a coupled interface-tracking/ interface-capturing technique for ......3 finite element formulation...

A coupled interface-tracking/ interface-capturing technique for free-surface flows

M. Watts, S. Aliabadi & S. Tu School of Engineering, Jackson State University, Jackson, MS, USA

Abstract

In this paper we will present a new computational approach to simulate free-surface flow problems efficiently. The finite element solution strategy is based on a combination approach derived from fixed-mesh and moving-mesh techniques. Here, the free-surface flow simulations are based on the Navier-Stokes equations written for two incompressible fluids where the impact of one fluid on the other one is extremely small. An interface function with two distinct values is used to locate the position of the free-surface in regions near the floating object, while mesh-moving is used to move the free-surface in regions where wave breaking is not expected. The stabilized finite element formulations are written and integrated in an arbitrary Lagrangian-Eulerian domain. In the mesh-moving scheme, we assume that the computational domain is made of elastic materials. The linear elasticity equations are solved to obtain the displacements for each computational node. The numerical example includes a 3D simulation of flow past a sphere with a Reynolds number of 5000 and a Froude number of 0.5. Keywords: finite element method, parallel simulation, free-surface flows, interface-tracking, and interface-capturing.

1 Introduction

The numerical simulations of free-surface flows require the coupling between the equations governing the dynamics of the fluid and the free-surface. Generally, there are two distinct approaches in the numerical simulation of free-surface flows. Depending on the physical characteristics of the problem, either “interface-tracking” or “interface-capturing” techniques are used. In the

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Computational Methods in Multiphase Flow III 353

interface-tracking technique, to absorb the motion of the free-surface, the nodal coordinates on the free-surface are usually moved with either the normal component of the fluid velocity to the free-surface or the fluid velocity itself [1]. The space-time finite element method is an example of the interface-tracking technique [2]. In the interface-tracking methods, the computational nodes need to be moved to account for the motion of the free-surface [3-8]. In the applications where the deformation of the free-surface is large, the moving-mesh methods usually result in element distortion. As the element distortion grows and becomes unacceptable, the generation of a new mesh and the projection of the solution from the old mesh to the new mesh is essential. In complex 3D applications, this procedure is extremely difficult and time consuming. In such cases, computations using the interface-capturing technique over fixed meshes are more desirable. Most of the interface-capturing techniques are based on the volume of the fluid (VOF) approach [1, 9-19]. In the VOF approach, the Navier-Stokes equations are solved over a non-moving mesh. An interface function with two distinct values serves as a marker identifying the location of the free-surface. This function is transported throughout the computational domain with a time-dependent advection equation. Generally, the accuracy of the interface-capturing techniques depends on how accurate the free-surface function is represented. Therefore, mesh resolution becomes a prime factor in determining the accuracy of this technique [1, 10]. Clearly, both techniques have inherent advantages and disadvantages specific to the class of problems being solved. In this article, we present a new finite element technique for the simulation of free-surface flows, which combines these two techniques, each eliminating the disadvantages of the other. This finite element method is based on advanced computational technologies we have developed in the past several years for simulation of free-surface flows. Our free-surface computational technology is very user friendly and has been applied to many applications including sloshing in tanker-trucks [1], waves interacting with marine vessels in motion [9], and flow in open channels [13]. Our free-surface flow solver has gone through extensive communication and performance optimization [10]. In our new finite element technique, we will keep track of the free-surface in the region close to the floating object using an interface function based on the VOF method. In the far field regions where wave breaking and mixing is not expected we use an interface tracking method. Figure 1 shows the subdomains where the interface-tracking and interface-capturing techniques will be used. Due to the presence of breaking waves, interface-capturing technique will be used in domain Ω A, as shown in Figure 1. Consequently, this volume will be substantially more refined than the rest of the computational domain. In domain Ω B shown in Figure 1, mesh-moving will be used to allow the free-surface to translate in the vertical direction only. In domain Ω A, the new finite element method is based on the implementation of the Interface-Sharpening/Global Mass Conservation (IS-GMC) [1, 3-8] flow solver in the Arbitrary Lagrangian-


354 Computational Methods in Multiphase Flow III

Eulerian (ALE) moving-mesh frame. The advantage of the IS-GMC over other VOF approaches is the enforcement of the conservation of mass not only locally, but also globally. In domain Ω B, the automatic mesh-moving scheme is also used to absorb the motion of the free-surface. This scheme is based on the solution of the linear elasticity equations [4-7]. Here, we assume that the computational domain is made of elastic materials. The displacement of the free-surface will be the boundary conditions of the elasticity equations. The linear elasticity equations are coupled with the governing equations of the fluid and the interface function equation, which are solved simultaneously.

Figure 1: Top view of generic ship.

The discretization of the finite element formulations results in a coupled, non-linear system of equations, which need to be solved at every time step. Using the Newton-Raphson iteration algorithm, we solve the linearized systems of equations with the Generalized Minimal RESidual (GMRES) update technique [3-8]. For very large systems of equations, we use a matrix-free iteration strategy to obtain the solution of the non-linear system. This vector-based computation strategy totally eliminates a need to form any matrices, even at the element-levels and therefore minimize the memory requirements in large-scale computations. This new finite element technique is implemented in parallel using the message passing interface (MPI) libraries and METIS [20] mesh partitioning package reducing the inter processors communication.

2 Governing equations

To simulate free-surface flow problems, we assume there are two fluids interacting with each other where the dynamic impacts of one fluid on the other one is very small. For example, in water flows, air is constantly interacting with water. Since the density of the air is almost 1000 times less than the water, its effect on the water is negligible. We consider the governing equations for two interacting fluids in the spatial domains Ω A and Ω B, and their boundaries Γ A and Γ B, respectively. Here we assume that the spatial domains and boundaries are functions of time, t. The two fluids are incompressible (e.g. air-water) and separated with an interface. Along the interface, the traction force is continuous (surface tension is negligible).

Ω A

Ω B



The governing equations are the incompressible Navier-Stokes equations written for two fluids in the Arbitrary Lagrangian-Eulerian (ALE) frame. These equations are:

0)( =⋅−

−⋅−+ σ∇∇ guuuu

mesht ζ∂∂ρ , (1)

0=⋅u∇ , (2)

where

(u)I εσ µ2p +−= , (3)

)(21 Tuu ∇∇ε += . (4)

Here u , meshu , p, ρ , g, and µ are the fluid velocity, mesh velocity, pressure, density, gravitational force, and dynamic viscosity, respectively. The strain tensor is denoted by ε and I represents the identity tensor. Eqns (1-2) are completed by an appropriate set of boundary and initial conditions. In domain Ω A, the interface function φ has two distinct values (0, 1) and is used to differentiate between the two fluids [1-8]. A time-dependent advection equation transports this function throughout the computational domain with the fluid velocity. This equation can be written as:

0)( =⋅−+ φ∂∂φ

ζ

∇meshtuu . (5)

Using φ, the density and viscosity can be calculate as:

BA ρφφρρ )1( −+= , (6)

BA µφφµµ )1( −+= , (7) where the subscripts A and B denote the fluid A and fluid B. Initially, φ is set to 0 in Fluid A and 1 in Fluid B. Our mesh-moving scheme will be used to allow the free-surface to move in the vertical direction only in regions where wave breaking and mixing is not expected, i.e. domain Ω B. The equation used to translate the interface nodes in the vertical direction is:



wηvηuη=

∂∂

+∂∂

+∂∂

yxt. (8)

where η is the free-surface elevation. Here, η is initially equal to the free-surface static height. In this approach, it is clear that the need for high-resolution grids in domain Ω B becomes less significant. Therefore, we can use significantly coarse mesh in domain Ω B. In our mesh-moving scheme, we assume that the computational domain is made of an elastic material. We solve linear elasticity equations to obtain the displacements for every computational node in domain Ω B. These equations are:

[ ] 0dId =+ )(2)( 21 ∇κ∇. . ∇ λλ , (9)

)(21 Tdd ∇∇κ += , (10)

where d is the displacement, κ is the strain tensor, and 1λ and 2λ are the linear elasticity coefficients. Here our objective is have a mechanism for moving the computational nodes around, rather than finding the exact displacements. To have a better control over the movements of nodes, we set the coefficients 1λ and 2λ inversely proportional to the volume of each element. This makes the smaller elements stiffer than the larger ones and allows the motion of the physical boundary to be absorbed more in the larger elements, which delays the mesh distortion.

3 Finite element formulation

We use classical Galerkin formulation for linear elasticity equations and stabilized finite element formulation for the Navier-Stokes and interface function equations. The stabilizations are based on the stabilized-upwind/Petrov-Galerkin (SUPG), pressure-stabilization/Petrov-Galerkin (PSPG), and the artificial diffusion techniques. The SUPG term allows us to solve flow problems at high speeds [1, 3-8] and the PSPG term eliminates instabilities associated with the use of linear interpolation functions for both pressure and velocity. The artificial diffusion stabilization technique is used for the interface function for over stabilization [1]. Later, we will see that this feature allows us to enforce the conservation of mass not only locally, but also globally. In the finite element formulations, we first define appropriate sets of trail solution spaces Sh

u, Shp, Sh

φ and Shd and weighing function spaces Vh

u, Vhp, Vh

φ and Vh

d for the velocity, pressure, the interface function and displacements, respectively. The finite element formulation of Eqns (1-2, 5, 8) can then be written as follows: for all wh ∈ Vh

u, qh ∈ Vhp,

hψ ∈ Vhφ, γh ∈ Vh

η and π h ∈ Vh

d, find uh ∈ Shu, ph ∈ Sh

p, φ h ∈ Shφ, ηh ∈ Sh

η and dh ∈ Shd such that:



[ ]

Ω

⋅−−⋅−+

⋅⋅−⋅−+Ω⋅+

Ω+Ω−⋅−+⋅

∑ ∫∫

∫∫

=ΩΩ

ΩΩ

dpt

qdq

dpdt

hhhhmesh

hh

ne

ee

hhp

hhmesh

hmhhp

hhhhhmesh

hh

h

)(])([

)()(

)(:)(])([

1

u,guuuu

w,wuuu

u,wguuuuw

σ∇∇

σ∇∇∇

σε∇

∂∂ρ

ρτ

∂∂ρ

Γ=Ω⋅⋅+ ∫∑ ∫ Γ=

Ωdd

uh

hhhne

ee c h.wuw ∇∇ ρτ

1, (11)

Eqn (11) is solved throughout the computational domain. In domain Ω A we solve the following equation:

0)(1

=Ω⋅+Ω

⋅−+ ∑ ∫∫

=ΩΩ

ddt

ne

ee

hhi

hhmesh

hh

h φψτφ∂

∂φψ ∇∇∇uu . (12)

However, in domain Ω B we solve:

Ω

−

∂∂

+∂

∂+

∂∂

∫Ω dyxt

hhhh w

ηv

ηu

ηγ

0wη

vη

uηγ

vγ

u1

=Ω

−

∂∂

+∂

∂+

∂∂

∂∂

+∂

∂+ ∑ ∫

=Ω

.dyxtyx

hhhne

ee

hh

IIτ (13)

[ ] 0)(2)()( 21 =Ω+∫Ω dhhh dId: ∇κ∇.πκ λλ , (14)

The parameters mτ , cτ , iτ , and IIτ are the stabilization parameters defined as:

21

2

2

22 422−

+

+

∆

=hhtm ρµτ

||u||, (15)

,||u|| zhc 2

=τ

=1

3/uRz

RuRu

<≤

33

(16)

||,u||2h

i =τ (17)

21

2222 vu22

−

++

∆

=htIIτ , (18)



where h is the element length, t∆ is the time increment and Ru is the cell Reynolds number [1]. In Eqn (11), the first three integrals together with the right-hand-side term represent the Galerkin formulation of the governing equations of fluid. The first and second series of element-level integrals in the same equation are the SUPG and PSPG stabilization for the momentum equation, and the least-square stabilization of the continuity equation, respectively. In Eqn (12), the first integral is the Galerkin formulation of the interface function equation and the second integral (element-level) is the artificial diffusion stabilization. The diffusive formulation for the interface function eliminates the numerical undershoots (below 0) and overshoots (above 1) of the interface function around the interface. In the IS-GMC algorithm, we recover the sharpness of the interface in such a way that the global conservation of mass for each fluid is enforced. In Eqn (13) the first integral represents the Galerkin formulation of the elevation equation. Eqn (14) is the Galerkin formulation of the linear elasticity equations. The element-level integral in the same equation is the SUPG stabilization for the elevation equation.

4 Numerical example

Here we simulate water flow past a sphere using our coupled interface-tracking/interface capturing technique. The mesh consists of approximately 13 million tetrahedral elements. The Reynolds number for this case is 5000, with a Froude number of 0.5. Figure 2 below, depicts the solution at time steps 1 and 10, respectively. Domain Ω A contains approximately 9 million unstructured elements, which includes almost 300,000 prism elements on the sphere surface. Domain Ω B contains approximately 4 million elements. The free-surface in domain Ω B consists of 65,000 triangular elements. Also, domain Ω B is approximately 7 times larger in volume than domain Ω A.

Figure 2: Flow past a sphere at time steps 1 and 10, respectively.



Acknowledgements

Work is funded by Northrop Grumman Ship Systems. The computations are carried out using Cray supercomputers located at the Army High Performance Computing Research Center. Partial support is provided by NASA –AMES.

References

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a coupled interface-tracking/ interface-capturing technique for ......3 finite element formulation...

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