Computers & Fluids 40 (2011) 315–323

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Computers & Fluids

journal homepage: www.elsevier .com/ locate /compfluid

On the modified dispersion-controlled dissipative (DCD) scheme for computationof flow supercavitation

Z.M. Hu a, H.S. Dou a, B.C. Khoo b,⇑a Temasek Laboratories, National University of Singapore, Singapore 117411, Singaporeb Department of Mechanical Engineering, National University of Singapore, Singapore 119260, Singapore

a r t i c l e i n f o

Article history:Received 4 January 2010Received in revised form 1 July 2010Accepted 1 October 2010Available online 8 October 2010

Keywords:Flow supercavitationDCD schemeFlux splittingInterfaceCavitation model

0045-7930/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compfluid.2010.10.001

⇑ Corresponding author.E-mail addresses: tslhz@nus.edu.sg (Z.M. Hu), mpe

a b s t r a c t

In the present study, a robust scheme initially proposed for capturing shock waves in gasdynamics, thedispersion-controlled dissipative (DCD) scheme is modified to simulate supercavitating flows in theframework of one-fluid model. Due to the stiffness of the equations of state (EOS), the Steger-Warmingmethod becomes inapplicable and the Lax–Friedrichs method should be used for the numerical flux split-ting. Following the formulation, the updated scheme is validated by a series of comparisons with theo-retical and experimental data. The validation shows that the modified DCD scheme performs very wellfor supercavitating flows in the presence of steep gradients of both density and EOS. Several one-fluidcavitation models and speed of sound (SoS) models are compared for the numerical investigation ofsupercavitating flows.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Bubbles or cavities, which are filled with vapor, may emerge inthe initially homogeneous liquid if the surrounding pressurereduces to its saturated vapor pressure (SVP). This kind of phasetransition process is referred to as cavitation. The driving mecha-nism for cavitation is not usually temperature related as for boilingbut a pressure drop generally controlled by the flow dynamics.Cavitation often happens at an extremely high speed where theliquid pressure can rapidly drop below the corresponding SVPalthough it can occur at any other speeds. In hydrodynamic applica-tions, cavitation is usually an unintended and undesirable phenom-enon: the cavitation bubbles typically do not sustain and mayimplode as the surrounding flow decelerates. The constant implo-sion of these small-scale cavitation bubbles causes material damage.As such, a key subject of decades of research on cavitation has beento prevent its inception or to ensure control of its evolution. On theother hand, the cavitation effects can be harnessed for usefulapplications. The induced cavity over an underwater object movingat a sufficient speed can extend as a single large bubble of vapor toenvelop the entire object, hence leading to flow supercavitation.Because the viscosity and density of vapor respectively are twoand five orders of magnitude lower than those of water, flow

ll rights reserved.

kbc@nus.edu.sg (B.C. Khoo).

supercavitation can lead to drastic reduction of the hydrodynamicdrag.

Research attention on flow supercavitation has been largelydrawn towards issues covering the unsteadiness of supercavitatingbubbles, the cavity instability due to vehicle planning or tail slap-ping, and the interaction between cavity and ventilation or propul-sion exhaust. Besides experimental and theoretical studies,numerical analysis has become a viable and promising approachfor understanding the fundamentals of supercavitating flows. Themain numerical difficulty in treating supercavitating flows, amongother multiphasic flows, arises in dealing with the dynamic inter-faces. The presence of large discontinuities in the thermodynamicproperties and equations of state (EOS) at material interfaces mayresult in numerical instabilities and nonphysical oscillations. Gener-ally, there have been developed two classes of methods to overcomethese difficulties. One of them is Sharp Interface Method (SIM) inwhich the interface is numerically treated as a sharp discontinuity.The other is Diffuse Interface Method (DIM) in which the interfaceis considered as a diffuse zone, like a contact discontinuity in gasdy-namic flows [1].

The popular class of SIM is based on the well known level-setfunction [2] to locate the interface in the framework of the frontcapturing methods (FCM) which have been successfully appliedin gasdynamic flows. To reduce numerical smearing and to preventnonphysical oscillations across the material interfaces, Fedkiwet al. proposed the original ghost fluid method (GFM) [3]. However,the original GFM was found to admit inaccuracies such as non-physical oscillations when especially applied to multi-fluid

316 Z.M. Hu et al. / Computers & Fluids 40 (2011) 315–323

problems, e.g., the impacting of a strong shock wave on a gas–water interface. As such, a modified ghost fluid method (MGFM)has been proposed by Liu et al. to deal with such limitations ofGFM with a series of successful applications [4,5]. The MGFMwas reported to satisfy the entropy condition across the materialinterfaces. A different SIM approach, front-tracking method(FTM), has been proposed to use explicit marker points to capturewaves and discontinuities [6,7]. However, this method also suf-fered from numerical diffusion or spurious oscillations at the inter-faces under some extreme conditions. In addition, as the fluidinterfaces are explicitly tracked by connected marker points, FTMbecomes very complicated for reconstructing complex interfacesand discontinuities. Unfortunately, none of these SIM methods isdeemed able to satisfactorily resolve the dynamically emerginginterfaces separating multiple fluids [1].

The Diffuse Interface Method (DIM) considers an interface as anumerically diffused zone. This method has been proved to be ableto solve the dynamical appearance of multi-fluid interfaces such asthe cavity boundary which is absent in an initially homogeneousliquid [1]. The challenge for this class of methods arises from deriv-ing a physically and mathematically consistent EOS for the multi-phasic fluids in the numerically diffused zone. Representativestudies using DIM are performed by Abgrall [8], Saurel and Abgrall[9], Murrone and Guillard [10], Coutier-Delgosha et al. [11], Liuet al. [12], and Goncalves and Patella [13] among others. Forcavitating flows, the two-phase methods can be further dividedinto three categories: one-fluid model or homogenous model,two-fluid model, and hybrid model.

The two-fluid model [14,15] assumes that both phases coexisteverywhere in the flowfield and each phase flow is governed byits own set of partial difference equations. Because the exchangeof mass, momentum and energy is explicitly treated in thisapproach, the two-fluid model can easily take into account thephysical details occurring at the interface. However, the two-fluidmodel is less often employed in unsteady cavitation flows becauseof the dearth of knowledge on the parameters associated withphase transition rates. A one-fluid model, on the contrary, treatsthe cavitating flow medium as a mixture of different fluids, how-ever, behaving as one. These models are based on the assumptionthat the phases are in locally kinetic and thermodynamic equilib-rium; i.e. unique velocity, temperature and pressure for bothphases. Therefore, one set of differential equations similar to thesingle-phase flow is used to govern the whole multi-fluid flow.Models of this kind have been extensively used to simulate cavitat-ing flows (for instance see [11–13,16–18] among others) becausethey are relatively easy to treat the dynamic inception and collapseof cavitating bubbles. However, physical phenomena relating tovaporization, condensation or other phase transition processesare assumed to complete instantaneously. Therefore, the one-fluidmodel cannot reflect any strong thermodynamic or kinetic non-equilibrium effects if present. Hybrid models are intermediatemodels between the one-fluid and two-fluid models. In these mod-els, each phase is governed by its own continuity equation with asource term accounting for the phase exchange rate. Other localkinetic and thermodynamic properties are treated the same wayas in one-fluid models. The hybrid models are employed in simu-lating cavitating flows (see [19–21] among others). Some difficul-ties lie in the determination of the source terms for phasetransition.

In the present study, a numerical scheme, dispersion-controlleddissipative (DCD), is modified to simulate supercavitating flows inthe framework of the one-fluid model and Diffuse Interface Meth-od (DIM). The DCD scheme was proposed by Jiang et al. intendingfor shock wave capturing [22,23]. The primary principle of thescheme aims at removing non-physical oscillation across strongdiscontinuities by making use of the dispersion characteristics of

the modified equation instead of adding artificial viscosity. TheDCD scheme has also been applied successfully to chemicallyreacting flows for compressible multi-component mixtures [24]and flows with strong shock wave interactions [25,26]. Such men-tioned applications have shown that the DCD scheme is fairlyrobust, computationally efficient, and capable of resolving strongdiscontinuities. The paper is organized as follows. In Section 2,we have the numerical models and the modified DCD scheme. Thisis followed by Section 3 on validation of the assembled code viacomparing with theoretical and experimental results. Several cav-itation models are compared for modelling supercavitating flows.Finally, a brief summary is given in Section 4.

2. Numerical models and DCD scheme formulation

2.1. Governing equations and numerical models

In the present study, the supercavitating flow is assumed to beisothermal, along with the locally homogeneous and equilibriumassumption. Therefore, the vapor and liquid phases share the samepressure, temperature and velocity while the energy equation canbe avoided in the governing equations. In two-dimensional Carte-sian coordinates, the compressible laminar Navier–Stokes (NS) sys-tem reads:

@U@tþ @F@xþ @G@yþ i

yS ¼ @Fv

@xþ @Gv

@yþ i

ySv ð1Þ

For planar and axisymmetric flows, the control parameter i takesvalue of 0 and 1, respectively. Here, the convective vectors U, F, Gand the geometric source term S are

U ¼ ½q;qu;qv�T ð2ÞF ¼ ½qu;qu2 þ p;quv �T ð3ÞG ¼ ½qv;quv ;qv2 þ p�T ð4ÞS ¼ ½qv ;quv ;qv2�T ð5Þ

In the above NS equations, the viscous vectors, Fv, Gv, and Sv, arenot listed for conciseness. The viscosity coefficient for the two-phase medium is l = alv + (1 � a)lw, where lv and lw are the vis-cosity coefficients for pure vapor and water at a given temperature.The void fraction, a ¼ q�qsw

qsv�qsw, varies from zero to unity between pure

liquid phase to pure vapor condition. Here qsv and qsw denote thevapor and water densities in the mixture and will be explainedlater. In the governing equations, q, p, u and v denote the averagedensity, pressure, velocities in x and y directions, respectively. Toclose system Eq. (1), an equation of state (EOS) is necessary todetermine pressure based on other flow variables. One of thechallenges in one-fluid methods is to define an appropriate EOSapplicable for the vapor, liquid phases and the mixture conditionin the numerical diffused zone. For instance, there are different ver-sions of barotropic EOS that directly link pressure p to density q formodelling cavitating or supercavitating flows. Examples are the cut-off model, Schmidt’s model [16], isentropic one-fluid model [12],sinusoidal model [13], and so on. Here, they are plotted in Fig. 1for an isothermal cavitation process in initially homogeneous waterat T0 = 295 K and p0 = 101,325 Pa. The Schmidt’s model [16] givesnegative pressure under this condition and is not represented onthe figure. Here, the saturation line is obtained using the formulaeprovided by IAPWS [27]. The abbreviations ‘sw’ and ‘sv’ correspondto the saturated water and saturated vapor conditions, respectively.Several of the one-fluid models which will be calibrated for flowsupercavitation in the present study are briefly documented asfollows.

The cut-off model of EOS, as shown by the solid line in Fig. 1,fixes the pressure of the two-phase mixture at the SVP value once

p(P

a)

10-2 10-1 100 101 102 103

10-4

10-2

100

102

104

106

108

0

Cut-off cavitation EOS

Tait’s water EOSSaturation line (IAPWS)

Barotropic vapor EOS

Sinusoidal cavitation EOSIsentropic cavitation EOS

ρ

psv

3(kg/m )

swFig. 1. Homogeneous EOS models for the simulation of cavitating flows.

(m/s

)

100

101

102

103

α

Model of speed of soundWallisSinusoidalFrozen

a

0 0.2 0.4 0.6 0.8 1

Fig. 2. Models for the speed of sound (SoS): Wallis [28], Sinusoidal [13] and ‘frozen’SoS.

Z.M. Hu et al. / Computers & Fluids 40 (2011) 315–323 317

the water flow reaches the saturated condition. If the Tait’s EOS isemployed for the pure water, the cut-off EOS can be written as

p ¼B0

qqsw

� �N� B0 þ A0; q > qsw

ps; qsw P q P qsv

psq

qsv

� �c; q < qsv

8>>><>>>: ð6Þ

where B0, A0 and N are model constants of the Tait’s EOS while c is thespecific heat ratio of vapor. ps, qsw and qsv denote the saturatedpressure, saturated water density and saturated vapor density at a gi-ven temperature, respectively. At T0 = 295 K, B0 = 3.07 � 108 Pa,A0 = ps = 2621 Pa, N = 7.15,c = 1.31, qsw = 997.76 kg/m3, qsv =0.0193 kg/m3.

In the isentropic one-fluid EOS [12], the two-phase mixture EOScan be calculated via Newton’s iteration algorithm from

q ¼ kqcavv þ qcav

w

pþB0�A0pcavþB0�A0

� ��1=Nþ k p

pcav

� ��1=c ð7Þ

where qcavv and qcav

w respectively denote the associated vapor andwater densities at the cavitation pressure pcav. k = a0/(1 � a0),where a0 is the known void fraction of the mixture at pcav. It shouldbe noted that the original isentropic model was developed assum-ing that the cavity boundary is a strong discontinuity, where ajumps from zero to a0 across the cavity interface. The flow is judgedas pure water when a < a0. However, the natural cavitation processis considered continuous for the DIM method as used in the presentwork, i.e., cavitation is assumed to start immediately when aexceeds zero. Otherwise, a negative pressure will occur if the Tait’sEOS as given in Eq. (6) is used when 0 < a < a0. On the other hand, aninfinite pressure will be obtained if the isentropic one-fluid model isemployed in this region. Therefore, unlike the original one, thecurrently used isentropic model as plotted in Fig. 1 switches tothe cut-off model when a < a0. Details on the original isentropicone-fluid model can be found in the literature [12].

The sinusoidal EOS was further developed by Goncalves andPatella [13] from its original version to avoid the infinite SoS. Thismodel in conjunction with the Tait’s EOS for the pure water can bedescribed as

p ¼B0

qqsw

� �N� B0 þ A0; p > ps þ Dp

ps þqsw�qsv

2

� �c2

min sin�1ðAð1� 2aÞÞ; ps þ Dp P p P ps � Dp

8<:ð8Þ

where the parameters A and cmin can be determined by the continu-ity relation between the pure water and the mixture for a given sat-uration condition. The quality of the problem-related parameter,DP, has to be smaller than ps to avoid negative pressure.

One of the purposes of cavitation model is to obtain a positivespeed of sound (SoS) a via ensuring that dp

dq > 0 inside the two-phase domain. The Wallis’s SoS [28] for the isentropic one-fluidEOS can be written as

a ¼ qa

qsva2vþ 1� a

qswa2w

� �� �1=2

ð9Þ

where the density of the two-phase mixture is expressed by linearlycombining the vapor phase density and the water phase density as

q ¼ aqsv þ ð1� aÞqsw ð10Þ

Here, av and aw are the speeds of sound of the pure vapor and waterat the given temperature, respectively. The SoS for the sinusoidalEOS can be computed as

a ¼ cminAffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� A2ð1� 2aÞ2q264

3751=2

ð11Þ

From Fig.1 and Eq. (6), it is clear that the cut-off model does not en-able a positive SoS and an artificial model of SoS should be strictlyprovided to get a hyperbolic system of governing equations.

The well known Wallis [28] and sinusoidal [13] models of SoSare depicted in Fig. 2 by the solid and dashed lines, respectively.Both models are widely used in the modelling of low-speedcavitating flows in hydrodynamics. However, these two SoS mod-els exhibit a non-monotonic variation with the void fractionwhich likely poses serious computational challenges. For exam-ple, to model high-speed supercavitating flows especially in thelight of applying the class of DIM for interface resolution, thesemodels can lead to numerical inconsistency; i.e., the flow insidethe numerically diffused zone may be hypersonic due to the verylow value of SoS. This can thus lead to a delay in the wave trans-mission through the interfaces [1]. It may be noted that bothWallis [28] and sinusoidal [13] models may not present any dif-ficulty when one employs the SIM method for interface resolu-tion. The SoS can alternatively switch between the water andvapor speeds of sound across the cavity boundaries because thereis a clear and sharp interface between phases in a SIM solution.To circumvent this difficulty associated with the employment of

318 Z.M. Hu et al. / Computers & Fluids 40 (2011) 315–323

DIM, a ‘frozen’ speed of sound is introduced. After substitutingthe isentropic vapor EOS and the Tait’s EOS into Eq. (10) onecan get

q ¼ aqsvpps

� �1=c

þ ð1� aÞqswpþ B0 � A0

B0

� �1=N

ð12Þ

Under the assumption that the composition of the two-phase mix-ture, i.e., a, does not vary with a infinitesimal change of pressure,the ‘frozen’ SoS then can be written as

a ¼ dpdq

� �1=2

¼ aqsvcpþ ð1� aÞqsw

Nðpþ B0Þ

� �1=2

ð13Þ

It may be noted that the present ‘frozen’ model of speed ofsound is different from the ‘frozen’ SoS defined for the non-equilibrium 6-equation model [1]. As plotted by a dashed–dottedline in Fig. 2, the ‘frozen’ SoS continuously and monotonouslyvaries from SoS of water at about 1483 m/s to that of vapor atabout 425 m/s as a increases from zero to unity. Other detailedproperties of the various one-fluid cavitation models and therelated discussions on the physical characteristics can be foundin [12].

One more technical difficulty in the numerical implementa-tion of compressible one-fluid model arises from the stiffness(or incompressibility) of the EOS for the pure water. From theinitial condition where T0 = 295 K and p0 = 101,325 Pa to thesaturated water state where ps = 2621 Pa, the change of waterdensity is extremely small from q0 = 997.81 kg/m3 to qsw =997.76 kg/m3. As a consequence, the pressure field is subjectedto round-off errors during the simulation of the flow cavitation.For numerical computations of low-speed cavitating flows likethe work of [13], the stiffness can be artificially eliminated viaemploying a very small value of SoS. However, this approach isnot feasible for high-speed cavitating flows where the compress-ibility is not negligible. In the present study, a filter function isintroduced to re-initialize the density of the water phase by q0

whenever the density change is less than a specified value asfollowing

q ¼q0; jq� q0j < 0:02ðq0 � qswÞq; else

�ð14Þ

Obviously, the re-initialization only works for the water phase inthe vicinity of the initial density.

2.2. Modification of DCD scheme for flow supercavitation

The DCD scheme [22,23] for the simulation of high-speed gas-dynamic flows is different from other conventional shock-captur-ing schemes. It satisfies the dispersion conditions to ensure thesolution is free of nonphysical oscillations across shock waves orcontact surfaces inside the complex flow-fields. The attractive fea-ture of the scheme is that it can systematically regulate the schemedissipation effect according to the discontinuity intensity. Thus,over- or under-dissipated solutions can be avoided without intro-ducing any free parameters which are generally problem depen-dant. More details on the DCD scheme can be found in thereview work [23].

The second-order DCD scheme for the convective part of Eq. (1)in two-dimensional curvilinear coordinates (n,g) can be written asfollows:

@ eU@s

!conv

i;j

¼ � 1Dn

Fiþ12� Fi�1

2

� �� 1

DgGjþ1

2� Gj�1

2

� �ð15Þ

where

Fi�12¼ eFþ

i�12;jþ eF�i�1

2;j

Gj�12¼ eGþ

i;j�12þ eG�i;j�1

2eFþiþ1

2;j¼ eFþi;j þ 1

2minmod eFþi;j � eFþi�1;j;

eFþiþ1;j � eFþi;j� �eGþ

i;jþ12¼ eGþi;j þ 1

2minmod eGþi;j � eGþi;j�1;

eGþi;jþ1 � eGþi;j� �eF�iþ1

2;j¼ eF�i;j � 1

2minmod eF�iþ2;j � eF�iþ1;j;

eF�iþ1;j � eF�i;j� �eG�i;jþ1

2¼ eG�i;j � 1

2minmod eG�i;jþ2 � eG�i;jþ1;

eG�i;jþ1 � eG�i;j� �The symbol ‘�’ indicates the respective vectors in the computa-tional space. It can be seen from the above equations that theDCD scheme belongs to the scheme family based on the flux split-ting method. In the original DCD scheme [22,23], the Steger-Warm-ing (SW) method was suggested to obtain the split fluxes eF� or eG�.For the gasdynamic flows with an ideal gas EOS, the method workswell and very efficiently [24–26]. Based on the Steger-Warmingmethod, the split flux for the flux vector, eF ¼ nxFþ nyG, in the com-putational space can be written as

eF� ¼ q2

~k�2 þ ~k�3ðu� kxaÞ~k�2 þ ðuþ kxaÞ~k�3ðv � kyaÞ~k�2 þ ðv þ kyaÞ~k�3

264375: ð16Þ

Here, k�i ¼ 12 ki �

ffiffiffiffiffiffiffiffiffiffiffiffiffik2

i þ eq� �

, (i = 1,2,3, 0 < e�1). The three eigen-

values of the Jacobian matrix eA ¼ @eF=@ eU are ~k1 ¼ nxuþ nyv ; ~k2 ¼nxuþ nyv � ha, and ~k3 ¼ nxuþ nyv þ ha, respectively. Here,

h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

x þ n2y

q, kx = nx/h and ky = ny/h where nx and ny are components

of transformation matrix between the physical and computationalcoordinate systems, i.e., (x,y) and (n,g). The speed of sound a canbe one of the forms as listed in the previous subsection. For cavitat-ing flows however, the above eigenvalue splitting method results inmathematical inconsistency, i.e., the flux eF cannot be recoveredsimply via eFþ þ eF�. The reason of the failure of SW method maybe associate with the stiffness of the barotropic EOS. On the otherhand, the Lax–Friedrichs flux splitting method is applicable andthe numerical flux can be written as follows,

eF� ¼ q2

~u� ~km

~uð~u� ~kmÞ þ nxp~vð~u� ~kmÞ þ nyp

264375; eG� ¼ q

2

~v � ~kn

~uð~v � ~knÞ þ gxp

~vð~v � ~knÞ þ gyp

264375: ð17Þ

Here, ~km and ~kn are the maximum absolute values of the eigenvaluesof the Jacobian eA ¼ @eF=@ eU and eB ¼ @eG=@ eU, respectively. It is easyto note that the flux eF can be recovered by eFþ þ eF�. The reason thatLax–Friedrichs flux splitting is applicable as opposed to the Steger-Warming approach for computing cavitating flows is that the split-ting operation is not required for the pressure term for the former.In general, the attractive feature of DCD scheme for computationalgasdynamics is that it can achieve non-oscillatory solutions withoutthe need of adding artificial viscosity or otherwise. This feature isalso very critical to capture the cavity boundaries accurately inthe presence of extremely sharp gradients of flow density or otherthermodynamic properties.

3. Numerical results and discussion

3.1. 1-D natural cavitation

A one meter-long tube is initially filled with pure water at theatmospheric pressure with density q1 = 997.81 kg/m3. An initial

Z.M. Hu et al. / Computers & Fluids 40 (2011) 315–323 319

velocity discontinuity is located at x = 0.5 m. The velocity is set asu = �100 m/s on the left and u = 100 m/s on the right. The cut-offcavitation model and the ‘frozen’ SoS model are used in thiscomputation. The results obtained on a uniform grid with 1000nodes are shown in Fig. 3 at time t = 1.85 ms. As strong rarefactionwaves propagate in the tube, evaporation occurs resulting in acavitation pocket. This indicates that the present algorithm is ableto handle the dynamic appearance of two interfaces that are absentinitially.

3.2. Flow supercavitation and validation

In the following computations for validation, the flow isassumed laminar and axisymmetric. The freestream temperatureof water is 295 K, and the cavitation process is assumed isother-mal. For the convective part of Eq. (1), the DCD scheme asdescribed above is used for all the computations. A second-ordercentral difference scheme is applied to discretize the viscous terms,and a third-order Runger–Kutta algorithm is used for the time-marching procedure. In addition, because of the presence of extre-mely steep gradients of density and EOS across the supercavityboundaries, a small CFL number is used, i.e., CFL = 0.1.

The first case for validation follows the experimental work ofHrubes [29] for high-speed supercavitating underwater projectiles.The experiment was conducted in a test range submerged at 4 mdepth in a fresh water tank. A disc cavitator of r = 0.71 mm in ra-dius was mounted at the head of the conical shaped projectile.

x (m)-0.4 -0.2 0 0.2 0.4

0

0.2

0.4

0.6

0.8

1

ρ/ρ ∞

Density at saturated condition

x (m)-0.4 -0.2 0 0.2 0.4

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

u/u ∞

Fig. 3. One-dimensional natural cavitation showing the profiles of density, speed o

The projectile travelled at 970 m/s corresponding to a Mach num-ber of 0.65 and a very small cavitation number r = 2 � 10�4. Afairly uniform and transparent cavity was obtained in the experi-ments [29] under such conditions. Here, the cavitation number isdefined as

r ¼ p1 � pc12 qwV2

1ð18Þ

where p1, pc, qw and V1 are the ambient water pressure, cavityvapor pressure, density of the water and the cruising speed of thecavitator, respectively. A multi-block axisymmetric grid is usedwith 137,000 total nodes over a computational domain of360r � 140r. The mesh is locally refined near the projectile surfaces.Fig. 4a and b shows a visual comparison between the computationalshadowgraph and experimental shadowgraph from Hrubes [29].The cut-off cavitation model supplemented with the ‘frozen’ SoS(see details on Figs. 1 and 2) is used in the simulation to obtainthe computational shadowgraph. The Munzer–Reichardt [30] theo-retical solution, Hrubes’s experimental data (obtained from theshadowgraph) [29] along with the numerical density contour areshown together in Fig. 4c. The coordinates are non-dimensionalizedby the radius of the cavitator, r. In general, the computational resultagrees well with experimental data and theory. It can be seen from(a) that a slight angle of attack in the test results in a slight asym-metry of cavity shape; this reflects one of the deficiencies in exper-imental research on the supercavitation of underwater projectiles.Therefore, the data in (c) which are obtained from the shadowgraph

x (m)-0.4 -0.2 0 0.2 0.4

0

200

400

600

800

1000

1200

1400

a(m

/s)

x (m)-0.4 -0.2 0 0.2 0.4

0

0.2

0.4

0.6

0.8

1

α

f sound, velocity and void fraction (Cavitation model: cut-off and ‘frozen’ SoS).

Fig. 4. Supercavitating flow over a high-speed underwater projectile, V1 = 970 m/s:(a) Experimental shadowgraph [29]; (b) numerical shadowgraph; (c) comparison ofcavity shape among theory, experiment, and the computation with cut-off model.

Fig. 5. (a) Experimental shadowgraph [21,29] and (b) numerical density contour fora transonic projectile travelling at Mach 1.03 (cavitation model: cut-off).

Fig. 6. Supercavity over a cylinder projectile travelling at 100 m/s (cavitationmodel: cut-off).

320 Z.M. Hu et al. / Computers & Fluids 40 (2011) 315–323

[29] entail experimental uncertainties. On the other hand, the def-inition of the cavity shape by the Munzer–Reichardt’s model [30]does not include any information regarding the diffused zone asfor the present DIM computations. As such, the comparison isbroadly qualitative rather than quantitative. Nevertheless, the com-putational cavity boundary as determined by the maximum densitygradient conforms better with the experiment than the theory. TheMunzer–Reichardt’s model for an axisymmetric cavity shape is anearly model based on the potential flow assumption and can be de-scribed as

RcðxÞ ¼ Rmax4x

Lmax 1� 4xLmax

� �24 351=2:4

ð19Þ

Here Rc, is the cavity radius at location x along the centerline of cav-ity originating from the cavitator nose. The maximum radius Rmax

and the length Lmax [31] of the cavity are respectively given as

Rmax ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCdðrÞ=r

pLmax ¼ 2r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCdðrÞ=r2 lnð1=rÞ

qð20Þ

where r, Cd and r are the cavitator radius, the cavitator drag coeffi-cient and the cavitation number, respectively. The hydrodynamicdrag coefficient for a disc-shaped cavitator at a zero angle of attackis given by Cd(r) = Cd0(1 + r), where Cd0 = 0.815.

Fig. 5 shows the cavity shape over a high-speed projectiletravelling at a much higher speed of 1530 m/s corresponding to atransonic Mach number of 1.03 in the water. The cavitation modelused in the computation is again the cut-off model. The numerical

flow structure looks fairly similar to that shadowgraph given byHrubes’s experiment [29]. The bow shock wave is well capturedin the simulation. Due to deformation of the transonic projectileand the lack of higher quality images, the experimental resultshown in (a) is of relatively low fidelity.

Fig. 6 shows the comparison of cavity shape for a cylinder pro-jectile travelling at a relatively low speed of 100 m/s. Here, the flowcondition corresponds to a small Mach number of about 0.07 and acavitation number r = 0.02. The computational results agree rea-sonably with the theoretical solution obtained via the Munzer–Reichardt’s model over a disc cavitator.

3.3. Cavitation model effects

Different EOS models are tested and compared for the supercav-itating flow over a high subsonic underwater projectile at V1 =970 m/s. First of all, it may be mentioned that an attempt to applythe sinusoidal model fails in which a severe density oscillation oc-curred. This failure indicates that the sinusoidal model may be not

p /

y/r

0.9 0.92 0.94 0.96 0.98 1 1.02

10

20

30

40

50

60

∞tot

Wallis SoS

Frozen SoS

0.5 Vρ ∞2

(a)

(b)

Fig. 8. Density contour (a) and total pressure ptot profile at x/r = 100 (b) showing thecomparison between the Wallis (lower half) and ‘frozen’ (upper half) SoS models(cavitation model: cut-off).

Z.M. Hu et al. / Computers & Fluids 40 (2011) 315–323 321

suitable or appropriate for high-speed supercavitating flowsalthough it has been proved to perform well in computations oflow-speed cavitating flows [13]. On the other hand, both the com-putational cavitation profiles obtained via the isentropic [12] andcut-off models in conjunction with the ‘frozen’ SoS are closer tothe experiment data than the Munzer–Reichardt’ theory as shownin Fig. 7. In this figure, each strip corresponds to the numerical dif-fused zone covering the range of void fraction 0.1 6 a 6 0.9,respectively. Moreover, the cut-off model results in a relativelynarrower diffused zone and shows better performance than theisentropic cavitation model.

To investigate the effects of SoS model on the cavity shape, forthe same flow condition as in Fig. 7, the cavity boundary obtainedvia Wallis’s SoS is compared to the ‘frozen’ SoS model solution aspresented in Fig. 8; the computation is based on the cut-off cavita-tion model. It is clear that the former cannot reach the correct cav-ity shape. In addition, the flow Mach number nonphysicallyexceeds 200 inside the numerical diffused zone for the Wallis’smodel solution due to the presence of extremely small speed ofsound. Fig. 8) shows the total pressure profiles at x/r = 100 for bothcomputations. The Wallis’s SoS simulation results in high-frequency oscillations in the water region. In the cavity, thereappears to be no oscillation. Saurel et al. pointed out that thenumerical diffused zone created by using an equilibrium SoS mod-el like the Wallis’s may have serious consequences regarding wavepropagation [1]. The speed of sound in the cavity which was care-fully measured in an experiment [32] was found to be betweenthose of the water and vapor thereby implying the ‘frozen’ SoS isapplicable. The obstructed exchange of information via wavepropagation/interaction between the cavity and the ambient waterflow appears to be the primary cause to the under-predicted cavityshape as shown in Fig. 8a. As such, the Wallis’s model of SoS isnumerically inapplicable to the present DIM computations ofhigh-speed supercavitating flows under the framework of theone-fluid cavitation model.

Fig. 9. Flow regime effects: planar flow (upper half) v.s. axisymmetric flow (lowerhalf) (V1 = 970 m/s; cavitation model: cut-off).

3.4. Flow regime effects

Next, the interest is to examine the effects of flow regime on thesupercavity geometry in a two-dimensional planar flow versus axi-symmetric flow. The computation is based on the same geometri-cal setup and flow conditions as for Fig. 7. The planar flowsimulation indicates a considerably larger cavity as shown on theupper half for a comparison to the axisymmetric flow solution de-picted on the lower half in Fig. 9. This great difference implies thatthe cavity shape is highly sensitive to the different flow regimes,i.e., whether planar or axisymmetric flow configuration. Our futurework will involve a full three-dimensional configuration where theunderwater projectile is mounted at different angles of attack; it is

x

y

0 100 2000

10

20

/r

/r

Projectile

Munzer Reichardt TheoryExperiments (J.D. Hrubes)Cut-off modelIsentropic model

0.1 α 0.9≤ ≤}

Fig. 7. Cavitation model comparison for the simulation of the supercavitating flowover a high-speed underwater projectile, V1 = 970 m/s (Experimental data [29];Cavitation model: cut-off v.s. isentropic model [12]; The radial coordinate ismagnified for clarity).

surmised that the effect on the cavity geometry is likely to be pro-found judging from the above preliminary study of planar versusaxisymmetric flow regime at the same zero angle of attack.

3.5. Partial cavitation

It should be pointed out that the laminar flow assumption willbreakdown at the rear portion of the cavity. In the cavity closureregion, very complicated flow phenomena associating with diversefluid physics occur, e.g., collapse of vapor bubbles as accompaniedby small scale implosions, collision and coalescence of droplets,condensation, and turbulence, etc. As such, the computationaldomain used in the above cases has to be appropriately selectedto exclude the closure region of a supercavity. However, a partialcavitation will occur over the projectile if its travelling speed isnot sufficiently high resulting in a cavity closure on the surfaceof the projectile. In such a circumstance, laminar flow simulationmay not be applicable. To demonstrate the limitation of the pres-ent numerical techniques, the computational result for a partialaxisymmetric cavitation flow at r = 0.3 is shown in Fig. 10. Thecase is water flow over a cylinder geometry with experimental datafrom Rouse and McNown [33]. Details of other flow conditions

s/d

Cp

0 2 4 6 8 10-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Rouse and McNown (1948)

σ

Computation

= 0.3

(b)

(a)

Fig. 10. Computational partial cavitation compared to experimental data [33]: (a)Surface pressure distribution; (b) Void fraction (V1 = 25.8 m/s, r = 0.3; Cavitationmodel: cut-off).

322 Z.M. Hu et al. / Computers & Fluids 40 (2011) 315–323

were not clearly specified in their report. Here, the water isassumed to be under the standard condition, i.e., q1 = 997.81 kg/m3 and p1 = 1 atm, in the present simulation. In Fig. 10,Cp ¼ p�p1

0:5q1V21

is the pressure coefficient, d is the diameter of thecylinder, and s is the surface distance from the nose. The cavity isnot well predicted as compared to the experimental data [33] atthe same cavitation number. In particular, the flow phenomenain the vicinity of the cavity closure have not been resolved duethe barotropic EOS used in the simulation. In previous studies onpartial cavitation where two-fluid [15,20] or hybrid [19,21] cavita-tion models were used, computations and experimental data agreereasonably. This may be attributed to the tunable phase transitionrate which is problem dependent as employed in their cavitationmodelling.

4. Conclusions

Based on a robust scheme which was initially proposed forshock-capturing in gasdynamics, the dispersion-controlled dissipa-tive (DCD) scheme is further modified in the present study for thepurpose of solving steep discontinuities existing in supercavitatingflows. The numerical flux splitting algorithm of the original DCDscheme is found inapplicable for the simulation of flow cavitationand should be replaced with a Lax–Friedrichs method. The modi-

fied DCD scheme is applied to several tests of supercavitatingflows. Comparisons with theoretical and experimental data helpverify the applicability of the revised DCD scheme for the numeri-cal simulation of supercavitating flows.

In the one-fluid cavitation models for the simulation of super-cavitating flows, the cut-off model and the isentropic model areable to reproduce the supercavity with acceptable accuracy ascompared to the experiments. On the contrary, the sinusoidal mod-el which has been qualified for the simulation of low-speed cavi-tating flows fails for the high-speed cavitating flows in thepresent study. Moreover, a ‘frozen’ model of speed of sound (SoS)as used in the present simulation performs well for high-speedcavitating flows while the well known Wallis’s model leads tosome numerical instabilities with an under-prediction of the cavityshape. Finally, further improvement and modification of the pres-ent numerical techniques are needed to accurately reproduce mul-tiple physical phenomena at the closure region of a partialcavitation flow; this is for the future work.

References

[1] Saurel R, Petitpas F, Berry RA. Simple and efficient relaxation methods forinterfaces separating compressible fluids, cavitating flows and shocks inmultiphase mixtures. J Comput Phys 2009;28:1678–712.

[2] Mulder W, Osher S, Sethian JA. Computing interface motion in compressiblegas dynamics. J Comput Phys 1992;100:209–28.

[3] Fedkiw RP, Aslam T, Merriman B, Osher S. A non-oscillatory Eulerian approachto interfaces in multimaterial flows (the ghost fluid method). J Comput Phys1999;152:457–92.

[4] Liu TG, Khoo BC, Yeo KS. Ghost fluid method for strong shock impacting onmaterial interface. J Comput Phys 2003;190:651–81.

[5] Liu TG, Khoo BC, Wang CW. The ghost fluid method for compressible gas–water simulation. J Comput Phys 2005;204:193–221.

[6] Unverdi SO, Tryggvason G. A front-tracking method for viscous,incompressible, multi-fluid flows. J Comput Phys 1992;100:25–37.

[7] Terashima H, Tryggvason G. A front-tracking/ghost-fluid method for fluidinterfaces in compressible flows. J Comput Phys 2009;228:4012–37.

[8] Abgrall R. How to prevent oscillation in multicomponent flow calculation. JComput Phys 1996;125:150–60.

[9] Saurel R, Abgrall R. A multiphase Godunov method for compressible multifluidand multiphase flows. J Comput Phys 1999;150:425–67.

[10] Murrone A, Guillard H. A five equation reduced model for compressible twophase flow problems. J Comput Phys 2005;202:664–98.

[11] Coutier-Delgosha O, Reboud JL, Delannoy Y. Numerical simulations inunsteady cavitating flows. Int J Numer Methods Fluids 2003;42:527–48.

[12] Liu TG, Khoo BC, Xie WF. Isentropic one-fluid modeling of unsteady cavitatingflow. J Comput Phys 2004;201:80–108.

[13] Goncalves E, Patella RF. Numerical simulation of cavitating flows withhomogenous models. Comput Fluids 2009;38:1682–96.

[14] Kunz RF, Boger DA, Stinebring DR, Chyczewski TS, Lindau JW, Gibeling HJ, et al.A preconditioned Navier–Stokes method for two-phase flows with applicationto cavitation prediction. Comput Fluids 2000;29:849–75.

[15] Lindau JW, Kunz RF, Boger DA, Stinebring DR, Gibeling HJ. High Reynoldsnumber, unsteady, multiphase CFD modeling of cavitating flows. J Fluids Eng2002;124:607–16.

[16] Schmidt DP, Rutland CJ, Corradini ML. A fully compressible, two-dimensionalmodel of small, high speed, cavitating nozzles. Atomization Spray1999;9:255–76.

[17] Venkikos Y, Tzabiras G. A numerical method for the simulation of steady andunsteady cavitating flows. Comput Fluids 2000;29:63–88.

[18] Shin BR, Iwata Y, Ikohagi T. Numerical simulation of unsteady cavitating flowsusing a homogenous equilibrium model. Comput Mech 2003;30:388–95.

[19] Ahuja V, Hosangadi A, Arunajatesan s. Simulations of cavitating flows usinghybrid unstructured meshes. J Fluids Eng 2001;123:331–40.

[20] Owis FM, Nayfeh AH. Computations of the compressible multiphase flow overthe cavitating high-speed torpedo. J Fluids Eng 2003;125:459–68.

[21] Neaves MD, Edwards JR. All-speed time-accurate underwater projectilecalculations using a preconditioning algorithm. J Fluids Eng 2006;128:284–96.

[22] Jiang ZL, Takayama K, Chen YS. Dispersion conditions for non-oscillatoryshock-capturing schemes and its applications. Comput Fluid Dynam J1995;2:137–50.

[23] Jiang ZL. On the dispersion-controlled principles for non-oscillatory shock-capturing schemes. Acta Mech Sinica 2004;20:1–15.

[24] Hu ZM, Jiang ZL. Wave dynamic process in cellular detonation reflection fromwedges. Acta Mech Sinica 2007;23:33–41.

[25] Hu ZM, Myong RS, Kim MS, Cho TH. Downstream flow conditions effects on theRR ? MR transition of asymmetric shock waves in steady flows. J Fluid Mech2009;620:43–62.

[26] Hu ZM, Wang C, Zhang Y, Myong RS. Computational confirmation of anabnormal Mach reflection wave configuration. Phys Fluids 2009;21:011701.

Z.M. Hu et al. / Computers & Fluids 40 (2011) 315–323 323

[27] Cooper JR, Dooley RB. The international associated for the properties ofwater and steam: revised release on the IAPWS industrial formulation1997 for the thermodynamic properties of water and steam. Switzerland:Lucerne; 1997.

[28] Wallism GB. One-dimensional two-phase flow. New York: McGill-Hill; 1969.[29] Hrubes JD. High-speed imaging of supercavitating underwater projectiles. Exp

Fluids 2001;30:57–64.[30] Munzer H, Reichardt H. Rotational symmetric source-sink bodies with

predominantly constant pressure distributions[sic]. ARE translation 1/

50. England: Aerospace Research Establishment; 1975 [as described in May,1975].

[31] Garabedian PR. Calculation of axially symmetric cavities and jets. Pac J Math1956;4:611–84.

[32] Wu XJ, Chahine GL. Characterization of the content of the cavity behind a high-speed supercavitating body. J Fluid Eng 2007;129:136–45.

[33] Rouse H, McNown JS. Cavitation and pressure distribution: head forms at zeroangle of yaw. Studies in engineering, vol. 32. Bulletin: State University of Iowa;1948.