Publications in scientific journals
- Lipp, M., & Helmig, R. (2020). A Locally-Refined Locally-Conservative Quadtree Finite-Volume Staggered-Grid Scheme. In G. Lamanna, S. Tonini, G. E. Cossali, & B. Weigand (Eds.), Droplet Interactions and Spray Processes (pp. 149--159). Springer International Publishing.
About this project
There is a broad variety of technical, environmental and medical systems, in which the coupling between free flow and flow through a porous medium plays a crucial role.
While the highly nonlinear multiphase processes in the porous-medium domain have less requirements on the temporal and spacial resolution, the nonlinear turbulent processes in the free-flow domain tend to have more. The complicated flow patterns that emerge e.g. in the case of interfaces with surface topology, require a high spatial resolution locally. This is the main focus of the research project. The fast turbulent processes require a high temporal resolution. Multi-rate time stepping can be realized by a domain decomposition approach as addressed in the Internal Project I-01. Within a domain decomposition framework, the Navier-Stokes equations can be solved in a decoupled way, which is another issue dealt with in this project.
We implemented a locally-refined quadtree finite-volume staggered-grid scheme for the Navier-Stokes equations monolithically coupled to the equations describing the porous-medium flow. The free flow can be a single-phase multi-component non-isothermal flow with variable density or viscosity. The porous medium may be described on the representative-elementary-volume scale or as a pore-network model.
We use a refinement ratio of 1:2 and choose to have two velocity degrees of freedom at the interface between coarse and fine grid cells. Furthermore, we use subsequent linear interpolations to complete stencils. There are two versions of our discretization: a conservative one and a non-conservative one. We can ensure local mass and momentum conservation by the following: There are interfaces at which one coarse control volume touches two fine ones. There, the momentum flux out of the coarse control volume is in general not equal to the flux into the fine ones. We can either enforce it to be by setting the coarse flux to the negative sum of the fine fluxes. This sometimes results in an unnatural mismatch between this coarse flux and the flux of the coarse control volume on the opposing face. Or we can leave the fluxes non-conservative. We found that depending on the test case, the one version is better or the other. In general, this conservation issue is closely related to the staggered grid arrangement.
Despite this challenge of conservation, we successfully applied the local refinement method for example to the test case of a supergaussian velocity distribution shown in the Figure below:
There, we get smaller pressure and velocity L2 errors with 6120 degrees of freedom distributed on a quadtree grid (as shown in the picture) than with 6165 degrees of freedom distributed on a 45x45 uniform grid.
On the topic of a decoupled solution of the Navier-Stokes equation, we successfully implemented a fully-implicit version of the well-known SIMPLE algorithm, while not doing the typical approximation of the inverse by the inverse of a diagonal matrix. Instead, we use iterative linear solvers to approximate the action of the inverse. This way, we don’t have to find underrelaxation factors for this nonlinear solver.