Abstract
We develop a Lagrangian Mesh-based method for simulating Navier-Stokes flows. It solves Navier-Stokes equations in Lagrangian frame of reference discretized by the Finite Elements Method with the use of dynamic meshing techniques.
Since the method uses Lagrangian frame of reference, it naturally resolves all kinds of moving or deformable boundaries and contact surfaces. Therefore it can be successfully applied to solve problems that include fluid/fluid of fluid/structure interaction where deformable domains occur.
Method features FEM as underlying numerical scheme. This allows the use of primary variables (velocities and pressure) in the contrast to the other Lagrangian particle methods where streamfunction/vorticity formulation is usually used. Since convective term is absorbed into the material derivative, method yields simpler discrete equations (although still nonlinear) than in the case of Eulerian or ALE methods.
In the method, nodes of the finite element mesh correspond to the fluid particles. Since the method is Lagrangian, particles/nodes move according to the computed velocity field. New nodal configuration is obtained each time step and therefore new finite element mesh as well. In other words, mesh "flows" according to the flow. It is obvious that after very few time steps, mesh gets distorded to the point that further computation is not possible. Maintenance techniques are introduced to keep the mesh in good quality but still moving according to the flow. Techniques include parallel retriangulation, refinement and coarsening.
We have developed a highly scalable parallel code and applied it to a set of standard test problems. We note that, to our knowledge, this is the first time a parallel Lagrangian mesh-based method is developed. We also note that it is the first time to apply Lagrangian mesh-based method to the flow problems other than the free-surface problems. Results show that Lagrangian mesh based method is concurrent to other commonly used CFD techniques.
More details about the method, the parallel algorithm, test problems and results can be found in the Report(to appear soon).
Method performance on some test problems
Poiseuille flow
We first tested our algorithm applying it to the Poisseulle flow. While it is simple, it served as excellent test for learning how dynamic mesh movement algorithm behavies. Existance of the exact solution gave us the possibility to check the numerical part of the method. Some results can be viewed here(will be available soon).
Flow around cylinder
Code has been successfully applied to simulate flow around stationary circular cylinder. Reynolds number used varied from 0 (Stokes flow) to 5000. Method shows no sign of numerical instabilities even for highly nonlinear flows. We include some results on the folowing pages. We note that method showed very good results even for the extremely course mesh (2800 nodes in fully developed stage) indicating natural adaptivity of the method. Nodes/particles concentrate in the areas of high gradients so fine resolution is achieved only at the places where it is needed and not everywhere in the domain like in fixed-grid methods. This is significant save in both memory and computing time requirements.
Deformable domains
We include some preliminary results for simulations of the fluid drops in the surrounding fluid. Circular drops of the heavy and viscous fluid are placed in the initially uniform flow coming from the left. This simulation demonstrates the ability of Langrangian methods to efficiently handle problems involving deformable boundaries. Note that our strategy for parallel dynamic meshes successfully handles complex problems involving contacts between the fluid drops. Our current effort focuses on incorporating the physical laws to better model the cell behaviour (surface tension, membrame modeling for plasma-cell blood flow modeling etc.) and to increase the number of cells in our simulations.
| Animation | File |
|---|---|
| 4 heavy, viscous drops in surrounding fluid. | 4drops (avi, 6MB) |
| Conference | File |
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| Supercomputing 2000, High performance Computing and Networking | PowerPoint format (8MB) |