Turbulence simulation

Modeling turbulence in the interior of the flow

If the Reynolds number is high the flow may become turbulent, which is characterized by the development of vortices in 3D on a range of scales. If we do not have a sufficiently fine mesh which can resolve all the turbulent scales, or if we use a 2D model, we will not be able to represent the turbulent flow explicitly.

Instead we can seek to modell the flow on a larger scale, typically defined by the mesh size. This is referred to as turbulence modelling. Turbulence modelling may either be viewed as a statistical average of the flow field (RANS - Reynolds averaged Navier-Stokes equations Links to an external site.) or a filtered flow field (LES - Large Eddy Simulation Links to an external site.). When a statistical average (RANS) or spatial filter (LES) is applied to the Navier-Stokes equations we get new equations for the averaged variables (RANS) or filtered variables (LES), which include a turbulence model that model the effect of the turbulent scales that are unresolved. 

The unresolved turbulent scales mainly consist of small vortices that on the large (resolved) scales are manifested as an increased viscosity (dissipation). You can compare with the effect of a Gaussian blur filter on an image with a lot of small scaled details. This increased viscosity is referred to as a turbulent viscosity, or eddy viscosity. In contrast to the regular (physical) viscosity which is a property of the fluid (air, water, blood,...), the eddy viscosity is a property of the flow (the degree to which it is turbulent).

In the context of finite element methods an approach to turbulence simulation has been developed based on regularisation of the flow through stabilisation of the residual of the Navier-Stokes equations Links to an external site.. With this approach, the resolved scale of the flow corresponds to the mesh, and the turbulence model corresponds to the residual based stabilisation method (e.g. GLS, SUPG or SD). That is, it is a form of implicit turbulence model, based solely on the Navier-Stokes equations, without any explicit turbulence model. 

Modeling turbulent boundary layers

In simulations of laminar flow, the natural boundary condition at a wall boundary is a no slip boundary condition, where all components of the velocity are identical to the velocity of the wall (e.g. zero). 

For a turbulent boundary layer, vortices develop near the wall and in a turbulence simulation using RANS or LES, the flow velocity is nonzero at the boundary (which represents that the average flow at a distance from the boundary is nonzero). This can be modelled either by a Dirichlet Boundary condition (assigning a tangential wall velocity) or with a Neumann boundary condition (assigning a wall shear stress). In both cases we prescribe a zero normal velocity, to avoid flow through the boundary. A zero normal velocity can be enforced as either strongly (as a Dirichlet Boundary condition on the normal component of the velocity) or weakly (as a penalty type term in the variational formulation). 

One way to implement a wall shear stress model is by using a skin friction boundary condition, where the wall shear stress is assumed to be proportional to the tangential velocity (strictly speaking, a Robin boundary condition).  

A penalty formulation of a skin friction boundary condition is demonstrated in the following template file. Links to an external site.