,which is more consistent to the observation experimentally.For the shedding cavity,the shape and scale have some errors.However,this value is already enough to activate the contribution of others models.DES model has a well performance on the accuracy for predicting the water entry of semi-closed cylinder.

The accuracy of numerical results is an important validation index of the turbulence models.By comparison with the experimental data,we take the errors analysis to evaluate the simulation accuracy of turbulence models.The relative error is defined as

where χCFDsignifies the variable by computation, χexpsignifies the variable by experimental measure.The accuracy of computational results is tested by comparing the relative errors of typical kinematics parameters including the sinking distance s,the axial velocity v,the maximum expanding radius amaxand the mean length of cavity b.The errors are shown in Tab.2 where the least errors are marked by bold fonts.According to comparisons between three turbulence models,excepting the error of mean length is larger at t=0.04 s,the results obtained by DES turbulence model manifest a higher accuracy.Above all,DES is a validity turbulence model for cavitating flows in water entry of semi-closed cylinder.

2.3 Vortex structure

For the water entry of a solid object,generally,cavitating flow and field structure are stable,the surface of cavity is smooth,and the closed point position is immobile.The shear flow is week leading to the vortex is hardly formed in cavity boundary-layer.So the distribution of viscosity layer is homogeneous,and the laminar flow is stable near the cavity wall.However,for the water entry of the semi-closed cylinder,the effect of the air spring motion in the opened hollow cell is violent to the cylinder motion and cavitating flows.A high flow velocity formed on the cavity surface by the water jet from the opening cell.In the domain of the cavity,the jet flow affects the flow around the cylinder and appears local circular rector of velocity.The local velocity circulation generates vortex near the cavity surface.It is one of the typical flow characteristics of local vortex flow pattern for water entry of semi-cylinder.

Fig.8 shows the flow structures at two typical time,liquid jet stage(a)and cavity shedding stage(b),to illustrate the vortex structure produced by the time-dependent disturbance flow.The second invariant of the velocity gradient tensor Q on the left and the streamlines on the right are shown in the same figure.The second invariant of the velocity gradient tensor is used to describe the vortex structure[22],which can be expressed as

where S is modulus of the mean rate-of-strain tensor and Ω is modulus of vorticity,ωk=?ui/?xj-?uj/?xi.It represents vortex flow which plays a dominant role when Q＞0,and it represents shear deformation which plays a dominant role when Q＜0.The vortex near the cavity wall focus mainly on the opening end and the trough of cavity.The mechanism of vortex production close to the opening end is the result of the difference of cavity expending speed between the wave crest and the wave trough,and the mechanism of vortex production lied to the trough of fluctuate cavity is the result of the re-entrant jet which created by local flow around the crest of cavity.The other action result of re-entrant jet is that the cavity emerges deep closure and local shedding more than once.The cavity shedding occur at the trough position and the vortex falls off with the shedding cavity.

Fig.8 Distributions of Q value near the cavity wall and streamlines(a)t=0.02 s;(b)t=0.08 s

According to the boundary-layer theory,the flow in boundary-layer field is shown in Fig.9.The pressure decreases along the flow direction and reverses at singularity.The flow velocity slows down and reverses to form re-entrant jet due to the pressure gradient inversion.

Fig.9 Schematic of boundary flow and re-entrant jet

The viscosity is predominant near the cavity surface,so it can be assumed that the flow pattern around the fluctuating cavity is similar with the boundary-layer flow.The liquid velocity is faster on the crest of cavity and lower on the trough of cavity.The boundary-layer is thicken gradually from the crest to the trough,re-entrant jet is formed at trough and supervene segmental is shedding many times at the trough of wave cavity.

In sight of energy transport,the kinetic energy of cylinder is converted into the pressure potential energy by compressing the air in the cell,then the potential energy releases into kinetic energy of the liquid in the form of a liquid jet.The process is over and over again,and forms a period disturbance source at the center of the opening end.The results are to generate a time-dependent separating flow in which the intensity and direction determine the diameter of cavity expansion,and change the stable flow structure in the cavity boundary-layer where laminar flow,turbulence flow and transition flow are concurrent,local velocity gradient and pressure gradient are notable and form multi-scale turbulence vortex.DES model can accurately predict the vortex,liquid jet and the cavity shedding.

3 Conclusions

In the present study,the time-dependent cavitating flow with perturbation during water entry is investigated by different turbulence models.The typical turbulence models,including k-ε model,SST model and DES model have been utilized to capture the special fluctuation of the cavity caused by the perturbation of the air flow.Then the numerical results were compared with experimental data to validate the turbulence models.

The water flows into the cell form an internal sealing cell for the semi-closed cylinder water entry.In the cell,the air flowing and the cylinder motion are coupled each other.The air flowing force on the closing end periodically to change the kinematic characteristic.The velocity and acceleration present fluctuant variation synchronized with the air compression and expansion.Excepting the influence for motion of the cylinder,the jet causes the flow field disturbance to form cavity wave and shedding periodically and to induce the vortex flow structure.The vortex appears at the position of wave trough,in which the formation mechanism of vortex is the boundary layer flow near the cavity surface.

DES model results perform best in all aspects in terms of kinematics parameters,cavity visualization and vortex structure.The highlight characteristic of DES model is the ability to capture the small-scale vortex in the boundary-layer of cavity and the pressure fluctuation.Therefore,DES model is applicable and efficient for the time-dependent cavitating flows with perturbation,such as the cavitating flow during water entry of the cylinder with an opening cell.