Section 1-7 s547

SCHMIDT, J., WALL, W.A., RAMM, E.

Transient fiee surface flows via a stabilized ALE finite element method

f i e surface problems appear in a wide range of industrial and engineering applications, e.9. when modelling sloshing of fluid in a container or when tmcking the free surface evolution in casting and molding processes. A finite element technique is presented to study time-dependent large free surface motions of viscous, incompressible fluids. The approach is based upon an arbitrary Lagmngean-Eulerian (ALE) representation of kinematics and field equations, i.e. continuum mechanical conservation laws. Both convective effects and equal-order interpolation for velocities and pressure are stabilized in a Galerkin least-squares sense. This leads to a fully stabilized finite element method (FEM) for the governing instationary incompressible Navier-Stokes equations. The algorithmic setup is complemented by a combination of the stabilized FEM with direct t ime integration procedures and fixed point-like itemtive schemes. The performance of the overall algorithm is demonstrated with the help of selected two-dimensional numerical examples.

1. Stabilized finite elements

The fluid behavior on the domain R is assumed to be governed by the instationary incompressible Navier-Stokes equations, written here in terms of velocity u and pressure p with appropriate boundary and initial conditions:

(1)

u = g on r s x ( O , T ) , n . a = h on r h x ( 0 , T ) and u = m in 51 for t=O (2)

au at -+u~Vu-2vV.cs(u)+Vp=f and V-u=O in Rx(0 ,T)

Due to the well-known numerical problems, e.g. spurious oscillations in the solution, that occur when these equations are solved using the standard FEM, stabilization terms are added to the standard variational formulation. To preserve consistency of the FEM, these terms are functions of the Euler-Lagrange equations evaluated elementwise. The stabilization terms in this approach are essentially based on [l] and are given in equation (4).

Introducing finite-dimensional function spaces VVh and P h and expanding the trial and weighting functions in terms of their finite element basis functions yields the semi-discrete matrix equations M ~ I + N(u)u + Ku = F. These equations are integrated in time using a one-step kscheme. The nonlinear term N(u)u is linearized and the system of equations is solved in a fixed point-like iteration scheme.

2. ALE formulation

In free surface problems, both large deformations and moving boundaries appear. Hence we would like to combine the advantages of both the Lagrangean and the Eulerian approach to describe our problems. This is accomplished by introducing a third, so-called reference domain, which is allowed to move arbitrarily and independently of spatial or material points. In the FE context the referential domain is represented by the moving FE mesh.

The material derivative of a quantity f , which is needed in our formulation of the conservation laws, is also referred to as the “ALE fundamental equation”: (x, t)l + c ig (y , t ) , where x denotes the material and y the spatial coordinates. c = u - uG is the convective velocity: defined as the difference between the particle velocity u and the grid velocity uG. The we& form of the stabilized ALE FE formulation for the Navier-Stokes equations with the stability parameters T [4] may now be written as follows: Find u E V and p E P such that V(w,q) E VO x P

=

BU ST = (V . u, T ~ ~ ~ ~ V * w) + (z I + c * Vu - 2vV - e(u) + V p - f , T,~*(C - Vw - 2vV - e(w) - Vq)), (4) e

3. Free surface problems - Two-fleld coupled problems

In free surface problems the motion of the FE mesh adjacent to the free surface is governed by the fluid motion. A

S 548 ZAMM Z. Angew. Math. Mech. 80 (2000) 52

local Lagrange formulation is used here, which requires that the grid velocity uG be equal to the fluid velocity u on the free surface. To determine the mesh motion needed in an ALE formulation the mesh is introduced as a separate field. It is modelled as a pseudo-structural system with artificial elastic properties, which leads to a stiffness matrix Km. The mesh displacements q are obtained by solving an elastostatic system of equations, Kmq = Fm, where the right hand side vector Fm results from the Dirichlet boundary conditions 4 I uG = u on the free surface boundary.

The overall staggered solution algorithm proceeds as follows: In every time step tn+l, first the fluid problem is solved on the mesh qn for the discrete velocities un+l and pressures pn+l. Then the mesh system is solved for the new mesh coordinates qn+l using the boundary condition qn+l I u:+~ = u,+1.

4. Numerical examples

rl(x,y) = asin(rac) 1.010

B 1.005- .- p = 1.0 ]*JO&

v ~ 0 . 0 1 C 0.995- i4

0.99 I I I I I I I I I , 1 I I , I , I , 1

0 1 2 3 4 5 6 7 8 9 10 time

t = o

-

Figure 1: Large amplitude sloshing due to initial deformed geometry

First we will solve a problem of large amplitude sloshing due to an initial deformed geometry. Figure 1 shows the initial geometry and pressure distribution, and the geometry and velocity vectors after one period of sloshing. Our results are identical to those in [2]. The “broken dam” or “collapsing column” problem is shown in Figure 2. For

1 6 8 lLzl end of simulation due to severe mesh distortion * remeshingneeded t at t - 0

Y

4 0.5 1.5 2.5 3.5 -

X time

Figure 2: Broken dam or collapsing column problem

this problem experimental results exist [3], which are reproduced quite nicely, as demonstrated in the plot of the column width versus time. Current research is concerned with the improvement of the presented method, e.g. to include remeshing and to imbed our algorithm into an approach for coupled fluid-structure interaction problems [5] .

Acknowledgements

Support of this research by the German National Science foundation DFG within the Sonderforachungsbereich 104 at Stuttgart University is gratefully acknowledged.

5. References

[l] FRANCA, L.P., FREY, S.L.: Stabilized finite element methods: 11. The incompressible Navier-Stokes equations. Comp.

[2] RAMASWAMY, B.: Numerical simulation of unsteady viscous free surface flow. J. Comput. Physics 90 (1990) 396430. [3] RAMASWAMY, B., KAWAHARA, M.: Lagrangian finite element analysis applied to viscous free surface fluid flow. Intl. J.

[4] WALL, W.A.: Fluid-Struktur-Interaktion mit stabilisierten Finiten Elementen. Ph.D.-Dissertation, Bericht Nr. 31, hstitut

[5] WALL, W.A., RAMM, E.: Fluid-structure interaction based upon a stabilized (ALE) finite element method. Computational

Meth. Appl. Mech. Eng, 99 (1992) 209-233.

Num. Meth. Fluids 7 (1998) 963-984.

fiir Baustatik, Universitat Stuttgart (1999).

Mechanics - New ’Ikends and Applications, E. Onate, S.R. Idelsohn (Eds.), CIMNE, Spain (1998).

Addresses: Schmidt, J., Wd, W.A., Ramm, E., Institut fiir Baustatik, Universitat Stuttgart, Pfaffenwddring 7, D-70550 Stuttgart, Germany. Homepage: http://www.uni-stuttgart.de/ibs