EDICT for 3D computation of two- uid interfaces q

Transcription

1 Comput. Methods Appl. Mech. Engrg. 190 (2000) 403±410 EDICT for 3D computation of two- uid interfaces q Tayfun E. Tezduyar a, *, Shahrouz Aliabadi b a Mechanical Engineering and Materials Science, Team for Advanced Flow Simulation and Modeling, Rice University-MS 321, 6100 Main Street, Houston, TX , USA b Department of Engineering, Clark Atlanta University, USA Received 23 July 1998; received in revised form 16 October 1998 Abstract We present the 3D implementation and applications of the enhanced-discretization interface-capturing technique (EDICT) in computation of unsteady ows with two- uid interfaces. In such computations, EDICT can be used as a very e ective method, which combines the exibility and e ciency of interface-capturing techniques with the accuracy provided by enhanced discretization at the interfaces. A stabilized nite element interface-capturing technique is used as the base formulation to solve, over a typically nonmoving mesh, the Navier±Stokes equations and an advection equation governing the interface function. To increase the accuracy in modeling the interfaces, we use nite element functions with multiple components at and near the interfaces, with each component coming from a di erent level of mesh re nement. With its parallel implementation on advanced high-performance computing platforms such as the CRAY T3E, EDICT is a powerful tool for the simulation of a complex, 3D unsteady ow problems with two uidinterfaces, including free surfaces. Ó 2000 Elsevier Science S.A. All rights reserved. 1. Introduction In ow problems with complex and very unsteady interfaces, especially in 3D simulations, interfacecapturing methods with xed meshes can be used as practical alternatives to interface-tracking methods, which require the mesh to move to track the interface. Typically the interface-capturing methods are more exible than the interface-tracking methods, but result in, for comparable mesh re nement levels, less accurate representation of the interface. The enhanced-discretization interface-capturing technique (EDICT) was designed to compensate for this de ciency without a major loss in the computational exibility and e ciency. The EDICT was rst introduced in [1] and described in more detail in [2,3]. In this paper, we present 3D implementation and applications of EDICT. The details of the EDICT concept was described in the earlier publications mentioned above, and therefore in this paper we provide only an overview of this concept. The underlying philosophy in EDICT is to enhance the spatial discretization around the interfaces to increase the accuracy in representing them. The base method is the stabilized- nite-element/interfacecapturing (SFE/IC) method, which was rst described in [4]. The SFE/IC starts with a volume of uid (VOF) approach [5]. An interface function with two distinct values serves as a marker identifying the two q This work was sponsored by the Army High Performance Computing Research Center under the auspices of the Department of the Army, Army Research Laboratory cooperative agreement number DAAH and contract number DAAH04-95-C The content does not necessarily re ect the position or the policy of the Government, and no o cial endorsement should be inferred. * Corresponding author. Tel.: ; fax: ; address: tezduyar@rice.edu (T.E. Tezduyar) /00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved. PII: S ( 0 0 )

2 404 T.E. Tezduyar, S. Aliabadi / Comput. Methods Appl. Mech. Engrg. 190 (2000) 403±410 uids. A time-dependent advection equation governs the evolution of the interface function. This advection equation and the Navier±Stokes equations are solved over a non-moving mesh using stabilized nite element formulations. The stabilization methods employed are the streamline-upwind/petrov±galerkin [6] and pressure-stabilizing/petrov±galerkin [7] formulations. Results from the SFE/IC method were also reported in [2]. Details of the SFE/IC method are given in [8]. To increase the accuracy in representing the interface, we use function spaces corresponding to enhanced discretization at and near the interface. A subset of the elements in the base mesh, Mesh-1, are identi ed as those at and near the interface (see Fig. 1). A more re ned mesh, Mesh-2, is constructed by patching together second-level meshes generated over each element in this subset (see Fig. 1). For each element in this subset there will be a unique second-level mesh. If an automatic mesh generator is used to generate that, the mesh will be generated only once and stored to be used later if that element needs a second-level mesh again. The trial and weighting functions for velocity and pressure will all have two components each: one coming from Mesh-1 and the second one coming from Mesh-2. To further increase the accuracy in representing the interface, we construct a third-level mesh, Mesh-3, for the interface function only. This is done by identifying a subset of the elements in Mesh-2 as those at and very near the interface (see Fig. 1). The construction of Mesh-3 (see Fig. 1) from Mesh-2 will be very similar to the construction of Mesh-2 from Mesh-1. The trial and weighting functions for the interface function will have three components, each coming from one of these three meshes. We re-de ne the subsets over which we build Mesh-2 and Mesh-3 not every time step but with su cient frequency to keep the interface within the zones covered by these subsets of elements. Although it would not be ``illegal'' for the interface to fall out of these zones, we attempt to estimate or keep track of for how many time steps of the simulation the interface will remain inside these zones. How many time steps one can carry out the simulation without re-de ning these subsets of elements will depend, among other things, on how ``wide'' we decide to keep these zones around the interface. Whenever we re-de ne these subsets, the mesh generation cost will not be a signi cant one. If we have to use an automatic mesh generator for the second- and third-level meshes, we will be able to use and re-use the meshes which were generated at the very beginning of the simulation and stored. One may want to eliminate Mesh-3 by not choosing to go to a third-level of re nement. It is also possible to design the second- and third-level meshes in such a way that the zones covered by these two meshes coincide. However, if we keep the Mesh-2 zone ``wider'' than Mesh-3 zone, we can chose to limit the existence (as an unknown) of the interface function to Mesh-2, and therefore solve for it only over the part of the computational domain covered by Mesh-2. If we do that, we have to make sure that the interface remains in Mesh-2 zone. Since our objective will be to keep the interface in Mesh-3 zone, this would also keep it in Mesh-2 zone, even if the interface occasionally falls slightly out of the Mesh-3 zone. The 3D EDICT has been implemented for parallel computations by using the MPI programming environment, and computations reported in this paper were carried out on the CRAY T3E. In Section 2 we review the governing equations. The base nite element formulation used with the EDICT is described in Section 3. The nite element functions used in EDICT are de ned in Section 4. In Section 5 we cover some of the implementation issues. The 3D numerical examples are presented in Section 6, and the concluding remarks in Section 7. Fig. 1. EDICT multi-level meshes.

3 2. Governing equations The governing equations are the time-dependent Navier±Stokes equations of incompressible ows, which we brie y review here. The space and time domains are denoted by X and 0; T respectively, and C denotes the boundary of X. The symbols q x; t, u x; t, p x; t, and f x; t represent the density, velocity, pressure, and the external forces (e.g. gravity), respectively. The Navier±Stokes equations we use here consist of the following momentum conservation equations and the incompressibility constraint: q ou ot u $u f T.E. Tezduyar, S. Aliabadi / Comput. Methods Appl. Mech. Engrg. 190 (2000) 403± $ r ˆ 0 on X 8t 2 0; T ; 1 $ u ˆ 0 on X 8t 2 0; T ; 2 where r p; u ˆ pi 2le u : 3 Here I is the identity tensor, l is the dynamic viscosity, and e u is the strain rate tensor de ned as e u ˆ1 2 $u $u T : 4 To model uid± uid interfaces, we consider two immiscible uids, A and B, with densities q A and q B and viscosities l A and l B. An interface function / serves as a marker identifying uid A and B with the de nition / ˆ {1 for uid A and 0 for uid B}. The interface between the two uids is approximated to be at / ˆ 0:5, and q and l are de ned as q ˆ /q A 1 / q B ; 5 l ˆ /l A 1 / l B : 6 The evolution of the interface function is governed by a time-dependent advection equation o/ u $/ ˆ 0 on X 8t 2 0; T : 7 ot In addition to the equations given so far in this section, we need to provide a set of boundary conditions, a divergence-free initial condition for the velocity eld, and an initial condition for the interface function. 3. Finite element formulation Corresponding to the velocity, pressure and interface function, respectively, let S h u n, Sh p n,and Sh / n be the trial function spaces, and let V h u n, Vh p n, and Vh / n be the test function spaces. The superscript h implies that these are nite-dimensional function spaces, and the subscript n implies that different time levels might have different spatial discretizations. The stabilized formulations of Eqs. (1), (2), and (7) can be written as follows: given u h n and /h n, nd u h n 1 2 Sh u n 1, ph n 1 2 Sh p n 1, and /h n 1 2 Sh / n 1, such that, 8wh n 1 2 Vh u n 1, 8qh n 1 2 Vh p n 1, and 8w h n 1 2 Vh / n 1 : w h n 1 q ouh u h $u h f dx e w h n 1 X ot : r ph ; u h dx q h n 1 $ uh n 1 dx X X Xn el s SUPG u h $w h n 1 s PSPG eˆ1 X e q $qh n 1 q ouh u h $u h f $ r p h ; u h dx ot Xn el s CONT $ w h n 1 q$ uh n 1 dx ˆ w h n 1 hh dc; 8 X e C eˆ1

4 406 T.E. Tezduyar, S. Aliabadi / Comput. Methods Appl. Mech. Engrg. 190 (2000) 403±410 w h n 1 X o/ h ot u h $/ h dx Xn el eˆ1 s / u h $w h o/ h n 1 u h $/ h dx ˆ 0; 9 X e ot where h h represents the Neumann-type boundary condition associated with the momentum equation. The stabilization parameters, s SUPG, s PSPG, s CONT and s /, are de ned as follows: s SUPG ˆ 2ku h 2 k 4m! 2 1=2 ; 10 h h 2 s PSPG ˆ s SUPG ; s CONT ˆ h 2 kuh kz; where z ˆ s / ˆ h 2ku h k ; 8 Re < u Re 3 u 6 3 ; : 1 Re u > where Re u is the cell Reynolds number. In Eq. (8), the rst three integrals, together with the right-hand-side, represent the Galerkin formulation of (1) and (2). The rst series of element-level integrals in the formulation are the SUPG and PSPG stabilization terms. The second series of element-level integrals are the least-squares stabilization terms based on the incompressibility constraint. In Eq. (9), the rst integral represents the Galerkin formulation of (7), while the series of element-level integrals are the SUPG stabilization terms. In time discretization, the time derivatives, ou=ot and o/=ot, and the functions u h, p h and / h are represented as follows: ou ot ˆ uh n 1 uh n ; Dt o/ ot ˆ /h n 1 /h n ; Dt u h 1 a u h n auh n 1 ; p h p h n 1 ; / h 1 a / h n a/h n 1 ; where Dt is the time step size between time levels n and n 1, and a is a time-integration parameter controlling the stability and accuracy of the integration. Normally, we set a ˆ 0:5. 4. Construction of the nite element functions Consistent with the function spaces described in Section 1, we construct the trial functions for the velocity, pressure and the interface function as follows: u h n ˆ u1 n u2 n ; p h n ˆ p1 n p2 n ; 19 20

5 T.E. Tezduyar, S. Aliabadi / Comput. Methods Appl. Mech. Engrg. 190 (2000) 403± / h n ˆ /1 n /2 n /3 n : 21 Here, superscripts 1, 2, and 3 imply correspondence to Mesh-1, Mesh-2, and Mesh-3. We note that only the interface function might have a third component. Similarly, the test functions for the velocity, pressure and the interface function are constructed as follows: w h n ˆ w1 n w2 n ; q h n ˆ q1 n q2 n ; w h n ˆ w1 n w2 n w3 n : 24 More detailed comments on the construction of these functions can be found in [2,3]. 5. Implementation issues The 2D EDICT was implemented [2,3] in a shared memory paradigm, and the 2D computations reported in [2,3] were carried out on an SGI power challenge. The 3D EDICT has been implemented using the message passing interface (MPI) libraries, and the computations reported here were carried out on a CRAY T3E. In our implementation in this paper, Mesh-1 is assumed to be unstructured and irregular, and Mesh-2 and Mesh-3 are to be obtained by successive subdivision of the Mesh-1 elements in the enhanced-discretization zones. This approach was chosen for convenience in mesh generation in this particular implementation, and does not imply that the EDICT requires that type of a relationship between Mesh-2, Mesh-3 and Mesh-1. In a typical nite element implementation, the nite element mesh is de ned with a set of nodes and elements, and the element connectivity data (i.e., the nodes forming an element) is part of the data structure (see Fig. 2, left) needed to carry out the computations. In the EDICT implementation here, we identify an element with its faces (see Fig. 2, right). In turn, the faces are represented by its edges, and the edges are represented by its two nodes (see Fig. 3). This sequence of representation allows us to form Mesh-2 and Mesh-3 on top of Mesh-1 quickly and e ciently. Also in this particular implementation of the EDICT, we assume that the Mesh-1 does not change throughout the computation. Therefore, prior to the computation, and only once, Mesh-1 is analyzed to determine the global arrays containing the face and edge data. These arrays are later localized for each element. Face connectivity is used to identify each element, edge connectivity is used to identify each face, Fig. 2. A tetrahedral element can be represented by four nodes (left) or four faces (right).

6 408 T.E. Tezduyar, S. Aliabadi / Comput. Methods Appl. Mech. Engrg. 190 (2000) 403±410 Fig. 3. A triangular face can be represented by three edges (left), with each edge represented by two nodes (right). and node connectivity is used to identify each edge. This process is a one-time cost and is carried out prior to the computation. To form Mesh-2 and Mesh-3, rst we identify the elements of Mesh-1 which are within the enhanceddiscretization zones. For every element in these zones, we ag its edges and faces for re nement. This is done through the use of localized face and edge arrays. Then the formation of Mesh-2 and Mesh-3 takes place in three loops: over the elements, faces and edges. In these loops, depending on the re nement ag, we insert nodes in the center of the elements, faces and edges. In this approach, the cost for the formation of Mesh-2 and Mesh-3 is negligible compared to the overall computation cost. 6. Numerical examples Sloshing in a container. A container is lled with half water and half air. It is suddenly subjected to the gravitational acceleration, and with both horizontal acceleration components at 0.2g. The base mesh, Mesh-1, consists of (45,000) hexahedral elements. For this mesh, at each time step, a coupled, non-linear equation system with over 236,000 unknowns is solved to obtain the solution. The computation was carried out on the CRAY T3E for a total of 50 time steps. We call this Solution-1. Fig. 4 shows for Solution-1, at t 0:15 s and from two different views, Mesh-1 and the air±water interface. Next we compute the problem with the EDICT, with only one level of enhancement, using a Mesh-2 on top of Mesh-1. Mesh-2 has about 85,000 hexahedral elements. For the enhanced discretization, at each time step, a coupled, non-linear equation system with around 740,000 unknowns is solved to obtain the solution. This computation was also carried out on the CRAY T3E and for the same duration as before. We call this Solution-2. Fig. 5 shows for Solution-2, at t 0:15 s and from two different views, Mesh-2 (on top of Mesh-1) and the air±water interface. Sloshing in a tanker during braking and turning. Here we are simulating sloshing in a partially- lled fuel tanker during braking and turning. The tanker, moving at 30 m/s, enters a curve with radius 920 m. At the Fig. 4. Sloshing in a container. Solution-1, at t 0:15 s and from two di erent views, Mesh-1 and the air±water interface.

7 T.E. Tezduyar, S. Aliabadi / Comput. Methods Appl. Mech. Engrg. 190 (2000) 403± Fig. 5. Sloshing in a container, Solution-2, at t 0:15 s and from two di erent views, Mesh-2 (on top of Mesh-1) and the air±water interface. Fig. 6. Sloshing in a tanker during braking and turning. Solution-1, at t 0:9 s, Mesh-1 and the air±fuel interface. Fig. 7. Sloshing in a tanker during braking and turning. Solution-2, at t 0:9 s, Mesh-2 (on top of Mesh-1) and the air±fuel interface. same time the driver brakes and slows the tanker down at the rate of 0.1g. The tanker has an elliptical crosssection. The dimensions of the minor and major axes are 1.5 and 2.0 m, respectively. Also, the tanker is partitioned into three compartments, and only one compartment is simulated. The length of each compartment is 1.0 m. The base mesh, Mesh-1, consists of 104,091 nodes and 100,000 hexahedral elements. Mesh-2 contains approximately 180,000 hexahedral elements. Solution-1 is obtained by using the base discretization, where all trial and weighting functions come only from Mesh-1. Solution-2 is obtained by using the EDICT, where all the trial and weighting functions come from Mesh-1 Mesh-2. Fig. 6 shows for Solution-1, at t 0:9 s, Mesh-1 and the air±fuel interface. Fig. 7 shows for Solution-2, at t 0:9 s,

8 410 T.E. Tezduyar, S. Aliabadi / Comput. Methods Appl. Mech. Engrg. 190 (2000) 403±410 Fig. 8. Sloshing in a tanker during braking and turning. The time histories of horizontal forces (in x and z directions) exerted on the container for Solution-1 and Solution-2. Mesh-2 (on top of Mesh-1) and the air±fuel interface. The time histories of the horizontal forces (in x and z directions) exerted on the container for Solution-1 and Solution-2 are shown in Fig Concluding remarks In this paper, we have presented the 3D implementation and applications of the EDICT in computation of unsteady ows with interfaces. As the base method, we use a stabilized nite element interface-capturing technique, with the spatial discretization carried out over a typically non-moving mesh. The equations solved are the time-dependent Navier±Stokes equations of incompressible ows and a time-dependent advection equation governing the evolution of the interface function. To increase the accuracy in approximating the interfaces, we use nite element functions with multiple components at and near the interfaces, with each component coming from a di erent level of mesh re nement. We have implemented this method on the CRAY T3E. With parallel implementation, the EDICT becomes a powerful computational tool that can be very e ectively used for simulation of complex, 3D unsteady ow problems with two uidinterfaces, including free surfaces. We have demonstrated this capability by applying the EDICT to two numerical examples with unsteady free-surface ows. References [1] T.E. Tezduyar, S. Aliabadi, M. Behr, enhanced-discretization interface-capturing technique, in: Y. Matsumoto, A. Prosperetti (Eds.), Proceedings of the ISAC '97 High Performance Computing on Multiphase Flows, vol. 1±6, Japan Society of Mechanical Engineers, [2] T.E. Tezduyar, S. Aliabadi, M. Behr, Parallel nite element computing methods for unsteady ows with interfaces, in: M. Hafez, K. Oshima (Eds.), Computational Fluid Dynamics Review 1998, World Scienti c, 1998, pp.643±667. [3] T.E. Tezduyar, S. Aliabadi, M. Behr, Enhanced-discretization interface-capturing technique (EDICT) for computation of unsteady ows with interfaces, Computer Methods in Applied Mechanics and Engineering 155 (1998) 235±248. [4] S. Aliabadi, T. Tezduyar, 3D simulation of free-surface ows with parallel nite element method, Computational Mechanics '97, Proceedings of International Conference on Computational Engineering Science, San Jose, Costa Rica, [5] C.W. Hirt, B.D. Nichols, Volume of uid (VOF) method for the dynamics of free boundaries, Journal of Computational Physics 39 (1981) 201±225. [6] A.N. Brooks, T.J.R. Hughes, Streamline upwind Petrov±Galerkin formulations for convection dominated ows with particular emphasis on the incompressible Navier±Stokes equations, Computer Methods in Applied Mechanics and Engineering 32 (1982) 199±259. [7] T.E. Tezduyar, Stabilized nite element formulations for incompressible ow computations, Advances in Applied Mechanics 28 (1991) 1±44. [8] S. Aliabadi, T.E. Tezduyar, Stabilized- nite-element/interface-capturing technique for parallel computation of unsteady ows with interfaces, Computer Methods in Applied Mechanics and Engineering 190 (2000) 243±261.