SUPG and VMS Stabilization

Stabilized Equations

Recalling the Galerkin Fromulation of the Navier-Stokes Equations:

$B^G = \int_\Omega \left ( R_m \cdot v \right )\;d\Omega + \int_\Omega \left ( R_c \cdot q \right )\;d\Omega$

Using same-order interpolation elements for velocity and pressure (e.g. Q1/Q1)

The SUPG (Streamline Upwind Petrov Galerkin) stabilzation term are:

$B^{SUPG} = \int_\Omega\bigg (\tau_m(u \cdot\nabla v +\nabla q)\cdot R_m \bigg )d\Omega+ \int_\Omega \tau_c (\nabla \cdot v)R_c \;d\Omega$

The extra terms added by the VMS:

$B^{VMS1} = \int_\Omega (u\cdot\nabla v')\cdot(\tau_m R_m) \;d\Omega$

$B^{VMS2} = -\int_\Omega \bigg (\nabla v\cdot(\tau_m R_m \otimes \tau_m R_m) \bigg )\;d\Omega$

Stabilization Parameters

The stabilization parameters \tau_m and \tau_c can be computed accordingly to a ScalarFormulation() or a TensorFormulation() and it can be specified in the code.

ScalarFormulation

Typically used with the SUPG.

$\tau_m = \bigg(\dfrac{2|u|}{h_e} +\dfrac{4\nu}{h_e^2} +\dfrac{2}{dt} \bigg)^{-1}$

$\tau_c = (u\cdot u)\tau_m$

Where h is the element dimension. It is the square root of the area for 2D case, and the cubic root of the volume for 3D case.

TensorFormulation

Typically used with the VMS.

$\tau_m =\bigg( \dfrac{4}{\Delta t^2} + u\cdot GGu + C_I \nu^2 G:G \bigg)^{-1/2}$

$\tau_c = (\tau_c g\cdot g)^{-1}$

Where G is the inverse of the gradient of the map cell.