Stability of preconditioned Navier–Stokes equations associated with a cavitation model

www.elsevier.com/locate/compfluid

Computers & Fluids 34 (2005) 319–349

Stability of preconditioned Navier–Stokes equationsassociated with a cavitation model

O. Coutier-Delgosha a,*, R. Fortes-Patella a, J.L. Reboud b,N. Hakimi c, C. Hirsch d

a LEGI/INPG, BP 53, 38041 Grenoble Cedex 9, Franceb ENISE, 58 rue J. Parot, 42023 St Etienne Cedex 2, France

c NUMECA, 5 av. F. Roosevelt, 1050 Brussels, Belgiumd VRIJE Universiteit Brussel Peinlaan 2, 1050 Brussels, Belgium

Received 22 July 2002; received in revised form 25 September 2003; accepted 13 May 2004

Available online 11 September 2004

Abstract

A 3D numerical model is proposed to simulate complex unsteady cavitating flows. The final objective is

to predict instabilities due to cavitation in turbopump inducers. It was previously applied to simpler two-

dimensional simulations such as a Venturi type section [Int. J. Numer. Meth. Fluids, in press] and foil sec-

tions [Int. J. JSME B 45(3) (2002)].

The model is based on the code FineTurbo significantly modified to take into account the cavitation

process. The numerical scheme consists in a time-marching algorithm initially devoted to compressible

flows. A low-speed preconditioner is applied to treat low Mach number flows. This numerical resolutionis coupled to a single-fluid model of cavitation. The evolution of the density is governed by a barotropic

state law proposed and validated previously by Delannoy and Kueny [Proc 1990 ASME Cavitation Multi-

phase Flow Forum 98 (1990) p. 153] and Coutier-Delgosha et al. [Int. J. Numer. Meth. Fluids 42 (2003)

527].

0045-7930/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compfluid.2004.05.007

* Corresponding author. Present address: ENSAM Lille/LML Laboratory, 8 bld Louis XIV, 59046 Lille Cedex,

France.

E-mail addresses: [email protected] (O. Coutier-Delgosha), [email protected] (R. Fortes-Patella), jean-

[email protected] (J.L. Reboud), [email protected] (N. Hakimi), [email protected] (C.

Hirsch).

mailto:[email protected]






320 O. Coutier-Delgosha et al. / Computers & Fluids 34 (2005) 319–349

The present work focuses on the numerical stability of the Navier–Stokes equations associated to the

barotropic state law. Fourier footprint representations are applied to several two-dimensional non-cavitat-

ing and cavitating flow field configurations, and the influence of the numerical and physical parameters on

the stability is investigated. The influence of the preconditioner is also discussed: a modification is proposed

in the two-phase areas, and its effect is tested in a two-dimensional Venturi type section flow configuration.

A significant improvement is obtained concerning both the convergence level and speed.

� 2004 Elsevier Ltd. All rights reserved.

1. Introduction

To gain further knowledge concerning cavitating flows in turbopump inducers, a three-dimen-sional model of steady and unsteady cavitation for inviscid or viscous fluids is proposed in thispaper. The numerical resolution is based on the code FineTurbo developed by Numeca Interna-tional. Important modifications have been brought in the initial solver to take into account thecavitation phenomenon. The time-averaged Reynolds equations are coupled to a physical modelof cavitation resulting from previous works performed in the LEGI laboratory (Grenoble,France). This model was presented and validated in 2D configurations of unsteady cavitatingflows by Delannoy and Kueny [6], Reboud et al. [17], and Coutier-Delgosha et al. [4,5]. The liq-uid/vapor mixture is controlled by a postulated barotropic state law that links the fluid density tothe pressure variations. The two-phase medium is thus considered as a single fluid in which thedensity varies from the liquid one to the vapor one, with respect to the local static pressure.The main numerical challenge of this approach results from the difficulty to manage both an al-most incompressible state in pure vapor or in the liquid phase, and a highly compressible state inthe liquid/vapor transition areas. The physical model is presented in Section 2 of the paper, andthe main features of the numerical model are reported in Section 3 with emphasis on the specialtreatment induced by the two-phase flow model.

The present work is devoted to the stability analysis of the numerical resolution in the case ofcavitating flow fields. The stability of the Navier–Stokes equations system in non-cavitating con-ditions was previously investigated by Hakimi [7] and the improvement due to the preconditionerin low compressible and fully incompressible flows was clearly demonstrated.

As a matter of fact, simulations including cavitation lead to treat simultaneously highly com-pressible areas (in the two-phase mixture) and almost incompressible flows (pure liquid and purevapor). The second case results in the use of a preconditioner (cf. Turkel [20,21] and Choi and Mer-kle [1]), so that the convergence rate does not slow down in liquid and in vapor. The influence of thispreconditioner on the stability of the resolution in the two-phase mixture is worth to be analyzed.

The stability analysis of numerical schemes is a powerful way to determine their properties andtheir eventual weakness, as explained by Hirsch [8]. Such an investigation concerns here the con-tinuity and momentum equations in the three spatial directions. For simplicity, we consider in thepresent work only two-dimensional problems.

The method applied hereafter is derived from the analysis proposed by Von Neumann, and de-picted in detail by Hirsch [8]. It is based on the Fourier decomposition of the residual operator. Itprovides a necessary stability condition, but not a sufficient one. The equations are linearized, andall non-constant parameters of the flow field are frozen. In other words, the stability analysis is

Nomenclature

c local speed of sound (m/s)Cmin minimum speed of sound in the mixture (m/s)Lref length of the Venturi type section (m)P local static pressure (Pa)Pref reference pressure = inlet pressure (Pa)Ptot total pressure (Pa)Pg dynamic pressure Ptot � P (Pa)Pvap vapor pressure (Pa)Tref Lref/Vref = reference time (s)v(u,v,w) local velocity (m/s)V control volume (m3)Vref reference velocity = inlet velocity (m/s)a local void fractionap preconditioning coefficient (standard value ap = �1)bp b0 · Vref preconditioning coefficient: pseudo-speed of sound (m/s)b0 constant in the expression of bp (standard value b0 = 3)l dynamic viscosity (Pas)m cinematic viscosity (m2/s)q aqv + (1�a)ql local density of the mixture (kg/m3)qv, ql vapor and liquid density (kg/m3)r ðP ref � P vapÞ=qV 2

ref cavitation numbers, Ds pseudo-time, pseudo-time-step (s)t, Dt physical time, physical time-step (s)

O. Coutier-Delgosha et al. / Computers & Fluids 34 (2005) 319–349 321

performed on a local configuration of the flow field. We focus in the present study on the effect ofthe numerical and physical parameters on stability, in order to detect non-stable configurations orlow convergence rate conditions. The results of the analysis are represented by Fourier footprintspopularized by Lynn and Van Leer [15]. The main features of the method and its interest are re-ported in Section 4: drawing the eigenvalues of the residual operator in the complex plane andoverlapping the stability contours of the time-integration scheme gives indications about the prop-erties of the numerical resolution.

The linearized Navier–Stokes equations system is written in Section 5 in the particular case ofthe barotropic state law, and the expression of the residual operator is obtained after the spatialdiscretization and the Fourier decomposition.

The stability analysis concerns here the dual time-step resolution, i.e. the convergence processinside each physical time-step. Nevertheless, the influence of the physical time derivatives on thisresolution is also considered. The physical time-step progression is performed through an implicittime integration scheme, second order accurate and unconditionally stable.

Stability of several non-cavitating and cavitating flow configurations is investigated in Section6, on the basis of the flow field conditions obtained in a two-dimensional Venturi section. It was


designed so that the flow is submitted to the same pressure field as in an inducer blade-to-bladechannel [13]. The geometry considered is characterized by a 4� divergence angle, leading to arather stable behavior. The main features of the numerical simulations and of the results are pre-sented here, more details can be found in [2]. The influence of the numerical and physical param-eters on both the stability and the convergence rate is investigated for the whole barotropic law(composed of the incompressible and the highly compressible areas).

The influence of the preconditioner is found to be of primary importance, and a modification isproposed in Section 7 to improve the convergence rate in the central part of the barotropic statelaw. Simulations of the cavitating flow in the Venturi type section are performed with the modifiednumerical model, and the results are discussed.

2. Physical model

To predict the development, arrangement, and collapse of vapor bubbles in the liquid flow,time-averaged Reynolds equations must be coupled to a cavitation model. In the present workwe apply a single fluid approach based on previous numerical and physical work performed forabout 15 years in the LEGI laboratory by Delannoy and Kueny [6], Reboud et al. [17], and Cou-tier-Delgosha et al. [4].

The two-phase flow is considered as only one fluid, characterized by a density q that varies inthe computational domain according to a postulated state law. When the density in a cell equalsthe liquid one ql, the whole cell is occupied by liquid, and if it equals the vapor one qv, the cell isfull of vapor. Between these two extreme values, the cell is occupied by a liquid/vapor mixture thatis still considered as one homogeneous single fluid. The void fraction a = (q � ql)/(qv � ql) corre-sponds to the local ratio of vapor contained in the mixture. In this simple model, the void ratio a isrelated to the state law, the fluxes between the phases are treated implicitly, and no supplementaryassumptions are required.

Such model assumes that locally (in each cell), velocities are the same for liquid and for vapor:in the mixture areas vapor structures are supposed to be perfectly carried along by the main flow.This hypothesis is often assessed for this problem of sheet-cavity flows, in which the interface isconsidered to be in dynamic equilibrium [14,16,18]. The momentum transfer between the phases isthus strongly linked to the mass transfer.

The barotropic state law q(P) is presented in Fig. 1. When the pressure is respectively higher orlower than the vapor pressure, the fluid is supposed to be purely liquid or purely vapor, accordingrespectively to the Tait equation [12] and to the perfect gas law. In this last case, temperature T0 isimposed, since thermal effects associated with vaporization and condensation are not consideredin the present study:

q ¼ PrpgT 0

ðperfect gas law for the pure vaporÞ ð1Þ

q ¼ qref �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP þ P 0

P ref þ P 0

n

rðTait law for the pure liquidÞ ð2Þ

where Pref and qref are reference pressure and density, and rpg is the constant for the perfect gas.

Cmin=1.5 m/s

Mixture

0

200

400

600

800

1000

1200

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Pressure (bar)

dens

ity (

kg/m

3 )

Perfect gas law Tait equation behaviour

∆pvap

(pure liquid)(pure vapour)

pvap

Cmin=2 m/s

Fig. 1. The barotropic state law q(P) for cold water (20�C).


For water, P0 = 3 · 108 and n = 7.The two fluid states are joined smoothly in the vapor-pressure neighborhood by the following

law:

q ¼ Ab þ Bb � sin1

Bb

P � P vap

C2min

!ð3Þ

with Ab and Bb constants depending from the connection between the three parts of the state law,at the points (Pl,ql), and (Pv,qv).

Ab ¼ ql þ qv

2and Bb ¼ ql � qv

2ð4Þ

It results in the evolution law characterized mainly by its maximum slope 1=C2min, where

C2min ¼ oP=oq. Cmin can thus be interpreted as the minimum speed of sound in the mixture.

3. Numerical model

FineTurbo is a three-dimensional structured mesh code that solves the time dependant Reyn-olds-averaged Navier–Stokes equations.

3.1. Equations

Time accurate resolutions use the dual time stepping approach proposed by Jameson [11].Pseudo-time derivative terms are added to the equations. They march the solution towards con-vergence at each physical time-step. The code resorts to a multigrid strategy to accelerate the con-vergence, associated with a local time stepping and an implicit residual smoothing.

This kind of resolution is devoted to highly compressible flows. In the case of low-compressibleor incompressible flows, its efficiency decreases dramatically. This well-known problem has beenaddressed by many authors and solved by introducing a preconditioner. In the present work, the


preconditioner of Hakimi [7] is used. It is based on the previous studies of Turkel [20,21], andChoi and Merkle [1], and it solves the equations for the gauge pressure, velocity, and gauge totalenthalpy. It consists in multiplying the pseudo-time derivatives by a preconditioning matrix C�1.Such modifications have no influence on the converged result, since these terms are of no physicalmeaning, and converge to zero.

Moreover, in our cavitation model thermal effects are neglected in the vaporization and con-densation phenomena, so temperature does not appear in the state law of the fluid (see Section2). As a consequence, the energy equation is disconnected from the others: the temperature fieldhas no influence on the resolution of the mass and momentum equations. Its resolution is thus ofno use, and it is omitted hereafter. The resulting governing equations, written in an integral formin a control volume V whose surface is S, are
Z Z Z
VC�1 oQ

osdV þ

Z ZSðFinv þ FvÞ � ndS ¼

Z Z ZVScedV � o

ot

Z Z ZVUdV ð5Þ

where Q = (Pg,u,v,w)t, U = (q,qu,qv,qw)t; Pg is the gauge pressure (denoted simply P hereafter);Finv and Fv are the inviscid and viscous fluxes across the frontier S of V (normal n); s and t arerespectively the pseudo and physical time-steps; ap and bp are the preconditioner coefficients,and Sce is the source term

C�1 ¼

1

b2p

0 0 0

apub2

p

q 0 0

apvb2

p

0 q 0

apwb2

p

0 0 q

26666664

37777775

3.2. Spatial discretization

These equations are discretized in their conservative form with a finite volume approach. Thediscrete form of Eq. (5) over a computational cell volume becomes:

V � C�1 oQ

osþXFaces

ðF � nÞ�DS ¼ V � Sce� VoU

otð6Þ

where (F Æ n)* is the numerical flux at the cell interfaces. A centred approximation is applied to theviscous fluxes, while the inviscid ones are calculated with a central convection scheme associatedwith artificial dissipation. The resulting expression of the numerical flux along the i-direction onthe right side of the cell is

ðF � nÞ�iþ1=2;j;k ¼ ½Finv � niþ1=2;j;k þ ½Fv � niþ1=2;j;k �Dartiþ1=2;j;k ð7Þ

The first right hand term is the centred approximation of the inviscid flux, and it is treated as

½Finv � niþ1=2;j;k ¼ 12ð½Finv � ni;j;k þ ½Finv � niþ1;j;kÞ ð8Þ


The artificial dissipation Dart is composed of two terms, respectively of second and fourth order,as initially proposed by Jameson et al. [10]. The following form of Dart leads to a central secondorder accurate convection scheme:

Dartiþ1=2;j;k ¼ 1

2d2iþ1=2;j;kK

iiþ1=2;j;kdQiþ1=2;j;k þ d4

i;j;kKiiþ1=2;j;kd

2Qi;j;k � d4iþ1;j;kd

2Qiþ1;j;k ð9Þ

dQiþ1=2;j;k ¼ Qiþ1;j;k �Qi;j;k ð10Þ

d2Qi;j;k ¼ Qiþ1;j;k � 2Qi;j;k þQi�1;j;k ð11Þ

where Ki is the spectral radius along the i-direction multiplied with the cell face area; d2 and d4 arecoefficients associated respectively to second order and fourth artificial dissipations. Their expres-sion is given by

d2iþ1=2;j;k ¼ e2

p maxðdi�1;j;k; di;j;k; diþ1;j;k; diþ2;j;kÞ þ e2q maxðci�1;j;k; ci;j;k; ciþ1;j;k; ciþ2;j;kÞ

d4iþ1=2;j;k ¼ maxð0; e4

p � d2iþ1=2;j;kÞ

ð12Þ

The variables di and ci are sensors to activate the second-difference dissipation in regions of stronggradients, and to de-activate it elsewhere. di (proposed initially by Jameson et al. [10]) applies inpressure gradients areas, such as shocks, and ci is a supplementary sensor added for cavitatingflows:

di;j;k ¼P iþ1;j;k � 2P i;j;k þ P i�1;j;k

P iþ1;j;k þ 2P i;j;k þ P i�1;j;k

and ci;j;k ¼

qiþ1;j;k � 2qi;j;k þ qi�1;j;k

qiþ1;j;k þ 2qi;j;k þ qi�1;j;k

ð13Þ

The standard values for e2p and e2

q are respectively 1 and 0.5. The second term has been added toensure the numerical stability in cavitating areas, without modifying the numerical scheme forother flow configurations. The standard value for e4

p is 0.1. The influence of these numericalparameters on the numerical stability is investigated in Section 6.

Note that all expressions above correspond to the numerical flux along the i-direction. Similartreatments are applied for the two other directions.

3.3. Time integration

The pseudo-time integration is made by a four-step Runge–Kutta procedure. The residual Rcan be defined as

R ¼ Sce� 1

V

XFaces

ðF � nÞ�DS ð14Þ

and R(Q(s)) stands for the expression of R obtained with the variable Q characterizing the flowfield at the pseudo-time s.

Then the four-stage scheme that marches the solution from Q(s) to Q(s + ds) can be written as

Q1 ¼ QðsÞ þ a1DsRðQðsÞÞ

Q2 ¼ QðsÞ þ a2DsRðQ1Þ


Q3 ¼ QðsÞ þ a3DsRðQ2Þ

Qðs þ DsÞ ¼ QðsÞ þ a4DsRðQ3Þ

with a1 = 0.125, a2 = 0.306, a3 = 0.587, a4 = 1.The physical time-derivative terms are discretized with a second order backward difference

scheme that ensures a second order accuracy in time:

o

ot

Z Z ZVUdV

� �nþ1

¼ 1:5Unþ1 � 2Un þ 0:5Un�1

DtV ð15Þ

3.4. Underrelaxation of the density

The two-phase structure of cavitation is fundamentally unsteady, particularly in the closurearea of the cavitation sheet [4]. The collapse of vapor structures is so sudden that particles of fluidmay turn from pure vapor to pure liquid during only one time-step. A smooth description of thisphenomenon would impose a physical time-step about hundred times smaller than the one usedcurrently. Considering the frequency of typical unsteady processes associated with cavitation,such as cavity self-oscillation, it appears that the use of a very small time-step is unreasonable,because of the total computational time it would induce. This particularity leads to severe numer-ical instabilities.

For this reason, an efficient control of the density time fluctuations and space distribution mustbe applied. Density variations are thus underrelaxed at each iteration i to prevent too suddenchanges in a single pseudo-time-step:

qiþ1 ¼ qi þ gðqðP iþ1Þ � qiÞ ð16Þ

where the standard value of underrelaxation coefficient g is 0.3 in the case of the 2D calculationreported in the present study. After such treatment on the density, pressure Pi+1 is updatedaccording to the barotropic relation q(P), so that the fluid state law is always respected.

Underrelaxing the density is the same as multiplying the local time-step Ds by g in the massequation. As a matter of fact, the convergence significantly slows down when underrelaxationis activated. Nevertheless, the computational time increases much more if a global reduction ofthe pseudo-time-step is applied also in the momentum equations. So underrelaxing the densitywas found to be a satisfactory choice to improve stability without being confronted to prohibitivecomputation times.

3.5. Additional features

• Turbulence model: we use for the calculations presented in this paper a classical Baldwin–Lomax turbulence model.

• Boundary conditions: they are based on a system of dummy cells. Classical incompressible typesof boundary conditions are applied: imposed velocities at the inlet, and an imposed static pres-sure at the outlet.


• Initial transient treatment: first of all, a steady step is carried out, with a pseudo-vapor pressurelow enough to ensure no presence of vapor in the whole computational domain. Then, thisvapor pressure is increased progressively during the early time steps, until the required cavita-tion number r is reached. Vapor appears progressively in the low static pressure regions duringthis transient. The cavitation number is then kept constant throughout the computation.

4. Stability analysis process

The Von Neumann stability analysis is detailed by Hirsch [8], and the Fourier footprint repre-sentations was initially proposed by Lynn and Van Leer [15]. Thus, only the main features of themethod are reported here. As precised in the introduction, the stability analysis will be performedin Section 6 in the case of a two-dimensional problems, so all equations are written hereafter intwo dimensions.

4.1. Fourier decomposition of the residual operator

The linearized Navier–Stokes equations can be written under the following form (source terms,mainly due to inertial and coriolis forces in the case of rotating machines, are not written here forsake of simplicity, excepted the physical time derivatives):

oQ

osþ A

oQ

oxþ B

oQ

oy¼ C

o2Q

ox2þD

o2Q

oy2þ E

o2Q

oxoy� oQ

otð17Þ

where Q(P,u,v) is the vector of primitive variables; s and t are respectively the pseudo andthe physical time steps; A, B are inviscid Jacobians, and C, D, E are viscous Jacobians relatedto primitive variables.

The convection scheme results in a transformation of the space derivatives, and it leads to anexpression of oQ

os in each cell as a function of the Qi,j values in the neighboring cells. Then the solu-tion is decomposed into an expression of the form:

Qi;j ¼X

u

Xw

Qi;jðsÞeIðiuþjwÞ

where I ¼ffiffiffiffiffiffiffi�1

pand u and w range from �p to p.

This Fourier transform results in a final expression of the form:

dQi;j

ds¼ ZQi;j ð18Þ

where Z is a complex matrix depending on u and w. It is characteristic of the initial Navier–Stokesequations, their eventual preconditioning, the convection scheme, and also the state law of thefluid. Eq. (18) is valid as well for the solution as for the error, defined as the difference betweenthe calculated solution and the exact solution of the discretized equations. To ensure stability,the error must decrease at each time-step. It can also be written under the form:

Ei;j ¼X

u

Xw

Ei;jðsÞeIðiuþjwÞ


Each mode also satisfies the Eq. (18):

Fig. 2

0.8,0.

dEi;j

ds¼ ZEi;jðsÞ ð19Þ

So, the numerical scheme will be unstable or stable, whether at least one mode will be amplified ornot, i.e. depending on the time integration scheme and the expression of the matrix Z.

The answer is obtained in two steps:First, the stability contour of the time integration scheme is determined, by considering the sca-

lar equation similar to Eq. (19): oaos ¼ z� a. The time integration scheme, i.e. the discretization of

the term oaos, is more or less stable: it depends on the way the variable a is updated at each new time-

step. The value of z Æ ds leads to a modulus G ¼ j anþ1

an j higher or lower than 1, leading respectivelyto an unstable or a stable time integration configuration. We obtain in the complex plane a rep-resentation of the stability contour by linking the complex numbers corresponding to G = 1. Fig. 2represents such a stability contour for the four-step Runge–Kutta time-integration scheme that isused in the numerical model.

For each mode, stability can be achieved only if matrix Z 0 = Z · ds is consistent with the timeintegration stability domain, i.e. if its eigenvalues are located inside this domain in the complexplane.

So, the second step consists in determining the eigenvalues of matrix Z 0, for all modes u and w,to overlap them with the stability areas of the time integration scheme. If some of them are outsidethese areas, the numerical resolution is potentially unstable. It will really become unstable if thecorresponding mode is excited in the flow field during the simulation.

Moreover, these eigenvalues correspond to characteristic velocities in the flow field, namelyconvection velocities or acoustics speeds. Therefore their value gives some piece of informationabout the propagation and the damping of the error, even if they belong to the stability domain.This information directly deals with the convergence speed of the calculation towards solution.

. Stability contour associated with a four steps Runge–Kutta scheme (iso-lines jGj = 0.1,0.2,0.3,0.4,0.5,0.6,0.7,

9,1).

Fig. 3. Frequency of the modes.


4.2. Fourier footprint representation

Hakimi [7] proposes a classification of matrix Z 0 eigenvalues into four groups based on thevalue of u and w, represented on Fig. 3.

Notation BuBw is devoted to low frequency modes in both x and y directions, HuHw is devotedon the contrary to high frequency modes in the two directions, and the two last notations corre-spond to intermediate cases.

The Fourier transform of the error contains modes of the four quadrants, characterized by theeigenvalues of the corresponding matrix Z 0. The best numerical scheme is the one that leads as fastas possible to the solution of the discretized equations. It means that all the errors modes shouldin the ideal case decrease rapidly at each new time-step. It can be obtained by the propagation ofthe error components out of the computational domain, or/and by their damping. The propaga-tion speed depends for each mode on the imaginary part of the corresponding eigenvalue [7]: ahigh imaginary part will lead to good propagation properties, while a very small one (eigenvalueson the real axis) will correspond to almost no propagation at all.

The error damping depends on the location of the eigenvalues in low or high amplification areasof the time integration scheme: the lowest the amplification is, the best the error component willbe damped. Considering that amplification usually equals 1 near the imaginary axis, it appearsthat an eigenvalue located near the origin will not be propagated neither damped. It will then con-tribute to limit the convergence of the numerical scheme towards solution.

This type of figure also gives information about the utility of using a multigrid treatment: indeed,error modes located in the BuBw quadrant, which usually have the poorest convergence properties,are moved in the HuHw quadrant when a coarser grid is used. It means that these modes will even-tually decrease on the coarse grid, if quadrant HuHw has better convergence properties.

5. Residual operator

The barotropic state law presented in Section 2 is associated with the Navier–Stokes equationsin a two dimensional problem. The first step to determine in this case matrix Z of the residualoperator consists in obtaining the expression of the Jacobians A, B, C, D, and E used in Eq. (17).


5.1. Determination of the Jacobians

The Navier–Stokes equations can be expressed under the following non-conservative form,resulting from the passage from 3D to 2D in Eq. (6) (only the physical time derivatives are writtenhere in the source term). Note that pressure pseudo-time derivatives in all equations are due to thepreconditioner [7]:

1

b2p

oPos

þ uoqox

þ qouox

þ voqoy

þ qovoy

¼ � oqot

apu

q � b2p

oPos

þ ouos

þ uouox

þ vouoy

þ 1

qoPox

¼ 4

3mo2uox2

þ mo2uoy2

þ m3

o2voxoy

� ouot

apv

q � b2p

oPos

þ ovos

þ uovox

þ vovoy

þ 1

qoPoy

¼ mo

2vox2

þ 4

3mo

2voy2

þ m3

o2u

oxoy� ov

ot

ð20Þ

Since density only depends on pressure (barotropic law), the speed of sound is defined as

c ¼

ffiffiffiffiffiffidPdq

sð21Þ

As a consequence, oqox and oq

oy can be replaced respectively by 1c2

oPox, and 1

c2oPoy .

The mass equation becomes:

oPos

þ ub2

c2

oPox

þ qb2 ouox

þ vb2

c2

oPoy

þ qb2 ovoy

¼ 0 ð22Þ

And the whole system can be written in the linearized form of Eq. (17):

oQ

osþ A

oQ

oxþ B

oQ

oy¼ C

o2Q

ox2þD

o2Q

oy2þ E

o2Q

oxoy� oQ

otð23Þ

with Q = (P,u,v) and matrix

A ¼

ub2p

c2qb2

p 0

1

q� apu2

qc2uð1 � apÞ 0

�apuvqc2

�apv u

266666664

377777775; B ¼

vb2p

c20 qb2

p

�apuvqc2

v �apu

1

q� apv2

qc20 vð1 � apÞ

266666664

377777775; C ¼

0 0 0

0 4m=3 0

0 0 m

264

375;

D ¼0 0 0

0 m 0

0 0 4m=3

264

375; E ¼

0 0 0

0 0 m=3

0 m=3 0

264

375

The final expression of the jacobians depends on the expression of c, issued from the differentparts of the barotropic state law q(P):


Pure liquid ðTait lawÞ : c ¼


s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin

qref

ðP ref þ P 0Þr

P þ P 0

P ref þ P 0

� �n�12n

ð24Þ

Pure vapor ðperfect gas lawÞ : c ¼


s¼

ffiffiffiffiffiffiffirT 0

pð25Þ

Two � phase mixture : c ¼


s¼ Cmin cos

P � P vap

Bb � C2min

!" #�12

ð26Þ

where P0 = 3 · 108 Pa, n = 7, Cmin is the minimum speed of sound in the mixture and Bb ¼ ql�qv

2as

defined in Section 2.

Actually, only two states of the fluid will be treated in the stability analysis:

• In the pure liquid or pure vapor areas, the Mach number is smaller than 0.03 so the fluid will beconsidered hereafter as fully incompressible.

• On the contrary in the liquid/vapor areas the fluid is highly compressible: Both theoretical esti-mations [9] and comparisons of the shape of sheets of cavitation obtained in experiments andby calculations [17] give a minimum speed of sound of about 1–3m/s. So the standard valueapplied in the present simulations is Cmin = 2m/s, which corresponds to a local Mach numberM = 5. This maximum Mach number is reached for an intermediate density value q = (ql + qv)/2, i.e. a void ratio a = 0.5. According to the central part of the barotropic state law (see Fig. 1),the corresponding pressure is P = Pvap.

For the stability analysis in the mixture configurations, two parameters are thus defined:

• Mmax = Vref/Cmin (characterizes the shape of the state law).• The cavitation number r ¼ P � P vap=ð1=2qlV

2refÞ is characteristic of the position of the consid-

ered configuration on the barotropic state law. r = 0 is the middle of the law (P = Pvap).

These two parameters define the cavitating configuration. Practically r = 0 will systematicallybe applied, so that the worst situation associated with cavitation is considered.

All the Jacobians of a cavitating configuration are now defined. The three remaining steps be-fore obtaining the final expression of matrix Z 0 are the application of the convective scheme in Eq.(17), the Fourier transform to obtain Z, and the determination of the local time-step ds to com-pute Z 0 = Z Æ ds.

5.2. Application of the convective scheme

The aim is to obtain a time differential equation, which may be simplified by Fourier transform.It is thus necessary to express all space derivatives in each cell as a function of Q in the neighbor-ing cells. This requires a convection scheme, which is presently a second order central schemeincluding fourth order and second order artificial dissipations (see Section 3).


Term A oQox can be written in the form:

A

2DxðQiþ1;j �Qi�1;jÞ �

1

DxðDart

iþ1=2;j �Darti�1=2;jÞ ð27Þ

The expression of Dart was given in Section 3.2. In the framework of the stability analysis d2 andd4 are supposed to be constant, and the following expression of Dart is obtained:

Dartiþ1=2;j ¼ 1

2d2KAðQiþ1;j �Qi;jÞ � 1

2d4KAðQiþ2;j � 3Qiþ1;j þ 3Qi;j �Qi�1;jÞ ð28Þ

KA is a diagonal matrix with the spectral radius of matrix A as elements. The spectral radius ofmatrix A is the maximal absolute value of its eigenvalues.

As a matter of fact, the second order dissipation is strongly non-linear because of the limitersdetailed in Section 3.2. Since the Fourier analysis is only valid for a linear model equation, suchkind of non-linear character cannot be taken into account. Nevertheless, in order to evaluate thegeneral influence of second order artificial dissipation on the numerical stability we will test in Sec-tion 6.4.3 the effect of applying it uniformly with no limiter.

The final expression of the space derivative along the i-direction becomes:

AoQ

ox¼ A

2DxðQiþ1;j �Qi�1;jÞ �

d2

2DxKAðQiþ1;j � 2Qi;j þQi�1;jÞ

þ d4

2DxKAðQiþ2;j � 4Qiþ1;j þ 6Qi;j � 4Qi�1;j þQi�2;jÞ ð29Þ

A similar expression is obtained for matrix B, and second order derivatives are discretized in acentral way.

5.3. Fourier transform

The Fourier transform is a powerful way to simplify the time differential equation. Qi,j is

decomposed into the form: Qi;j ¼P

u

PwQi;jðtÞeIðiuþjwÞ, I ¼

ffiffiffiffiffiffiffi�1

p, u and w ranging from �p to

p. This formulation is introduced into Eq. (29), then each mode is considered separately. Simpli-

fying by eI(iu+jw), a relation between Qi;jðtÞ anddQi;jðtÞ

ds is deduced:

dQi;jðtÞds

¼ ZQi;jðtÞ ð30Þ

with

Z ¼ �IA

Dxsin u þ B

Dysin w

� �� d2

DxKAð1 � cos uÞ � d2

DxKBð1 � cos wÞ

� 2d4

DxKAð1 � cos uÞ2 � 2d4

DyKBð1 � cos wÞ2 � 2C

Dx2ð1 � cos uÞ

� 2D

Dy2ð1 � cos wÞ � E

DxDysin u sin w � 1:5

DtI ð31Þ


5.4. Local time-step

The pseudo-time-step (that will be called hereafter time-step) is computed locally, according tothe flow configuration. Then it is applied to the three conservation equations.

The inviscid time-step Dsinv is defined on the basis of the CFL condition by the followingexpression:

1

Dsinv

¼ ðq1 þ q2ÞCFL � V

ð32Þ

where q1 and q2 are the spectral radii associated to the local propagation directions n1 and n2.They are obtained by the resolution of jAnix + Bniy � kIj = 0 with ni(nix,niy), i = 1,2.

The viscous time-step Dsv is given by the following expression [22]:

1

Dsv

¼ 8m

CFLv � V 2ðn2

i þ n2j þ 2ni � njÞ ð33Þ

The final local time-step Ds is obtained by taking the lowest of these two values:

1

Ds¼ 1

minðDsinv þ DsvÞð34Þ

Matrix Z 0 = Z · Ds is then completely defined.

6. Stability analysis

The stability analysis is investigated in configurations derived from non-cavitating and cavitat-ing calculations performed in a 2D Venturi type section. The results are detailed in [2], and onlythe main features are presented here in Section 6.1. Standard values of the numerical and physicalparameters to be tested are reported in Section 6.2, and then the stability of non-cavitating andcavitating configurations is successively investigated in Section 6.3.

6.1. Results of the 3D NS time-accurate calculation

Numerical simulations are performed on a Venturi type section characterized by small conver-gent–divergent angles (4.3–4�) and a small contraction ratio at the throat (cf. Fig. 4). The shape ofthe Venturi bottom downstream of the throat simulates an inducer blade suction side with beveledleading edge geometry [2]. According to experimental observations reported by Stutz and Reboud[19], a stable cavity is obtained, with only small-scale fluctuations in its downstream part.

A 250 · 90 cells mesh is used. Such grid size is adequate to simulate the steady or unsteadybehavior of sheet cavitation in the Venturi type section, as reported in [2,4]. It is mainly charac-terized by a considerable grid contraction at the Venturi throat, so that the inception and thegrowing of the cavitation sheet can be correctly predicted.

Time-accurate computations have been carried out, and a satisfactory convergence rate was ob-tained for a large range of cavitation numbers. The stable cavitating behavior is correctly simu-lated by the model: in all the analyzed cases we obtain, after a transient fluctuation of thecavity length, a quasi-steady behavior of the cavitation sheet, which globally stabilizes [2].

Fig. 4. Mesh of the geometry (grid size: 250 · 90). (a) General view and (b) zoom at the throat.

Fig. 5. Pressure and density fields. Stable cavity for r = 0.4 and Vref = 10.8m/s.


Fig. 5 presents the shape of the liquid/vapor cavity in the reference simulation. It corresponds toa cavitation number r = 0.4. The inlet velocity equals 10.8m/s. We obtain in this case a stabilizedcavity whose length is about 45mm (that is to say Lcav/Lref = 0.18). Eighteen seconds of simula-tion were performed, that is to say 600 Tref, where Tref is a reference time, corresponding to thetime necessary for the flow field to cover the length Lref of the profile, with a speed Vref. Note thatthe downstream part of the cavity slightly fluctuates in time, so its precise shape may vary in time(Fig. 5 corresponds to T/Tref = 600).

6.2. Numerical and physical parameters

To analyze the stability of the numerical scheme, the eigenvalues and eigenvectors of matrix Z 0

must be calculated for each Fourier mode. They are computed numerically with Matlab. Jaco-bians A, B, C, D, E are expressed as functions of non-dimensional numbers characteristic ofthe flow configuration that is analyzed. The physical and numerical parameters applied in the pre-vious 3DNS simulation are presented in Table 1.

Table 1

Parameters to be tested

Parameter Description Reference value

Physical parameters

v/u Ratio of the velocities in directions i and j 1

Dx/Dy Ratio of the cell sizes in directions i and j 1

M = Vref/c Local Mach number 10�6 (incompressible)

Re ¼ Lref � V ref

mReynolds number 1

r Cavitation number defined previously 0

Numerical parameters

CFL It has a strong influence on the computation

of the local time-step (Eq. (32)). Actually, it is a scaling

coefficient of the time-step, and thus also a scaling coefficient

of Z 0 eigenvalues. In other words, the lower the CFL is,

the smaller the time-step will be, and the more

stable the numerical scheme will be

1

e4p Coefficient of fourth order artificial dissipation 0.1

e2p Coefficient of second order artificial dissipation

(limiter based on the pressure)

0 (not activated)

e2q Coefficient of second order artificial dissipation

(limiter based on the density)

0 (not activated)

Ds/Dt Ratio of the pseudo and the physical time steps 0.1

g Density underrelaxation 1 (not activated)


6.3. Stability analysis in non-cavitating conditions

For each configuration, two Fourier footprints are drawn:

• The left one presents the position of the eigenvalues, with a CFL adjusted so that they are alllocated inside the stability contour jGj = 1, but as close as possible. It gives an estimation of themaximum CFL available to ensure the numerical scheme stability.

• The right one is the same illustration with CFL = 1 and a scale adapted to each configuration.This figure gives more precise information concerning propagation and damping of all errormodes. Thus, it indicates the convergence limitation and speed expected in the considered con-figuration. The eigenvalues are drawn in four different colors and also denoted by differentsigns, according to the frequency of the corresponding mode: blue �� for HuHw, red �.� forHuBw, green �*� for BuHw, and black �+� for BuBw (see Fig. 3). It makes appear which problem-atic modes could be removed by an appropriate multigrid treatment, and which modes cannotbe removed.

We consider first a non-cavitating incompressible configuration (actually, the study is per-formed with a Mach number M = 10�6), in order to demonstrate the improvement associatedwith the preconditioner in that case. We apply here the reference values of the parameters pre-sented in Table 1. The results without any preconditioning are presented in Fig. 6.

Fig. 6. Stability analysis in an incompressible non-preconditioned configuration.


The left-hand figure reveals that the stability bound is obtained for a CFL equal to 4: this is thelimit to ensure the location of all the eigenvalues inside the stability contour of the time integra-tion scheme. This value is not much different from the stability bound in the case of a single advec-tion equation, which was found to be equal to 3 with the present Runge–Kutta integrationscheme.

Nevertheless, the right-hand figure makes appear some important expected convergence limita-tions: as a matter of fact, a lot of eigenvalues are clustered on the real axis, which indicates thatthe corresponding modes will not be propagated at all. This is an effect of the very high disparitybetween the fluid velocity and the speed of sound (M = 10�6), which results in the same disparitybetween the convective eigenvalues and the acoustic ones. The time-step is thus imposed by theacoustic eigenvalues, and it is very small, compared to the one adapted to convective eigenvalues.Consequently, many time-step are necessary to obtain a slight progression of the flow. The con-vective modes of the errors can almost not propagate, which results in a dramatic decrease of theconvergence speed. Moreover, such limitation cannot be removed by a multigrid treatment, sinceeigenvalues related to modes of the four quadrants (see Fig. 3) are clustered on the real axis. Thisproblem has been addressed by many authors [1,20,21] and solved by introducing apreconditioner.

The same analysis is performed in the similar preconditioned case, all parameters being con-stant. Results are presented on Fig. 7. A spectacular difference with Fig. 6 can be observed: con-vective eigenvalues are now kept away from the real axis, which allows a satisfactory propagationof convective modes. Moreover, the limit CFL increases up to 8.2, so the preconditioning alsoallows the use of larger CFL number.

The global effect of the preconditioner appears here clearly positive. The next step consists inanalyzing the cavitating configurations obtained in the 2D Venturi type section.

6.4. Stability analysis of cavitating configurations

Stability is investigated in Section 6.4.1 in a configuration as close as possible to the calculationconditions applied in 6.1. Then, the influence of numerical and physical parameters on the stabil-ity is studied in further sections.

Fig. 7. Stability analysis in an incompressible preconditioned configuration.


6.4.1. Standard configuration

Calculations were performed with the following parameters (second order artificial dissipation,and density underrelaxation are not considered yet):

• d4 = 0.1;• CFL = 3, preconditioned equations;• Re = 106;• Barotropic state law including incompressible and highly compressible states;• ql = 1000 and qv = 10;• Maximum slope of the barotropic law defined by: Cmin = 2m/s;• Almost regular grid: Dx/Dy = 5;• Velocities in the direction of the mesh: v/u = 0.

The ratio Ds/Dt is quite difficult to estimate precisely, because it depends on the local speed ofsound and also on the cell size. The physical time-step Dt classically equals Lref/100Vref where Lref

is the length of the profile while Vref is the upstream liquid flow velocity. Presently Lref = 0.224mand Vref = 10.8s so Dt = 2 · 10�4 s. Values of the pseudo-time-step Ds are different in the mixtureand the incompressible parts of the flows. A range of the possible values for Ds/Dt is thus givenhereafter in the two cases. The lowest value corresponds to the smallest cells, while the highest oneis imposed to 1 in the largest cells:

Incompressible flow: 0.1 6 Ds/Dt 6 1.Liquid/vapor mixture: 0.01 6 Ds/Dt 6 1.

In the present case, Ds/Dt = 0.1 is applied. This question will be discussed in Section 6.4.2.Fourier footprints obtained in that configuration are represented on Fig. 8: the upper part cor-

responds to the fully incompressible state (pure vapor or pure liquid) while the bottom part con-cerns the liquid/vapor mixture, for r = 0 (void ratio a = 0.5). The numerical scheme will besatisfactory in that configuration only if stability and convergence are ensured in both cases.

Fig. 8. Stability in the case of the barotropic state law. Calculation conditions applied in the reference simulation of the

2D Venturi type section (CFL = 3, as in the computations).


The incompressible part of the barotropic state law is correctly treated: the equation precondi-tioning leads to nice convergence properties, characterized by eigenvalues that are kept away fromthe imaginary axis, and also mostly from the real axis. CFL = 3 ensures here the necessary con-dition for stability.

On the contrary, the compressible part of the barotropic law leads to rather poor propagationproperties, since many of the eigenvalues are clustered near the real axis or around the origin, whichis the most disadvantageous situation. This result is consistent with the residuals behavior during thecomputation: vapor apparition generates a notable increase of the residuals, and they oscillate with-out decreasing until the end of the computation. In other words, drastic convergence criteria are notsatisfied by a cavitating computation, and only a limited convergence level can be reached.

It can be remarked that multigrid treatments are not well efficient in this case, since modes of allquadrants (see Fig. 3) are partially clustered on the real axis. Actually, performing cavitatingsimulations with several grid levels does not substantially improve the convergence rate.

6.4.2. Influence of the physical time derivativesThe value Ds/Dt = 0.1 applied in the previous section depends on the cell size. In the case of the

very little cells located in the expected cavitation area, i.e. just after the Venturi throat, close to the

Fig. 9. Influence of the physical time-step: Ds/Dt = 1. (a) Incompressible state and (b) two-phase mixture.


wall, the ratio is still lowered, and no particular effect on the location of the eigenvalues is ob-served: they are only slightly moved on the right. On the contrary if we consider larger cells,the ratio Ds/Dt = 1 may be reached. The corresponding Fourier footprints are drawn in Fig. 9:the main effect of increasing the ratio Ds/Dt is the global shift of all eigenvalues on the left. Asa result, stability is not ensured anymore in the liquid/vapor medium. Such configuration canbe easily avoided in the simple configuration of the Venturi type section, since the area of cavita-tion is relatively well known, and the grid can be adapted to obtain small cells in this part of thecomputational domain. This is much more difficult in the case of complex 3D turbopump inducergrids: cavitation is obtained in several parts of the domain, and applying a grid contraction in allof them leads to a prohibitive number of cells. Moreover, the general size of the cells in such gridsis larger than in the present calculation, so the accuracy is not so good. As a result, some non-physical low-pressure cavitating areas may transiently or permanently appear during the calcula-tions, making them abruptly diverge. To avoid these spurious divergences, one solution wouldthus consist in applying a global pseudo-time-step instead of a local one. In this case the lowestof all pseudo-time-steps is used, and the difficulty vanishes.

6.4.3. Influence of the physical parametersFour parameters were tested, namely the maximum slope of the barotropic state law Cmin, the

density ratio ql/qv, the Reynolds number, and the velocity direction.The density ratio and the Reynolds number are tested respectively in the range [10 1000] and

[105 108], and they are found to have almost no influence on the Fourier footprint. Nevertheless,applying a value qv = 0.1 (which is the physical value for the vapor density) instead of qv = 10leads to severe instabilities that can be only partially avoided by increasing the second order arti-ficial density coefficient. The reason for this inefficiency of artificial dissipation will be discussed inthe next section. However, the use of qv = 10 is not a limitation, since previous calculations per-formed with the same physical model [4] have shown that the density ratio has almost no influenceon the results, so far ql/qv � 1.

The Fourier footprints relative to the influence of Cmin are presented in Fig. 10 (Cmin = 1m/sand Cmin = 0.5m/s are tested, to be compared to the reference case Cmin = 2m/s).

Fig. 10. Influence of Cmin. Upper part: Cmin = 1m/s and Bottom: Cmin = 0.5m/s.


It can be seen that as Cmin decreases, i.e. as the compressibility increases in the intermediatevoid ratio configurations, eigenvalues that were previously around the real axis are progressivelyclustered on it. It removes any propagation for the corresponding modes. Moreover, stability isprogressively endangered: the Fourier footprint keeps its original shape, but it grows slowlyand it crosses the stability bound of the time integration scheme when Cmin is about 1m/s.

This behavior illustrates the difficulties encountered in cavitating computations when Cmin islowered down to 1m/s.

The Fourier footprints relative to the influence of the flow angle are presented on Fig. 12: thisparameter has almost no influence in the incompressible case, so only the two-phase mixture caseis presented. The result corresponds to a diagonal flow with respect to the cell (configuration 2 inFig. 11), and it has to be compared to the reference case, corresponding to a mesh aligned flow(Fig. 8, configuration 1 in Fig. 11).

A slight modification of the eigenvalue location can be noticed, but they are still partially clus-tered on the real axis. CFL = 3 leads here to an unstable configuration. That result suggests thatany perturbation of the flow field may result in a possible destabilization. For example, self-oscil-lation of cavitation sheets implies a high vorticity level and velocities in all directions. The resultobtained in that simple case (Fig. 12) suggests that such configuration may lead to severe reduc-tion of stability.

Fig. 11. Flow angle.

Fig. 12. Influence of the velocity direction. Flow angle 45�, two-phase mixture.


6.4.4. Influence of numerical parametersInfluences of both fourth and second order dissipation coefficients d4 and d2, of the density

underrelaxation, and of the mesh aspect ratio are successively investigated.A reduction of the fourth order artificial dissipation (e4

p ¼ 0:03 instead of 0.1) is tested for boththe incompressible flow and the two-phase mixture. The Fourier footprints are presented on Fig.13. In comparison with the reference case (Fig. 8), the reduction of d4 has almost no influence onthe numerical scheme stability, but it leads to a reduction of the convergence speed: actually,eigenvalues are in that case closer to the imaginary axis, which is associated with a lowest generaldamping of the error modes. This result is consistent with the 3DNS simulation, which does notconverge correctly with a too small dissipation coefficient value (not only in cavitating configura-tion, but also for a one-phase fluid).

To investigate the influence of the second order dissipation, non-linear limiters are omitted. Sod2 depends only on the values of the coefficients e2

p and e2q, as detailed in Section 3.2. Inside the

sheet of cavitation, the mixture is characterized by low pressure gradients and high density gra-dients. So only the value of e2

q (whose standard value is 0.5) is of interest here. We consider here-after the most critical configuration, i.e. the passage in two contiguous cells from pure vapor topure liquid. Such situation may happen during the phase of bubble collapse, even with very smallcells. It results in Eq. (13) in ci,j,k = 1, and finally d2 = 0.5. This value is applied in the stabilityanalysis and the result is presented in Fig. 14a. The increase of d2 clearly leads to an unstable con-figuration. If e2

q is increased up to 1 (for example to circumvent divergences that are encountered

Fig. 13. Stability analysis with e4 = 0.03. Upper figures: incompressible case, lower figures: two-phase mixture.


when the ratio qv/ql is lowered) the general tendency is the same (Fig. 14b) and the CFL should bedrastically decreased to ensure the numerical stability. So the conclusion is that the second orderdissipation should be used circumspectly because of its destabilizing effect.

Nevertheless e2q ¼ 0:5 was systematically applied in our calculations, which probably explains

that we should also underrelax the density variations, as explained in Section 3.4. To investigatethe influence of underrelaxation on the stability, we presently multiply the pseudo-time-step by g(see Eq. (16)) in the first equation of the linearized system (23). The previous case of Fig. 14a(d2 = 0.5) is then considered again with g = 0.3. Actually this is the standard value applied inthe simulation presented in Section 6.1. The corresponding Fourier footprints are drawn inFig. 15a: it can be seen that the necessary condition on stability is here recovered. However, ifg = 0.3 is applied with d2 = 1 (Fig. 15b), underrelaxation is not so efficient. So the second orderdissipation coefficient should not be increased, even in association with the underrelaxation ofthe density variations.

The effect of the mesh aspect ratio was also tested: cells larger in the main flow direction wereconsidered, to explore the effect of mesh distorsion in boundary layers or in 3D grids. Results arepresented in Fig. 16 for an aspect ratio Dx/Dy = 500 instead of 5.

Fig. 14. Influence of the second order artificial dissipation: (a) d2 = 0.5 and (b) d2 = 1.

Fig. 15. Influence of the density underrelaxation: (a) d2 = 0.5, g = 0.3 and (b) d2 = 1, g = 0.3.

Fig. 16. Influence of mesh aspect ratio: DxDy ¼ 500.



Although all eigenvalues remain inside the stability bound, some of them are now clustered onthe real axis. This is a general effect of high aspect ratio, which systematically generates stiffnessbetween eigenvalues of matrix A and B in Eq. (23). A high aspect ratio will always decrease theconvergence rate. Multigrid treatments are not efficient in that case to improve the convergence,since eigenvalues of all quadrants are clustered on the real axis.

7. Analysis of the preconditioner

The preconditioner modifies the equations so that incompressible or low compressible flow con-ditions can be treated by numerical resolutions based on the dual time stepping approach. As amatter of fact, it was clearly confirmed in Section 6.3 that its use greatly improves the numericalscheme properties. In the present section, the efficiency of the preconditioner is investigated in thecase of a two-phase mixture characterized by a high compressibility.

7.1. Preconditioner efficiency in highly compressible flow conditions

It was pointed out in Fig. 8 that a part of the eigenvalues is clustered around the real axis, whichis unfavorable to the convergence speed and rate. The same flow field configuration is presentlytested with the non-preconditioned numerical scheme, and the corresponding Fourier footprintsare drawn in Fig. 17:

Fourier footprints show in that case very nice convergence properties: the numerical scheme isstable with CFL = 3, and the eigenvalues are now correctly located. The stiffness between convec-tive and acoustic eigenvalues is completely removed. Only modes from BuBw quadrant (in black)are still too close to the origin, in a low damping area, but that problem can be removed by anappropriate multigrid treatment [22].

It shows evidence that the preconditioner is not well adapted to highly compressible two-phasemixtures. Anyway, it is necessary, since the main part of the flow field is purely liquid, i.e. almostincompressible. To improve convergence in the liquid/vapor areas, preconditioning should be pro-

Fig. 17. Stability analysis in the two-phase mixture with no preconditioning.


gressively removed, to almost neutralize it in that configuration. In this context, a deeper under-standing of the mechanism that clusters eigenvalues on the real axis is necessary.

7.2. Analysis of the eigenvalues stiffness

The general formulation of eigenvalue resulting from the Navier–Stokes equations associatedwith the central part of the barotropic state law can be obtained from:

jAnx þ Bny � kIj ¼ 0 with nðnx; nyÞ a propagation direction

7.2.1. With no preconditioningThe three resulting eigenvalues are

k1 ¼ v � nk2 ¼ v � nþ cjnjk3 ¼ v � n� cjnj

ð35Þ

For an intermediate void ratio a = 0.5(P = Pvap), c = Cmin and Cmin belongs to the range [1,3], asreported in Section 5.1. So for this flow configuration the three eigenvalues are of the same mag-nitude order.

7.2.2. With preconditioningThe expression of eigenvalues associated with propagation direction n becomes:

k1 ¼ v � n

k2 ¼1

2v � n 1 � ap þ

b2p

c2

!þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv � n 1 � ap þ

b2p

c2

! !2

þ 4b2p n2 � ðv � nÞ2 1

c2

� �vuut264

375

k3 ¼1

2v � n 1 � ap þ

b2p

c2

!�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv � n 1 � ap þ

b2p

c2

! !2

þ 4b2p n2 � ðv � nÞ2 1

c2

� �vuut264

375

ð36Þ

Stiffness between the acoustic eigenvalues k2, k3 and the convective eigenvalue k1 depends mainlyon the term b2

p=c2, which equals b2

p=C2min for r = 0.

Parameters have the following values for the 2D simulation of the Venturi type section:

• Cmin = 2m/s minimal speed of sound in the middle of the barotropic law.• bp = b0 · Vref = 3 · 10.8 = 32.4m/s.

Thus, we obtain b2p=C

2min ¼ 262 � 1.

Acoustic eigenvalues are much higher than the convective one. This problem is similar to theincompressible configuration without any preconditioner. It explains why the convective eigen-values are clustered on the real axis on the Fourier footprint (Fig. 8).


To reduce this stiffness, a progressive removal of the preconditioner is applied in the two-phaseareas. Such treatment was implicitly proposed previously by Choi and Merkle [1] and Turkel[20,21], since their preconditioning coefficients depend on the Mach number. However, cavitatingsimulations imply very rapid changes of speeds of sound as well in space as in time, and such flowconditions are not reported in these previous studies. In practice, using varying preconditioningcoefficients in cavitating flow calculations may also produce some spurious effects: stability is verysensitive to the way the preconditioner is progressively removed.

7.3. Effect on the calculation

The modification of the preconditioner is applied in the case of the Venturi type section, to testits effects on the calculation of the cavitating flow field. It is not expected that the stability rangewill be increased (the stability domain remains almost the same with and without the modifica-tion, see Figs. 8 and 17), but only that the convergence speed and rate will be improved.

The calculation is performed successively with the standard numerical model (unchanged pre-conditioner in both incompressible and two-phase areas), and with the modified preconditioner(the local density value is checked, and preconditioner coefficients are progressively modified inthe cavitating areas).

To ensure a smooth evolution of parameters ap and bp, the following expressions are applied inthe whole flow field, according to the local void ratio a:

ap ¼�1 if a ¼ 0 or a ¼ 1

� j2a � 1j if 0 < a < 1

�bp ¼ minðc; b0 � V refÞ

ð37Þ

The expression for bp ensures for all flow conditions the same order of magnitude for the threeeigenvalues k1, k2, and k3. In low compressible or incompressible configurations, bp is maintainedat the constant standard value b0 Æ Vref, which limits the weight of the term 4b2

p � n2 in k2 and k3,while in highly compressible regions, bp is reduced down to the physical speed of sound, to controlthe term b2

p=c2.

The condition for each time-step to be converged is a maximum residual decreasing below 6.5(this value corresponds to a satisfactory convergence level of the initial stationary calculation),and a maximum number of 100 pseudo-time-steps per iteration is imposed.

The physical and numerical parameters were given in Section 6.4.1. They are identical for bothcalculations (r = 0.4, Dt = 2 · 10�4, Vref = 10.8m/s). The simulations are performed during 30,000physical time steps, i.e. 6 s. In both cases, the sheet of cavitation progressively stabilizes after atranscient oscillation, and then it only slightly fluctuates in its rear part. The cavity shapes anddensity distributions obtained at t = 6s are close from each other (Fig. 18). Table 2 presents aquantitative comparison between the two results: both the mean value and the standard deviationof the vapor and cavity volumes during the last 10,000 physical time steps are reported. It con-firms that the two calculations almost lead to the same results.

The main difference between the two computations is their convergence history: Fig. 19 showsin both cases the total number of pseudo-time steps that was necessary to perform the simulationsduring 6s, as well as the associated evolution of the mass equation residual.

Fig. 18. Density distribution in the cavity: (a) standard preconditioner and (b) modified preconditioner. Calculation

conditions: Vref = 10.8m/s, r = 0.4, mesh size 250 · 90, Dt = 2 · 10�4 s.

Table 2

Comparison between the two calculations

Mean vapor

volume

(·10�4 m3)

Mean cavity

volume

(·10�3 m3)

Standard deviation

of the vapor volume

(·10�4 m3)

Standard deviation

of the cavity volume

(·10�4 m3)

Standard preconditioner 13.8 5.1 1.6 6.8

Modified preconditioner 13.6 5.0 1.3 6.8

Time averaging of the results 4s < t < 6s. r = 0.4, Dt = 2 · 10�4, Vref = 10.8m/s.

Fig. 19. Evolution of the mass equation residual in the two calculations.


A significative convergence improvement is obtained with the modified preconditioner: only35,000 pseudo-time-steps are necessary (instead of 140,000) to perform the 30,000 physical timesteps, and the convergence level is remarkably better (the final residual equals 5, instead of 6.5in the standard case). Since the convergence criteria at each physical time-step had been fixed hereto 6.5, the solution is converged at each time-step in the case of the modified preconditioner (afterthe initial transcient), which is not the case with the standard model.

The interest of the modification proposed in the model to treat the two-phase flow fields is thusclearly confirmed by this confrontation.


8. Conclusion

A 3D model for unsteady cavitation was presented in this paper. The cavitation physical modelis based on a single-fluid approach, and the two-phase areas are considered as a single fluid, whosedensity is managed through a postulated barotropic state law. The association of this state lawwith the numerical resolution applied in the code FineTurbo may result in particular stability con-ditions. A complete stability analysis was thus presented in this paper. The two different states of acavitating flow field (almost incompressible in the pure liquid and pure vapor, and highly com-pressible in the mixture) were considered, and the influence of different numerical and physicalparameters on both convergence and stability was investigated. Three main conclusions can bedrawn. The first one concerns the grid, which must be as well very fine in all two-phase areas(to avoid spurious divergences) as characterized by a small aspect ratio (to prevent convergencefrom slowing down). The second one concerns the second order artificial dissipation, which canonly be used circumspectly (because of its destabilizing effects). Finally, the interest of underrelax-ing the density was confirmed, even if it cannot guarantee stability if the two previous items arenot respected. It was also found that the preconditioner had to be adapted in the two-phase areasto enhance the convergence level and speed. A modification was proposed, and a test case per-formed on a Venturi type section clearly showed its interest. Further applications on 3D cavitat-ing turbopumps and/or pronounced unsteady behaviors will take into account the conclusions ofthis study.

Acknowledgements

This research was supported by a doctoral grant from the Education French Ministry MERTand by SNECMA Moteurs DMF. The authors wish also to express their gratitude to CNES (Cen-tre National d�Etudes Spatiales, France) for the continuous support.

References

[1] Choi D, Merkle C. The application of preconditioning in viscous flows. J Comput Phys 1993;105:207–23.

[2] Coutier-Delgosha O, Fortes-Patella R, Reboud JL, Hakimi N, Hirsch C. Numerical simulation of cavitating flow

in 2D and 3D inducer geometries. Int J Numer Meth Fluid, in press.

[3] Coutier-Delgosha O, Reboud JL, Fortes-Patella R. Numerical study of the effect of the leading edge shape on

cavitation around inducer blade sections. Int J JSME B 2002;45(3):678–85.

[4] Coutier-Delgosha O, Reboud JL, Delannoy Y. Numerical simulations in unsteady cavitating flows. Int J Numer

Meth Fluids 2003;42:527–48.

[5] Coutier-Delgosha O, Fortes-Patella R, Reboud JL. Evaluation of the turbulence model influence on the numerical

simulations of unsteady cavitation. J Fluid Eng 2003;125:38–45.

[6] Delannoy Y, Kueny JL. Two phase flow approach in unsteady cavitation modelling. In: Proc of the 1990 ASME

Cavitation and Multiphase Flow Forum, vol. 98, 1990. p. 153–8.

[7] Hakimi N. Preconditioning methods for time dependent Navier–Stokes equations. PhD Thesis, Vrije Universiteit

Brussels, Belgium, 1997.

[8] Hirsch C. Numerical computation of internal and external flows. John Wiley and Sons; 1990.

[9] Jakobsen JK. On the mechanism of head breakdown in cavitating inducers. J Basic Eng, Trans ASME

1964:291–305.


[10] Jameson A, Schmidt W, Turkel E. Numerical solutions of the Euler equations by finite volume methods using

Runge–Kutta time-stepping schemes. AIAA Paper 81-1259, 1991.

[11] Jameson A. Time dependant calculations using multigrid, with application to unsteady flows past airfoils and

wings. AIAA paper 91-1596, 1991.

[12] Knapp RT, Daily JT, Hammit FG. Cavitation. McGraw Hill; 1970.

[13] Kueny JL, Reboud JL, Desclaux J. Analysis of partial cavitation: image processing and numerical prediction. In:

Proc of the 1991 ASME Fluids Engineering Summer Conf, vol. 116, 1991. p. 55–60.

[14] Kunz R, Boger D, Chyczewski T, Stinebring D, Gibeling H. Multi-phase CFD analysis of natural and ventilated

cavitation about submerged bodies. In: Proc of the 3rd ASME/JSME Joint Fluids Engineering Conf, San

Francisco, USA, 1999.

[15] Lynn JF, Van Leer B. Multi-stage schemes for the Euler and Navier–Stokes equations with optimal smoothing.

AIAA Paper 93-3355-C, 1993.

[16] Merkle CL, Feng J, Buelow PEO. Computational modeling of the dynamics of sheet cavitation. In: Proc of the 3rd

Int Symp on Cavitation, Grenoble, France, 1998. p. 307–15.

[17] Reboud JL, Stutz B, Coutier O. Two phase flow structure of cavitation: experiment and modeling of unsteady

effects. In: Proc of the 3rd Int Symp on Cavitation, Grenoble, France, 1998.

[18] Song C, He J. Numerical simulation of cavitating flows by single-phase flow approach. In: Proc of the 3rd Int Symp

on Cavitation, Grenoble, France, 1998. p. 295–300.

[19] Stutz B, Reboud JL. Two-phase flow structure of sheet cavitation. Phys Fluids 1997;9(12):3678–86.

[20] Turkel E. Preconditioning methods for solving the incompressible and low speed compressible equations. J

Comput Phys 1987;72:277–98.

[21] Turkel E. Review of preconditioning methods for fluid dynamics. J Appl Numer Math 1993.

[22] Zhu ZW. Multigrid operation and analysis for complex aerodynamics. PhD Thesis, Vrije Universiteit Brussel,

Belgium, 1996.

Date post:	17-Nov-2023
Category:	Documents
Upload:	independent
View:	0 times
Download:	0 times

Stability of preconditioned Navier–Stokes equations associated with a cavitation model

Documents