Variational Multiscale Immersed Boundary Method for Laminar and Turbulent flows*

The ability to solve laminar and turbulent flows around complex geometric shapes without the need of boundary fitted meshes is of great interest in engineering analysis and design. For shape design optimization where the revised design in the form of STL files can be directly embedded in the flow domain without complex remeshing of the updated geometry can substantially reduce the time and efforts in the preprocessing phase in CFD.

This talk presents an immersed boundary method for weak enforcement of Dirichlet boundary conditions on surfaces that are immersed in the stationary background discretizations. An interface stabilized form is developed by applying the Variational Multiscale Discontinuous Galerkin (VMDG) method at the immersed boundaries. The formulation is augmented with a variationally derived ghost-penalty type term. The weak form of the momentum balance equations is embedded with a residual-based turbulence model for incompressible turbulent flows. A significant contribution in this work is the variationally derived analytical expression of the Lagrange multiplier for weakly enforcing the Dirichlet boundary conditions at the immersed boundary. In addition, the analytical expression for the interfacial stabilization tensor emerges that accounts for the geometric aspects of the cut elements produced when the immersed surface geometry traverses the underlying mesh. A unique attribute of the fine-scale variational equation is that it also yields a posteriori error estimator that can evaluate the local error in weak enforcement of the essential boundary conditions at the immersed boundaries. The method is shown to work with meshes comprised of hexahedral and tetrahedral elements. Numerical test cases with increasing levels of complexity are presented to validate the method on benchmark problems, and turbulent features of flows are analyzed.