level set based parallel computations of unsteady free surface flowsfrey/papers/interfaces... ·...

European Conference on Computational Fluid DynamicsECCOMAS CFD 2006

P. Wesseling, E. Oñate and J. Périaux (Eds)c© TU Delft, Delft The Netherland, 2006

LEVEL SET BASED PARALLEL COMPUTATIONS OFUNSTEADY FREE SURFACE FLOWS

E. Marchandise∗, P. Geuzaine† and J.F Remacle∗

∗ Université catholique de LouvainDepartment of Civil Engineering

Place du Levant 11348 Louvain-la-Neuve, Belgium

e-mail: [email protected]† Cenaero

Avenue Jean Mermoz 306041 Gosselies, Belgium.

e-mail:[email protected]

Key words: Free surface, Parallel Computing, Level Set, Stabilised Finite ElementMethod

Abstract. The simulation of free surface flow is a computationally demanding task. Real

life simulations require often the treatment of unstructured grids to capture the complex

geometries and the use of parallel supercomputers for the efficient solution.

For that aim, we have developed an efficient parallel free surface flow solver based on a

full tetrahedral discretization. An implicit stabilized finite element method is used for solv-

ing the unsteady incompressible two-phase flow in three-dimensions. A pressure stabilized

Petrov Galerkin technique is used to avoid spurious pressure modes, while an upwind finite

volume discretization is used to discretize the advective fluxes in a stable manner. The

interface between the fluid phases is captured with the level set method implemented with

a quadrature-free discontinuous Galerkin method. The parallel implementation is based

on the MPI message-passing standard and is fully portable.

We show the effectiveness of the method in the simulation of complex 3D flows, such

as the flow past a cylindrical and the flow in a partially-filled tank of a car that suddenly

brakes.

1 INTRODUCTION

The study of free surface flows covers a wide range of engineering and environmentalflows, including molten metal flow, sloshing in containers, wave mechanics, open channelflows, flows around a ship or structure.

The main challenge for solving time-dependent free-surface flow problems in threedimensions is to provide an accurate representation of the interface that separates the two

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E. Marchandise, P.Geuzaine and J.F Remacle

different fluids. This involves the tracking of a discontinuity in the material propertieslike density and viscosity.

The principal computational methods used to solve incompressible two-phase flowsare the front tracking methods [10, 12], and the front capturing methods (volume offluid [11], [24] and level set [22, 30]).

A successful approach to deal with two phase flows, especially in the presence of topo-logical changes, is the level set method (Osher and Sethian [22]). Application of levelsets in two-phase flow calculations have been extensively described by Sussman, Smerekaand Osher in [29–31]. The level set function is able to represent an arbitrary numberof bubbles or drops interfaces and complex changes of topology are naturally taken intoaccount by the method. The level set function φ(~x, t) is defined to be a smooth functionthat is positive in one region and negative in the other. The implicit surface φ(~x, t) = 0represents the current position of an interface. This interface is advected by a velocityfield u(~x, t) that is, in case of two-phase flows, the solution of the Navier-Stokes equations.The elementary advection equation for interface evolution is:

∂tφ + u · ∇φ = 0 . (1)

In [19], we have developed a high order quadrature free Runge-Kutta discontinuousGalerkin (DG) method to solve the level set equation (1) in space and time. The methodwas compared with classical Hamilton-Jacobi ENO/WENO methods [6, 13, 23, 29] andshowed to be computationally effective and mass conservative. Besides, we showed thatthere was no need to reinitialize the level set.

Level sets represent a fluid interface in an implicit manner. The main advantageof this approach is that the underlying computational mesh does not conform to theinterface. Hence, discontinuous integrals have to be computed in the fluid formulationbecause both viscosities and densities are discontinuous in all the elements crossed by theinterface. The most common approach is to define a zone of thickness 2� in the vicinityof the interface (|φ| < �) and to smooth the discontinuous density and viscosity over thisthickness [20,21,30,35,36]. Smoothing physical parameters in the interfacial zone may bethe cause of two problems. The first one is the introduction of non physical densities andviscosities in the smoothed region, leading to possible thermodynamically aberrations [7].The second problem is the obligation to keep the interface thickness constant in time. Toensure that the smoothed region has a constant thickness, one has to reinitialize the levelset so that it remains a distance function. In this work, we rather adopt a discontinuousapproach [28,34] to compute the discontinuous integrals. The use of a recursive contouringalgorithm [25] allows to localize the interface accurately. Consequently, we are able tocompute the discontinuous integrals with a very high level of accuracy.

For the computation of the incompressible two-phase Navier-Stokes equations, variousnumerical methods have been developed. Among them are the projection methods [4,17],stabilized finite element methods [14, 32] and artificial compressibility methods [5], [26].

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A key feature of stabilized methods is that they have proved to be LBB stable and tohave good convergence properties [15] [8].

In this work, we present a stabilized finite element method for computing flows in bothphases [18] and combine it with a discontinuous Galerkin level set method [19] for com-puting the interface motion. The overall algorithm avoids the cost of the renormalizationof the level set as well as the introduction of a non physical interface thickness and ex-hibits good mass conservation properties. This numerical technology is implemented inthe parallel mode using mesh partitioning and the message passing interface (MPI).

The outline of this paper is as follows: we first present the governing equations inSection 2. Section 3 is devoted to the description of our computational method. Wepresent the Navier-Stokes solver and the coupling with the discontinuous Galerkin methodfor the level set equation. In section 4, we show the effectiveness of the method in thesimulation of complex 3D problems, such as the flow past a cylindrical and the flow in apartially-filled tank of a car that brakes suddenly.

2 GOVERNING EQUATIONS

We consider that the interface is represented by the iso-zero of the level set functionφ. The level set function φ is positive in the liquid (denoted fluid 1) and negative in thegas (denoted fluid 2).

The governing equations describing the motion are then given by the non dimensionalNavier Stokes equations:

∂tu + u · ∇u = −∇pρ(φ)

+1

ρ(φ)Re∇ ·

(

µ(φ)(∇u + ∇uT ))

+g

Fr2(2)

∇ · u = 0. (3)

and by the level set equation :

∂tφ + ∇ · (uφ) = 0. (4)

From the level set function one can derive the density ρ(φ), viscosity µ(φ), the normal~n and the curvature κ. Density and viscosity are written as

ρ(φ) = ρ2(1 − H(φ)) + ρ1H(φ) (5)µ(φ) = µ2(1 − H(φ)) + µ1H(φ) (6)

where H(φ) is the Heaviside function that is equal to 1 in the liquid and 0 in the gas.Here u is the velocity field, p the pressure field, ρ and µ are the density and dynamic

viscosity of the fluid all in non-dimensional forms; these are variables in whole domainbut constant and in general different in each phase. g denotes the gravitational field.

The key flow parameters are then the ratios of density and viscosity (ρ1/ρ2, µ1/µ2),the Reynolds number Re = ρRURLR/µR , the Froude number Fr = UR/

√gLR . The

subscript R denotes the reference value.

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3 COMPUTATIONAL METHOD

To derive the finite element discretization, we assume that we have some appropriatefinite dimensional function spaces for the trial (Sh

u,Shp ,Shφ) and weighting (Vhu,Vhp ,Vhφ)

function spaces corresponding to the velocity, pressure and the level set function.Concerning the function spaces, we use linear continuous approximations for the ve-

locity and pressure and piecewise continuous approximations of order p on each elementfor the level set. The physical domain Ω of boundary Γ is discretized into a collection ofNe elements (Ωe) called a mesh. The same computational mesh is used for the NavierStokes solver and the Interface solver.

3.1 The Navier-Stokes solver

We employ a pressure stabilized pressure finite element (PSPG) [32] method for thediscretization of the Navier Stokes equations. For the stabilization of the upwind term,we follow the work of Barth and Selmin [2] and use a finite volume stabilization with thefinites volumes being the median cells of the mesh .

The stabilized finite element formulation of Eqs. (2) and (3) can be written as follows:find uh ∈ Sh

uand ph ∈ Shp such that ∀vh ∈ Vhu and ∀qh ∈ Vhp :

∫

Ωe

vh · (uht + uh · ∇uh + ∇ph

ρ− g

Fr2) dv +

∫

Ωe

∇vh · µρRe

(∇u + ∇uT ) dv (7)+

∫

Ωe

qh∇ · uh dv + ST =∫

Γ

vh · hh ds

where hh denotes the Neumann-type boundary condition associated with the momentumequations hh = σ · n on the interface Γ, where σ = −pI + µ(∇u + ∇uT ) is the stresstensor.

The stabilization term ST

ST =∑

e

(

τe∇qh,R(ph,uh))

Ωe(8)

contains the residual of the momentum equation

R(ph,uh) = uht + (uh · ∇)uh − 1

ρRe∇ · µ(∇u + ∇uT ) + ∇p

h

ρ− g

Fr2. (9)

The stabilization parameter τ� is of order O (h2e/ν) in the diffusion dominated case andof order of O (he/‖u‖) in the advection dominated case [32, 33].

For the numerical solution of Eq.(7), a second-order three-point backward differencescheme is employed for the time-integration. An inexact Newton method based on a finitedifference Newton-Krylov algorithm [9] is used to solve at each time step the system ofnonlinear equations. The iterative solution of the large sparse linear system of equationsthat arises at each Newton iteration is solved by the GMRES method preconditioned bythe RAS [3] algorithm.

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3.1.1 Discontinuous density and viscosity

As the density and the viscosity are constant in each phase but may be discontinu-ous across the interface (the interface does not conform with the mesh), some integralsof Eq. (7) are discontinuous. Those discontinuous integrals are computed exactly bynumerical integration:

∫

Ωe

vh∇phρ

dv =

∫

V 1

vh∇phρ1

dv +

∫

V 2

vh∇phρ2

dv (10)

∫

Ωe

∇vh µρRe

∇uh dv =∫

V 1

∇vh µ1ρ1Re

∇uh dv +∫

V 2

∇vh µ2ρ2Re

∇uh dv (11)

where the subscripts 1 and 2 denote the fluid 1 and the fluid 2.

3.2 The Interface solver

We employ a quadrature free discontinuous Galerkin level set method [19] for thediscretization of the level set equation.

The discontinuous finite element formulation of Eqs.̃refeq:levelset2 can be written asfollows: find φh ∈ Shφ such that ∀φ̂h ∈ Vhφ :

∫

Ωe

φ̂h · ∂tφ dv +∫

Ωe

∇φ̂h · (uhφ) dv −∫

∂Ωe

φ̂h · f ds (12)

where f = (φu · n) is the normal trace of the fluxes and is chosen to be the upwind flux.The resulting system is solved in time using an explicit Runge-Kutta method.

More details of implementation of the discontinuous Galerkin method as well as super-convergence analysis can be found in [19].

3.3 Summary of the algorithm

Our computational approach for numerical modeling of two-phase flows can be sum-marized as follows. First we initialize all the variables (velocity u, pressure p, level setφ). Then,

For each time tn, n = 0, 1, 2...:

• First advance implicitly the Navier-Stokes equations from tn−1 until tn and find avelocity un. This equation is solved implicitly for the velocity and pressure variablesbut uses the previous position of the level set φn−1 to identify the fluid variable suchas viscosity and density. To go to step 2, project the velocity field u onto the dofsof the interface solver.

• Second advance explicitly the level set (Eq. (4)) from tn−1 until tn (using severalsmaller time steps) with a linear velocity varying in time between un−1 until un andfind a new position for the level set φn. Then project the level set function φ ontothe dofs of the fluid solver.

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• Third go back to step 2, with this new position φn and repeat those steps until‖φn−1 − φn‖ < eps.

4 NUMERICAL RESULTS

In this section, we show the effectiveness of the method in the simulation of complex 3Dproblems, such as the flow past a cylindrical and the sloshing in a tanker during braking.

Our methods were implemented in Standard C++, compiled with INTEL icc v8.0, andrun on the NEC/Clustervision Linux cluster installed at CENAERO.

The iterations are carried out with a GMRES update technique with a Krylov spaceof 60.

4.1 3D Dam break with a cylindrical obstacle

To show the ability to simulate 3D free surface flows, we consider the breaking of acubic water column in a domain containing a cylindrical obstacle .

The computational domain is described by the length L = 6 m and the height H =1.5 m . The height and the width of the water column are hl = 1 m and hw = 1.5 m. Thecylinder is located 1.3 m downstream the water column and has a diameter of 0.4 m. Themesh depicted in fig. 1 is unstructured and made of 30, 820 nodes.

Figure 1: Computational mesh for the dam break problem made of 30.820 nodes.

Figure 2 shows snapshots of the water surface position at selected times. We seethat from the time the water front reaches the cylinder, the flow shows clearly threedimensionality.

Table 1 compares the relative error of mass conservation at t = 1s and t = 2s fordifferent meshes and different order of polynomials for the level set. The mass errors are

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Figure 2: Perspective view of the free surface for the dam break with a cylindrical obstacle at the selectedtimes t = 1s and t = 2s.

calculated by:

�V =V (tf ) − V (0)

V (0)(13)

where V (tf ) is the total volume of the liquid at the final time tf .From Table 1 we see that mesh refinement as well as increasing the order p, improves the

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Table 1: Mass fluctuation of the method for the 3D dam break with cylindrical obstacle computed attime t = 1s and t = 2s on different computational meshes and with different polynomial orders p toapproximate the level set

Mesh order Mass loss Mass loss #CPU Elapsed Time# nodes p (t = 1s) (t = 2s)12,500 1 7.4 % 14.3 % 2 1h3012,500 2 3.4% 6.2 % 2 3h1012,500 3 2.1% 3.1 % 4 15h1030,820 2 1.16% 0.86 % 8 13h1074,016 2 0.62% 0.11 % 24 6h45

mass conservation. We show that those results are obtained within reasonable computa-tional time. The computational time may however be dramatically reduced by performingmesh adaptation near the interface. We are currently working on this topic.

4.2 Sloshing in a partially filled tank that brakes suddenly.

Sloshing is known as the oscillation of the free surface of a liquid in an externallyexcited container. Sloshing of liquids in moving containers is of practical concern in manyengineering applications, such as fuel sloshing in aircraft tanks as a result of sharp flightmaneuvers, liquid sloshing in large storage tanks due to earthquakes and fluid oscillationin tanker trucks traveling on highways.

Partially filled liquid tanks undergoing accelerated motions are susceptible to slosh-induced loads, which may affect the dynamic behavior of the liquid-retaining structure,and may even be severe enough to cause structural damage.

Accurate numerical simulation of the sloshing phenomena is very important in thedesign of the tank. It allows fuel system validation before actual prototyping and indus-trialization and therefore, reducing time and costs for their development.

Large amplitude liquid sloshing in partially filled tank vehicles has been studied in [1,16, 27].

The geometry of the tank considered here is a simplified geometry of a vehicle tank.The computational mesh (fig. 3 is made of 605, 923 tetrahedral elements and 106, 384nodes).

It is assumed that the vehicle is moving at a constant velocity and it suddenly brakeswith a constant deceleration equal to 5 m/s2 for 0.5 seconds. Afterwards, it moves againwith a constant velocity.

Since the inertial force due to the acceleration of the tank is very large compared tothe viscous forces in the fluids, a slip-wall boundary condition was imposed on the tankwalls.

The level of water is 12 cm measured from the bottom of the fuel tank. This representsabout 45 liters of fuel.

The pictures in fig. 4 show at different instants the sloshing and pressure distribution.

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Figure 3: Computational mesh for the vehicle tank made of 106, 384 nodes (left) and cut in the mesh(right).

The computation was carried out with 32 processors and took about 19h to simulate 2s.

5 CONCLUSIONS

A unified approach for the numerical simulation of three dimensional two-phase flowshas been presented. The approach relies on an implicit stabilized finite element approxi-mation for the Navier-Stokes equations and discontinuous Galerkin method for the levelset method.

Such a combination of those two numerical methods results in a simple and effective al-gorithm that allows to simulate diverse flow regimes presenting large density and viscosityratios (ratio of 1000).

Two advantages of the method are:

• Simplicity and flexibility: The stabilized method utilizes linear elements for theunknowns of velocity and pressure and a discontinuous Galerkin method of higherorder for the level set unknown. The tetrahedral meshes can be structured orunstructured;

• Accuracy: The overall scheme is second order accurate in space and time. Besideswe do not need to introduce any artificial parameter as interface thickness. Theinterface can be localized precisely which enables us to compute accurately theintegrals with a discontinuous density and/or viscosity. Furthermore, the methodexhibits excellent conservation properties using high order polynomials for the levelset.

For our future applications we will focus on mesh adaptivity to achieve higher resolutionof the interface while reducing memory and CPU time.

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XYZ

XYZ

Figure 4: Sloshing during a vehicle tank during braking. The images show at different instants (t =0.1 s, 0.2 s, 0.4 s) , the free surface position (left in red) and the pressure distribution (right).

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INTRODUCTIONGOVERNING EQUATIONSCOMPUTATIONAL METHODThe Navier-Stokes solverDiscontinuous density and viscosity

The Interface solverSummary of the algorithm

NUMERICAL RESULTS3D Dam break with a cylindrical obstacleSloshing in a partially filled tank that brakes suddenly.

CONCLUSIONS

level set based parallel computations of unsteady free surface flowsfrey/papers/interfaces... ·...

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