AIAA 2001-0279 Preconditioning Algorithms for the

AIAA 2001-0279

Preconditioning Algorithms for the Computation of Multi-Phase Mixture Flows

Sankaran Venkateswaran University of Tennessee

Jules W. LindauApplied Research LabThe Pennsylvania State University

Robert F. KunzApplied Research LabThe Pennsylvania State University

Charles L. MerkleUniversity of Tennessee

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Preconditioning Algorithms for the Computation of Multi-Phase Mixture Flows

Sankaran Venkateswaran,*

University of Tennessee, Tullahoma, TN 37388

Jules W. Lindau,† Robert F. Kunz,††

Applied Research Laboratory, Penn State University, University Park, PA 16802

Charles L. Merkle**

University of Tennessee, Tullahoma, TN 37388

ABSTRACT

Preconditioned time-marching algorithms are devel-oped for a class of isothermal compressible multi-phasemixture flows, relevant to the modeling of sheet- andsuper-cavitating flows in hydrodynamic applications.Using the volume fraction and mass fraction forms ofthe multi-phase governing equations, three closelyrelated but distinct preconditioning forms are derived.The resulting algorithm is incorporated within an exist-ing multi-phase code and several representative solu-tions are obtained to demonstrate the capabilities of themethod. Comparisons with measurement data suggestthat the compressible formulation provides an improveddescription of the cavitation dynamics compared withprevious incompressible computations.

INTRODUCTION

Multi-phase flows are encountered in a wide range ofapplications involving heat exchange, cavitation, sprays,porous media, etc. The computation of multi-phaseflows has received growing research attention in recentyears, due in part to the evolving maturity of single-phase algorithms. There remain, however, several physi-cal and modeling challenges. A primary issue is thestrong coupling of acoustic phenomena [1-5] due to thefact that the speed of sound in two-phase mixtures canbe extremely low compared to the sound speeds in theindividual component phases. Thus, multi-phase flowsare frequently characterized by local regions, whereinthe flow may be transonic or even supersonic with the

1

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Copyright © 2001 by the American Institute of Aeronautics andAstronautics, Inc. All rights reserved.

* Research Associate Professor, Member AIAA.† Research Associate, Member AIAA.†† Research Associate, Member AIAA.** H.H. Arnold Chair of Excellence in Computational Mechanics, Member AIAA.

presence of shocks, although the bulk of the flow mayremain incompressible. This situation presents a uniquechallenge to the design of CFD algorithms. The devel-opment of appropriate numerical schemes for suchmulti-phase problems is the subject of the present paper.

There are many levels of modeling that may be uti-lized in multi-phase computations [6]. In general, onemay distinguish between methods that employ an Eule-rian framework for both phases and those that employEulerian for the gas-phase and Lagrangian for the liq-uid-phase. In the Eulerian-Eulerian framework, the sim-plest approach is to employ a single continuity equationfor both phases, with the fluid density being describedas a continuous function varying between the vapor andliquid phases [7,8]. At a more detailed level of model-ing, separate continuity equations for the liquid andvapor phases are employed along with appropriate masstransfer terms to represent the phase-change phenomena[9-12]. The gas-liquid interface is, however, assumed tobe in dynamic and thermal equilibrium and, conse-quently, mixture momentum and energy equations areused. This model is usually referred to as the homoge-neous mixture model [6] and is the level of modelingconsidered in this paper. The model is appropriate formodeling cavitation in naval hydrodynamics applica-tions, the area of primary interest to us. Finally, we notethat full multi-fluid modeling, involving separatemomentum and energy equations for each of the phases,has also been utilized for certain classes of multi-phaseflow [6], but is not addressed here.

The crucial requirement of multiphase algorithms isthe ability to accurately and efficiently span both incom-pressible and compressible flow regimes. For single-phase applications, time-marching techniques have longbeen established as the method of choice for high-speedcompressible flows, while artificial compressibility orpreconditioning techniques have enabled the extensionof these methods to the incompressible and low-speedcompressible regimes [13-16]. Preconditioning methods

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essentially maintain proper conditioning of the control-ling time-scales of the time-marching system by intro-ducing appropriate pseudo-time derivatives. Indeed, it iswidely recognized that the careful selection of thesederivatives is crucial for ensuring efficiency and accu-racy over a wide range of Mach numbers, Reynoldsnumbers and Strouhal numbers [15,16]. In this paper,we seek to extend the preconditioning formulation tomulti-phase mixture flows.

Several researchers have previously reported precon-ditioning formulations for multi-phase mixtures. Merkleet al. employed a two-species formulation, using massfraction as the dependent variable [9]. Kunz et al. alsoemployed a multi-species formulation, but used volumefraction as the dependent variable [10]. Both these for-mulations assumed constant densities for both liquidand vapor phases and did not account for compressibil-ity effects in the two-phase mixture region. Ahuja et al.have developed a multi-phase algorithm, including com-pressibility effects in the component phases [12]. In thispaper, we review these formulations and, in the case ofthe first two methods, we show that they can readily beextended to the compressible situation. In particular, wederive all three formulations using a common frame-work, facilitating comparative study. We find that thethree approaches are, in fact, nearly the same with onlyminor differences between them.

The focus of this paper is on the isothermal multi-phase system, in which the densities of the fluids areassumed to be functions of the pressure, but not the tem-perature. Under this assumption, the energy equation isnot solved and only the continuity and momentum equa-tions are considered. It should be noted that the systemis still compressible because the pressure dependence ofthe densities gives rise to finite acoustic speeds. The iso-thermal compressible model serves as a useful interme-diate step between the incompressible model and thefully compressible system, and development of the fullycompressible model is currently underway. Neverthe-less, we observe that the isothermal assumption is gen-erally valid for the class of hydrodynamic cavitationproblems considered here.

Our primary interest lies in sheet- and super-cavitat-ing flows encountered in naval hydrodynamics applica-tions. These flowfields are characterized by largedensity ratios between the liquid and vapor states. Fur-ther, at the Reynolds numbers of the interest, the flowsare fully turbulent. Moreover, most problems alsoexhibit large-scale unsteadiness because of cavity reen-trant jets and cavity pinching [11]. Accordingly, the pre-conditioning formulation derived in this paper isincorporated into a time-accurate, Reynolds-averaged,multi-phase Navier-Stokes code (UNCLE-M, see ref.

[10] for details). The code is applied to a hierarchy oftest cases, both simple one-dimensional and more prac-tical multi-dimensional flows. In particular, we considercavitating flows over several axisymmetric ogive con-figurations for which measurement data are available.These computational test cases are used to demonstratethe overall capabilities of the formulation.

The paper is organized as follows. We begin with theequations of motion for two-phase homogeneous mix-ture flows. The equations are presented both in their vol-ume fraction and mass fraction forms. We examine theeigenvalues of these systems and determine their behav-ior under limiting circumstances. We then use thisinsight to derive preconditioning forms that maintainwell-conditioned behavior in both compressible andincompressible flow regimes. These investigationsreveal that the preconditioning formulation is not uniqueand several distinct (but closely related) forms may bearrived at depending upon the precise version of thegoverning equations used in the analysis. In the Resultssection, we perform a series of computations of simpli-fied and practical test problems to assess the perfor-mance capabilities of the formulations. Finally, weconclude with a summary and a brief description of cur-rent and future work.

EQUATIONS OF MOTION

The governing equations for two-phase flow are cus-tomarily written in terms of volume fraction variables.For the purposes of the theoretical derivation, weemploy only the one-dimensional inviscid flow equa-tions, although, in practical implementation, the fullmulti-dimensional Reynolds-averaged equations areused.

(1)

(2)

(3)

with the mixture density being defined as:

(4)

Note that the individual phasic densities, and aredefined as mass of the phase/volume occupied by thatphase.

The above system is closed by the phasic equations ofstate:

and (5)

∂ρ̃vαv

∂t---------------

∂ρ̃vαvu

∂x------------------+ 0=

∂ρ̃lαl

∂t-------------

∂ρ̃lαlu

∂x-----------------+ 0=

∂ρu∂t

--------- ∂ρu2

∂x------------ ∂p

∂x------+ + 0=

ρ ρ̃lαl ρ̃vαv+=

ρ̃v ρ̃l

ρ̃v ρ̃v p( )= ρ̃l ρ̃l p( )=

2

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where the individual phasic densities are assumed to befunctions of pressure only. In our previous work [10,11],these quantities were taken to be constant, representinga mixture of incompressible phases.

The above equations may equivalently be written interms of mass fraction variables as well. This form iscustomarily used in the case of gaseous mixtures, but isin fact equally valid for multi-phase mixtures.

(6)

(7)

with the momentum equation having the same form asEqn. 3.

Note that it is sometimes useful to define the individ-ual phasic densities, and ,whichare given as the mass of phase/volume occupied by themixture. Note that these phasic densities are distinctfrom those introduced earlier and are related by the fol-lowing expressions:

(8)

Further, it is clear that the mixture density, .It is straightforward to see that the mass and volumefraction forms are in fact identical.

In the case of gaseous mixtures, it is customary toreplace one of the phasic (or species) continuity equa-tions by an overall continuity equation (obtained bysumming up the individual continuity equations):

(9)

However, for multi-phase flows, where significant den-sity differences may exist between the phases, this pro-cedure is prone to error. It is thus advisable to use all theindividual continuity equations as shown above.

Alternately, some researchers prefer to define an over-all mixture volume continuity equation, obtained bydividing the phasic continuities by the respective phasicdensity and then summing them up:

(10)

In this paper, we develop the preconditioning formula-tion for the system with all the phasic continuity equa-tions, but we note that the overall mixture continuityform may be derived from it.

EIGENVALUE ANALYSIS

Volume Fraction FormThe system given in Eqns. 1-3 may be expressed in

the following vector form:

(11)

where:

(12)

Note that, for the isothermal system under considerationhere, the functions and represent thereciprocal of the squares of the speed of sound in thetwo individual phases. Also note that:

(13)

To determine the eigenvalues of the above two-phasesystem, we define the Jacobian:

(14)

The system eigenvalues are then given by the eigen-values of :

(15)

∂ρYv

∂t------------

∂ρYvu

∂x----------------+ 0=

∂ρYl

∂t-----------

∂ρYlu

∂x---------------+ 0=

ρv ρYv= ρl ρYl=

ρv ρ̃vαv= ρl ρ̃lαl=

ρ ρv ρl+=

∂ρ∂t------ ∂ρu

∂x---------+ 0=

1

ρ̃v

-----∂ρ̃vαv

∂t--------------- 1

ρ̃l

----∂ρ̃lαl

∂t------------- 1

ρ̃v

-----∂ρ̃vαvu

∂x------------------ 1

ρ̃l

----∂ρ̃lαlu

∂x-----------------+ + + 0=

Γα∂Qα∂t

---------- ∂E∂x------+ 0=

Qα

p

αv

u

= E

ρ̃vαvu

ρ̃lαlu

ρu2

p+

=

Γα

αv∂ρ̃v

∂p--------

αv

ρ̃v 0

αl∂ρ̃l

∂p--------

αv

ρ̃– l 0

u∂ρ∂p------

αv

u ρ̃v ρ̃l–( ) ρ

=

∂ρ̃v ∂p⁄ ∂ρ̃l ∂p⁄

∂ρ∂p------

αv

αv∂ρ̃v

∂p--------

αv

αl∂ρ̃l

∂p--------

αv

+=

Aα∂E

∂Qα----------

uαv∂ρ̃v

∂p--------

αv

uρ̃v 0

uαl∂ρ̃l

∂p--------

αv

uρ̃– l 0

1 u2∂ρ∂p------+

αv

u2 ρ̃v ρ̃l–( ) ρu

= =

Γα1–

Aα( )

Γα1–Aα

u 0 ρc2

0 u ρc2αlαv1

ρ̃l

----∂ρ̃l

∂p--------

αv

1

ρ̃v

-----∂ρ̃v

∂p--------

αv

–

1ρ--- 0 u

=

3


where the sound speed is given by the following expres-sion:

(16)

which is the standard mixture rule for the sound speedof two-phase mixtures.

It can be readily seen that the eigenvalues of theabove system are given as:

(17)

which has the familiar form of the single-phase com-pressible system.

Mass Fraction FormThe mass-fraction equation system may be expressed

in the following vector form:

(18)

where:

(19)

The Jacobian is given by:

(20)

The system eigenvalues are then given by the eigen-values of :

(21)

and the sound speed is defined by the following expres-sion:

(22)

It can be easily shown that the property Jacobians aregiven as:

(23)

(24)

These expressions specify these properties in terms ofknown properties of the two phases. Note that theexpression for the speed of sound is the same as in Eqn.16.

It is also evident that the eigenvalues of the abovematrix system are:

(25)

PRECONDITIONING FORMULATION

Examination of the eigenvalues of the multi-phasesystem reveals that the system reverts to the standardsingle-phase eigenvalues when only one phase ispresent. Under such circumstances, for the liquid phaseas well as for the vapor phase at low speeds, the systembecomes poorly conditioned for efficient time-march-ing. Moreover, it is well known that standard discreteformulations (such as Roe’s flux difference scheme)become inaccurate under such stiff conditions. Precon-ditioning or artificial compressibility methods are theestablished approach for alleviating such stiffness prob-lems [13-16] within a time-marching algorithmic frame-work. These methods re-scale the time-derivativesselectively and thereby ensure that the system eigenval-ues become well-conditioned.

For the multi-phase system, when both liquid andvapor phases co-exist, the mixture sound-speed is suchthat the local Mach number may become transonic oreven supersonic at relatively low speeds [1-5]. It istherefore important that the preconditioning formulationis capable of handling the low Mach numbers in thepure liquid and vapor phases as well as the transonic/supersonic Mach numbers in the mixture region. In the

1

c2----- ρ

αl

ρ̃l

-----∂ρ̃l

∂p--------

αv

αv

ρ̃v

------∂ρ̃v

∂p--------

αv

+

=

λ Γα1–Aα( ) u u c±,=

Γy

∂Qy

∂t---------- ∂E

∂x------+ 0=

Qα

p

Yv

u

= E

ρYvu

ρYlu

ρu2

p+

=

Γα

Yv∂ρ∂p------

Yv

ρ Yv∂ρ∂Yv---------

p

+ 0

Yl∂ρ∂p------

Yv

ρ– Yl∂ρ∂Yv---------

p

+ 0

u∂ρ∂p------

Yv

u∂ρ∂Yv---------

p

ρ

=

Ay

Ay∂E

∂Qy----------

uYv∂ρ∂p------

Yv

ρu uYv∂ρ∂Yv---------

p

+ ρYv

uYl∂ρ∂p------

Yv

ρ– u uYl∂ρ∂Yv---------

p

+ ρYl

1 u2∂ρ∂p------+

Yv

u2 ∂ρ∂Yv---------

p

ρu

= =

Γy1–

Ay( )

Γy1–Ay

u 0 ρc2

0 u 0

1ρ--- 0 u

=

1

c2----- ∂ρ

∂p------

Yv

=

∂ρ∂p------

Yv

ραl

ρ̃l

-----∂ρ̃l

∂p--------

αv

αv

ρ̃v

------∂ρ̃v

∂p--------

αv

+

=

∂ρ∂Yv---------

p

ρ2 1

ρ̃l

---- 1

ρ̃v

-----– =

λ Γα1–Aα( ) u u c±,=

4


present section, we examine several potentialapproaches for preconditioning the multi-phase system.

Formulation IThe volume fraction form given in Eqn. 11 may be

preconditioned by the introduction of pseudo propertiesinto the matrix premultiplying the time-derivatives.Thus, we have:

(26)

with the preconditioning matrix being defined as:

(27)

The pseudo-properties are defined so as to render theeigenvalues well-conditioned. One possible choice issuggested by the Kunz et al. formulation for incom-pressible two-phase mixtures [10]. Accordingly, wedefine:

, (28)

(29)

The term ‘‘V’’ is some characteristic velocity scale, typ-ically the local convective velocity under inviscid con-ditions.

The system Jacobian, , is now given as:

(30)

where

(31)

(32)

(33)

The eigenvalues of the preconditioned two-phase sys-tem are given as:

(34)

Since , we note that the ‘‘acoustic’’ eigen-values in the above expression are of the same order asthe particle speed, thereby ensuring well-conditionedeigenvalues at all speeds.

The above formulation reduces to the Kunz et al. for-mulation for incompressible mixtures. It is further evi-dent that it reduces to Chorin’s artificial compressibilityfor single-phase incompressible flows [13] and to stan-dard preconditioning for single-phase compressibleflows [15]. However, unlike the single-phase precondi-tioning system, this formulation does not automaticallyswitch the preconditioning off for supersonic flows.While this may be viewed as a disadvantage of themethod, the effect appears to be small at moderatesupersonic Mach numbers.

Finally, we note that the above preconditioning defini-tion may be formally derived using a perturbation pro-cedure applied to the governing equations in the volumefraction form (Eqns. 1-3). The procedure is not pre-sented here for reasons of brevity, but it is similar to thatdescribed for single-phase flow in Ref. [15].

Formulation IIIn their model for isothermal compressible mixtures,

Ahuja et al. [12] use a similar form of the precondition-ing matrix as in Eqn. 27, but they define the pseudo-properties in a different fashion:

, (35)

where and V is some characteristic veloc-ity scale.

Interestingly, this preconditioning formulation yieldsthe same eigenvalues as those in formulation I. This is aconsequence of the following relationship being true:

(36)

For single-phase compressible flows, formulation IIalso reduces to the standard preconditioning formula-tion. For two-phase mixtures, it has the further advan-tage of automatically turning preconditioning off atsupersonic speeds (unlike formulation I). However,

Γαp ∂Qα

∂t---------- ∂E

∂x------+ 0=

Γαp

αv∂ρ̃v′∂p

----------αv

ρ̃v 0

αl∂ρ̃l′∂p

----------αv

ρ̃– l 0

u∂ρ′∂p--------

αv

ρ̃v ρ̃l– ρ

=

∂ρ̃v'∂p

----------αv

ρ̃v

ρ----- 1

V2------=

∂ρ̃l'∂p

---------αv

ρ̃l

ρ---- 1

V2------=

∂ρ̃'∂p-------

αv

αv∂ρ̃v'∂p

----------αv

αl∂ρ̃l'∂p---------

αv

+=

Γα1–Aα

Γα1–Aα

uc′( )2

c2----------- 0 ρ c′( )2

X u Y

1ρ--- 0 u

=

1

c′( )2------------ ρ

αl

ρ̃l

-----∂ρ̃l′∂p

----------

αv

αv

ρ̃v

------∂ρ̃v′∂p

----------

αv

+

1

V2------= =

X ρu c′( )2αlαv

ρ̃lρ̃v

-----------∂ρ̃v

∂p--------

αv

∂ρ̃l′∂p

----------

αv

∂ρ̃l

∂p--------

αv

∂ρ̃v′∂p

----------

αv

–

=

Y ρ c′( )2αlαv1

ρ̃l

----∂ρ̃l′∂p

----------

αv

1

ρ̃v

-----∂ρ̃v′∂p

----------

αv

–

=

u12--- u 1

c′( )2

c2------------+

u2 1c′( )2

c2------------–

24 c′( )2+±,

c′( )2V2=

∂ρ̃v'∂p

----------αv

1M2-------

∂ρ̃v

∂p--------

αv

=∂ρ̃l'∂p---------

αv

1M2-------

∂ρ̃l

∂p--------

αv

=

M2

V2

c2⁄=

1

c′( )2------------ ρ

αl

ρ̃l

-----∂ρ̃l′∂p

----------

αv

αv

ρ̃v

------∂ρ̃v′∂p

----------

αv

+

1

c2M2------------- 1

V2------= = =

5


implementation of the method in the incompressiblelimit is a little clumsy because the Mach number tendsto zero in this limit. In practice, this is not a major issuesince the incompressible sound speed may arbitrarily beset to a large number.

It is interesting to note that formulation II may also bederived from a perturbation analysis when one of thephasic continuity equations is replaced by the overallmixture continuity equation. Again, for reasons of brev-ity, we do not provide the details here.

Formulation IIIA third formulation may be derived by starting with

the mass fraction form of the governing equations. Wemay write the preconditioning system as:

(37)

with the preconditioning matrix being defined as:

(38)

where

(39)

and, as usual, V is some characteristic velocity scale.

The above formulation is identical to the precondi-tioning used for gaseous mixtures [15] and is similar tothat proposed by Merkle et al. for two-phase flows [9].It reduces to Chorin’s artificial compressibility forincompressible single-phase flows and to the standardpreconditioning form for compressible single-phaseflows. For multiple phases, the system is well-condi-tioned at low Mach numbers and automatically switchesoff the preconditioning at supersonic Mach numbers. Infact, it can be shown that the eigenvalues of the aboveformulation are the same as the eigenvalues of formula-tions I and II (see Ref. [15]).

Although the three formulations are not identical, theypossess the same eigenvalues and are closely related.Our experience suggests that the three approaches per-form equally well in practice. Thus, the decision ofwhich one to use may well depend upon the particularform of the equations or the solution variables used inthe candidate code.

NUMERICAL METHODOLOGY

The preconditioning system for isothermal compress-ible multi-phase flows has been incorporated into theUNCLE-M code. This code was originally developedfor incompressible flows at Mississippi State University[17] and later extended to two-phase mixtures by Kunzet al. [10]. The code is structured, multi-block, implicitand parallel with Roe-upwind flux-difference splittingfor the spatial discretization and Gauss-Seidel relaxationfor the inversion of the implicit operator.

Physical modeling of the phase transformation fromliquid to vapor is modeled by a source term that is pro-portional to the product of the liquid volume fractionand the difference between the local pressure and thevapor pressure. For transformation of vapor to liquid, asimplified form of the Ginzburg-Landau potential isused [10]. A high Reynolds number two-equation turbu-lence model with wall functions is used for turbulenceclosure.

For time-accurate unsteady solutions, a dual-time pro-cedure is used. The physical time terms are discretizedusing second-order one-sided differencing. The discretealgebraic equations at each physical time level aredriven to convergence by a set of pseudo-time deriva-tives, which are preconditioned for maximizing conver-gence and accuracy.

The UNCLE-M code represents the phasic continuityequations in volume fraction form and, for that reason,only formulations I and II were incorporated into thecode. In both cases, the characteristic velocity parameterwas supplied as user-specified value, usually expressedas some multiple of the free-stream velocity. Morerefined definitions of this parameter based upon localvariables will be the subject of future study. As men-tioned earlier, both formulations performed comparablyfor the test cases studied here.

RESULTS

Inviscid Flow in Straight DuctAs a first example, we consider the simple case of

inviscid two-phase mixture flow in a two-dimensionalstraight duct. Figure 1 shows convergence results forthree different Mach numbers of 0.01, 0.55 and 2.Results are given for the standard equations without anypreconditioning and for two forms of preconditioning (Iand II). The results reveal that, at the low Mach numberof 0.01, convergence is not obtained with the non-pre-conditioned equations because of the inherent stiffnessof the original time-marching system (at low speeds).On the other hand, efficient convergence is realized withboth preconditioning forms tested. Even at the higher

Γyp∂Qy

∂t---------- ∂E

∂x------+ 0=

Γαp

Yv∂ρ′∂p--------

Yv

ρ Yv∂ρ∂Yv---------

p

+ 0

Yl∂ρ′∂p--------

Yv

ρ– Yl∂ρ∂Yv---------

p

+ 0

u∂ρ′∂p--------

Yv

u∂ρ∂Yv---------

p

ρ

=

∂ρ′∂p-------

Yv

1c′( )2

------------ 1V2------= =

6


subsonic Mach number of 0.55, the two preconditioningforms are observed to slightly out-perform the non-pre-conditioned system. Finally, at the supersonic Machnumber, all three systems converge at virtually the samerate. We recall that at supersonic Mach numbers, formu-lation I does not turn off the preconditioning, while for-mulation II does turn off the preconditioning. The lattercase hence becomes identical to the non-preconditionedcase for supersonic flows. We note that the results offormulation I are also very nearly the same, indicatingthat the preconditioning effect is very slight for moder-ately supersonic flows.

Shock-Tube ProblemAs a second example, we consider the unsteady two-

phase shock tube problem, investigated both experimen-tally and theoretically by Campbell and Pitscher [5].The problem can be modeled as a shock wave movinginto a stationary and non-condensable, non-vaporizablegas/liquid mixture. Assuming that the liquid is incom-pressible and the gas is perfect, an exact expression maybe obtained for the shock speed [5]:

(40)

where the subscript 1 denotes conditions in front of theshock, the subscript 2 denotes conditions behind it and

is the shock speed.

Figure 2 shows the results for two different pressureratios of 2 and 6. In each case, the predicted results aftera given period of time are compared with the theoreticalshock location and very good agreement is obtained. Wenote here that the time-accurate results are obtainedusing dual time-stepping. The characteristic velocityparameter in the definition of the preconditioning isspecified based upon the time-scales associated with theshock motion. The results demonstrate that the two-phase preconditioning formulation is capable of resolv-ing acoustic/compressibility effects very well.

Natural Cavitation on Axisymmetric BodiesThe application of principal interest to us involves the

modeling of sheet cavitation around axisymmetric bod-ies at high Reynolds numbers. Because of their impor-tance to naval hydrodynamics applications, numerousexperimental and computational studies of cavity flow-fields around bodies of different shapes have been car-ried out. Rouse and McNown have documented steadyand time-averaged measurements of relevant cavitationparameters for various forebody shapes [18]. May hasassembled cavity shape and size parameters for a widerange of cavitation numbers [19]. Stinebring et al. havedocumented the unsteady cycling behavior of several

axisymmetric cavitators [20]. Recently, Kunz et al. andLindau et al. have modeled sheet cavity flowfields in avariety of configurations using an incompressible mix-ture model [10,11]. They have made extensive compari-sons with experimental data and have determined thestrengths and shortcomings of their model. In this sec-tion, we perform similar computations with the currentisothermal compressible model.

Figures 3 and 4 show flowfield snapshots of theunsteady cavitating flow over a 0-caliber and a 1/4-cali-ber ogive. The Reynolds numbers are approximately

and the cavitation number is 0.3. In both setsof figures, corresponding results using the incompress-ible mixture model have also been included to facilitatedirect comparison.

It is evident that the flowfields are rich in complexity.We first consider the 0-caliber results in Fig. 3. Bothincompressible and compressible results show the cav-ity forming at or very near the leading edge of the ogivebody. Further, because of the sharp turning of theincoming flow, a recirculating bubble is also formed inthis cavity region. The incoming liquid then flows overthe bubble and rejoins the surface of the body down-stream of the cavity and the recirculating bubble. Thepressure recovery in this downstream region then setsup a liquid rentrant jet which shoots into the cavity. Aswe follow the snapshots through a complete period, weobserve that this rentrant jet appears to progress deeperinto the cavity until some form of cavity pinchingoccurs that causes the jet to retract until the cycle cancommence again. The unsteady cycle thus seems to beclosely co-ordinated with the re-entrant jet and the cav-ity pinching process.

Figure 3 shows a marked difference between the com-pressible and incompressible results in the extent towhich the liquid reentrant jet penetrates the cavity. Inthe compressible case, the liquid jet traverses only aboutone-half of the cavity length before the jet is pinched offand begins to retract. The incompressible case alsoshows the reentrant jet getting pinched off; however, theliquid bubble appears to remain intact within the cavityuntil it reaches the leading edge. In fact, at t=0.5, therentrant liquid appears to have pushed the cavity down-stream of the leading edge.

This difference between the compressible and incom-pressible results is also observed in Fig. 4 for the 1/4-caliber case. In fact, the difference appears even moremarked with the compressible case showing the liquidrentrant jet barely reaching one-half length into the cav-ity, while in the incompressible result, it again appearsto traverse most of the cavity length.

u12 p2

p1-----c1

2=

u1

1.4 105×

7


These trends in the behavior of the reentrant jetappear to control the unsteady dynamics of the flow-fields. Figure 5 shows the instantaneous (pressure) dragcoefficient histories for all four cases. The drag coeffi-cient essentially is a measure of the total pressure on theface of the ogive body. The fluctuations in this quantitythen result from the upstream propagation of pressuredisturbances generated by the cavity pulsations.

Examination of the 0-caliber drag coefficient historiesfirst reveals that the incompressible amplitudes are sig-nificantly higher, a fact that is reflected by the moreextreme cavity distortions that are evident in the incom-pressible snapshots in Fig. 3. Further, the dominantmode of the compressible result shows a slightly higherfrequency than the incompressible result. Again, thisresult appears to correlate with the distance traversed bythe reentrant jet in Fig. 3. For instance, the compressiblecase shows shorter reentrant jet penetration and, hence,a higher frequency.

Drag coefficient results for the 1/4-caliber ogive arealso given in Fig. 5. Here, both incompressible and com-pressible amplitudes are observed to be small, in accor-dance with the more stream-lined cavity shapes evidentin the snapshots in Fig. 4. However, there is now a moresignificant difference in the cycling frequency, with thecompressible case showing a much higher frequency.Again, this result correlates well with the jet penetrationdistance in Fig. 4. The incompressible case shows thereentrant jet reaching much deeper into the cavity and,hence, has a smaller cycling frequency.

The frequency data for the above cases are comparedwith measurement data obtained by Stinebring et al.[20] in Fig. 6. While all the computed data lie within thebounds of the experimental data, it is interesting that theexperimental data for the hemispherical forebody indi-cates a higher Strouhal frequency than the correspond-ing result for the 0-caliber ogive. We point out that thistrend is in agreement with the higher frequency obtainedfor the 1/4-caliber ogive using the compressible model.On the other hand, the incompressible model predictsroughly the same Strouhal frequency for the two shapes.

The above comparison certainly suggests that thecompressible model may be capturing the dynamics ofthe cavity more correctly than the incompressiblemodel. In particular, it would appear that the penetrationof the liquid jet (in Fig. 4) is over-predicted by theincompressible model. It is not immediately apparentwhy this may be the case. We speculate that the com-pressible nature of the vapor phase may be causing thepressure within the cavity to increase somewhat duringthe jetting process, thereby weakening the penetration ofthe jet. In the incompressible case, on the other hand, thevapor phase is treated as an incompressible fluid, which

may readily accommodate the incoming jet by expand-ing the size of the cavity as a whole. More detailedinvestigations are clearly necessary to reach definitiveconclusions regarding these effects.

Extensive time-averaged data are also available tocharacterize various cavity parameters. Figure 7 shows acomparison of arithmetically averaged data for the 0-caliber and 1/4-caliber ogive cases with average surfacepressure measurements from Rouse and McNown [18].The incompressible results are also included for com-parison. In all cases, the computed data generally agreewith the experiments. Although there are some differ-ences between the compressible and incompressibleresults, both sets appear to compare well with the data.

Other parameters of relevance include the cavitatordiameter, d, cavity length, L, and cavity diameter ( ).Some ambiguity is inherent in both the experimentaland computational definition of these parameters. Cav-ity closure location is difficult to define due to unsteadi-ness and its dependence upon the afterbody diameter.Accordingly, the cavity length is customarily defined astwice the distance from the cavity leading edge to thelocation of the maximum cavity diameter. Also, the cav-ity diameter (in the computations) is determined byexamining the contour and determining itsmaximum radial location.

Figure 8 compares computed results with experimen-tal data from May [19]. Figure 8a shows the finenessratio, , plotted versus the cavitation number,while Fig. 8b plots the quantity, . At thepresent time, we have computed data using the isother-mal compressible mixture model only for a cavitationnumber of 0.3. We observe that the results agree wellwith the data at this condition and are again in goodagreement with the incompressible results as well.

In summary, we note that the compressible results forthe ogive configurations are in general agreement withprevious incompressible results in the time-averagedquantities. Both sets of results also agree well with aver-aged experimental data. The instantaneous results doshow some significant differences in the amplitudes andperiods of the fluctuations. The compressible resultscorrectly predict the higher frequency in the 1/4-calibercase compared to the 0-caliber case. Additional resultsand investigations are needed to confirm the validity ofthese trends.

Other ApplicationsIn this section, we briefly consider two other applica-

tions of interest that involve the modeling of two-phasecompressibility effects. Figure 9 shows an underwatersupersonic projectile. Both computational results and acorresponding photograph of an actual test are included

dm

αl 0.5=

L dm⁄L d CD( )⁄

8


in the figure. The flow Mach number for the case shownis 1.03 and the liquid to gas density ratio is nominally1000. The experiments and the computations show thepresence of a bow shock upstream of the nose. In addi-tion, because of the high velocity, the cavitation numberis about . Consequently, most of the flow immedi-ately adjacent to the body is completely vaporized as isthe downstream wake portion.

The second example shown in Fig. 10 is the plumeflowfield of an underwater rocket exhaust. The plumeexhaust is supersonic and is slightly under-expanded. Itis surrounded by a co-flowing secondary subsonic gasstream, which in turn is surrounded by a liquid waterfree-stream flow. Again, the nominal liquid to gas den-sity ratio is 1000. Figure 10 shows the shock functionfield, which exhibits the classic expansion pattern. Inparticular, the interaction of the compressible gas streamwith the incompressible liquid is demonstrated first bythe contraction and then by the expansion of the gasstream. In addition, the interface between the liquid andgas phases is comprised of a two-phase mixture, whichis also fully supersonic due to the low magnitude of themixture sound speed.

Both of the above examples involve supersonic Machnumbers in the bulk flow for at least one of the phases.The current isothermal assumption is clearly inadequateto fully represent the dynamics of these flowfields. Nev-ertheless, we have presented preliminary results usingthe isothermal formulation. Detailed modeling using afully compressible two-phase formulation (including theenergy equation) will be the subject of future research.

CONCLUSIONS

Multi-phase mixture flows present a unique challengeto CFD algorithms because of the simultaneous exist-ence of incompressible flow in the liquid phase, low-speed compressible flow in the vapor phase and tran-sonic and supersonic flows in the two-phase mixtureregion. The CFD algorithm has to be efficient and accu-rate over all these Mach number regimes.

We have developed a preconditioned time-marchingalgorithm for the computation of multi-phase mixtureflows. The preconditioning formulation introducespseudo-time derivatives, which automatically adapt tokeep the system well-conditioned in the incompressibleas well as compressible regimes and, thereby, ensurethat proper accuracy and optimal efficiency are main-tained.

Three closely related but distinct preconditioning for-mulations have been derived for isothermal compress-ible flows. The differences arise from the precise formof the governing equations---volume or mass fraction---

used in the derivation. However, we have shown that allthree systems possess identical eigenvalues and, there-fore, should perform comparably in practice.

The preconditioning algorithm has been incorporatedinto an existing multi-phase code (UNCLE-M). Simpletest cases such as an inviscid straight duct and shocktube are used to verify the formulation. The code is thenapplied to more practical application problems.

The principal focus of the paper is the modeling ofsheet- and super-cavitating flowfields in hydrodynamicsapplications. Computational results are obtained forflow over 0-caliber and 1/4-caliber ogives and com-pared with experimental data as well as previouslyobtained incompressible results. The computed com-pressible and incompressible results agree well witheach other and with the experiments for time-averageddata.

The unsteady results, however, show some markeddifferences between the compressible and incompress-ible cases. The flowfields are characterized by unsteadyeffects caused by cavity reentrant liquid jets and cavitypinching. In the compressible case, the cavity reentrantjets appear to be relatively short-lived compared withthe incompressible case. In the latter case, the reentrantjet persists almost until it reaches the leading edge of thecavity. In turn, these effects lead to higher frequenciesfor the compressible model, which is in agreement withmeasured Strouhal frequency data. Thus, it appears thatcompressibility effects may need to be accounted to cor-rectly describe the cavity dynamics. Additional investi-gations are necessary to confirm these findings.

Additional computations have also been performedfor underwater supersonic projectile and underwaterrocket plume flowfields. In these examples, the bulkflow of one of the phases is supersonic, suggesting theneed for a fully compressible formulation (including theenergy equation). The development of preconditioningalgorithms for the fully compressible two-phase systemwill be the subject of future research.

ACKNOWLEDGMENTS

This work is supported by the Office of NavalResearch, contract #N00014-98-0143, with Dr. KamNg as contract monitor.

104–

9


REFERENCES

[1] Eddington, R., ‘‘Investigation of Supersonic Phe-nomena in a Two-Phase (Liquid Gas) Tunnel,’’ AIAAJournal, Vol. 8, no. 1, January, 1970.

[2] Witte, J., ‘‘Mixture Shocks in Two-Phase Flow,’’Journal of Fluid Mechanics, Vol. 36, Part 4, pp. 639-655, 1969.

[3] Van Wijngaarden, L., ‘‘One-Dimensional Flow ofLiquids Containing Small Gas Bubbles,’’ AnnualReview of Fluid Mechanics,’’ pp. 369-396, 1972.

[4] Reisman, G. E., Wang, Y. C. and Brennan, C. E.,‘‘Observations of Shock Waves in Cloud Cavitation,’’Journal of Fluid Mechanics, vol. 355, pp. 255-283,1998.

[5] Campbell, I. J. and Pitscher, A. S., ‘‘Shock Wavesin a Liquid Containing Gas Bubbles,’’ Proceedings ofthe Royal Society London Ser. A, pp. 243-534.

[6] Toumi, I., Kumbaro, A. and Paillere, H., ‘‘Approx-imate Riemann Solvers and Flux Vector SplittingSchemes for Two-Phase Flow,’’ Von Karman InstituteLecture Series, 1999-03, March, 1999.

[7] Wesseling, P., ‘‘Non-Convex Hyperbolic Sys-tems,’’ Von Karman Institute Lecture Series, 1999-03,March, 1999.

[8] Liou, M.-S. and Edwards, J. R., ‘‘AUSM Schemesand Extensions for Low Mach and MultiphaseFlows,’’ Von Karman Institute Lecture Series, 1999-03, March, 1999.

[9] Merkle, C. L., Feng, J. Z. and Buelow, P. E. O.,‘‘Computational Modeling of the Dynamics of SheetCavitation,’’ Proceedings of the 3rd InternationalSymposium on Cavitation, Grenoble, 1998.

[10] Kunz, R. F., Boger, D. A., Stinebring, D. R., Chy-czewski, T. S., Lindau, J. W., Gibeling, H. J., Ven-kateswaran, S. and Govindan, T. R., ‘‘APreconditioned Navier-Stokes Method for Two-PhaseFlows with Application to Cavitation Prediction,’’Computer and Fluids, Vol. 29, pp. 849-875, Novem-ber, 2000.

[11] Lindau, J. W., Kunz, R. F., Boger, D. A., Stine-bring, D. R. and Gibeling, H.J., ‘‘Validation of HighReynolds Number, Unsteady Multi-Phase CFD Mod-eling for Naval Applications,’’ Proceedings of ONR23rd Symposium on Naval Hydrodynamics, Val deReuil, France, Sept 17-22, 2000.

[12] Ahuja, V., Hosangadi, A., Ungewitter, R. andDash, S. M., ‘‘A Hybrid Unstructured Mesh Solverfor Multi-Fluid Mixtures,’’ AIAA 99-3330, 14thComputational Fluid Dynamics Conference, Norfolk,VA, June, 1999.

[13] Chorin, A. J., ‘‘A Numerical Method for SolvingIncompressible Viscous Flow Problems,’’ Journal ofComputational Physics, Vol. 2, pp. 12, 1967.

[14] Choi, Y.-H. and Merkle, C. L., ‘‘The Applicationof Preconditioning to Viscous Flows,’’ Journal ofComputational Physics, Vol. 105, pp. 207-223, 1993.

[15] Venkateswaran, S. and Merkle, C. L., ‘‘Analysisof Preconditioning Methods for the Euler and NavierStokes Equations,’’ Von Karman Institute LectureSeries, 1999-03, March, 1999.

[16] Venkateswaran, S. and Merkle, C. L., ‘‘Effi-ciency and Accuracy Issues in Contemporary CFDAlgorithms,’’ AIAA Fluids 2000 Conference, 2000-2251, Denver, CO, June 2000.

[17] Taylor, L.K., Arabshahi, A., and Whitfield, D. L.,‘‘Unsteady Three-Dimensional IncompressibleNavier-Stokes Computations for a Prolate SpheroidUndergoing Time-Dependent Maneuvers,’’ AIAA 95-0313, 1995.

[18] Rouse, H. and McNown, J. S., ‘‘Cavitation andPressure Distribution, Head Forms at Zero Angle ofYaw,’’ Studies in Engineering Bulletin, 32, State Uni-versity of Iowa, 1948.

[19] May, A., ‘‘Water Entry and the Cavity-RunningBehavior of Missiles,’’ Naval Sea Systems CommandHydroballistics Advisory Committee TechnicalReport, 75-2, 1975.

[20] Stinebring, D. R., Billet, M. L. and Holl, J. W.,‘‘An Investigation of Cavity Cycling for Ventilatedand Natural Cavities,’’ TM 83-13, The PennsylvaniaState University, Applied Research Laboratory, 1983.

10


FIGURES

Figure 1: Convergence history for inviscid flow in a straight duct. History shown with two types of

preconditioning and no preconditioninga) Mach 0.01b) Mach 0.55c) Mach 2.0.

0 200 400 600 800 1000−12

−10

−8

−6

−4

−2

0

2

cycles

log[

R] none

Kunz

Ahuja

a)

none

Form I

Form II

0 200 400 600 800 1000−12

−10

−8

−6

−4

−2

0

2

log[

R]

cycles

none

Kunz

Ahuja

b)

none

Form I

Form II

0 200 400 600 800 1000−12

−10

−8

−6

−4

−2

0

2

cycles

log[

R]

none

Kunz Ahuja

c)

none

Form IForm II

Figure 2: Mixture shock tube computations and comparison with theory of Campbell and Pitscher [5].

Shock moves from left to right with initial location at 0. Model solution shown at even time intervals beginning after initial state. Fluid to right of shock is at rest with

liquid to gas density ratio of 1000, and αl=0.95.a) Pressure ratio 2.b) Pressure ratio 6.

−1 −0.5 0 0.5 1 1.5 2 2.5 3950

955

960

965

970

975

Ideal shock location.

shock tube position

a)

ρ

−1 −0.5 0 0.5 1 1.5 2 2.5 3950

955

960

965

970

975

Ideal shock location.

shock tube position

b)

ρ

11


Figure 3: Volume fraction contours and streamlines. Snapshots span an approximate modeled cycle. 0-

caliber ogive at ReD=1.46x105 and σ=0.3.a) Isothermal formb) Incompressible form

a)

t=0.0

t=0.4

t=0.8

t=1.2

t=1.6

t=2.0

b)

t=0.0

t=0.5

t=1.0

t=1.5

t=2.0

t=2.5

Figure 4: Volume fraction contours and streamlines. Snapshots span an approximate modeled cycle. 0.25-

caliber ogive at ReD=1.36x105 and σ=0.3.a) Isothermal formb) Incompressible form

a)

t=0.0

t=0.2

t=0.4

t=0.6

t=0.8

t=1.0

b)

t=0.0

t=0.6

t=1.2

t=1.8

t=2.4

t=3.0

12


Figure 5: Drag coefficient histories for 0-caliber and 1/4 caliber ogive shapes modeled with incompressible and

isothermal forms.a) 0-caliber ogive based on isothermal model.b) 0-caliber ogive based on incompressible model.c) 1/4-caliber ogive based on incompressible and

isothermal models

−4 −2 0 2 4−5

0

5x 10

−3

CD

Am

plitu

de

time

a)isothermal model

−4 −2 0 2 4−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

CD

Am

plitu

de

time

b)incompressible model

−5 −4 −3 −2 −1 0 1 2 3−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

time

CD

Am

plitu

de

incompressible model

isothermal model

c)

Figure 6: Strouhal frequency comparison modeling results and data from Stinebring [20].

Figure 7: Comparison of time averaged results with data of Rouse and McNown [18].

a) 0-caliberb) 1/4-caliber

0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

σ

Str

hemisphere ReD

=(0.35−1.55)x106

0−cal ReD

=(0.96−1.46)x105

Isothermal 0−caliberIncompressible 0−caliberIsothermal 1/4−caliberIncompressible 1/4−caliber

0 2 4 6 8 10−0.5

0

0.5

1

1.5

Cp

s/d

isothermalmodel

incompressiblemodel

0 1 2 3 4 5 6 7−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

s/d

Cp isothermal

model

incompressiblemodel

13


Figure 8: Comparison of isothermal and incompressible model results to the data of May [19].

10−2

10−1

10010

−2

10−1

100

101

102

L/d

m

σ

Isothermal 0−calIsothermal 1/4−calIncompressible 0−calIncompressible 1/4−caldata for various forebody shapes

10−2

10−1

10010

0

101

102

103

104

σ

L/(

dCD1/

2 )

Isothermal 0−calIsothermal 1/4−calIncompressible 0−calIncompressible 1/4−caldata for various forebody shapes

Figure 9: Isothermal model result and photograph of supersonic underwater projectile. Model result contains

flow field density contours (red high blue low) and projectile surface colored by pressure (blue high red

low).

wake bowshock

Figure 10: Cartoon vehicle and 3-stream, axisymmetric aft flow region. Center jet (diameter=1) surrounded by gas at free stream velocity (outer diameter=2). Liquid to gas density ratio 1000. Shock function:

Mixturedensity

Shockfunction

V ρm∇•

14


Documents

AIAA 2001-0279 Preconditioning Algorithms for the