Simulation of three-dimensional cavitation behind a disk using various ...

Applied Mathematical Modelling 40 (2016) 542–564

Contents lists available at ScienceDirect

Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Simulation of three-dimensional cavitation behind a disk using various turbulence and mass transfer models Ehsan Roohi ⇑, Mohammad-Reza Pendar, Amin Rahimi High Performance Computing (HPC) Laboratory, Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, P.O. Box 91775-1111, Mashhad, Iran

a r t i c l e

i n f o

Article history: Received 29 March 2014 Received in revised form 9 June 2015 Accepted 23 June 2015 Available online 2 July 2015 Keywords: Disk cavitator LES turbulence model Mass transfer model OpenFOAM Volume of fluid (VOF) Zwart model

a b s t r a c t In this study, we performed numerical investigations of the cavitating and supercavitating flow behind a three-dimensional disk with a particular emphasis on detailed comparisons of various turbulence and mass transfer models. Simulations were performed using the OpenFOAM package and flows at three different cavitation numbers, (r = 0.2, 0.1, and 0.05) were considered. Large eddy simulation (LES) and k—x shear stress transport turbulence approaches were coupled with various mass transfer model types (e.g., Kunz, Schnerr–Sauer, and Zwart models). The Zwart mass transfer model was added to the standard OpenFOAM package. A compressive volume of fluid method was used to track the interface between the liquid and vapor phases. Our numerical results in terms of the cavity length, diameter, and drag coefficient compared fairly well with experimental data and a broad set of analytical relations. Moreover, this study provides a better understanding of the cavitation dynamics behind disk cavitators. Our results indicate that the most accurate solutions will be obtained by applying an LES turbulence approach combined with the Kunz mass transfer model. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction Cavitation is a multi-phase and complex physical phenomenon, which occurs when the local liquid pressure becomes lower than its saturated vapor pressure [1]. This phenomenon appears often over marine vehicle applications such as submarines and marine propeller blades. To increase the performance of submarines by reducing viscous drag, underwater vehicles usually operate in cavitating conditions [2]. Cavitation is an unsteady, three-dimensional (3-D), and discontinuous or periodic phenomenon, which occurs during the formation, growth, and rapid collapse of bubbles [3]. A dimensionless number characterizes this process, i.e.,

r ¼ ðP1  P# Þ=0:5qU 21 , which is called the cavitation number [1]. If the moving body is accelerated to high speeds, supercavitation will occur, which refers to a long cavity that extends more than the body length and that closes in the liquid. There is a constant movement of a re-entrant liquid jet in the cavity closure section. Studying the cavitating flow behind a 3-D disk cavitator has been an interest of the scientific community in this field for many years. Challenging issues that need to be considered during the numerical simulation of a 3-D cavitation are: the

⇑ Corresponding author. Tel.: +98 (51) 38805136; fax: +98 (511) 8763304. E-mail address: [email protected] (E. Roohi). http://dx.doi.org/10.1016/j.apm.2015.06.002 0307-904X/Ó 2015 Elsevier Inc. All rights reserved.

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

Nomenclature Ae, Al parameters in near wall length scales B unresolved transport term in LES Ce, CK LES empirical constant coefficients Cd drag coefficient Cl parameter in near wall length scales CDKx positive portion of the cross-diffusion Cdest, Cprod Kunz mass transfer model constants Cd0 constant in the drag coefficient for a disk cavitator Cc constant between 0 and 1 Co courant number (C = U  dt/dx) D cavity diameter d cavitator diameter D rate of strain tensor ~D eddy diffusivity tensor D E constant coefficient Fe, Fc Zwart two empirical coefficients F1 turbulence Function (given by Eq. (15)) G filter function I unit tensor I turbulent intensity k kinetic energy L cavity length l liquid ll , le near wall length scales L characteristic length (disk diameter) d turbulence length scale _ m mass transfer rate between the phases n0 initial number of bubbles p pressure Re Reynolds number Rb radius of bubbles RB radius of a nucleation site rnuc nucleation site volume fraction S viscous stress tensor S strain rate us friction velocity mSGS subgrid scale viscosity t1 mean flow time U velocity magnitude uþ non-dimensional velocity Xj components of the Cartesian coordinate y distance between surfaces yw wall distance yþ non-dimensional wall distance r cavitation number 1 free stream value / volumetric flux j Von Karman Constant v velocity vector # vapor q density s wall shear stress x specific dissipation rate l viscosity D filter width c volume fraction sij shear stress tensor

543

544

lk

b ; rx2 ; log vis w –

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

viscosity of the vortex

a; b; cl constant coefficients for the k–x SST turbulence model logarithmic region viscous region wall averaging

selection of an appropriate mass transfer model, a technique for solving the advection equation of the free surface, and the application of an appropriate turbulence model. Kunz et al. [4], Singhal et al. [5], and Merkle et al. [6] proposed semi-analytical mass transfer models for cavitation. Sauer [7] and Yuan et al. [8] suggested a cavitation mass transfer model with some changes and improvements based on the classical Rayleigh equation. Zwart et al. [9] derived a mass transfer model based on the simplified Rayleigh-Plesset equation, which was divided into two parts: one for the liquid phase condensation and another for the vapor phase production. Senocak and Shyy [10] suggested an analytical model based on the balance of mass–momentum around the cavity interface. Chen et al. [11] applied a computer code including several homogenous-equilibrium cavitation models and a local linear k–e turbulence model to consider cavitation around a disk cavitator. The cavity shape and profiles were found to agree well with the analytical solutions and an experimental relation over a broad range of cavitation numbers. Typically, the volume of fluid (VOF) technique is utilized to solve the advection equation of the liquid volume fraction and to predict the cavity interface accurately. The VOF technique has been shown to be an appropriate tool for the numerical simulation of free-surface flows [12]. VOF can capture the cavity shape and track the cavity interface in an accurate manner. A review of the literature shows that VOF has been employed extensively for treating this class of problems [13–17]. Shang [13], Passandideh-Fard and Roohi [14], Frobenius and Schilling [15], Wiesche [16], and Bouziad et al. [17] all used the VOF method to simulate cavitation over various geometries. Cavitating flows occur under highly unsteady conditions at large Reynolds number. Thus, the selection of an appropriate turbulence model is one of the essential issues that must be addressed to achieve an accurate treatment of cavitation. Various approaches such as the large eddy simulation (LES) and k–x shear stress transport (SST) turbulence model have been utilized to incorporate turbulence effects in cavitating flows [3,18–21]. Compared with other turbulence models, LES can estimate the details of small-scale flow structures in cavitating flows with better fidelity. Kunz et al. [22] developed a cavitation model based on multiphase computational fluid dynamics and they examined the validity of the model for the flow around submerged axisymmetric objects with steady and transient natural cavitation. Numerical predictions showed that the pressure distribution, drag coefficient, and cavity shape agreed well with experimental data. Passandideh-Fard and Roohi [14] performed transient two-dimensional (2-D)/axisymmetric simulations of the cavitating flows behind disk and cone cavitators. Nouri et al. [23] used the finite volume technique to simulate the developing cavitation behind a disk with the Kunz cavitation model and by considering LES as the turbulence model. They compared their computational results with experimental data and the accuracy of their results demonstrated the capacity of combined cavitation and turbulence models for predicting cavity characteristics. Baradaran-Fard and Nikseresht [24] simulated the unsteady 3-D cavitating flows around a circular disk and a cone cavitator. Reynolds Averaged Navier–Stokes (RANS) equations and an additional transport equation for the liquid volume fraction were solved using the finite volume approach together with the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm. A k–x SST model was used for modeling turbulent flows and the results agreed well with experimental data and analytical relations. In contrast to [23,24], the present study uses 3-D simulations with a much more accurate numerical model, i.e., the VOF algorithm, as well as various sets for cavitation and advanced turbulence models. Guo et al. [25] simulated the cavitating flow around an underwater projectile with natural and ventilated cavitation based on the homogeneous equilibrium flow model, a mixture model for the transport equation, and a local linear low-Reynolds number k–e turbulence model. They showed that the morphology of the ventilated cavity was similar to that of the natural cavity at the same cavitation number, except at the tail of the cavity. Shang et al. [26] validated numerical simulations of the cavitating flow over a sphere based on experimental data. Roohi et al. [27] used the LES turbulence approach to investigate the cavitating flows over hydrofoils at various cavitation numbers using the OpenFOAM package. Morgut et al. [28] compared the Kunz, Zwart, and Full Cavitation Model mass transfer models using RANS turbulence equations over a hydrofoil. Shang [13] simulated cavitation around cylindrical objects such as a submarine within wide ranges of cavitation numbers from 0.2 to 1.0, where the k–x SST turbulence model, VOF method, and Sauer mass transfer model were employed to capture the cavitation mechanisms. Ji et al. [29–31] used the LES turbulence approach and Sauer mass transfer model to investigate the flow characteristics over various types of hydrofoils. In this study, we validated the ability of the open source package OpenFOAM to simulate the cavitation and supercavitation flow behind a disk cavitator, for which experimental and analytical data are widely available [2]. The VOF technique is employed to track the interface between the liquid and vapor phases. The VOF model implemented in OpenFOAM considers the effect of the surface tension force over the free surface. Both the LES and k–x SST turbulence model are used to simulate the cavitating flows behind the disk. Furthermore, we compare our results with those obtained using the Kunz, Sauer, and Zwart mass transfer models. In contrast to previous studies that focused on cavitation behind a disk, no 3-D simulations

545

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

have been reported for disk cavitation simulations based on the combination of accurate surface reconstruction by VOF, highly accurate turbulence models such as k–x SST and LES, and a comprehensive set of cavitation models, i.e., Kunz, Sauer, and Zwart. In addition, we provide detailed comparisons of the results obtained using the aforementioned turbulence and mass transfer models, which have not been reported in previous studies. 2. Mathematical model 2.1. Governing equations The governing continuity and momentum equations for a homogeneous mixture multiphase flow are given as:

@ t ðqv Þ þ r  ðqv  v Þ ¼ rp þ r  s; @ t q þ r  ðqv Þ ¼ 0:

ð1Þ

The rate-of-strain tensor is expressed as:

D¼

1 ðrv þ rv T Þ: 2

ð2Þ

A phase change from liquid to vapor occurs under cavitation, so multiphase flow modeling should be employed to describe the flow. In this study, we consider a ‘‘two-phase mixture’’ method, which uses a local vapor volume fraction transport equation and source terms for the mass transfer rate between the two phases due to cavitation, as follows:

_ @ t c# þ r  ðc# mÞ ¼ m;

ð3Þ

_ is the mass transfer rate between the phases. The mixture density q and the viscosity where m follows:

l are defined as

l ¼ cl ll þ ð1  cl Þl# ;

ð4Þ

q ¼ ql cl þ ð1  cl Þq# ;

ð5Þ

where the subscripts l and # indicate the liquid and vapor phases, respectively. 2.2. Mass transfer modeling We utilized three different mass transfer models in the current study, i.e., Kunz, Schnerr–Sauer, and Zwart. The semi-analytical cavitation model proposed by Kunz et al. [4] is implemented in the standard OpenFOAM package [32]. In this model, there are two different source terms for vaporization and condensation (i.e., bubble growth and collapse) during cavitation as follows:

C prod q# MinðP  P # ; 0Þc# C dest ð1  c# Þc2# MaxðP  P# ; 0Þ @ c# ~ þ r  ðc# v Þ ¼  : jP  P# j @t ql ð0:5ql u21 Þt1 ql t1

ð6Þ

The Kunz model assumes a moderate rate of constant condensation to reconstruct cavitation with a suitable accuracy. Cdest and Cprod are two empirical coefficients, the values of which were set to 1000 and 75, respectively, to ensure the best overall agreement with the experimental data obtained from various geometries [27]. The characteristic time (mean flow time) is defined as t1 = Dcavitator/U1, where Dcavitator is the diameter of the cavitator and U1 is the free-stream velocity. Pl and P # are the liquid pressure and vapor pressure, respectively. The Kunz model predicts the cavity region quite accurately, especially in the closure area, where this is a continuous flow of the re-entrant liquid jet and detachment of small vapor structures from the cavity cloud [27]. A mass transfer model was derived by Schnerr–Sauer, as follows [33].

@ c# ~ qq 3 þ r  ðc# v Þ ¼ # l ð1  c# Þc# Rb @t q

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 jp#  pj : 3 ql

ð7Þ

This model is a function of the bubble diameter and bubble numbers per volume unit. In Eq. (7), Rb is the radius of the bubbles and it is assumed to be a function of the vapor volume fraction, which can be expressed as follows:

Rb ¼



3 c# 4pn0 1  c#

13 :

ð8Þ

546

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

The Schnerr–Sauer model is based on the Rayleigh–Plesset equation and it requires the estimation of the initial number of bubbles (n0), the value of which was set to 1.6  109 [32]. Our literature review showed that there is not a unique value for n0 (see [8,15]). In fact, this parameter is adjusted by comparing the numerical results with experimental data or analytical solutions of cavitating flows. In this study, we used the default value in the OpenFOAM package for n0. Schnerr–Sauer solutions agree well with analytical solutions for cavity parameters using this default value (see Section 4.3). According to Eq. (8), the bubble radius is a function of the local vapor volume fraction at each location. This model ignores bubble interactions, non-spherical bubble geometries, and local mass–momentum transfer. The cavitation model suggested by Zwart et al. is governed by the following mass transfer equation [9]:

8 qffiffiffiffiffiffiffiffiffiffiffiffi c# Þq# 2ðP# PÞ > < Fe 3rnuc ð1 P < P# ; RB 3ql _ ¼ m qffiffiffiffiffiffiffiffiffiffiffiffi > 3 c q 2ðPP Þ # : Fc # # P > P# ; RB 3q

ð9Þ

l

where rnuc is the nucleation site volume fraction, RB is the radius of a nucleation site, and Fe and Fc are two empirical coefficients for the evaporation and condensation processes, respectively. According to [9], by default, the aforementioned coefficients are set as: rnuc = 5.0  104, RB = 1.0  106, Fe = 50, Fc = 0.01. This model is based on the simplified Rayleigh–Plesset equation for bubble dynamics [28]. In the above equation, the expressions for condensation and evaporation terms are not symmetric, and thus for evaporation, c# is replaced by rnuc ð1  c# Þ, which indicates that the nucleation site density decreases as the vapor volume fraction increases. 2.3. The k–x SST turbulence model We used the k–x SST model as one of the turbulence models. Menter [34] developed the k—x SST model to efficiently blend the accurate formulation of the k—x model in the near-wall region and the k—e model at the far field. In this model, the turbulence kinetic energy and specific dissipation rate, respectively, are given as follows:

@ @ @ ðqkÞ þ ðqkuj Þ ¼ @t @xj @xj @ðqxÞ @ðquj xÞ @ þ ¼ @t @xj @xj



lþ



lþ





lt @k @ui þ sij  b qkx; rk3 @xj @xj 





ð10Þ



lt @ x x @u 1 @k @ x þ a3 sij i  b3 qx2 þ ð1  F 1 Þ2q : rx3 @xj k @xj xrx2 @xj @xj

ð11Þ

The model coefficients ða3 ; b3 ; rk3 ; rx3 Þ are linear combinations of the corresponding coefficients of k—x and modified k—e turbulence models, which are calculated as follows:

a3 ¼ F 1 a1 þ ð1  F 1 Þa2 ;

w ¼ F 1 wkx þ ð1  F 1 Þwke ;

ð12Þ

rk1 ¼ 2; rx1 ¼ 2; b ¼ 9=100 k—e : a2 ¼ 0:44; b2 ¼ 0:0828; rk2 ¼ 1; rx2 ¼ 1=0:856; C l ¼ 0:09

k—x : a1 ¼ 5=9; 

sij ¼ lt 2sij  S¼

qffiffiffiffiffiffiffiffiffiffiffi 2sij sij ;

b1 ¼ 3=40;

 2 @uk 2 dij  qkdij 3 @xk 3

ð13Þ

  1 @ui @uj ; þ 2 @xj @xi

ð14Þ

sij ¼

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

F 1 ¼ tanhðC4 Þ;

ð15Þ !

C ¼ min max

pffiffiffi k 500m ; b xy xy2

! 4qrx2 k ; ; CDK x y2

ð16Þ

where y is the distance to the next surface and CDK x is the positive portion of the cross-diffusion term of Eq. (11):

  1 @k @ x CDK x ¼ max 2qrx2 ; 1020 : x @xj @xj

ð17Þ

This model benefits from the Wilcox k—x and the superiority of the Launder–Spalding k—e model, but it cannot predict the starting point and the amount of flow separation from smooth surfaces due to the over-prediction of the eddy-viscosity, i.e., the transport of the turbulence shear stress is not considered properly. By adding a limiter to the formulation of the eddy-viscosity, an appropriate transport behavior can be obtained:

547

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

lt ¼ q

k ; maxðx; SF 2 Þ

ð18Þ

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

F 2 ¼ tanhðC22 Þ;

ð19Þ

with the function C2 as:

! pffiffiffi 2 k 500m : ; b xy xy2

C2 ¼ max

ð20Þ

If the underlying assumptions do not hold for the free shear flow, F2 restricts the limiter to the wall boundary layer. Following Menter et al. [35], OpenFOAM uses a blending function that depends on y+ for the near-wall treatment. The solutions for x in the linear and the logarithmic near-wall region are as follows:

xlog ¼

1 us ; 0:3j y

xVis ¼

6m : 0:075y2

ð21Þ

The equations above are rewritten in terms of y+ to obtain a smooth blending function.

x1 ðyþ Þ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2Vis ðyþ Þ þ x2log ðyþ Þ:

ð22Þ

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

uVis s ¼

us ¼

U1 ; yþ

ulog s ¼

1

j

U1 ; lnðyþ Þ þ c

ð23Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4  log 4 4  þ us uVis : s

ð24Þ

This formulation expresses the relation between the velocity near the wall and the wall shear stress. A zero flux boundary condition is applied for the k-equation, which is correct for both the low-Re and the logarithmic limit. This model exploits the robust near wall formulation of the underlying k–x model and switches automatically from a low-Reynolds number formulation to a wall function treatment based on the grid density [35]. 2.4. Large eddy simulation The LES turbulence approach is based on computing the large, energy-containing structures that are resolved on the computational grid while the smaller sub-grid eddies are modeled. RANS models are based on solving an ensemble average of the flow properties, but LES typically allows for medium to small scale, transient flow structures. The cavitating flows are unsteady, so this feature of the LES is an important property that helps to capture the mechanisms that govern the dynamics of the formation and shedding of the cavity [36–38]. The LES equations are theoretically derived from Eq. (1) [39]. In the 0

ordinary LES, all of the variables, i.e., f, are split into grid scale (GS) and subgrid scale (SGS) components, i.e., f ¼ f þ f , where f ¼ G  f is the GS component, G = G(X, D) is the filter function, and D = D(x) is the filter width [40]. OpenFOAM utilizes the top-hat filter as follows [41]:

( Gðx; DÞ ¼

1 D

:

  if x 6 D2 ;

0 : otherwise:

ð25Þ

In OpenFOAM, the filter width D is set as equal to the grid spacing [41]. The top-hat (box) filter is an implicit filter that depends on the grid spacing, which in turn determines that the smallest scales are retained. All of the scales are modeled below the filter width D, the SGS. In this study, we used ‘‘smooth’’ delta, as described below. After the fluid moves from a coarse mesh area into a fine mesh area, or vice versa, an unphysical non-equilibrium state will exist for finite levels of resolved and unresolved turbulence [41]. Eddies that move from a coarse mesh to a finer mesh will be resolved more accurately on the finer grid, but the SGS turbulent length scale is directly related to the cell size, which is the same as the filter size, so there will be an abrupt decrease in the SGS viscosity. OpenFOAM applies smoothing to the distribution of the SGS turbulent length scale (or filter width D) near the grid refinement boundaries to alleviate this problem, i.e., D is smoothed using a biased wave scheme [41]. This scheme smoothes the distribution by increasing the SGS length scale for cells that neighbor larger cells, so the value of D cannot be smaller than the cell-derived value (see Fig. 1). The gradient of the smoothed distribution is fixed by an adaptable coefficient, CDs, as follows:

D ¼ maxðDP ; DN =C DS Þ;

ð26Þ

548

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

Fig. 1. Schematic showing the distribution of a smoothed filter width on a typical one-dimensional mesh [41].

where P and N denote the current cell and the neighbor cell, respectively. In practice, the value of CDS  1.15 is set so the equilibrium is obtained over 4–5 cells from the refinement boundary [42]. The LES equations, which are obtained after convolving the continuity and momentum equations, by the G = G(x, D) filtering, can be expressed as:

@ t ðqv Þ þ r:ðqv  v Þ ¼ rp þ r:ðs  BÞ; @ t q þ r:ðqv Þ ¼ 0;

ð27Þ

where the rate-of-strain tensor is expressed as D ¼ 12 ðrv þ rv T Þ. B is the unresolved transport term, which can be decomposed exactly as [43]:

B¼q





v  v  v  v þ B~

;

ð28Þ

~ needs to be modeled. The most common subgrid modeling approaches utilize an eddy or subgrid viscosity, mSGS, where B where mSGS can be computed with a wide variety of methods. In eddy-viscosity models,

B¼

2 q kI  2lkD~ D ; 3

ð29Þ

~ D is the SGS eddy diffusivity. In the current study, SGS where k is the SGS kinetic energy, l is the SGS eddy viscosity, and D terms are modeled using the one equation eddy-viscosity model (OEEVM). In order to obtain k, the OEEVM uses the following equation:

 kÞ @ðq ~ þ r:ðlrkÞ þ q  kv~ Þ ¼ B:D  e; þ r:ðq @t

ð30Þ

e ¼ ce k3=2 =D;

ð31Þ

pffiffiffi

lk ¼ ck q D k;

ð32Þ

where Ce and Ck are both empirical constant coefficients, which are set as 1.048 and 0.094, respectively, in the OpenFOAM package. In low-Reynolds number regions near walls, OpenFOAM applies Wolfshtein’s wall damping model to these equations to ensure the correct near-wall treatment [41]. This model employs Eqs. (31) and (32), but it replaces the length scale:

e ¼ ce k3=2 =le ; pffiffiffi

lk ¼ ck q ll k;

ð33Þ ð34Þ

where the length scales le and ll include the necessary damping effects in the near-wall region in terms of viscous wall units, as follows:

h i þ le ¼ C l yw 1  ey =Ae ;

ð35Þ

h i þ ll ¼ C l yw 1  ey =Al ;

ð36Þ

where C l ¼ jC 3=4 l , Al = 20–30, Ae = 2Cl [41]. It should be noted after we compared the computational costs of OEEVM with the most popular SGS model, i.e., Smagorinsky, we found that the computational costs decreased by up to 30%. However, it should also be noted that because

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

549

there are no experimental pictures of the cavity shape to compare the vapor shedding, fluctuating cavity behavior, or reentrant jet with the numerical solutions obtained from the OEEVM or Smagorinsky SGS models, then it is appropriate to use a more accurate SGC approach, although it incurs higher computational costs. By contrast, if the aim of computation is to compare the general properties of the cavity cloud such as the length, diameter, and pressure drag coefficient, then the simulation should be performed with less expensive SGS models because these parameters are insensitive to the SGS model employed. A blending function is employed in OpenFOAM’s LES to manage different values of y+ for the first grid cell and its influence on the near-wall function selection [41,44]. The blending function is given by Spalding Law as follows [41]:

yþ ¼ uþ



1 juþ 1 1 2 3 e  1  juþ  ðjuþ Þ  ðjuþ Þ ; E 2 6

ð37Þ

where j = 0.4187, E = 9, yþ ¼ yus =m, and uþ ¼ u=us . The main advantage of using this unified wall function is that the first off-the-wall grid point can be located in the buffer or viscous regions (y+ < 30) without the loss of accuracy that is inherent in the logarithmic profiles, which limit validity [41].

2.5. Volume of fluid model In this study, the VOF method [45] is adapted to capture the interface between the liquid and vapor phases. c1 is the volume fraction of fluid 1 defined by the following for:

8 fluid 1 > : 0 < c1 < 1 at the interface

ð38Þ

Assuming that the two fluids are incompressible, the transport equations of each volume fraction c1 and c2 are given as follows:

@ ci ~ þ r  ðv i ci Þ ¼ 0; i ¼ 1; 2: @t

ð39Þ

It is sufficient to only consider the transport equation for the volume fraction c1 :

@ c1 ~ þ r  ðv 1 c1 Þ ¼ 0: @t

ð40Þ

The velocity v1 is needed to solve this transport equation. In the original VOF method used by Hirt and Nichols [45], the velocity was assumed to be equal to the mixed velocity v.

@ c1 ~ þ r  ðv c1 Þ ¼ 0; @t

ð41Þ

v ¼ c1 v 1 þ c2 v 2 ¼ c1 v 1 þ ð1  c1 Þv 2 :

ð42Þ

Weller [46] first developed and implemented a conservative form of Eq. (41) in the OpenFOAM package, where he defined a compression velocity vc as follows:

 rc ~ v c ¼ min C c jv j; max ðjv jÞ : jrcj

ð43Þ

Using the definition of the relative velocities vr between phases c1 and c2 , and the definition of mixed velocity (Eq. (42)), we can obtain:

c1 v 1 ¼ c1 v þ c1 ð1  c1 Þv r :

ð44Þ

As a result, the compressive velocity formulation of the transport equation of c1 is obtained by inserting Eq. (44) into Eq. (40):

@c þ r  ðc ~ v Þ þ r  ½~ v c cð1  cÞ ¼ 0: @t

ð45Þ

where the explicitly fixed relative velocity (vr) appears in Eq. (44) in terms of the compression velocity vc given by Eq. (43). The last term in the equation above is known as artificial compression.

550

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

3. Numerical method 3.1. OpenFOAM validation Before simulating the cavitating flow behind a disk, we evaluated the accuracy of the OpenFOAM package by simulating an incompressible, turbulent, non-cavitating flow. Experimental investigations of the unsteady flow behavior and vortex shedding behind a non-cavitating disk at various Reynolds numbers were reported previously in [47]. In the present study, we set the Reynolds number equal to Red = 2.8  104. Fig. 2a shows the computational domain around a disk with a diameter of 0.1 m. This value was selected according to the geometrical data for the water tunnel used in [47]. In a typical time step, Fig. 2b compares the experimental and numerical solutions for the normalized velocity contours taken through the mid-plane of the disk. As shown in the figure, the agreement is adequate, especially for the velocity contours behind the disk. In addition, the predicted Strouhal number was around 0.14, which is within 5.4% of the experimental value reported at this Reynolds number [47]. 3.2. Solution domain, boundary, and initial conditions The 3-D computational domain and boundary conditions are shown in Fig. 3, where the disk is placed at the center of the water tunnel. The diameter of the disk is 2.54 cm. Two important non-dimensional numbers are considered, i.e., the Reynolds (Re) and cavitation (r) numbers. For the inlet free stream, these numbers can be obtained as follows:

Fig. 2a. Computational domain around the disk considered for the non-cavitating flow validation.

Fig. 2b. Comparison of the normalized velocity contour behind a vertical disk. Left: Experimental results [47]. Right: Numerical simulation obtained in this study at a typical time step of Red = 2.8  104.

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

551

Fig. 3. Computational domain and boundary conditions.

Re ¼

q  U1  D ; l

P1  P#

r¼

0:5qU 21

;

ð46Þ

ð47Þ

where P0 is the vapor pressure, P1 is the free stream vapor pressure, and U1 is the free stream velocity, which we set at 10 m/s. In all cases, we considered the Reynolds number as Re = 5  105. The outer boundary is located at a radial distance of 40D from the flow axis and it is treated as an impervious free-slip boundary. It should be noted that OpenFOAM does not provide the option of a fixed-pressure boundary condition, which would allow the flow to enter in free-stream conditions or leave the solution domain according to local mass continuity. The fixed-pressure condition has a significant economical advantage because it allows the outer boundary to be located much closer to the flow axis. To initialize k and x at the inlet, the turbulent intensity and mixing length were set as follows:

I ¼ 0:16 ðReÞ1=8 ;

ð48Þ

d ¼ 0:07L:

ð49Þ

Thus, the values of k and x can be computed as follows:

k¼

3 ðuav g IÞ2 ; 2

ð50Þ

1=2

x¼

k

1=4

Cl d

:

ð51Þ

The values obtained by the formula above for k and x at the inlet were also used to initialize the solution domain. At the inlet boundary, fluctuations in the large scale motion were neglected in the LES. For a more detailed description of the discretization schemes employed in two-phase flow solvers of OpenFOAM, see [48]. 3.3. Computational mesh and time discretization 3.3.1. Grid independency study The disk is not geometrically complex, so we used structured quadrilateral meshes. One-quarter of the geometry was considered to reduce the computational costs. The inflow conditions were steady and the geometry was axisymmetric, but the flow was actually 3-D and unsteady due to vortex shedding behind the disk at the Reynolds number investigated. The cavity length and the required domain length increase with a decrease in the cavitation number, so the simulation of the full 360-degree disk would be very time consuming for lower cavitation numbers. Thus, from a hydrodynamic engineering viewpoint, the essential aim of cavitation simulation is to accurately predict the cavity length, diameter, and drag coefficient. This can be achieved in a suitable manner using one-quarter of the geometry. Details of the grids investigated are reported in Table 1. The grid was refined in both the axial and normal directions. The mesh size near the disk has a critical effect on the accuracy of the results and the shape of the cavitation clouds. The effects of using three different grid sizes, as described in Table 1, on the cavity length, diameter, and drag coefficient over one period of cavitation are shown in Table 2, which

552

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

Table 1 Grid details for the cases investigated in this study. Grid

Region

Grid 1 (400,000) Grid 2 (900,000) Grid 3 (1,500,000)

Front surface

Upper surface

Behind surface

On the disk along the radius

Environment of the disk

Thickness of the disk

70 90 90

70 100 120

500 740 930

10 10 10

2 2 3

8 10 12

Table 2 Investigations of the effects of three different grid sizes on the cavity length/cavitator diameter, cavity diameter/cavitator diameter, and drag coefficient behind the disk, r = 0.05. Lcavity/dcavitator

Grid 1 Grid 2 Grid 3

Dcavity/dcavitator

CD

Simulation

Reichardt’s theory

Simulation

Reichardt’s theory

Simulation

Reichardt’s theory

42.9 37.7 37

32.73 32.73 32.73

4.26 4.33 4.3

4.26 4.26 4.26

0.882 0.938 0.924

0.882 0.882 0.882

demonstrate that Grid 2 provided accurate solutions for the cavity length while Grid 1 provided better accuracy in terms of the cavity diameter and drag coefficient. Therefore, to ensure better accuracy, we performed our simulations using Grid 2. We used a fine grid to obtain an appropriate y+, i.e., for a typical simulation case, the maximum value of y+ was 236 and the mean value of y+ was around 7.225. For 30 < y+ < 300, i.e., adjacent to the disk region, OpenFOAM uses wall functions. Fig. 4 shows the grid produced around the disk. The distance between the disk and outlet was set as 120D in order to ensure that there was a suitable distance between the outlet and cavity region. The concentration of cells is denser near the disk surface. Ò The simulation was performed in parallel using four cores of an Intel Core™ i7-2600K CPU equipped with 16 GB of memory RAM. Table 3 shows the total computational costs with a simulation time of 200 ms for r = 0.1. In this simulation, the calculations were performed so the Courant number did not exceed 0.175 for both of the turbulence models. It should be noted that the time step was computed automatically during the simulation according to the specified Courant number. Moreover, we found that that lower Courant number values did not change the main simulation results. 3.4. Solution procedure and convergence Our numerical solution followed the PIMPLE algorithm. Fig. 5 shows the PIMPLE flowchart, which is a merged PISO-SIMPLE algorithm for solving the pressure–velocity coupling.

Fig. 4. Left: close-up view of the mesh near the disk surface. Right: schematic of the computational domain.

553

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564 Table 3 Details of the computational cost for a typical test case at r = 0.1. Turbulence–mass transfer model

LES-Sauer

LES-Kunz

k—x SST-Sauer

Run time

44 h

62 h

39 h

START

Solve mass transfer equations Solve turbulence equations

Solve momentum equations

Number of iterations for inner pimple loop (PISO)

Number of iterations for outer pimple loop (SIMPLE)

Solve pressure correction equation

Pressure correction for NonOrthogonal cell

Number of iteration for pressure correct equation over NonOrthogonal cells

NO

YES

END Fig. 5. PIMPLE flowchart.

The PIMPLE algorithm facilitates a more robust pressure–velocity coupling where a SIMPLE outer-corrector loop is coupled with a PISO inner-corrector loop. This algorithm has better numerical stability for larger time steps or higher Courant numbers compared with PISO. Typically, one PISO and SIMPLE iterations are required per time step. However, we employed two PISO iterations and one SIMPLE iteration. Fig. 6(a) shows the convergence history of the residuals for the liquid volume fraction (alpha), pressure (p), and turbulence kinetic energy (k) for the entire run time of a typical test case at r = 0.1. The residual of each parameter was defined as the normalized difference between the current and previous value of that parameter. Fig. 6(b) and (c) show the variations in the residuals at the beginning and end of the simulations, respectively. The residuals increased at the beginning of each time step and then decreased by 2–3 orders of magnitude. Fig. 6(b) shows that at the beginning of the simulation, i.e., the first 10 time step, the parameters r and k needed only one iteration per time step to reach the pre-set threshold value, whereas pressure required four iterations per time step. It should be noted that the peaks in the p residuals occurred at the start of the iterations and the minimum residual occurred at the end of iterations at each time step. Moreover, the magnitude of the residuals was relatively high at the beginning of the simulation compared with the end, i.e., Fig. 6(c) shows that the residuals for p and r reached 106 and 1013, respectively. 4. Results and discussion We studied the cavitation characteristics of the disk cavitator at three different cavitation numbers: 0.2, 0.1, and 0.05. Cavity growth and detachment occurred under these regimes. Therefore, the primary task of suitable turbulence and mass transfer models is obtaining accurate predictions of the cavitation shape, dynamics, growth, and detachment mechanisms.

554

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

Fig. 6. Convergence of the residuals for a typical LES-Kunz simulation, r = 0.1: (a) entire run time, (b) first time steps, and (c) last time steps.

4.1. Comparison of various mass transfer/turbulence models 4.1.1. Cavity shape Figs. 7–9 show 3-D views of the cavitating flow behind the disk at r = 0.2, 0.1, and 0.05, respectively. We employed the LES turbulence model combined with the Kunz, Sauer, and Zwart mass transfer models. A reduction in the cavitation number led to a steady cavity shape. Both the cavity length and diameter increased as the cavitation number decreased. The Kunz model predicted a fluctuating and irregular cavity shape with unsteady behavior, whereas the Zwart model predicted a

Fig. 7. 3-D cavitation behind a disk using the LES with: (a) Kunz and (b) Zwart models; r = 0.2, t = 72 (ms).

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

555

Fig. 8. 3-D cavitation behind a disk using the LES with: (a) Kunz and (b) Zwart models; r = 0.1, t = 80 (ms).

Fig. 9. 3-D cavitation behind a disk using the LES with: (a) Kunz and (b) Zwart models; r = 0.05, t = 187 (ms).

steady, regular, and smooth cavity shape. Thus, the Zwart model could not predict the cavitation behavior as accurately as the Kunz model even when it was utilized together with the LES model. Videos of cavitation growth at r = 0.05 for all of cavitation models considered in this study are provided as supplementary information. Figs. 10–12 show 2-D views of the vapor volume fraction contour in the z-plane, where we compared the Kunz, Sauer, and Zwart mass transfer models. The main difference between these models was in terms of the behavior of re-entrant jets. The

556

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

(a) Kunz model, t=10 (ms)

(b) Kunz model, t=22 (ms)

(c) Sauer model, t=10 (ms)

(d) Sauer model, t=22 (ms)

(e) Kunz model,

(f) Sauer model,

(g) Zwart model,

t=83 (ms)

t=83 (ms)

t=83 (ms)

(h) Kunz model,

(i) Sauer model, t=91 (ms)

(j) Zwart model, t=91 (ms)

t=91 (ms) Fig. 10. Contour of the vapor phase (cavity region) for r = 0.2.

(a) Kunz model, t=81 (ms)

(b) Sauer model, t=81 (ms)

(c) Zwart model, t=81 (ms)

(d) Kunz model, t=87 (ms)

(e) Sauer model, t=87 (ms)

(f) Zwart model, t=87 (ms)

Fig. 11. Contour of the vapor phase (cavity region) for r = 0.1, LES model.

re-entrant jet length in the Sauer model was longer than that in the Kunz model, which caused some instability in the cavity shape. The Zwart model predicted the shortest length of the re-entrant jet. Thus, reducing the cavitation number led to a reduction in the re-entrant jet strength. Another difference was that the cavity cloud detached from the disk and lost its regularity and smoothness over time. This behavior was attributable to the cavity cloud instability at higher cavitation numbers. As shown in Fig. 12, the cavity cloud continuously covered the whole region behind the disk at t = 1 (ms) in the solution of the Kunz model, whereas it was divided into top and bottom branches in the Sauer model solution. As the cavity clouds grew over the time, the two branches connected with each other. In addition, the re-entrant jet developed as the cavity increased in size. Our simulation showed that a steady supercavity formed t = 40, 60, and 120 (ms) for cavitation numbers of r = 0.2,

557

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

(a) Kunz model, t=1 (ms)

(b) Sauer model, t=1 (ms)

(c) Zwart model, t=1 (ms)

(d) Kunz model, t=112 (ms)

(e) Sauer model, t=112 (ms)

(f) Zwart model, t=139 (ms)

(g) Kunz model, t=196 (ms)

(h) Sauer model, t=196 (ms)

(i) Zwart model, t=187 (ms)

Fig. 12. Contour of vapor phase (cavity region) for r = 0.05, LES model.

0.1, and 0.05, respectively. Two behaviors were observed as the steady supercavity formed: (a) inward movement of the reentrant jet and (b) separation of the subtle vapor bubbles into the main flow due to re-entrant jet movement. Fig. 13 shows the difference between the LES solution obtained when combined with the Kunz and Sauer mass transfer models, and using the k–x SST model in conjunction with the Sauer mass transfer model. At r = 0.2, the LES solutions exhibited instability and the cavity cloud lost its uniformity, whereas k–x SST predicted a smooth and regular shape attached to the disk. In addition, the k–x SST model predicted a weak re-entrant jet and it was unable to model the instabilities and cavity separation at the closure point of the cavity cloud. Thus, the selection of an appropriate turbulence model was more crucial at higher cavitation numbers, where the flow behavior was typically unsteady. As the cavitation number increased, the re-entrant jet became weaker and the cavity shape was smoother. 4.2. Pressure field Figs. 14 and 15 show the pressure contours obtained using various turbulence and mass transfer models. As the flow reached the disk, the pressure increased due to the formation of the frontal stagnation point. Behind the disk, the flow

(a)k-ω SST, Sauer model, t=91 (ms) σ = 0.2

(b) LES, Sauer model, t=91 (ms)

(c) LES, Kunz model, t=91 (ms)

(d)k-ω SST, Sauer model, t=87 (ms) σ = 0.1

(e) LES, Sauer model, t=87 (ms)

(f) LES, Kunz model, t=87 (ms)

(g)k-ω SST, Sauer model, t=196 (ms) σ = 0.05

(h) LES, Sauer model, t=196 (ms)

(i) LES, Kunz model, t=196(ms)

σ = 0.2

σ = 0.1

σ = 0.05

σ = 0.2

σ = 0.1

σ = 0.05

Fig. 13. Comparison of the cavity cloud with the LES and k–x SST turbulence models.

558

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

(a) LES, Kunz model,

(b) LES, Zwart model,

(c) k-ω SST, Sauer model,

(d) LES, Sauer model,

t=72 (ms) σ = 0.2

t=72 (ms) σ = 0.2

t=77(ms) σ = 0.2

t=77(ms) σ = 0.2

(e) LES, Kunz model,

(f) LES, Zwart model,

(g) k-ω SST, Sauer model,

(h) LES, Sauer model,

t=85 (ms) σ = 0.1

t=85 (ms) σ = 0.1

t=87(ms) σ = 0.1

t=87(ms) σ = 0.1

Fig. 14. Pressure contours obtained by different mass transfer/turbulence models,

r ¼ 0:2; 0:1.

(a) Sauer model, t=1 (ms)

(b) Kunz model, t=1 (ms)

(c) Zwart model, t=1 (ms)

(d) Sauer model, t=10 (ms)

(e) Kunz model, t=10 (ms)

(f) Zwart model, t=10 (ms)

(g)Sauer model, t=139 (ms)

(h) Kunz model, t=139 (ms)

(i) Zwart model, t=139 (ms)

(j) Sauer model, t=187 (ms)

(k) Kunz model, t=187 (ms)

(l) Zwart model, t=187 (ms)

Fig. 15. Pressure contours obtained by three mass transfer models and the LES solution,

r ¼ 0:05.

559

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

separated at the sharp edges and the resulting drop in pressure created a vaporous cavity area. A pressure gradient was created at the interface between the vapor phase and liquid phase, which was due to the pressure difference between the two phases and it was normal to the interface. In addition, cavity shedding and the condensation of cavity bubbles cause a high-pressure difference at the closure point of the cavity region. A sharp interface was visible around the cavity domain, which was attributable to the application of the VOF model. The pressure levels were similar with the Sauer and Zwart models. Fig. 14 shows that the Zwart model predicted a lower pressure at the closure point of the cavity cloud compared with the Kunz model. This behavior was a result of the lower rate of vapor condensation in this model. Moreover, the region with a constant vapor pressure expanded to a greater extent in the solution of the Zwart model compared with the Kunz model solution, whereas the pressure distribution was almost non-uniform in the Sauer/LES model solutions. Fig. 15 shows the pressure contours obtained by various mass transfer models at three time steps. At the starting point of cavity formation, the solutions of the Kunz and Zwart models were similar whereas the Sauer model predicted a dissimilar pressure contour because of the formation of two cavity cloud pieces (Fig. 12(b)). At t = 10 ms, the pressure contours of the Zwart model had a stronger re-entrant jet compared with the other models and the cavity length was shorter in this model. At t = 187 ms, the Sauer model exhibited more instability inside the cavity and the Zwart model had the smoothest pressure contour. A high-pressure region was observed at the closure point of the cavity, which was stronger than that in the Zwart model solution. 4.3. Cavity characteristics We compared the three dimensionless parameters computed in our simulations with those obtained from experiments and analytical relations (Figs. 16–18 and Tables 4–6). To validate our results, we considered a broad set of analytical equations. The cavitation number is the primary parameter in the following formulae [49]. May formula:

 L pffiffiffiffiffiffi ¼ C d 1:24r1:123  0:6 : d

ð52Þ

Reichardt’s relations:

  L r þ 0:008 d ; ¼ d rð1:7r þ 0:066Þ D

0:5 Cd ; 0:5 rð1  0:132r Þ

ð54Þ

50 45 Cavity lenght/Cavitator diameter, (L/d)

d ¼ D

ð53Þ

Richardt’s theory May theory Serberiakav theory Garabedian theory Experimental LES- Kunz LES- Sauer K- ω- Sauer LES- Zwart

40 35 30 25 20 15 10 5 0

0

0.05

0.1

0.15

Cavitation Number,

σ

0.2

Fig. 16. Comparison of the normalized cavity length for various cavitation numbers.

0.25

560

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

Cavity Diameter/Cavitator diameter, (D/d)

5

Richardt’s theory Garabedian theory Experimental LES- Kunz LES- Sauer K- ω- Sauer LES- Zwart

4.5

4

3.5

3

2.5

2

1.5

0

0.05

0.1 0.15 Cavitation Number, σ

0.2

0.25

Fig. 17. Comparison of the normalized cavity diameter for various cavitation numbers.

1.4 Richardt’s theory Garabedian theory Experimental LES- Kunz LES- Sauer K- ω- Sauer LES- Zwart

1.3 1.2

Drag Coefficient, (CD)

1.1 1 0.9

0.8

0.7

0.6

0

0.05

0.1

0.15

Cavitation Number,

σ

0.2

0.25

Fig. 18. Comparison of the drag coefficient for various cavitation numbers.

Table 4 Comparison of the error percentage for all of the computed parameters at r = 0.2. Parameter

L/d (Error%) D/d (Error%) CD (Error%)

Method LES Kunz

LES Sauer

k–x SST Sauer

LES Zwart

6.07 3.03 4.26

26.14 2.16 7.44

9.44 7.36 8.63

39.46 3.03 20.93

561

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564 Table 5 Comparison of the error percentage for all of the computed parameters at r = 0.1. Parameter

Method

L/d (Error%) D/d (Error%) CD (Error%)

LES Kunz

LES Sauer

k–x SST Sauer

LES Zwart

2.04 3.87 8

24.52 3.55 13.42

33.19 15.81 15.15

38.40 15.48 38.7

Table 6 Comparison of the error percentage of all of the computed parameters at r = 0.05. Parameter

Method

L/d (Error%) D/d (Error%) CD (Error%)

LES Kunz

LES Sauer

k–x SST Sauer

LES Zwart

0.21 7.75 9.3

15.3 1.64 10.43

44.33 10.8 10.32

43.14 14.55 11.9

C d ¼ C d0 ð1 þ rÞ;

ð55Þ

where Cd0 was considered to be 0.84 for the disk cavitator. Garabedian relations:

L ¼ d D ¼ d

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi C d ln r1

r rffiffiffiffiffiffi Cd

r

ð56Þ

;

ð57Þ

:

Serebryakov relations [50]:

L ¼ d

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C d0 ð1 þ rÞ ln 1:5 r

r

;

ð58Þ

where k = 0.95. L/d and D/d are the ratios of the cavity length and cavity diameter relative to the cavitator diameter, respectively. Figs. 16 and 17 show the cavity lengths and diameters at various cavitation numbers, where the parameters were normalized by the cavitator diameter. We compared the numerical results with the experimental data and analytical relations, which demonstrated their suitable accuracy. As mentioned earlier, the length and diameter of the cavity increased as the cavitation number decreased. Fig. 16 shows that the Kunz model combined with the LES turbulence approach obtained the most accurate prediction of L/d. Fig. 17 shows that both the Kunz and Sauer models combined with the LES obtained relatively accurate solutions. The experimental data and analytical/numerical solutions differed because the effects of the water tunnel walls on the experimental results were not considered [25]. Fig. 18 shows the variation in the drag coefficient versus the cavitation number, which indicate that the Kunz model combined with the LES model predicted the most accurate solutions. Tables 4–6 show the errors associated with different models, which are compared with the Reichardt analytical relations. These tables show that the Zwart model yielded the maximum error in the cavity length but its accuracy was acceptable for the cavity diameter. The most accurate solutions for the cavity length and drag coefficients were obtained with the Kunz/LES approach, while the Sauer/LES model predicted the cavity diameter most precisely. 5. Conclusions In this study, we employed a finite volume solver together with the VOF interface capturing method, the LES or k–x SST turbulence model, and the Kunz, Sauer, and Zwart mass transfer models to capture the unsteady cavitation and supercavitation flow behind a 3-D disk cavitator at various cavitation numbers. The simulations were performed using the OpenFOAM package. The main innovation of the present study is that we provided a detailed comparison of the different

562

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

sets of turbulence and mass transfer models. In addition, the Zwart mass transfer model was added to the OpenFOAM framework. We studied the unsteady growth of the cavity while considering the pressure behavior. We validated our numerical results using experimental data and analytical relations for the cavity length, diameter, and drag coefficient, which demonstrated their suitable accuracy. We found that the most accurate solutions for the cavity length and drag coefficient were obtained by applying the LES turbulence model together with the Kunz mass transfer model. The Sauer model combined with the LES turbulence model obtained the most accurate predictions of the cavity diameter results. Acknowledgements The authors acknowledge financial support from the Ferdowsi University of Mashhad under grant No. 30000. The authors also acknowledge Dr Eugene de Villiers from ENGYS, UK, and Dr Santiago Marquez Damian, from CIMEC-CONICET, Argentina for very helpful discussions about turbulence modeling in OpenFOAM. We also acknowledge Mr. Omid Ejtehadi, PhD Researcher at Gyeongsang National University (GNU), and Dr. Craig White, from Glasgow University, UK, for reading and improving the language of the paper. We also acknowledge the anonymous reviewers of this paper for very helpful suggestions, which improved the quality of the manuscript.

Appendix A One of the improvements employed in this study is the inclusion of the Zwart mass transfer model in the OpenFOAM package. The Zwart solver was written as follows: alpha 1: Liquid volume fraction rho 1: liquid density rho 2: vapor density

C c ¼ 0:01 C v ¼ 50

Foam::tmp Foam::phaseChangeTwoPhaseMixtures::SchnerrSauer::pCoeff ( const volScalarField& p ) const { volScalarField limitedAlpha1(min(max(alpha1_, scalar(0)), scalar(1))); return (3 * rho2()) * sqrt(2/(3 * rho1())) *(1.0/dNuc_)/(sqrt(mag(p  pSat()) + 0.01 * pSat())); }

Foam::Pair Foam::phaseChangeTwoPhaseMixtures::SchnerrSauer::mDotAlphal() const { const volScalarField& p = alpha1_.db().lookupObject (‘‘p’’); volScalarField limitedAlpha1(min(max(alpha1_, scalar(0)), scalar(1))); volScalarField pCoeff(this->pCoeff(p)); return Pair ( Cc_⁄pCoeff ⁄ max(p  pSat(), p0_), Cv_⁄0.0005 ⁄ pCoeff ⁄ min(p  pSat(), p0_) ); }

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

563

Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.apm. 2015.06.002. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]

C.E. Brennen, Cavitation and Bubble Dynamics, Oxford University Press, Oxford, UK, 1995. M. Self, J.F. Ripken, Steady state cavity studies in free-jet water tunnel, St. Anthony Falls Hydrodynamic Laboratory Report No. 47, 1955. G. Wang, S.M. Ostoja, Large eddy simulation of a sheet/cloud cavitation on a NACA0015 hydrofoil, Appl. Math. Model. 31 (2007) 417–447. R.F. Kunz, D.A. Boger, D.R. Stinebring, T.S. Chyczewski, J.W. Lindau, H.J. Gibeling, A preconditioned Navier–Stokes method for two-phase flows with application to cavitation, Comput. Fluids 29 (2000) 849–875. N.H. Singhal, A.K. Athavale, M. Li, Y. Jiang, Mathematical basis and validation of the full cavitation model, J. Fluids Eng. 124 (2002) 1–8. C.L. Merkle, J. Feng, P.E.O. Buelow, Computational modelling of the dynamics of sheet cavitation, in: Proceedings of the Third International Symposium on Cavitation, (CAV1998), Grenoble, France, 1998. J. Sauer, Instationaren kaviterende Stromung – Ein neues Modell, baserend auf Front Capturing (VoF) and Blasendynamik (PhD thesis), Universitat Karlsruhe, 2000. W. Yuan, J. Sauer, G.H. Schnerr, Modelling and computation of unsteady cavitation flows in injection nozzles, J. Mech. Ind. Eng. 2 (2001) 383–394. P. Zwart, A.G. Gerber, T. Belamri, A two-phase model for predicting cavitation dynamics, in: ICMF 2004 International Conference on Multiphase Flow, Yokohama, Japan, 2004. I. Senocak, W. Shyy, Evaluation of cavitation models for Navier–Stokes computations, in: Proceeding of FEDSM 02, ASME Fluid Engineering Division Summer Meeting, Montreal, Canada, 2002. Y. Chen, C.J. Lu, L.P. Xue, Validation of HEM-based cavitation for cavitation flows around disk, in: New Trends in Fluid Mechanics Research, Proceedings of the 5th International Conference on Fluid Mechanics, Shanghai, China, 2007. M. Passandideh-Fard, E. Roohi, Coalescence collision of two droplets: bubble entrapment and the effects of important parameters, in: Proceeding of the 14th Annual (International) Mechanical Engineering Conference, Isfahan, Iran, 2006. Z. Shang, Numerical investigations of supercavitation around blunt bodies of submarine shape, Appl. Math. Model. 37 (20) (2013) 8836–8845. M. Passandideh-Fard, E. Roohi, Transient simulations of cavitating flows using a modified volume-of-fluid (VOF) technique, Int. J. Comput. Fluid Dyn. 22 (2008) 97–114. M. Frobenius, R. Schilling, R. Bachert, B. Stoffel, G. Ludwig, Three-dimensional unsteady cavitating effects on a single hydrofoil and in a radial pumpmeasurement and numerical simulation, in: Proceedings of the 5th International Symposium on Cavitation, Cav03-GS-9-005, Osaka, 2003. S. Wiesche, Numerical simulation of cavitation effects behind obstacles and in an automotive fuel jet pump, Heat Mass Transfer 41 (2005) 615–624. A. Bouziad, M. Farhat, F. Gunning, K. Miyagawa, Physical modelling and simulation of leading edge cavitation, application to an industrial inducer, in: Proceedings of the 5th International Symposium on Cavitation, Cav03-OS-6-014, Osaka, Japan, 2003. D. Li, M. Grekula, P. Lindell, A modified k–x SST turbulence model to predict the steady and unsteady sheet cavitation on 2D and 3D hydrofoils, in: Proceedings of the 7th International Symposium on Cavitation, CAV2009, 2009. T. Huuva, Large Eddy Simulation of Cavitating and Non-cavitating Flow (PhD thesis), Chalmers University of Technology, Sweden, 2008. D. Liu, S. Liu, Y. Wu, H. Xu, LES numerical simulation of cavitation bubble shedding on ALE 25 and ALE 15 hydrofoils, J. Hydrodyn. 21 (6) (2009) 807– 813. N.X. Lu, R.E. Bensow, G. Bark, LES of unsteady cavitation on the delft twisted foil, J. Hydrodyn., Ser. B 22 (5) (2010) 784–791. R.F. Kunz, T. Chyczewski, D. Boger, D. Stinebring, H. Gibeling, T.R. Govindan, Multi-phase CFD analysis of natural and ventilated cavitation about submerged bodies, in: Proceedings of FEDSM 99, 3rd ASME/JSME Joint Fluids Engineering Conference, 1999. N.M. Nouri, M. Moghimi, S.M.H. Mirsaeedi, Numerical simulation of unsteady cavitating flow over a disk, Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci. 224 (2010) 1245–1253. M. Baradaran Fard, A.H. Nikseresht, Numerical simulation of unsteady 3D cavitating flows over axisymmetric cavitators, Sci. Iran. Trans. B: Mech. Eng. 19 (2012) 1258–1264. J. Guo, C. Lu, Y. Chen, Characteristics of flow field around an underwater projectile with natural and ventilated cavitation, J. Shanghai Jiaotong University (Science) 16 (2) (2011) 236–241. Z. Shang, D.R. Emerson, G.U. Xiaojun, Numerical investigations of cavitation around a high speed submarine using OpenFOAM with LES, Int. J. Comput. Methods 9 (2012) 1250040–1250054. E. Roohi, A.P. Zahiri, M. Passandideh-Fard, Numerical simulation of cavitation around a two-dimensional hydrofoil using VOF method and LES turbulence model, Appl. Math. Model. 37 (2013) 6469–6488. M. Morgut, E. Nobile, Numerical predictions of cavitating flow around model scale propellers by CFD and advanced model calibration, J. Rotating Mach. 2012 (2012) 618180. B. Ji, X.W. Luo, R.E.A. Arndt, X.X. Peng, Y.L. Wu, Large eddy simulation and theoretical investigations of the transient cavitating vortical flow structure around a NACA66 hydrofoil, Int. J. Multiph. Flow 68 (2015) 121–134. B. Ji, X.W. Luo, R.E.A. Arndt, Y.L. Wu, Numerical simulation of three dimensional cavitation shedding dynamics with special emphasis on cavitation vortex interaction, Ocean Eng. 87 (2014) 64–77. B. Ji, X.W. Luo, Y.L. Wu, X.X. Peng, Y.L. Duan, Numerical analysis of unsteady cavitating turbulent flow and shedding horse-shoe vortex structure around a twisted hydrofoil, Int. J. Multiph. Flow 51 (2015) 33–43. OpenFOAM, . G.H. Schnerr, J. Sauer, Physical and numerical modeling of unsteady cavitation dynamics, in: Proceedings of 4th International Conference on Multiphase Flow, New Orleans, USA, 2001. F.R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA J. 32 (8) (1994) 1598–1605. F. Menter, J. Carregal Ferreira, T. Esch, B. Konno, The SST turbulence model with improved wall treatment for heat transfer predictions in gas turbines, in: Proceedings of the International Gas Turbine Congress, Tokyo, Japan, IGTC2003-TS-059, 2003, pp. 2–7. R.E. Bensow, G. Bark, Simulating cavitating flows with LES in OpenFOAM, in: J.C.F. Pereira, A. Sequeira (Eds.), Lisbon European Conference on Computational Fluid Dynamics, ECCOMAS CFD, Portugal, 2010. N.X. Lu, Large Eddy Simulation of Cavitating Flow on Hydrofoil (Licentiate of Engineering thesis), Department of Shipping and Marine Technology, Chalmers University of Technology, Göteborg, Sweden, 2010. P. Sagaut, Large Eddy Simulation for Incompressible Flows, third ed., Springer, New York, 2006. C. Fureby, F. Grinstein, Large eddy simulation of high-Reynolds number free and wall-bounded flows, J. Comput. Phys. 181 (2002) 68–97. S. Ghosal, An analysis of numerical errors in large-eddy simulations of turbulence, J. Comput. Phys. 125 (1996) 187–206. E. De Villiers, The Potential of Large Eddy Simulation for the Modelling of Wall Bounded Flows (PhD thesis), Department of Mechanical Engineering, Imperial College, London, UK, 2006. A.G. Kravchenko, P. Moin, R. Moser, Zonal embedded grids for numerical simulations of wall-bounded turbulent flows, J. Comput. Phys. 127 (1996) 412–423.

564

E. Roohi et al. / Applied Mathematical Modelling 40 (2016) 542–564

[43] R.E. Bensow, C. Fureby, On the justification and extension of mixed methods in LES, J. Turbul. 8 (2007) N54. [44] S.M. Damiána, M.N. Nigro, Comparison of single phase laminar and large eddy simulation (LES) solvers using the OpenFOAM suite, in: Eduardo Dvorkin, Marcela Goldschmit, Mario Storti (Eds.), Computacional, vol. XXIX, Buenos Aires, Argentina, 2010, pp. 3721–3740. [45] F.H. Hirt, B.D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys. 39 (1981) 201–225. [46] H.G. Weller, A new approach to VOF-based interface capturing methods for incompressible and compressible flow, Technical Report, OpenCFD Ltd, 2008. [47] J.J. Miau, T.S. Leu, T.W. Liu, J.H. Chou, On vortex shedding behind a circular disk, Exp. Fluids 23 (1997) 225–233. [48] J. Klostermann, K. Schaake, R. Schwarze, Numerical simulation of a single rising bubble by VOF with surface compression, Int. J. Numer. Meth. Fluids 71 (8) (2013) 960–982. [49] A. May, Water entry and the cavity-running behavior of missiles, Final Technical Report NAVSEA Hydroballistics Advisory Committee, Silver Spring, MD, 1975. [50] V. Serebryakov, Supercavitation flows with gas injection prediction and drag redaction problems, in: Fifth International Symposium on Cavitation (CAV2003), Osaka, Japan, CAV 03-OS-7-003, 2003.