Local penalty methods for flows interacting with moving solids at high Reynolds numbers

www.elsevier.com/locate/compfluid

Computers & Fluids 36 (2007) 902–913

Local penalty methods for flows interacting with moving solidsat high Reynolds numbers

Stephane Vincent *, Tseheno Nirina Randrianarivelo,Gregoire Pianet, Jean-Paul Caltagirone

TRansferts, Energetique, FLuide, Ecoulements (TREFLE), UMR 8508, Site ENSCPB – Universite Bordeaux 1,

16, Avenue Pey-Berland, 33607 Pessac Cedex, France

Received 17 November 2005; received in revised form 5 April 2006; accepted 25 April 2006Available online 20 November 2006

Abstract

An original numerical modelling of multiphase flows interacting with solids in unsteady regimes is presented. Based on the generalizedNavier–Stokes equations for multiphase flows and Volume of Fluid (VOF) formulations, an Uzawa minimization algorithm is imple-mented for the treatment of incompressibility and solid constraints. Augmented Lagrangian terms are added in the momentum equationsto speed the convergence of the iterative solver. Defining a priori the penalty parameters which are dedicated to incompressibility andsolid constraints is difficult, or impossible, as soon as the flow involves more than one phase and inertia becomes predominant comparedto viscous and gravity forces. The Lagrangian penalty terms are calculated automatically according to an original local estimate of thevarious physical contributions. Numerical validations have been carried out for single particle settling in confined media and viscous flowthrough a fixed Cubic Faced Centered array. A very good agreement is obtained between experimental, theoretical and numerical results.Extension to unsteady free surface flow interacting with particles is illustrated with the simulation of a dam break flow over movingobstacles.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Multiphase flows involving gas, liquid and solid phasesare of importance in academic and industrial researchworks dealing with environment, coastal flows, fluidizedbeds, polymer design or food processing. Experimentalstudies dedicated to the measurements of local characteris-tics associated with free surface or interfacial flows inter-acting with solids are difficult to lead. For example, thepopular Particle Image Velocimetry (PIV) (see [1] or [16])is restricted to the characterisation of fluids with adaptedtranslucency. Generally, the presence of interfaces induces

0045-7930/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compfluid.2006.04.006

* Corresponding author. Tel.: +33 5 40 00 27 07; fax: +33 5 40 00 66 68.E-mail addresses: [email protected] (S. Vincent), [email protected]

(T.N. Randrianarivelo), [email protected] (G. Pianet), [email protected](J.-P. Caltagirone).

optical difficulties that makes the PIV techniques unusablein most configurations.

The Direct Numerical Simulation (DNS) is an impor-tant alternative tool to understand and analyse such com-plex flows. Over the past 15 years, numerous numericalmodelling have successfully proposed to study free surfaceor interfacial flows at small space scale. The majority ofthem are based on the one fluid formulation of theNavier–Stokes equations [25] with appropriate methodsto track interface. However, as soon as a solid phase inter-acts with the flow, new conservation equations have to beconsidered and specific numerical treatments must bedeveloped.

In the framework of fictitious domains and penaltynumerical approaches [10,14,24], the conservation equa-tions for solid and the interaction between solid and fluidare treated by introducing specific terms in the Navier–Stokes equations. In addition, minimization procedures

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S. Vincent et al. / Computers & Fluids 36 (2007) 902–913 903

and Lagrangian terms accumulating the incompressibilityand indeformability constraints are introduced in thenumerical algorithms. The objective of the present studyis to propose an original numerical method for solving par-ticulate multiphase flows. The present approach is based onthe generalisation of the Adaptive Augmented Lagrangian(AAL) approach first published in [32]. A single model for-mulation is utilized whatever the local phase considered. Apenalty method based on the augmented Lagrangianmethod of Fortin and Glowinski [9] has been adopted fortreating the incompressibility constraint. The present workalso introduces a generalization of the AAL approach toparticulate flows.

The article is structured as follows: In the second sec-tion, the governing equations, the approximation and solu-tion methods as well as the new penalty techniquesdedicated to solving the interactions between multiphaseflows and solids are presented. A 3D validation of thenew numerical modelling is carried out in the third sectionfor flow through a cubic faced centered array of fixedspheres. The following sections illustrate the interest ofthe generalized augmented Lagrangian method to solveparticulate flows. The settling of a spherical particle inthree dimensions at a Reynolds number value of 280 isinvestigated. Then, 2D simulations are led on the dambreak flow over moving particles. The last section is dedi-cated to concluding remarks.

2. Conservation equations and mathematical approximation

2.1. One fluid model

We consider the incompressible Navier–Stokes equa-tions for a Newtonian fluid in their generalized formulation(see for example [27,25,30]) for isothermal non-misciblemultiphase flows with surface tension. The model is basedon the convolution of the motion equations in each phasewith a color function C that is used to distinguish betweendifferent phases. For a two-phase problem, C = 1 in thefirst fluid and C = 0 in the other one. The jump relationsproviding mass and momentum conservation at the inter-face are verified by the model since no sliding betweenthe fluids occurs. Indeed, under this assumption, the veloc-ity field is continuous through the interface and the flow isincompressible. In a domain X of boundary C, let u be thevelocity vector, p the pressure, t the time, g the gravity vec-tor, ~l the dynamic viscosity, ~q the density, r the surfacetension coefficient, p the local curvature of interface, ni

the unit normal to interface and di the Dirac function indi-cating interface, we have

~qou

otþ u � ru

� �¼ �rp þr � ð~l½ruþrTu�Þ þ ~qgþ rpnidi

r � u ¼ 0

ð1Þ

In the following sections, the divergence of stress tensor�$p + $ Æ (l[$u + $Tu]) will be referred to as r � ��c. More-over, no details will be given concerning the treatment ofboundary conditions as the penalty methods proposed inthis article do not depend on their choice. Neumann,Dirichlet or periodic boundary conditions have been usedsuccessfully.

The interface between the phases is moving under fluidflow interactions. Its evolutions are modelled thanks toan advection equation for C

oCotþ u � rC ¼ 0 ð2Þ

To solve the system (1) and (2), closure equations areneeded in order to estimate the variations of the densityand viscosity fields as well as the curvature and normalto the interface with respect to C. The evolutions of thephase average ~q and ~l are usually related to the local colorfunction thanks to discontinuous averages. If two fluidphases are considered, the physical properties of phases 1and 2 are q1, l1, q2 and l2 respectively in the related sub-domains X1 and X2. We have

~q ¼ q2 and ~l ¼ l2 if C P 0:5

~q ¼ q1 and ~l ¼ l1 elseð3Þ

If two fluid and one solid phases are considered, the phys-ical properties of phases 1, 2 and 3 are q1, l1, q2, l2, q3 andl3 respectively in the related subdomains X1, X2 and X3.We define C2 = 1 in X2 and C3 = 1 in X3 such that

~q ¼ q3 and ~l ¼ l3 if C3 P 0:5

~q ¼ q2 and ~l ¼ l2 if C2 P 0:5

~q ¼ q1 and ~l ¼ l1 else

ð4Þ

The surface tension force is modelled by the ContinuumSurface Force CSF of Brackbill et al. [2] using the gradientsof the phase function:

nidi ¼rCkrCk

p ¼ �r � ni ¼ �r �rCkrCk

� � ð5Þ

The one-fluid model (1)–(5) describes the unsteady evolu-tions of the velocity, pressure and phase function fieldseverywhere in the physical domain. The main difficultyconsists in building a numerical method for treating simul-taneously both the incompressibility and the solid con-straints in the whole domain with the same discretizationand solvers.

2.2. Standard penalty methods

The aim of this article is to present an original methodfor treating in the same solver and with the same set ofequations free surface flows in interaction with solids.The idea is to split the stress tensor in the Navier–Stokesequations into four parts, associated respectively to

904 S. Vincent et al. / Computers & Fluids 36 (2007) 902–913

incompressibility, elongation, pure shearing and rotation.This formulation was previously presented in [3 and 21].In Cartesian coordinates, the stress tensor ��c reads

��c¼�pþkr�u 0 0

0 �pþkr�u 0

0 0 �pþkr�u

264

375

þj

ouox 0 0

0 ovoy 0

0 0 owoz

264

375þ f

0 ouoy

ouoz

ovox 0 ov

ozowox

owoy 0

264

375�g

0 ouoy� ov

oxouoz� ow

ox

ovox� ou

oy 0 ovoz� ow

oy

owox� ou

ozowoy � ov

oz 0

2664

3775

¼ð�pþkr�uÞ ��Idþj��CðuÞþ f ��HðuÞ�g��XðuÞ ð6Þ

where ��Id is the identity tensor of R3 and ��C, ��H and ��X respec-tively pseudo-tensors of elongation, pure shearing androtation. The standard viscous stress tensor for a Newto-nian fluid is recovered by stating k ¼ � 2

3~l, j ¼ 2~l,

f ¼ 2~l and g ¼ ~l.First proposed by Fortin and Glowinski [9] for simulat-

ing incompressible single phase flows, the formulation (6)allows to build a penalty method based on augmentedLagrangian techniques. The idea is to replace the solvingof Navier–Stokes equation system by the minimization ofa new equation under the constraint of incompressibilityin fluid zones and deformation in solid parts of the flow.Practically, an Uzawa algorithm is implemented [21], inwhich the following equations are solved iteratively toobtain velocity and pressure

~qunþ1 � un

Dtþ un � runþ1

� �

¼ �rpn þr½kðr � unþ1Þ� þ r � j��Cðunþ1Þ þ f ��Hðunþ1Þh

�g��Xðunþ1Þiþ ~qgþ rpnnn

i dni

pnþ1 ¼ pn � kr � unþ1 ð7Þ

until k$ Æ un+1k < �1 in the fluid zones and kr � ½j��Cðunþ1Þþf ��Hðunþ1Þ � g��Xðunþ1Þ�k < �2 in the solid zones. As a defini-tion, �1 and �2 are small thresholds controlling the mini-mization of the constraints. Eq. (7) are presented insemi-discrete form with n the time index corresponding totime nDt and Dt the time step.

In the fluid phases, a Standard Augmented Lagrangian(SAL) method of Fortin and Glowinski [9] is obtained bymaking k! +1. An incompressible constraint is thusdirectly imposed during the solving of the momentum con-servation thanks to term $[k($ Æ un+1)]. Indeformability canbe ensured in solid media by a similar penalty method, act-ing on the elongation, shearing and rotation viscosities. Itis assumed that j, as well as f and g, tends to infinity.The main difficulty of the penalty numerical method is tochoose the numerical value of k, j, f and g. These numer-ical parameters must be several orders of magnitude largerthan the larger term in the Navier–Stokes equations. How-ever, they must keep a sufficiently small value in order to

maintain an as good as possible conditioning of the linearsystem.

In SAL approach, k is chosen constant in the range1 < k < 10,000. With such a method, multiphase freesurface or interfacial flows have been successfully simulated[29 and 30] with Reynolds number less than 10 and largeinterfacial structures compared to the size of the grid aswell as the one of the computational domain. Concerningparticulate flows, penalty methods called Standard SolidPenalty (SSP) approach, which uses constant l between1000 and 100,000 in the solid zones, allows to accuratelysolve Stokes flow interacting with moving particles [24].However, as soon as the unsteadiness of the flow increasesas well as the complexity of the fluid–solid interactions,numerical failures are noticed in the flow solutionsprovided by SAL and SSP approaches [32]. In the follow-ing section, an automatic criterion is proposed for deter-mining the magnitude of the penalty coefficients k, j, fand g according to the physical characteristics of theproblem.

2.3. The adaptive augmented Lagrangian method generalized

to various constraints

The key point when simulating multiphase flows withpenalty methods lies in setting correctly the magnitude ofthe penalty coefficients. Instead of choosing an empiricalvalue at the beginning of the simulations, these coefficientsmust be locally adapted in time and space to the phase theybelong to. Indeed, the topology of the multiphase mediumis varying at each time step. It was observed that anadapted penalty parameter in a gas phase is not suitablefor example for a liquid phase. Concerning the incompress-ibility treatment, this property has been first demonstratedby Vincent et al. [32]. In the present paper, we intent toextend the idea of setting k locally to all the viscous penaltycoefficients.

In a first step, we recall the main results presented in [32]concerning the Adaptive Augmented Lagrangian (AAL)approach. The non-dimensional form of the momentumequations are first considered to identify the magnitudeof k. If L0, t0, ui, uv, C0 and p0 are respectively associatedto reference space length, time, velocity associated to iner-tia and viscosity, volume fraction and pressure, one itera-tion step m of the penalty method SAL (7) concerningincompressibility is:

~qL2

0ui

t0uv

u�;m � un

Dt

� �þ ~q

u2i L0

uvðu�;m�1 � rÞu�;m �rðkr � u�;mÞ

¼ ~qL2

0

uvg� p0L0

uvrpn þr � ½~lðru�;m þrTu�;mÞ� þ r

uvpnidi

ð8Þ

As expected, k is comparable to a viscosity coefficient. Inthe framework of penalty method dedicated to incompress-ibility, it is then defined as


kðt;MÞ¼K max ~qðt;MÞL20ui

t0uv;~qðt;MÞu

2i L0

uv;~qðt;MÞL

20

uvg;

p0L0

uv;~lðt;MÞ; r

uv

� �;

10<K < 1000 ð9Þ

As explained in [32], the choice (9) allows to defined wellbalanced values of k in all phases. The AAL parameter isat least 10–1000 times the order of magnitude of the mostimportant term between inertia, viscosity, pressure or grav-ity in both X1 and X2 domains. Compared to the SALapproach, the AAL method prevents from too weak valuesof k which does not ensure $ Æ u = 0 as well as from exces-sive values of k for which the momentum conservation isnot verified.

When multiphase flows involving fluid and solids areconsidered, the divergence free property must be verifiedin the fluid media Xf = X1 [ X2 according to Eq. (9). More-over, the elongation, pure shearing and rotation viscositiesare used to impose indeformability constraint in the solidzones Xs. The adaptive augmented Lagrangian techniqueis then extended to these penalty terms, as done in the fluiddomain Xf

kðt;MÞ ¼ K max ~qðt;MÞ L20ui

t0uv; ~qðt;MÞ u

2i L0

uv;

�

~qðt;MÞ L20

uvg;

p0L0

uv; ~lðt;MÞ; r

uv

�

jðt;MÞ ¼ gðt;MÞ ¼ 2fðt;MÞ ¼ 2~l; 10 < K < 1000

ð10Þ

whereas in Xs

kðt;MÞ ¼ � 2

3�l

jðt;MÞ ¼ K2

max ~qðt;MÞ L20ui

t0uv; ~qðt;MÞ u

2i L0

uv;

�

~qðt;MÞ L20

uvg;

p0L0

uv; kðt;MÞ; r

uv

�

gðt;MÞ ¼ K2

max ~qðt;MÞ L20ui

t0uv; ~qðt;MÞ u

2i L0

uv;

�

~qðt;MÞ L20

uvg;

p0L0

uv; kðt;MÞ; r

uv

�

fðt;MÞ ¼ K max ~qðt;MÞ L20ui

t0uv; ~qðt;MÞ u

2i L0

uv;

�

~qðt;MÞ L20

uvg;

p0L0

uv; kðt;MÞ; r

uv

�10 < K < 1000

ð11Þ

with �l a mean viscosity representative of the fluid part ofthe multiphase flow. Classically, �l is chosen equal toP

ili

i . This value has a little effect on the numerical solutionas it is always several orders of magnitude less than j, gand f.

Finally, the time-marching procedure for a penaltymethod treating fluid–solid interactions, which includes

the Generalized AAL (GAAL) method proposed in (10)and (11), is the following:

� Step 1: defines initial values L0, t0, ui, uv, C0, p0, u0 andp0 and boundary conditions on C.� Step 2: knowing un, pn and constraint thresholds �1

and �2, estimates the values u*,m and p*,m with the Uzawaalgorithm (7) associated to the local estimate of k, j, gand f defined in expressions 10,11, so that un+1 = u*,m

and pn+1 = p*,m when m verifies k$ Æ u*,mk < �1 andkr � ½j��Cðu�;mÞ þ f ��Hðu�;mÞ � g��Xðu�;mÞ�k < �2.� Step 3: advects interface between all phases by solvingEq. (2) and define new physical characteristics q and laccording to (3).� Step 4: iterates n in steps 2 and 3 until the physicaltime is reached.

As unsteady multiphase flows with moving interfaces aredealt with in this paper, the same time step Dt is chosen insteps 3 and 4 to verify the consistency of the numericalsolution with the physical model (1)–(5). As explicit inter-face tracking algorithms are used in step 3 (see details inthe following section), a CFL stability criterion is suitableto lead stable simulations with the previous algorithm. Thischoice penalizes the numerical efficiency of step 2 whichis discretized in a fully implicit manner. Concerning thechoice of the dimensionless parameters used in 10,11, theyare all chosen at the beginning of the calculations accord-ing to the physical characteristics of the problem, exceptfor velocities ui and uv for which different strategies canbe adopted. They will be discussed in Sections 3,4.1 and4.2 dedicated to validation and illustration of multiphaseflow problems.

2.4. Discretization and solvers

The approximation of the motion and color functionequations are widely presented and discussed by Vincentand Caltagirone in previous works [29 and 30]. It is basedon implicit finite volume discretization on fixed Cartesianstaggered grids. This Eulerian numerical approach is com-monly used in the literature [6,13,15,27] or [23] for its effi-ciency in tracking various interface shapes as well asparticulate flows and for its easy programming.

From a general point of view, the space derivatives of theNavier–Stokes equations are discretized with a second ordercentered scheme. In the same way, the curvature and normalterms of the surface tension force are approximated thanksto centered flux on the color function [30]. In the Navier–Stokes equations, implicit discretizations are used for theapproximation of the standard inertial and viscous termsas well as the penalty terms, which induces the determina-tion of the inverse of a matrix. As the matrices resultingfrom the previous discretizations are non-symmetrical, thelinear system is solved with an iterative bi-conjugate gradi-ent stabilized BiCGSTAB II [33], preconditioned under amodified incomplete MILU method [11].

Table 1Definition of the penalty parameters considered for the flow through aCFC array of particles at Re = 100

Penalty k ls ui uv L0 K

GAAL Eqs. (10) and (11) l1 �uðtÞ Um Rp 100


The phase function (2) is numerically considered as theVolume of Fluid (VOF) in each grid cell [13]. A PiecewiseLinear Interface Construction PLIC [35 and 17] is imple-mented for the VOF treatment of the advection equationof the phase function C. This geometrical technique hasproved accurate behavior with a numerical diffusion thatdoes not exceed the cell size. With a divergence free velocityfield, it provides almost zero computer error on the volumeof fluid. Other efficient techniques on fixed grids could havebeen used for the advection equation, such as a Markermethod ([6] and [7]), a level set approach [27] or a TotalVariation Diminishing TVD interface capturing ([28] or[29]).

3. Validation of GAAL method on the flow through a fixed

CFC array

Averaged two-fluid models used for the simulation ofindustrial fluidized beds require an accurate expression ofthe interactions between discrete and continuous phases.One way to treat these interactions is to model themthrough a drag coefficient multiplier which takes intoaccount the particles volume fraction. It is generally formu-lated as the ratio between the drag force exerted on all theparticles and its value for an isolated sphere. Numerousexperimental studies have been carried out to determinethe relationship between drag coefficient and particle vol-ume fraction. Various empirical correlations have beenproposed for the expression of the drag coefficient Cd

defined as:

Cd ¼F d

12qSU 2

m

ð12Þ

where S ¼ pR2p is the apparent surface of the sphere of ra-

dius Rp, Fd is the magnitude of the drag force and Um theaverage fluid velocity. The validity domain of these corre-lations remains already variable. They are widely presentedin the work of Massol et al. [18].

The study of the drag force exerted on particles in anordered arrangement is helpful to evaluate the validity ofliterature correlations. Cubic Face Centered (CFC) arraycontaining 4 spheres has been chosen in the present studyto evaluate the GAAL method. Our numerical model forsolving particle motion has been utilized to simulate theflow through a fixed ordered array of spheres. This prob-lem also offered us the opportunity to test the ability ofGAAL method to deal with explicit interactions betweenseveral particles. Numerical experiments have been carriedout for various solid fractions and Reynolds numbers inthe CFC array of solid spheres [22]. The typical CFC arrayis depicted in Fig. 1(left). The flow is generated by imposinga pressure gradient according to one of the face of the cube.Periodic boundary conditions have been imposed at eachface. The flow is characterized by its Reynolds numberdefined as Re ¼ 2q1RpUm

l1, where q1 and l1 are respectively

the fluid density and viscosity. Re numbers ranging from

10 to 300 have been studied to evaluate the influence offluid velocity on the value of the drag force exerted onpacked spheres. It is to be noticed that the mean phasevelocity Um is known a priori as soon as the particle Rey-nolds number Re is chosen.

On a 70 · 70 · 70 fixed Cartesian grid, one example offlow simulation through a CFC array at Re = 100 is con-sidered here. The penalty parameters used in the GAALmethod are presented in Table 1. The cubic calculationdomain is 0.038 [m] long and the radius Rp is rangingbetween 0.00547 [m] and 0.01253 [m] according to theexpected solid fraction. A mesh refinement study has dem-onstrated that increasing the number of points did not sig-nificantly improve the computed value of Cd. Thecomparison of GAAL and SAL/SSP numerical results toliterature data are presented in Fig. 1(right). First observa-tions show that our numerical results are very close toexisting correlations of Wen and Yu [34]. As previouslydemonstrated in the work of Massol et al.[18], even if itis commonly used to evaluate drag force in fluidized beds,the law of Ergun [8] is not well adapted to the Reynoldsconsidered here. For Re = 100, a boundary layer isdetached at the rear of the spheres and a toroidal recircu-lating zone appears downstream the spheres. The top rightand bottom pictures of Fig. 2 depict the isoline of x-compo-nent of velocity in a diagonal slice plane parallel to thepressure gradient and the streamlines in a median planeparallel to �rp. Attached and detached boundary layersare clearly observed. The local differences between SAL/SSP and GAAL penalty approaches are not negligible evenif the calculated drag forces are in good agreement.

Concerning the comparison between SAL/SSP andGAAL methods, the steady flow are almost the same interms of drag force and flow structure (see Figs. 1 and 2).However, if we have a look at the convergence of the Rey-nolds number presented in Fig. 2(top left), we observe thatthe steady state is reached earlier with the GAAL method,which means that the calculation time is divided by 3 whenusing this method. A comparison of CPU time, memoryrequirements, iterative solver residual of physical behaviorof numerical solutions is shown in Table 2 using SAL/SSPand GAAL approaches. It can be observed that the GAALdoes not have any memory over cost. Values of mean pres-sure gradient �rp, level of BICGSTAB II residual, fulfill-ment of incompressibility and indeformability obtainedusing GAAL penalty are better than SAL/SSP method.The computational cost with GAAL involves an increaseof 8% due to the estimation of the local viscosity penaltycoefficients. The same numerical performances of GAALare found in the problems considered in Section 4.

Fig. 1. Flow through a CFC array of fixed spheres – topology of the spheres in the periodic calculation domain (left) and comparison betweendimensionless drag forces obtained with analytical solutions and simulations (right).

Fig. 2. Evolution of the Reynolds number during convergence to the steady state (top left), comparison of the x-component of the velocity in a diagonal2D slice for Cp = 0.2 (top right) and stream lines in a median slice obtained with the GAAL penalty method for Cp = 0.05 (bottom). The flow comes fromthe left in the 2D slices.


Table 2Comparison between SAL/SSP and GAAL penalty methods on a 70 · 70 · 70 grid for Cp = 0.2 and Re = 100

CPU time (s) Memory (Mo) BICG residual k$ Æ uk �rp

SAL/SSP penalty 1.086 · 105 884 1.1 · 10�5 3.1 · 10�6 1.1156GAAL penalty 1.208 · 105 887 2.9 · 10�8 7.1 · 10�7 1.1787


4. Simulation of moving particle flows

In this section, we illustrate the interest the reader mayfind in using the generalized adaptive augmented Lagrang-ian method. Then two relevant problems that involvestrong fluid–particle coupling are dealt with. Note thatattention is paid to the behavior of previous approachessuch as standard augmented Lagrangian method (SAL)and standard solid penalty method (SSP). A huge effortis made in this section to explain the way in which theGAAL achieves better numerical results in terms of consis-tency and physical meaning.

4.1. A rigid sphere settling in a square tank

The sedimentation of a solid sphere is considered in asquare tank with dimensions large enough to ensure thehypothesis of an unbounded domain. The resulting radialcontainment does not exceed 10�3. This test case has beenexperimented by Mordant and Pinton [19]. Our studyfocuses on the configuration involving a steel sphere ofdiameter dp = 8 · 10�4 [m] and density qs = 7710 [kg m�3].In any case the particle is released into water at T = 25[�C], with a zero initial velocity. At these conditions, thefluid viscosity and density are respectively l1 = 8.9 · 10�4

[Pa s] and q1 = 1000 [kg m�3] and considered as constantssince the rheology stands well within the Newtonian law.

The reproducibility is checked as several analog mea-surements yield a single set of results with an error less than1%. The maximum diameter-based Reynolds number isfound to be Rep = q1U1dp/l1 = 280 and the correspondingparticle Stokes number is St = Repqs/9q1 = 240. Thesewere calculated from the terminal settling velocity U1observed during experiments and measured equal to0.316 ± 0.9% [m s�1]. Furthermore, the particle velocitytime series, measured in the range Rep 2 [40,8000], wereshown to be correctly fitted by the single exponentialrelation:

UpðtÞ ¼ U1 1� e�3tst

� �ð13Þ

with st the time required for the particle to reach 95% ofU1. In the present configuration, st = 0.108 [s].

Table 3Definitions of the simulation sets considered for a spherical particle settling a

Set k ls ui

SAL/SSP 100 1 Not useGAAL Eqs. (10) and (11) l1 �uðtÞ

Our interest in the present test case lies in providing reli-able results concerning strong interactions between a fluidand a moving solid body. The 3D numerical simulationsare implemented in a parallelepipedic domain of heightH = 0.064 [m] as the horizontal square section length sizesL = 0.004 [m]. A 50 · 50 · 800 Cartesian grid is considered.Initially, the sphere is released 0.0596 [m] away from thetank’s bottom. Note that the radial particle containment(0.2) exceeds largely the one from experiments in order toprevent excessive numerical consumption. However, asRep and St admit large values, this would have little effecton the particle trajectory, which was observed to be recti-linear and vertical during experiments.

As presented in Table 3, two sets of penalty parametershave been chosen in the simulations for illustrating thebehavior of our numerical method: (i) in the set SAL/SSP, the constant value of k is calculated from the initialflow characteristics; (ii) in the set GAAL, the velocity ui

associated to inertia is not chosen constant, since the mag-nitude of the velocity in the fluid ranges between 0 and U1during the acceleration phase. Then a relevant choice for ui

lies in estimating the average velocity magnitude in thefluid uðtÞ at each time step.

As it could be seen in both Figs. 3 and 4, large differ-ences occur between the standard augmented lagrangian/penalty method (SAL/SSP) and the new GAAL approach.The SAL/SSP calculations have been realized by choosingthe most adapted constant values of k and ls in order to geta physically acceptable numerical solution during the accel-eration phase. Fig. 3 shows discrepancies in topologybetween two simultaneous flow field slices. First, at a timet ’ st, it is found that the distance covered by the particlewith set SAL/SSP is less than half the one obtained viathe GAAL method. Then the choice for SAL/SSP methodsleads to an over-estimation of the lateral flow field exten-sion compared to the typical boundary layer thicknessexpected at this Rep regime.

Classically, the SSP constant penalty viscosity is set at alow value that does not affect the liquid/solid coupling (see[20]). When strong interface constraints occur, this value ofls is generally under-estimated, which means that we tendto give too much importance to the incompressibility con-straint compared to the constraint of no deformation in the

t Rep = 280

uv L0 K

d Not used Not used Not usedU1 dp 10

Fig. 3. Distributions of velocity magnitude in a plane containing thesphere center and the settling axis. Simulation sets SAL/SSP (left) andGAAL (right) - snapshot taken at t/st = 0.94.

Fig. 4. Particle velocity time series. Results from experiments andsimulation sets SAL/SSP and GAAL are presented.


solid. As a consequence, the balance between related con-tributions in the Navier–Stokes equations is not ensuredcorrectly and the fluid/solid interactions diverge from thephysical solution. This phenomenon is well illustrated inFig. 4, where the evolution of the particle velocity obtainedvia SAL/SSP is compared to the experimental time series.The dynamic equilibrium between both phases is respecteduntil t ’ st = 0.26 [s], then an increasing discrepancy isshown. We observe that the relative error stabilizes at animportant level (roughly 30%). The way in which the pen-alty method is used is disastrous in this case, since the final-state equilibrium between drag and buoyancy forces differsfrom reality. We have experimented analog penalty param-

eters for lower Rep regimes proposed by Mordant [19].Error levels are still not acceptable (roughly 10%), but sim-ulations fit better the experimental data, and this confirmsthat the SAL/SSP inefficiency appears with inertial regimesand subsequent strong particle/fluid couplings.

With the new GAAL method, the penalty of the diver-gence as well as the solid behavior are well balanced intime. With a dynamical treatment of the constraint field,this provides suitable particle/flow interactions. In Fig. 3,the GAAL approach provides a velocity field topology thatmakes much more sense, physically speaking, than the pre-vious one. In Fig. 4 is depicted the unsteady particle veloc-ity Up(t). The experimental reference is accurately predictedduring the acceleration phase as well as the equilibriumphase. The relative error on the terminal settling velocityis now about 1.4%. In the case of the GAAL set, the choiceof a constant uv and of ui being a function of time is deter-minant. Indeed, uv is used to estimate the penalty parame-ter as denominator and ui is near to 0 due to the spheredropped without initial velocity. By using uv = ui togetherwith the GAAL set leads the simulations to converge to thecorrect terminal velocity U1. However, these numericalparameters did not allow an acceptable restitution of theacceleration phase. On the contrary, by choosing uv = U1,the whole dynamics of the particle are accurately simu-lated. The penalty parameters are prevented from gettinghigh values and from the dramatic consequences thischoice could have as the conditioning of the linear systemdegenerates.

On the one hand, even with an optimal choice of con-stant penalty parameters, the SAL/SSP approach doesnot allow accurate simulation of particle flows at highReynolds numbers. On the other hand, the optimal imple-mentation of the GAAL method consisting in usingui ¼ �uðtÞ and uv constant different from ui leads to a correctdirect numerical simulation of unsteady moving particleproblems.

4.2. Dam break on solid objects

Among the various examples of complex fluid/structureinteractions, the free surface flow induced by a dam breakinteracting with moving particles in a tank is interesting forits possible experimental study and potential application inenvironment and ocean engineering. Two-dimensional sim-ulations are considered in this work but the behavior ofpenalty methods is straightforward in 3D. A rectangularvein of 0.2 [m] height and 0.5 [m] length is considered.The vein is full of air and oil, whose physical characteristicsare q0 = 1.1768 [kg m�3], q1 = 1000 [kg m�3], l0 = 1.85 ·10�5 [Pa s] and l1 = 0.1 [Pa s] respectively. The liquid ini-tially lies at rest in X1 defined by the points M(x,z) suchthat

0 6 x 6 0:1 and 0 6 z 6 0:15

0:1 < x 6 0:5 and 0 6 z 6 0:05

ParticlesStill fluid level

Fig. 5. Definition sketch of dam break flow interacting with circularparticles.


Solid cylindrical particles of density qs = 900 [kg m�3] andradius Rp = 0.01 [m] are placed downstream the dam. Theinitial vertical position of their center is 0.05 m and itshorizontal positions are 0.2, 0.25, 0.3, 0.35, 0.4 and 0.45[m]. A definition sketch of the particulate dam break prob-lem is presented in Fig. 5.

A fixed regular 500 · 200 Cartesian grid is used to per-form the simulation with the GAAL penalty methods. Aconstant time step Dt equal to 0.0001 [s] is also utilized.Inviscid analysis of the dam break flow has still beenachieved thanks to the nonlinear characteristic theoryapplied to the shallow-water equations (see for example[31]). A characteristic speed Us of the shock wave whichgenerates the breaking jet can be obtained. With H the

Fig. 6. Dam break flow in a 2D tank – left column: interaction with circular pabottom, the free surface and particle shapes are presented for times t = 0.05,

initial height of the dam and g = 9.81 [m s�2] the intensityof the gravity, the characteristic velocity reads Us ¼

rticles; right column: standard free surface flow configuration – from top to0.1, 0.15, 0.2, 0.25, 0.3 and 0.35 [s].

Fig. 7. Dam break flow in a 2D tank – comparison between simulated freesurfaces at time 0.015 [s] and 0.025 [s] for a particle free surface flow (solidline) and a standard dam break flow (dashed line).


ffiffiffiffiffiffiffigHp

¼ 0:99½m s�1�. Based on this velocity, a Reynoldsnumber Re ¼ q1UsH

l1can be built for the dam break flow.

In our configuration, Re = 990.In Fig. 6, the physical behavior of the free surface inter-

acting with the particle is first considered and compared tothe dam break flow without solids. Until time t = 0.15 [s], ajet forms downstream the dam, as observed in the experi-ments of Stansby et al. [26] for dam break flow on wet bot-toms (Fig. 7). No particular effect of the particles isobserved at this instant. From t = 0.2 [s] to t = 0.35 [s],the jet breaks on the forward face of the liquid, while abore forms and propagates upstream. In this configuration,due to a density ratio qs

q1¼ 0:9 almost equal to 1, the parti-

cles behave as floating bodies. The free surface is weaklyaffected by their presence. However, the solid tends toretain the liquid and modifies slightly the jet breaking.An analogy can be made with nearshore structures usedto damp wave breaking to protect beaches and preventthem from erosion.

Fig. 8. Dam break flow in a 2D tank – free surface and particle shapes are presented at t = 0.014 [s] – GAAL (top) and SAL/SSP (middle) simulation ofparticle free surface interaction – zoom on one particle for GAAL (bottom left) and SAL/SSP (bottom right) approaches.

Table 4Definition of the penalty parameters considered for the dam break flow over circular particles at Re = 990

Set k ls ui uv L0 K

SAL/SSP 1000 10,000 Not used Not used Not used Not usedGAAL Eq. (10) and (11) l1 �uðtÞ Us 0.1 100


Our main goal is to show the interest of the GAAL pen-alty approach to deal with free surface particulate flows. Acomparison between the SAL/SSP and the GAAL penaltymethods is provided in Fig. 8. The penalty parameters arepresented in Table 4. According to the work of Ritz andCaltagirone [24] on SAL/SSP, a value ls = 10,000 [Pa s]is used in the particles to prevent them from any deforma-tion. From a computational point of view, the SAL/SSPinduces a strong alteration of the conditioning of the linearsystems. After 1500 time iterations, the SAL/SSP simula-tion diverges. At time t = 0.14 [s], we can notice that theconstant values of the penalty coefficients generates spuri-ous velocities in the air. This drawback was previouslyobserved and explained in [32] on free surface flows.Indeed, for an adapted constant value of k to the liquidmedia, only $ Æ u = 0 is solved in the air medium. The flowtends to follow the Cartesian grid directions. In the GAALsimulation, the flow field is well balanced and regular.Moreover, the continuity of the velocity field is verifiedacross fluid/fluid and fluid/solid interfaces.

As a conclusion, we have demonstrated that dealingwith the direct numerical simulation of a three phasedam break flow interacting with particle is possible byusing an augmented Lagrangian like penalty method.However, it is necessary to locally adapt the penalty coeffi-cient. The new GAAL method is an interesting approach toautomatically satisfy this numerical requirement.

5. Conclusions

A new augmented Lagrangian like penalty method ded-icated to the numerical simulation of particulate flows hasbeen presented in the framework of fictitious domains. It isbased on the generalization of the Adaptive AugmentedLagrangian approach [32] to multiphase flows involvinggas, liquid and solid phases. The Generalized AdaptiveAugmented Lagrangian GAAL has been successfully vali-dated on the flow through a CFC array of spheres. It hasbeen demonstrated that a faster convergence was obtainedcompared to standard SAL/SSP method. Two physicalconfigurations have been considered to illustrate the inter-est of GAAL for solving multiphase flows involving fluidsolid interactions. In both cases, the use of GAAL is neces-sary to reach a physically acceptable numerical solutionthanks to a penalty method. The numerical performanceof GAAL is comparable or better to standard penaltymethods in terms of iterative solver residual or memoryrequirements. However, a 8% increase of the computa-tional cost is observed.

The main drawback of the GAAL method consists in thechoice of the right set of dimensionless parameters requiredfor an automatic estimate of the penalty terms. Theseparameters can be difficult to deal with, for example whenirregular grids are utilized or when turbulent multiphaseflows involving boundary layers and large space and timescales are considered. Keeping the philosophy of providingan automatic estimate of k, j, g and f, future works will bedevoted to a priori scanning of the linear system for definingthe penalty terms. This version of the GAAL approach willnot require any user defined parameter. The penalty param-eters will be built in order to maintain an as good as possibleconditioning of the linear system.

Acknowledgements

We gratefully thank Pierre Lubin and Mohamed Radyfor their critical reading of the article. We also addressour deep acknowledgments to the IDRIS and CINESFrench computer resource centers that allowed us to leadour simulations through project ter-2237.

References

[1] Adrian RJ. Particle-imaging techniques for experimental fluidmechanics. Ann Rev Fluid Mech 1991;23:251.

[2] Brackbill JU, Kothe BD, Zemach C. A continuum method formodeling surface tension. J Comput Phys 1992;100:335.

[3] Caltagirone J-P, Vincent S. Tensorial penalisation method for solvingNavier–Stokes equations. CR Acad Sci serie II b 2001;329:607–13.

[6] Daly BJ. Numerical study of two-fluid Rayleigh–Taylor instability.Phys Fluids 1967;10:297.

[7] Daly BJ. Numerical study of the effect of surface tension on interfaceinstability. Phys Fluids 1969;12:1340.

[8] Ergun S. Fluid flow through packed columns. Chem Eng Prog1952;48:89–94.

[9] Fortin M, Glowinski R. Methodes de lagrangien augmente. Appli-cation a la resolution numerique de problemes aux limites. Dunod;1982.

[10] Glowinski R, Pan TW, Hesla TI, Joseph DD, Periaux J. A fictitiousdomain approach to the direct numerical simulation of incompress-ible viscous flow past moving rigid bodies: application to particulateflow. J Comput Phys 2001;169:363–426.

[11] Gustafsson I. On first and second order symmetric factorizationmethods for the solution of elliptic difference equations. ChalmersUniversity of Technology, 1978.

[13] Hirt CW, Nichols BD. Volume of fluid (vof) methods for thedynamics of free boundaries. J Comput Phys 1981;39:201.

[14] Khadra K, Angot P, Parneix S, Caltagirone J-P. Fictitious domainapproach for numerical modelling of Navier–Stokes equations. Int JNumer Methods Fluids 2000;34:651–84.

[15] Lafaurie B, Nardone C, Scardovelli R, Zaleski S, Zanetti G.Modelling merging and fragmentation in multiphase flows withSURFER. J Comput Phys 1994;113:134.


[16] Lawson, NJ, Rudman M, Guerra A, Liow J-L. The application ofparticle image velocimetry and computational fluid dynamics to thebreak-up of a large bubble in a liquid bath. In: Two-phase flowmodelling and experimentation, Edizioni ETS, Pisa, vol. 3, 1999. p.1413.

[17] Li J. Piecewise linear interface calculation. CR Acad Sci serie II b1995;320:391–6.

[18] Massol A, Simonin O, Poinsot T. Steady and unsteady drag and heattransfer in fixed arrays of equal sized spheres. Technical Report,CERFACS, Toulouse, France, TR/CFD/04/13, 2004.

[19] Mordant N, Pinton J-F. Velocity measurements of settling sphere.Eur Phys JB 2000;18:343–52.

[20] Pianet G, Ten Cate A, Derksen JJ, Arquis E. Assessment of the 1-fluid method for the DNS of particulate flows: sedimentation of asingle sphere at moderate to high Reynolds numbers. Comput Fluids2007;36:359–75.

[21] Randrianarivelo TN, Pianet G, Vincent S, Caltagirone J-P. Numer-ical modelling of solid particle motion using a new penalty method.Int J Numer Meth Fluids 2005;47:1245–51.

[22] Randrianarivelo TN, Vincent S, Simonin O, and Caltagirone JP.Numerical experiments on heat and mass transfer in fluidized beds.In: 4th Int conf on computational heat and mass transfer, Cachan,Paris, May 2005.

[23] Rider WJ, and Kothe DB. Stretching and tearing interface trackingmethods, AIAA paper 95-1717, 1995.

[24] Ritz J-B, Caltagirone J-P. A numerical continuous model for thehydrodynamics of fluid particle systems. Int J Numer Meth Fluids1999;30:1067–90.

[25] Scardovelli R, Zaleski S. Direct numerical simulation of free surfaceand interfacial flows. Annu Rev Fluid Mech 1999:531–67.

[26] Stansby PK, Chegini A, Barnes TCD. The initial stages of dam breakflow. J Fluid Mech 1998;374:407–24.

[27] Sussman M, Smereka P. Axisymmetric free boundary problems.J Fluid Mech 1997;341:269.

[28] Sweby PK. High resolution schemes using flux limiters for hyperbolicconservation laws. SIAM J Numer Anal 1984;21:995.

[29] Vincent S, Caltagirone J-P. Efficient solving method for unsteadyincompressible interfacial flow problems. Int J Numer Meth Fluids1999;30:795.

[30] Vincent S, Caltagirone J-P. A One Cell Local Multigrid method forsolving unsteady incompressible multi-phase flows. J Comput Phys2000;163:172.

[31] Vincent S, Caltagirone J-P, Bonneton P. Numerical modelling of borepropagation and run-up on sloping beaches using a Mac-CormackTVD scheme. J Hydraul Res 2001;39:1–9.

[32] Vincent S, Caltagirone J-P, Lubin P, Randrianarivelo TN. Anadaptative augmented Lagrangian method for three-dimensionalmulti-material flows. Comput Fluids 2004;33:1273–89.

[33] Van Der Vorst HA. Bi-CGSTAB: a fast and smoothly convergingvariant of Bi-CG for the solution of non-symmetric linear systems.SIAM J Sci Stat Comput 1992;13:631.

[34] Wen C, Yu Y. Mechanics of fluidization. Chem Eng Prog Symp Ser1966;62:100–11.

[35] Youngs DL. Time-dependent multimaterial flow with large fluiddistortion. In: Morton KW, Baines MJ, editors. Numerical methodsfor fluid dynamics. New York: Academic Press; 1982. p. 273–85.

Local penalty methods for flows interacting with moving solids at high Reynolds numbers

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