Theory for the Large Eddy Simulation Interfaces
Large Eddy Simulations offer an alternative approach to turbulent flow simulations as compared to the RANS approach (see Theory for the Turbulent Flow Interfaces). In LES, the larger three-dimensional, unsteady eddies are resolved, whereas the effect of the smaller eddies is modeled. This requires the simulations to be three-dimensional and time dependent. The current implementation of the LES interfaces is suitable for incompressible flow only. The velocity and pressure fields are divided into resolved and unresolved scales. Denoting the fields containing all scales with capital letters, the decomposition can be expressed as
(3-155)
Inserting Equation 3-155 into the incompressible form of the Navier–Stokes equation and the continuity equation yields
(3-156)
Next, Equation 3-156 is projected onto the finite element sub-spaces of the resolved velocity and pressure scales. Denoting the test functions for these scales by v and q, respectively, the projection can be expressed as
(3-157)
In Equation 3-157, f is the applied traction force on the boundary ∂Ω of the spatial domain Ω, and the stresses in the last term on the right-hand side are the resolved nonlinear advection term, the two cross-stresses, and the Reynolds stress. By assuming that the unresolved scales are orthogonal to the inner-product space of the resolved scales, the unresolved time derivative and viscous terms vanish. Another assumption made in the derivation of Equation 3-157 is that u' vanishes on ∂Ω (see Ref. 1 for further details).
In the Residual Based Variational Multiscale (RBVM) method, the unresolved velocity and pressure scales are modeled in terms of the equation residuals for the resolved scales
(3-158)
where the momentum and continuity equation residuals are given by
(3-159)
and the intrinsic time-scales are given by
(3-160)
Here, C1 is a constant depending on the temporal scheme, C2 a constant depending on the shape of the element, and C3 a constant depending on both the order of the shape functions and the shape of the element. G is the covariant metric tensor. When the Use dynamic subgrid time scale option is selected, the first term under the square-root in Equation 3-160 is replaced by an estimate of the time-derivative based on the resolved scales.
In certain cases, the residual-based Reynolds-stress contribution has been found to be too small (see Ref. 2). For this reason, the Residual Based Variational Multiscale with Viscosity (RBVMWV) method adds a residual based viscosity term to the right-hand side of Equation 3-157,
(3-161)
where is a model constant, and h is a measure of the element size.
For the Smagorinsky model, the Reynolds stress term is replaced by,
(3-162)
where,
(3-163)
and is the projection of S onto the space of constant shape functions. This corresponds to a further decomposition of the resolved scales into large resolved scales and small resolved scales, in which only the latter are affected by the Reynolds stress (see Ref. 3). Since the term, Equation 3-162, identically vanishes for linear shape functions, the Dicretization must be set to at least P2-P2 in order to include the Reynolds stress contribution in the Smagorinsky model (see Discretization).
Note that all three models require equal-order interpolation for velocity and pressure.
Furthermore, adequate resolution of wall-layers, , and convective time scales, , is essential in order to obtain accurate results in LES. Here, uτ is the friction velocity, hw the thickness of the first mesh-cell next to the wall, and hU the mesh size in the streamline direction.
Wall Boundary condition
Low Reynolds Number Wall Treatment
When Wall treatment is set to Low Re, a Dirichlet condition is imposed on the velocity field at the walls. Adequate resolution of the wall layers requires that , where hw is the thickness of the mesh cells next to the wall and is the friction velocity based on the tangential stress τw at the wall.
Automatic Wall Treatment
When Wall treatment is set to Automatic, a Dirichlet condition is imposed on the wall-normal velocity component and a traction force is applied in the tangential direction opposite to the local velocity vector. To evaluate the magnitude, , of the traction force, the Reynolds number based on the magnitude of the tangential velocity and the normal distance, y, to the wall,
is evaluated halfway between the wall and the first vertexes inside the domain, at y = δw. Asymptotic solutions to the L-VEL equation (Equation 3-75) can be found for low and high values of the Reynolds number. For low values of the Reynolds number (inside the viscous wall layer),
and for large values of the Reynolds number (inside the logarithmic layer),
The two expressions are blended according to
and the friction velocity is finally obtained from
The Automatic option should be used with caution since boundary layers, especially on smooth surfaces, in many cases need to be resolved down to the top of the viscous wall layer (). When the point of boundary-layer separation is known, such as for the flow around bodies with sharp edges, the Automatic option may be used to reduce the number of DOFs in the model.
Temporal resolution
Adequate resolution of the convective time scale requires that
where hU is the mesh size in the streamline direction. Larger values of Δt may lead to damping of turbulence and in some cases even convergence issues. The built-in variable spf.dt_CFL may be used to limit the maximum time step.