investigation of cavitation around 3d hemispherical head...

Ocean Engineering 112 (2016) 287–306

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Ocean Engineering

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

Mohammad-Reza Pendar, Ehsan Roohi n

High Performance Computing (HPC) Laboratory, Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, P.O. Box91775-1111, Mashhad, Iran

a r t i c l e i n f o

Article history:Received 18 January 2015Accepted 7 December 2015

Keywords:Conical cavitatorHemispherical head-form bodySupercavitationVolume-of-Fluid (VOF) methodOpenFOAM

x.doi.org/10.1016/j.oceaneng.2015.12.01018/& 2015 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ98 51 38805136; faxail address: [email protected] (E. Rooh

a b s t r a c t

In this paper, cavitation and supercavitation around 3D hemispherical head-form body and a conicalcavitator were simulated. Dynamic and unsteady behaviors of cavitation were solved using large eddysimulation (LES) and k-ω SST turbulence models, as well as Kunz and Sauer mass transfer models. Inaddition, the compressive volume of fluid (VOF) method is used to track the cavity interface. Simulationis performed under the framework of the OpenFOAM package. The main contribution of this work is topresent a correlation between the cavity length and diameter for hemispherical head-form bodies for thefirst time. Moreover, we provide a detailed comparison between different turbulence and mass transfermodels over a broad range of cavitation numbers, especially in small cavitation numbers, includingσ¼0.07, 0.05, 0.02 for two cases, which is not reported previously. Our numerical results are comparedwith the available experimental data and a broad set of analytic relations for the cavity characteristicssuch as cavity length and diameter with suitable accuracy. Discussions on boundary layer separation andre-entrant jet behavior, which play a significant role in the bubble shedding in the cavity closure region,are presented.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Cavitation is a multi-phase and complex physical phenomenondefined as the formation of vapor bubbles within a liquid whenthe pressure locally drops below the saturated vapor pressure(Brennen, 1995). As a useful phenomenon, cavitation attracted theattention of many researchers in the past decades. Cavitationusually appears over underwater vehicles such as underwatervehicles, submarine, hydrofoils and marine propeller blades. For-mation of cavity cloud significantly increases the performance ofthe marines by reducing the viscous drag. Cavitation is a three-dimensional and periodic phenomenon that exhibits unsteadydynamic behaviors such as periodic shedding of the cavity cloudand growth and rapid collapse of bubbles (Wang and Ostoja,2007). Cavitation could be categorized by a dimensionless num-ber; i.e., σ ¼ ðP1�PϑÞ=0:5ρU21 that is called cavitation number.When one decreases the cavitation number, i.e., via increasing thevelocity of the moving body further, supercavitation will occurwhich consists of a long and steady cavity region.

Precise simulation of cavitation phenomenon requires anaccurate mass transfer model, a surface reconstruction scheme for

: þ98 51 38763304.i).

capturing the sharp cavity interface as well a suitable turbulencemodel. Different kinds of mass transfer model can be used forcavitation modeling. Famous cavitation models based on semi-analytical approaches were derived by Merkle et al. (1998), Kunzet al. (2000), Sauer (2000); Yuan et al. (2001) and Singhal et al.(2002).

Among different surface reconstruction schemes, Volume ofFluids (VOF) method was extensively used to describe the phasetransition mechanism between liquid and vapor phases that bothexist in cavitating flows. For instance, Passandideh Fard and Roohi(2008), Shang, (2013), Roohi et al. (2013), Yu et al. (2014) and Kimand Lee (2015) used VOF method to simulate cavitation for dif-ferent sets of geometries.This method predicts the cavity interfaceaccurately.

Selection of an appropriate turbulence model is another crucialissue for accurate simulation of the cavitation because the cavi-tation is an unsteady phenomenon usually occurring in highReynolds number flows. Two turbulence approaches, large eddysimulation (LES) and k-ω SST have been most widely used tosimulate cavitating flow. LES regularly allows for medium-scale tosmall-scale energy transfer that can capture flow mechanismswith much detail for accurate prediction of the cavitation.

Literature survey shows that numerical simulation of cavi-tation attracted the attention of researchers during the lastdecades. Kunz et al. (2000) considered cavitation around

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Nomenclature

B Unresolved transport term in LESCε,CK LES empirical constant coefficientsCe,Cc Schnerr–Sauer mass transfer model constantsCDkω Positive portion of the cross-diffusionCdest, Cprod Kunz mass transfer model constantsCd0 Constant in the drag coefficient for a conical cavitatorCx, Cx0 Functions (given by Eqs. (36) and (37))D Cavity diameterd Cavitator diameterD Rate of strain tensor~DD Eddy diffusivity tensorF1, F2 Turbulence Functions (given by Eqs. (14) and (18))G Filter functionH Height of the cone cavitatorI Unit tensork Kinetic energyL Cavity lengthl Liquid_m Mass transfer rate between the phasesn0 Initial number of bubblesP PressureR radius of Cylinder

Re Reynolds numberRb Radius of bubbless Viscous stress tensort1 Mean flow timeU Velocity magnitudeνSGS Subgrid scale viscosityXj Components of the Cartesian coordinatey Distance the between surfaceσ Cavitation number1 Free stream valueαπ cone half-angleϕ volumetric fluxρ Densityϑ Vaporδ Re-entrant jet lengthv Velocity vectorμ ViscosityΔ Filter widthγ Volume fractionω Vorticityμk Viscosity of the vortexβ�,σω2,α, β, cμ Constant coefficients for the k-ω sst

turbulence model– Averaging

M.-R. Pendar, E. Roohi / Ocean Engineering 112 (2016) 287–306288

submerged objects for the axisymmetric and steady state con-dition with multiphase computational fluid dynamics. Theyconveyed different parameter such as pressure distribution,drag coefficient and cavity shape and compared with theexperimental data. Baradaran Fard and Nikseresht, (2012)simulated unsteady 3-D cavitating flows around a cone and diskcavitator. RANS equations and an additional transport equationfor the liquid volume fraction are solved using a finite volumeapproach through the SIMPLE (Semi-Implicit Method for Pres-sure Linked Equations) algorithm. For the implementation of theturbulent flow, k-ω SST model was used. The results are in goodagreement with experimental data and analytical relations. Guoet al. (2011) simulated the cavitating flow around an underwaterprojectile with natural and ventilated cavitation based on thehomogeneous equilibrium flow model, a mixture model fortransport equation and a local linear low-Reynolds-number k-εturbulence model. Shang (2013) simulated cavitation around thecylindrical submarine. They used K-ω SST for turbulence model,VOF method for the cavity interface reconstruction and theSauer model for mass transfer to capture the cavitationmechanisms within broad ranges of cavitation numbers from0.2 to 1.0. Park and Hyung (2012) simulated high-speed super-cavitating flows around a 2-D symmetric wedge-shaped cavi-tator and hemispherical head form body using an unsteadyReynolds-averaged Navier–Stokes equations solver based on acell-centered finite volume method. The computed result com-pared with an analytical solution and numerical results using apotential flow solver. Yu et al. (2014) simulated dynamic beha-viors of cavitation over a 3-D projectile at the cavitation numberσ¼0.58 based on LES, k�μ transport equation and VOF methodwith the Kunz model for the mass transfer. Evolution of cavi-tation in simulation was consistent with the experimental. Chenet al. (2015) investigated the collapse regimes of the cavitationon the submerged vehicles navigating with continuous decel-eration in the range of 0.2rσr0.5. A homogeneous equili-brium cavitation model that combined with the pressure–velocity–density coupling algorithm was used to simulate the

cavitating flows. There are some recent works witch reportcavitation phenomena using advanced turbulence models. Forexample, Decaix and Goncalves (2013) used a compressible,multiphase, one-fluid RANS solver to study turbulent cavitatingflows. Ji et al. (2013) simulated cavitating turbulent flow aroundhydrofoils by using the Partially-Averaged Navier–Stokes (PANS)method and a suitable mass transfer cavitation model. Theirpredicted cavity characteristic compared well with experi-mental data. Zhang and Khoo (2014) developed a pressure-based compressible-medium numerical method to performcomputations of the cavitating flow. They demonstrated thattheir method is capable of simulating the dynamics of unsteadycavitating flow. Ji et al. (2014) investigated numerically thestructure of the cavitating flow around a twisted hydrofoil usinga mass transfer cavitation model and a modified RNG k-ε modelwith a local density correction for turbulent eddy viscosity.Cavity structures and the shedding frequency agreed fairly wellwith experimental observations. Goncalves and Charriere (2014)proposed an original formulation for the mass transfer betweenphases to study one-dimensional inviscid cavitating tube pro-blems. Numerical results are given for various inviscid cases andunsteady sheet cavitation developing along venturi geometriesand compared with experimental data. Ji et al. (2015) studiedthe behavior of cavities around a NACA66 hydrofoil numericallyby using Large Eddy Simulation (LES) coupled with a homo-geneous cavitation model. Various fundamental mechanismsgoverning the complex cavitating flow behaviors, including thecavitation shedding dynamic evolution, cavitation–vortexinteraction and cavitation excited pressure fluctuation, wereexamined and summarized.

In this research, we consider cavitation and supercavitationover hemispherical head-form body and a conical cavitator usingan open source package, that is, OpenFOAM. We used Kunz andSauer mass transfer models combined with both of the LES and k-ω SST turbulence models to simulate cavitating flows. A com-pressive velocity form of the volume of fluid (VOF) method isemployed to track the interface of liquid and vapor phases.

M.-R. Pendar, E. Roohi / Ocean Engineering 112 (2016) 287–306 289

Additionally, we present a correlation for the cavity length anddiameter for hemispherical head-form body and provide a detailedcomparison of different turbulence and mass transfer models overa wide range of cavitation numbers especially in very low cavita-tion numbers such as σ¼0.07,0.05, 0.02.

2. Mathematical model

2.1. Governing equations

Continuity and momentum equations for a homogeneousmixture multiphase flow are given as follow:

∂t ρv� �þ∇: ρv� v� �¼ �∇pþ∇:s;

∂tρþ∇: ρv� �¼ 0 ð1Þ

The rate-of-strain tensor, D, is given as:

D¼ 12ð∇vþ∇vT Þ ð2Þ

where viscous stress tensor defined as s¼2μD.A multiphase flowmodeling should be used to describe a phase

change from liquid to vapor that happens under cavitation. In thisstudy, we consider a “two-phase mixture” method. This methoduses a local vapor volume fraction transport equation togetherwith a source term for the mass transfer rate between the twophases due to cavitation as follows:

∂tγþ∇: γv� �¼m: ð3Þ

The mixture density ρ and viscosity μ are given as follows:

μ¼ γlμlþ 1�γl� �

μϑ ð4Þ

ρ¼ ρlγlþ 1�γl� �

ρϑ ð5Þ

2.2. Mass transfer modeling

In this work, we employed two different mass transfer modelsKunz and Schnerr-Sauer. Kunz et al. (2000) recommended a semi-analytical cavitation model and currently is one of the masstransfer models implemented in the OpenFOAM. This model hastwo different terms for vapour production and destruction:

∂γ∂t

þ ∇!U ðγvÞ ¼ CdestρϑMinðPl�Pϑ;0Þγρlð0:5ρlu21Þt1

þCprodð1�γÞγ2

ρlt1ð6Þ

The first term in the right-hand expresses vapor productionthat is proportional to the pressure drop from the vapor pressureand reduces the amount of liquid phase. The other term considerscondensation and is proportional to the third power of the volumefraction (function γ increases). Cdest and Cprod are two empiricalcoefficients, whose values were set to 1000 and 75 in the Open-FOAM, according to the employed discretization of the equationsand so as to procure the best overall agreement with analyticalrelations and experimental data from various geometries (Roohiet al., 2013; Decaix and Goncalves, 2013; Roohi et al., 2016). Thecharacteristic time (mean flow time) is defined t1¼Dcavitator/U1.Kunz model reconstructs the cavity boundary quite correctly,especially in the closure region where there is a continuous flow ofthe re-entrant liquid jet and detachment of small vapor structuresfrom the cavity cloud (Yu et al., 2014). Kunz's model assumes amoderate rate of constant condensation to reconstruct this phe-nomenon quite accurately.

A mass transfer model was extracted by Sauer as follows (Sauer,2000; Schnerr and Sauer, 2001):

∂γϑ∂t

þ ∇!U ðγϑvÞ ¼ρϑρlρ

ð1�γϑÞγϑ3Rb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23

pϑ�p��

ρl

sð7Þ

This model is a function of bubble numbers per volume unitand bubble diameter. In Eq. (7), Rb is assumed to be the same forall of the bubbles given by:

Rb ¼3

4πn0γϑ

1�γϑ

� �1=3ð8Þ

The final expression of the Schnerr Sauer model is written inEq. (9).

_mþ ¼ ceρϑρlρ γϑð1�γϑÞ 3Rbffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðPϑ�PÞ

3ρl

qPoPϑ

_m� ¼ ccρϑρlρ γϑð1�γϑÞ 3Rbffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðP�PϑÞ

3ρl

qP4Pϑ

8<: ð9Þwhere Ce and Cc are empirical coefficients that are related toprocesses of condensation and vaporization, and the recom-mended value is 1 for both of them. In usual cases, the typicalbubble size Rb is 1�10�6 m in water. Schnerr–Sauer model isconstructed from the Rayleigh–Plesset equation and requiresestimation of the initial number of bubbles (n0), whose value wasset as 1.6�109 in OpenFOAM in order to obtain a suitable agree-ment with experimental results or analytical relations for differentgeometries (Roohi et al., 2013; Decaix and Goncalves, 2013; Roohiet al., 2016). According to Eq. (8), bubble radius is a function of thelocal vapor volume fraction at each location.

This model ignores bubble interactions, non-spherical bubblegeometries, and local mass–momentum transfer. It considered thebalance of forces over spherical bubbles. This issue becomesimportant in predicting cavity region, especially in the case ofsuper cavitation (Wu et al., 2005).

2.3. K-ω SST turbulence model

K-ω shear stress transport (SST) is one of the turbulencemodels employed in this study. Menter (1994) developed K-ω SSTeffectively to blend the accurate formulation of the K-ω model inthe near-wall region and, K-ε model in the far field. Turbulencekinetic energy and specific dissipation rate are calculated as fol-lows:

∂∂tðρkÞþ ∂

∂xjðρkujÞ ¼

∂∂xj

μþ μtσk3

� �∂k∂xj

� �þτij

∂ui∂xj

�β�ρkω: ð10Þ

∂ðρωÞ∂t

þ∂ðρujωÞ∂xj

¼ ∂∂xj

μþ μtσω3

� �∂ω∂xj

� �þω

kα3τij

∂ui∂xj

� ��β3ρω2

þð1�F1Þ2ρ1

ωσω2∂k∂xj

∂ω∂xj

ð11Þ

The model coefficients such as α3;β3;σk3;σω3 are linear com-binations of the corresponding coefficients of the K-ω and mod-ified K-ε turbulence models as follows:ψ ¼ F1ψ kωþ 1�F1ð Þψ kε; α3 ¼ F1α1þ 1�F1ð Þα2 ð12Þ

k�ω : α1 ¼ 5=9; β1 ¼ 3=40; σk1 ¼ 2; σω1 ¼ 2; β� ¼ 9=100k�ε : α2 ¼ 0:44; β2 ¼ 0:0828; σk2 ¼ 1; σω2 ¼ 1=0:856;Cμ ¼ 0:09

τij ¼ μt 2sij�23∂uk∂xk

δij

� ��23ρkδij ð13Þ

S¼ffiffiffiffiffiffiffiffiffiffiffiffi2sijsij

q; sij ¼

12

∂ui∂xj

þ∂uj∂xi

� �ð14Þ

where S denotes the magnitude of the strain rate and sij is thestrain rate tensor.


F1 ¼ tanhðΓ4Þ ð15Þ

Γ ¼ min maxffiffiffik

p

β�ωy;500νωy2

!;4ρσω2kCDkωy2

!ð16Þ

CDKω ¼ max 2ρσω21ω

∂k∂xj

∂ω∂xj

;10�20� �

ð17Þ

This model mixed the Wilcox K-ω and the Launder–Spalding K-ε model advantages; however, it cannot predict the starting pointand the amount of the flow separation from smooth surfaces, dueto the over-prediction of the eddy-viscosity, i.e., the transport ofthe turbulence shear stress is not correctly taken into account.Adding a limiter to the formulation of the eddy-viscosity, anappropriate transport behavior could be obtained:

μt ¼ ρk

max ω; SF2ð Þð18Þ

where S is an invariant measure of the strain rate, and F2 is ablending function as follows:

F2 ¼ tanhðΓ22Þ ð19Þ

Γ2 ¼ max2ffiffiffik

p

β�ωy;500νωy2

!ð20Þ

When the underlying assumptions are not correct for free shearflow, F2 restricts the limiter to the wall boundary layer.

Following Menter et al. (2003), OpenFOAM uses a blendingfunction depending on yþ for the near- wall treatment. Thesolution for ω in the linear and the logarithmic near-wall regionare:

ωlog ¼1

0:3κuτy;ωVis ¼

6ν0:075y2

ð21Þ

The above equations are rewritten in terms of yþ to deduce asmooth blending function:

ω1ðyþ Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2Visðyþ Þþω2log ðyþ Þ

qð22Þ

A similar formulation is used for the velocity near the wall:

uVisτ ¼U1yþ

;ulogτ ¼U1

1κ lnðyþ Þþc

ð23Þ

uτ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðuVisτ Þ4þðulogτ Þ4

4q

ð24ÞThis formulation expresses the relation between the velocity

near the wall and the wall shear stress. A zero flux boundarycondition is applied for the k-equation, which is correct for boththe low-Re and the logarithmic limit. This model exploits therobust near wall formulation of the underlying k-ω model andswitches automatically from a low-Reynolds number formulationto a wall function treatment based on the grid density (Menteret al., 2003).

2.4. Large eddy simulation

Large eddy simulation (LES) turbulence approach is based oncomputing the large, energy-containing structures that are deter-mined on the computational grid, while the smaller sub-grideddies are modeled. LES equations are theoretically derived fromEq. (1) (Fureby and Grinstein, 2002). In LES, all variables, i.e., f, aresplit into grid scale (GS) and subgrid scale (SGS) components, i.e.,where f ¼ G � f is the GS component, G¼G(X,Δ) is the filter func-tion, and Δ¼Δ(x) is the filter width (Ghosal, 1996). The LESequations can be expressed as:

∂t ρv� �þ∇: ρv � v� �¼ �∇pþ∇: s�Bð Þ;

∂tρþ∇: ρv� �¼ 0 ð25Þ

where the rate-of-strain tensor is expressed as D¼ 12 ∇vþ∇vT� �

and unresolved transport term B can be exactly decomposed as(Bensow and Fureby, 2007):

B¼ ρ:ðv � v�v � vþ ~BÞ ð26Þwhere ~B needs to be modeled. The most common subgrid mod-eling approaches utilize an eddy or subgrid viscosity, νSGS, whereνSGS can be computed with a wide variety of methods. In eddy-viscosity models,

B¼ 23ρkI�2μk ~DD ð27Þ

In the current study, sub-grid scale terms are modeled using“one equation eddy viscosity” model. It should be noted once wecompared the most popularly employed SGS model, i.e., stan-dard “Smagorinsky” with “one equation eddy viscosity” in manyaspects such as the computational costs and cavity cloud shape.it is observed that computational costs could decrease up to30%; however, one important point should be noted: as thereare not any experimental pictures of the cavity shape to com-pare the vapor shedding, fluctuating cavity behavior, or re-entrant jet with numerical solutions obtained from one equa-tion eddy viscosity model or Smagorinsky SGS model, it isappropriate to use a more accurate SGC approach even though itrequires higher computational costs. However, if the aim ofcomputation is to compare general properties of the cavitycloud such as length, diameter, and pressure drag coefficient,the simulation should be performed with less expensive SGSmodels as these parameters are insensitive to the employed SGSmodel. In order to obtain k, one-equation eddy-viscosity model(OEEVM) uses the following equation:

∂ ρk� �þ∇: ρk ~v� �¼ �B: ~Dþ∇:ðμ∇kÞþρε ð28Þε¼ cεk3=2=Δ ð29Þ

μk ¼ ckρΔffiffiffik

pð30Þ

where Cε and Ck are set as 1.048 and 0.094 in OpenFOAM,respectively.

A blending function is employed in the OpenFOAM's LES tomanage different values of yþ for the first grid cell and its influ-ence in near-wall function selection (De Villiers, 2006; Damiánaand Nigro, 2010). The blending function is given by Spalding Lawas follows (Spalding, 1961):

yþ ¼ uþ1E

eκuþ �1�κuþ �1

2ðκuþ Þ2�1

6ðκuþ Þ3

� ð31Þ

where κ¼0.4187, E¼9, yþ ¼ y0uτ=ν and uþ ¼ u0=uτ . The principaladvantage of using such a unified wall function is that the first off-the-wall grid point can be located in the buffer or viscous regions(yþo30) without the loss of accuracy inherent in the logarithmicprofiles limited validity (De Villiers, 2006).

2.5. Volume of fluid model

The volume of fluid (VOF) method is adapted to reconstruct theinterface between the liquid and vapor phases. γ is the volumefraction of fluid 1 determined in the following form:

γ1 ¼1 fluid 1 Liquidð Þ0 fluid 2 Vaporð Þ

0oγo1 at the interface

8><>: ð32Þ


OpenFOAM uses the following compressive-velocity (vc) formof the VOF equation:

∂γ∂t

þ∇U γ v!

�

þ∇U ½ v!cγð1�γÞ� ¼ _m ð33Þ

where _m is given by the selected mass transfer model (Kunz andSauer), vc is the compressive velocity term suggested in Rusche(2002). The additional term in Eq. (33) is only active at the inter-face region and acts as a surface compression term. It improves theinterface resolution and limits the smearing of the interface. Formore details about compressive velocity VOF approach, see Refs.(Roohi et al., 2013; Klostermann et al., 2013).

3. Numerical method

3.1. OpenFOAM validation

Before simulating the cavitating flow behind a disk, we evalu-ate the accuracy of the OpenFOAM package in simulating anincompressible, turbulent, non-cavitating flow. Experimentalinvestigation of unsteady flow behavior and vortex sheddingbehind a non-cavitating disk at various Reynolds numbers werereported in Miau et al. (1997). Here, we set the Reynolds numberequal to Red¼2.8�104. Fig. 1 shows the computational domainaround the disk with a diameter of 0.1 m. This value is chosen inaccordance with the geometrical data of the water tunnel usedin Miau et al. (1997). For a typical time step, Fig. 2 compares theexperimental and numerical solution for normalized velocitycontours taken through the mid-plane of the disk. As the figureshows, suitable agreement is observed, especially for velocitycontours behind the disk for LES turbulence model. Compared toRANS, LES is a more viable too for predicting unsteady flows. TheLES approach has become increasingly popular in the CFD appli-cation because of this unique nature. Using RANS models, cavi-tating flows often exhibit significant unsteadiness and thereforerequire considerable resolution in time and space. In addition, thepredicted Strouhal number for LES was around 0.14, which iswithin 5.4% of the experimental value (Miau et al., 1997) reportedat this Reynolds number.

3.2. Solution domain and boundary conditions

Fig. 3-a shows the 3D computational domain of the hemi-spherical head-form body with the diameter of D¼0.2 m. It hasthe following geometrical parameters: the computational domainis 100D in length: body length of 80D and distance between thedomain inlet and the head of the body is 20D, and 40D in height.

Fig. 1. Computational domain around the disk considered for the non-cavitatingflow validation.

Fig. 3-b shows the 3D computational domain of cone cavitatorwith the diameter of D¼0.8284 m and the height of H¼1 m. Thecomputational domain is 120D in length (the length behind thecavitator is 100D and distance in front of the cavitator is 20D) and52D in height. The boundary conditions of both cases are similar toeach other as illustrated in Fig. 3. Two non-dimensional numbers:Reynolds number ðRe¼ ρ:U1:D=μÞ and cavitation numberðσ ¼ ðP1�PϑÞ=0:5ρU21Þ are considered. Reynolds numbers forhemispherical head-form body and conical cavitator are con-sidered 4.4�106 and 1.8�107, respectively.

3.3. Computational mesh and time discrimination

3.3.1. Grid dependency studyAs Figs. 4 and 5 illustrate, we used structured quadrilateral

meshes for both cases in this paper. We considered the geometryas three-dimensional although both of geometries are axisym-metric, and the inflow conditions are steady; however, the flow isthree-dimensional and unsteady due to the vortex sheddingbehind the body at the investigated Reynolds/cavitation numbers.Accuracy of results strongly depends on the mesh size near thebody and especially around the cavity closure regions. In the other

Fig. 2. Comparison of the normalized velocity contour behind a vertical disk.

Fig. 3. (a) The computational domain and boundary conditions hemispherical head-form body. (b) The computational domain and boundary conditions of cone cavitator.


word, since the interaction between the near body flow and cavityis crucial; the mesh near the wall of the test body should be wellrefined. Fig. 4-c shows that there is a dense mesh at the length ofL¼40D, 25D, 15D for cavitation numbers σ¼0.02, σ¼0.05 andσ40.1 respectively. The grid size is progressively increased in theother regions, where the variations of flow proportion are small.This method helps to save the computational cost and decreasethe total cell numbers.

Table 1 compares simulation results for conical cavity para-meters (length and diameter) obtained using different grid sizeswith the analytical data. The grid used for σ40.07 cases arerelatively coarse compared to those used for σ¼0.02. From Table 1,it is observed that Grid 2 (which is 1�106 cells for σ40.07 and2.4�106 cells for σ¼0.02) provides close solutions for predictingdifferent characteristic of the cavity behind the conical cavitator.Therefore, we performed our simulations using Grid 2. For hemi-spherical head-form cylinder, we performed our simulations using2.4�106 cells for σ¼0.02 and 1.5�106 cells for σ40.05. Themaximum values of yþ were 241 and 275, and the mean values ofyþ were 8.59 and 7.9 for σ¼0.02 in conical cavitator case andhemispherical head-form cylinder case, respectively. For30oyþo300 at most positions, i.e., adjacent to the body region,OpenFOAM uses wall functions.

Table 2 compares the required time to reach steady state con-dition for different models for two cavitation numbers for the cone

cavitator. Simulations were performed in parallel using 4 and 12cores of Intel

s

Core™ i7-2600K CPU equipped with 16 GB memoryRAM in σ¼0.07 and 0.02, respectively. In these simulations, cal-culation is performed in a manner that Courant number in thecomputational domain does not surpass 0.175 for σ¼0.07 and 0.09for σ¼0.02.

3.4. Solution convergence and procedure

Fig. 6 shows the residuals convergence history of the liquidvolume fraction (alpha) and pressure (p) for the five last time stepsfor a typical test case at σ¼0.07. The iterative convergence ofunsteady flow problems depends on time step and number ofiterations per time step. Residual of each parameter is defined asthe normalized difference between the current and previous valueof that parameter. Residuals increase at the beginning of each timestep and then drop by two to three orders of magnitude.

In this work, we use PIMPLE algorithm. It is a merged PISO-SIMPLE algorithm for solving the pressure–velocity coupling. Thisalgorithm, as shown in Fig. 7, enables a more robust pressure–velocity coupling by coupling a SIMPLE outer-corrector loop with aPISO inner-corrector loop.is based on a coupling of a SIMPLE outer-corrector loop with a PISO inner-corrector loop. PIMPLE shows abetter numerical stability for larger time-steps or higher Courantnumbers or larger time-steps. Typically PISO and SIMPLE iterations

Table 1Effect of different grid sizes on the cavity length/cavitator diameter and cavity diameter/cavitator diameter behind the conical cavitator.

Meshfor σ40.07

Numberof cells(�106)

Lcavity/dcavitator Dcavity/dcavitator

Simulation(LES/Sauer)

Reichardt'sTheory


Reichardt'sTheory

Grid1 0.75 15.21 13.19 2.20 2.19Grid2 1 12.20 13.19 1.96 2.19Grid3 1.5 13.01 13.19 2.20 2.19

Meshforσ¼0.02

Numberof cells(�106)

Lcavity/dcavitator Dcavity/dcavitator


Reichardt'sTheory


Reichardt'sTheory

Grid1 1.8 67.25 55.53 4.30 3.96Grid2 2.4 61.46 55.53 3.91 3.96Grid3 3 59.42 55.53 3.92 3.96

Fig. 4. The computational structured meshes around hemispherical head-form body, figures show close-up view of the mesh near the body and refined mesh at differentlocations around the body.


Fig. 5. The computational structured meshes around conical cavitator, figures show close-up view of the mesh near the body and refined mesh at different locations aroundthe body.

Table 2Details of the computational cost of investigating test cases.

Turbulence and mass model Steady time (ms) Run time (h)

σ¼0.07 k-ω SST/Sauer 184 42Courant¼0.175 k-ω SST/Kunz 180 46Grid2Core numbers¼4 LES/Sauer 120 35

LES/Kunz 104 43σ¼0.02 k-ω SST/Sauer 450 58Courant¼0.09Grid2Core numbers¼12 LES/Sauer 433 79


Fig. 6. Convergence of residuals for five last time steps at σ¼0.07.

Fig. 7. PIMPLE flowchart.


σ=0.5

σ=0.4

σ=0.2

σ=0.1

σ=0.05

σ=0.02

rx

Fig. 8. 3D cavitation growth over hemispherical head-form cylinder, k-ω SST/Sauer models.


are required per time step. Here, we employed two PISO iterationsand one SIMPLE iteration.

4. Results and discussion

4.1. Hemispherical head-form body

Fig. 8 illustrates 3D views of isosurfaces of the volume fractioncontours of the cavitating flow over hemispherical head-form

Fig. 9. Comparisons of pressure coefficients with the experimental data (May,1975).

body over a broad range of cavitation number from σ¼0.5 toσ¼0.02, where the latter is reported for the first time in the openliterature. The k-ω SST turbulence model with Sauer mass transfermodel were employed. A considerable increase in the cavity lengthis observed at the lowest cavitation number. Frames in Fig. 8indicate that the growth of the cavity length could be obtainedfrom an inverse power law relation of the cavitation number. Atσ¼0.02, cavity has a fluctuating shape and shows an unsteadybehavior while the cavity shape is steady, regular and smooth atthe other cavitation numbers.

Fig. 10. Comparison of different turbulence and mass transfer models.

Fig. 11. Pressure coefficient over a broad range of cavitation number.


The pressure distribution, cp¼(p�p0)/0.5ρU21, on the hemi-spherical head-form body surface for various cavitation numbers(0.02rσr0.5) from numerical solutions (k-ω SST/Sauer, k-ω SST/Kunz, LES/Sauer, LES/Kunz) is compared with reported experi-mental data (Rouse and McNown, 1948) as shown in Figs. 9–11.The horizontal coordinate in these figures is the cavity length fromthe nose of the sphere non-dimensionalzed by the radius of thecylinder. The simulated results agree well with experimental data.It confirms that the models and numerical procedures used in thispaper are accurate for cavitation simulations. The pressure over-shoot at the cavity closure is more prominent in the computationalresults as the cavitation number decreases. Fig. 9 shows a generalvalidation of the employed schemes compared withexperimental data.

Fig. 10 shows the pressure coefficient distribution for variouscavitation models at σ¼0.1 and 0.2. Kunz model shows less sen-sitivity to the turbulence model, but Sauer model strongly inter-acts with the turbulence model. The maximum value of cp in k-ωSST/Sauer model has higher amount than other models. However,these models do not show a significant difference with each other.

Fig. 11 illustrate cp value over a broad range of cavitationnumbers with k-ω SST and Sauer cavitation model. An impor-tant point that can be found, is that a maximum overshoot in cpdistribution is brightly observed at different cavitation numbers.The level of overshot could be a function of the grid size,numerical discretization scheme, turbulence model andcavitation model.

Figs. 12–14 illustrate detailed features of the cavitating flowaround the hemispherical head-form body. Fig. 12 depicts thevolume fraction of the liquid phase for different turbulence and masstransfer models from 0.02rσr0.5. As soon as the cavity cloudreaches its fully developed condition, it shows a periodic behavior.The re-entrant jet is generated at the cavity closure. The re-entrantjet formed due to the adverse pressure gradient and shear stress. Weconsider the re-entrant jet in terms of two aspects: length andstrength. LES predicts a longer length for the re-entrant jet, but k-ωSST model predicts a weak re-entrant jet in all cases, an exception isσ¼0.1 case, where the re-entrant jet for two models are approxi-mately the same. However, it should be mentioned that turbulenceand mass transfer models simultaneously affect on the re-entrant jet

length and strength. As the cavitation number decreases, re-entrantjet becomes weaker. For the LES/Sauer model, the length of the re-entrant jet, non-dimensionalized by the cavity length (δ/L) at dif-ferent cavitation numbers of σ¼0.5, 0.2, 0.1, 0.05 are 0.356, 0.331,0.247 and 0.126, respectively. Frames (h–i) in Fig. 12 show theboundary-layer separation in the re-entrant jet region in the LESmodel. The separation point lactation is indicated by point C. Forinstance, separation point position is located at (X/R)¼12.88 in LES/Sauer solution. These observation demonstrates that selection of anappropriate turbulence model is a crucial issue in the correct pre-diction of the cavitation physics.

Fig. 13 illustrates cp contours for the same cases that werereported in Fig. 12. The pressure levels inside the cavity areobserved to stay close to the vapor pressure, giving an absolutevalue of cp approximately equal to the cavitation number σ. cpdistribution of each model has roughly the same behavior insidethe cavity cloud at different cavitation numbers. The value of cpat the cavity closure region with k-ω SST and Sauer are cp¼0.41,0.43, 0.28, 0.22 at σ¼0.2, 0.1, 0.05, 0.02, respectively. Themaximum overshoot in cp distribution is observed at σ¼0.1.

Fig. 14 shows velocity contours and streamlines for the samecases that reported in Fig. 12. In LES/Sauer solutions (right frames),the vortices are distributed from a single point as indicated by anarrow in each frame. On the left of the arrow, the flow direction isreversed from the streamwise direction; however, in the k-ω SST/Sauer solution, there are either two separate vortices (σ¼0.2) orone elongated vortex. The vortex strength is higher in k-ω SST/Sauer solution.

Fig. 15 shows the growth of the cavity length at differentcavitation regimes. The L/R values obtained from two numericalapproaches (k-ω SST/Sauer, LES/Sauer) agree with each other.The evolution of cavitation can be described in three mainstages: (1) the flow speed is relatively high over the cylinderbody. Once the pressure falls below the saturation vapor pres-sure, water begins to vaporize. Then, the cavity forms and growsgradually. (2) Re-entrant jet appears and flow moves back to theforepart of the body and cause the primary cavity to separatefrom the body. Then, the bubbles shedding start and bubblesmoves towards the tail of the body. Cavity cloud becomessmaller. (3) New vaporous cavity region starts growing again(after t4440 ms). This is the fluctuating characteristic of thecavitation evolution.

Figs. 16 and 17 show the cavity length and diameter at differentcavitation numbers. The parameters are normalized by the cylin-der diameter. Based on the current numerical solutions, we deduceinverse power law correlations for the cavity length and diameterover the hemispherical body as follows:

LR¼ 0:28σ1:60

ð0:02rσr0:5Þ ð34Þ

dR¼ 0:8σ0:4

ð0:02rσr0:5Þ ð35Þ

The cavity length and diameter exponentially increase withdecreasing the cavitation number. The results of different modelsare quite close to each other.

4.2. Conical cavitator

In this section, we consider cavitation behind a conical cavi-tator. Fig. 18 shows 3D view of the cavity cloud behind a conicalcavitator at cavitation number σ¼0.07 obtained using differentturbulence and mass transfer models. All models show nearly thesame length and diameter for the cavity clouds. Fig. 19 illustrate3D view of the cavitating flow in σ¼0.02. The cavity has a roughly

k-ω SST, Sauer σ=0.5 LES, Sauer σ=0.5

k-ωSST, Sauer σ=0.2 LES, Sauer σ=0.2


-ωSST, Sauerσ=0.02

LES, Sauerσ=0.02Fig. 12. Volume fraction contours for different turbulence and mass transfer models at 0.02rσr0.5.


LES, Sauer σ=0.1

LES, Kunz σ=0.1

K-ω SST, Sauer σ=0.1

K-ω SST, Kunz σ=0.1

Fig. 12. (continued)






k-ω SST, Sauer σ=0.02

LES, Sauer σ=0.02

Fig. 13. pressure coefficient contour for different mass transfer/turbulence models 0.02rσr0.5.






Fig. 14. Contour of velocity with streamline for different mass transfer/turbulence models 0.02rσr0.5.


pulsating surface in the k-ω SST/Sauer solution while LES/Sauerpredicts smooth and regular shape. Two models show a steadybehavior with a very long cloud length at σ¼0.02. The movie ofcavitation growth at σ¼0.02 is provided as the supplementarymovie of this paper.

Supplementary material related to this article can be foundonline at http://dx.doi.org/10.1016/j.oceaneng.2015.12.010.

The contours of volume fraction are illustrated in Fig. 20 as a2-D section in the z plane (z-axis is shown in Fig. 18) at differentcavitation numbers from various mass transfer and turbulencemodels. All models show nearly the same length and diameter forthe cavity clouds. A sharp interface is visible in the cavity domain

which is due to using an accurate VOF method. The re-entrant jetin the k-ω SST model is weaker than the LES model. It is becauseof that the k-ω SST model fails to properly predict the onset andamount of the flow separation and re-entrant jet from smoothsurfaces with suitable details, due to the over-prediction of theeddy-viscosity, i.e., the transport of the turbulent shear stress isnot properly taken into account (Baradaran Fard and Nikseresht,2012). In LES model, however, separation of scales within theflow is accomplished by a low-pass filtering of the Navier–StokesEquations. In contrast with RANS approaches, which are based onsolving for an ensemble average of the flow, LES naturally andconsistently allows for medium- to small-scale, transfer. This is

http://dx.doi.org/10.1016/j.oceaneng.2015.12.010

Fig. 17. Curve fitting for the cavity diameter over the hemispherical body.

Fig. 16. Curve fitting for the cavity length over the hemispherical body.

Fig. 15. Temporal growth of the cavity length.


an important property in order to be able to capture themechanisms governing the dynamics of the cavity such as re-entrant jet. LES/Sauer combination indicates a sharp re-entrantjet that is stronger compared to the LES/ Kunz model. Therefore,cavity closure region is unsteady in this model in comparisonwith other models. K-ω SST/Kunz solutions show a rough cavityboundary with a weak re-entrant jet. The frames in Fig. 20indicate that re-entrant weakens by decreasing the cavitationnumber.

Fig. 21 illustrates contours of cp for the same cases reported inFig. 20. Ahead of the cone, pressure increases due to the formationof the frontal stagnation point. Behind the body, the flow separatesat the sharp edges of the cone and the resultant drop in pressurecreates a vaporous cavity region.

Fig. 22 shows velocity contours with streamline for the samecases that reported in Fig. 20. Evidently, combination of turbulence

and mass transfer models determines the streamline and strengthdifferently. Vortex strength or circulation, defined as

RωUdA, i.e.,

product of the vorticity normal to the surface enclosed by thecavity region is higher in the LES/Sauer solution. In LES/Sauermodel, vortices are distributed from a division point. In k-ω SST/Kunz solution, there are two elongated vortex behind the cavitator,but at σ¼0.2, there are four strong vortices behind the cavitatorand in the cavity closure region.

Figs. 23 and 24 show predicted maximum cavity length anddiameter non-dimensionalized by the cavitator diameter. Weevaluate our result with an extensive set of Reichardt, Garabedianand Logvinovich (May, 1975) analytical relations. The set of ana-lytical relations are given as follows:

Reichardt's relations: This relation relates the steady cavitylength to the cavitation number:

Ld¼ σþ0:008σ 1:7σþ0:066ð Þ ð36Þ

Reichardt also derived a theoretical formula for the cavitydiameter that is a function of the drag coefficient and cavitationnumber.

dD¼ Cd

σ 1�0:132σ0:5� �" #0:5

ð37Þ

Also he presented the drag coefficient as follows:

Cd ¼ Cd0 1þσð Þ ð38Þ

where Cd0 is considered as 0.30275 for conical cavitators.Garabedian relations: Garabedian obtained asymptotic

relations for main properties of the cavities past cavitatorsas follows:

Ld¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCd ln 1σ

qσ

ð39Þ

Dd¼

ffiffiffiffiffiffiCdσ

rð40Þ

Logvinovich relations: Logvinovich derived formulas for prop-erties of the cavities depending on the cavitation number as

Fig. 18. 3D cavitation behind the conical cavitator, σ¼0. 07. a. K-ω SST, Sauer b. K-ω SST, Kunz c. LES, Sauer d. LES, Kunz.

Fig. 19. 3D cavitation behind the conical cavitator, σ¼0. 02. a. K-ω SST, Sauer b.LES, Sauer.


follows:

LDn

¼ 1:1σ

þ 4ð1�2αÞ1þ144α2

� ffiffiffiffiffiffiffiffiffiffiffiffiffifficx ln

1σ

rð41Þ

DDn

¼ffiffiffiffiffifficxκσ

rð42Þ

κ ¼ 1þ50σ1þ56:2σ ð43Þ

cx ¼ cx0þð0:524þ0:627αÞσ; 0rαr0:25;112

rαr12

ð44Þ

cx0 ¼ αð0:915þ9:5αÞ; 0oαo12

ð45Þ

where α is the cone half-angle. It should be noted that above-mentioned analytical relations were derived based on the poten-tial flow theories. A general agreement between numericalapproaches and analytical relation is observed, especially at lowercavitation numbers. However, the figures indicate that there isdeviation between analytical relations and numerical simulationsat higher cavitation numbers. It is expected as the analyticalrelations were derived from potential flow theory assuming fullydeveloped cavitation.

5. Conclusion

In this work, we investigated cavitation and supercavitationaround 3D hemispherical head-form body and a conical cavitatorover a wide range of cavitation numbers. A compressive VOFmethod was employed to track the interface between liquid andvapor phases. Benefiting from large eddy simulation (LES) and k-ωSST turbulence models as well as Kunz and Sauer mass transfermodels, we simulated cavitation accurately. Simulation is per-formed under the framework of the OpenFOAM package. One ofthe main contributions in this work is to present a correlationbetween the cavity length and diameter for hemispherical head-

Fig. 20. Volume fraction contours for the conical cavitator in a broad range of cavitation numbers.


form body. Secondly, we provided a detailed comparison of dif-ferent turbulence and mass transfer models in very low cavitationnumbers such as σ¼0.07, 0.05, 0.02 for both considered cases. Ournumerical results are compared with the experimental data forhemispherical head-form body and an extensive set of analyticalrelations for the conical cavity characteristics such as cavity length

and diameter. Combined models (k-ω SST/Sauer, k-ω SST/Kunz,LES/Sauer, LES/Kunz) predict approximately close results for gen-eral parameters of the cavity such as non-dimensional cavitylength and diameter. However, we showed that these modelsdiffer in predicting the structure of the re-entrant jet, distributionof pressure in the flow field, velocity contours and vortices in the

Fig. 21. Contour of pressure in different cavitation numbers.

Fig. 22. Contour of velocity with streamline.


Fig. 23. Comparison of the numerical cavity length/cavitator diameter withanalytical data.

Fig. 24. Comparison of the numerical cavity diameter/cavitator diameter withanalytical data.


cavity cloud, boundary layer separation, as well as cavitycloud shape.

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Investigation of cavitation around 3D hemispherical head-form body and conical cavitators using different turbulence and...IntroductionMathematical modelGoverning equationsMass transfer modelingK-ω SST turbulence modelLarge eddy simulationVolume of fluid model

Numerical methodOpenFOAM validationSolution domain and boundary conditionsComputational mesh and time discriminationGrid dependency study

Solution convergence and procedure

Results and discussionHemispherical head-form bodyConical cavitator

ConclusionReferences

investigation of cavitation around 3d hemispherical head...

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