Variational multiscale method stabilization parameter calculated from the strain-rate tensor

The stabilization parameters of the methods like the Streamline-Upwind/Petrov-Galerkin, Pressure-Stabilizing/Petrov-Galerkin, and the Variational Multiscale method typically involve two local length scales. They are the advection and diffusion length scales, appearing in the expressions for the advective and diffusive limits of the stabilization parameter. The advection length scale has always been in the flow direction. The diffusion length scales in use have mostly been just element-geometry-dependent, but there is good justification for also making that direction-dependent, so that the spatial variation of the solution is taken into account somehow. The length scale in the solution-gradient direction, which was introduced in 2001, was intended for making sure that near solid surfaces, the element length in the surface-normal direction is selected even if that is not the minimum element length. It was also intended for making sure that in a 2D computation with a 3D mesh, there would be no dependence on the element length in the third direction. With the same objectives, and with better invariance properties, we are now introducing the direction-dependent diffusion length scale calculated from the strain-rate tensor. We accomplish those objectives, get invariance with respect to switching to a different inertial reference frame, and the element length in the surface-normal direction, even when the surface is undergoing rotation, is selected as the diffusion length scale.

Automated summary

The stabilization parameters of certain methods involve two local length scales - advection and diffusion length scales. The advection length scale is always in the flow direction, while the diffusion length scale is typically dependent on the element geometry. However, there is a justification for making the diffusion length scale also direction-dependent to account for spatial variation of the solution. To achieve this, a direction-dependent diffusion length scale calculated from the strain-rate tensor is introduced.