Turbulence parameterisation

This chapter describes the turbulence models that are available in Firedrake-Fluids.

Large Eddy Simulation (LES)

The UFL implementation of all LES models can be found in the file les.py.

Smagorinsky model

The model calculates an eddy viscosity \(\nu^\prime\)

\[\nu^\prime = \left(C_s\Delta_e\right)^2|\mathbb{S}|,\]

where \(C_s\) is the Smagorinsky coefficient which is typically between 0.1 and 0.2 , and \(\Delta_e\) is some measure of the element size. Here it is given by the square root of the element’s area in 2D, or cube root of the element’s volume in 3D. The term \(|\mathbb{S}|\) is the modulus of the strain rate tensor \(\mathbb{S}\):

\[\mathbb{S} = \frac{1}{2}\left(\nabla\mathbf{u} + \nabla\mathbf{u}^{\mathrm{T}}\right).\]

The eddy viscosity \(\nu^\prime\) is added to the background/physical viscosity of the fluid, thereby contributing to the stress term in the momentum equation.