Residual-Based Large Eddy Simulation with Isogeometric Divergence-Conforming Discretizations

Isogeometric divergence-conforming discretizations have recently arisen as an attractive candidate for approximation of the incompressible Navier-Stokes problem. By construction, isogeometric divergence-conforming discretizations yield discrete velocity fields which are pointwise divergence-free, and as a consequence, they admit discrete balance laws for mass, momentum, angular momentum, energy, vorticity, enstrophy, and helicity. It has been demonstrated in previous work that isogeometric divergence-conforming discretizations are simultaneously more accurate and more stable than classical mixed methods when applied to the direct numerical simulation of incompressible fluid flow. In this chapter, we present two new residual-based large eddy simulation methodologies specifically designed for isogeometric divergence-conforming discretizations. The first methodology arises from a structure-preserving variational multiscale analysis of the incompressible Navier-Stokes equations. The second methodology combines ideas from variational multiscale analysis and large eddy simulation methodologies employing an eddy viscosity, yielding a residual-based eddy viscosity method. We develop quasi-static and dynamic models for both methodologies. Numerical results illustrate the new methodologies yield improved results as compared with standard eddy viscosity based approaches when applied to a transitional flow problem.