Large Eddy Simulation (LES)

Large Eddy Simulation (LES) is an inherently transient technique in which the large scales of the turbulence are directly resolved everywhere in the flow domain, and the small-scale motions are modeled.

One justification for the LES technique is that by modeling “less” of the turbulence, and explicitly solving for more of it, the error in the turbulence modeling assumptions is not as consequential. Furthermore, it is hypothesized that the smaller eddies are self-similar and thus lend themselves to simpler and more universal models. The downside of the approach is the computational expense, which, although less than direct numerical simulation, is still nonetheless excessive.

In contrast to the RANS equations, the equations that are solved for LES are obtained by a spatial filtering rather than an averaging process. Each solution variable $ϕ$ is decomposed into a filtered value $\tilde{ϕ}$ and a sub-filtered, or sub-grid, value $ϕ'$ :

Figure 1. EQUATION_DISPLAY

ϕ = \tilde{ϕ} + ϕ'

(1381)

where $ϕ$ represents velocity components, pressure, energy, or species concentration.

The spatial filtering removes the smaller eddies—associated with higher frequencies—and thereby reduces the range of scales that must be resolved. LES filtering can be either explicit or implicit. Explicit filtering applies a filter function (such as box or Gaussian) to the discretized Navier-Stokes equations. The filtering of the generic instantaneous flow variable $ϕ (t, x)$ is defined as:

Figure 2. EQUATION_DISPLAY

\tilde{ϕ} (t, x) = ∭_{- \infty}^{\infty} G (x - x', Δ) ϕ (t, x') d x'

(1382)

where $G (x, Δ)$ is the filter function characterized by a filter width $Δ = {(Δ_{x} Δ_{y} Δ_{z})}^{1 / 3}$ .

In Simcenter STAR-CCM+, implicit filtering is used. For this approach, the computational grid determines the scales of the eddies that are filtered out. Implicit filtering takes full advantage of the grid resolution and is, in general, computationally less expensive than explicit filtering.

Inserting the decomposed solution variables into the Navier-Stokes equations results in equations for the filtered quantities. The filtered mass, momentum, and energy transport equations can be written as:

Figure 3. EQUATION_DISPLAY

\frac{\partial ρ}{\partial t} + \nabla\cdot (ρ \tilde{v}) = 0

(1383)

Figure 4. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ \tilde{v}) + \nabla\cdot (ρ \tilde{v} \otimes \tilde{v}) = - \nabla\cdot \tilde{p} I + \nabla\cdot (\tilde{T} + T_{SGS}) + f_{b}

(1384)

Figure 5. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ \tilde{E}) + \nabla\cdot (ρ \tilde{E} \tilde{v}) = - \nabla\cdot \tilde{p} \tilde{v} + \nabla\cdot (\tilde{T} + T_{SGS}) \tilde{v} - \nabla\cdot \tilde{q} + f_{b} \tilde{v}

(1385)

where:

$ρ$ is the density.
$\tilde{v}$ is the filtered velocity.
$\tilde{p}$ is the filtered pressure.
$I$ is the identity tensor.
$\tilde{T}$ is the filtered stress tensor.
$f_{b}$ is the resultant of the body forces (such as gravity and centrifugal forces).
$\tilde{E}$ is the filtered total energy per unit mass.
$\tilde{q}$ is the filtered heat flux.

The filtered equations are rearranged into a form that looks identical to the unsteady RANS equations. However, the turbulent stress tensor now represents the subgrid scale stresses. These stresses result from the interaction between the larger, resolved eddies and the smaller, unresolved eddies and are modeled using the Boussinesq approximation as follows:

Figure 6. EQUATION_DISPLAY

T_{SGS} = 2 μ_{t} S - \frac{2}{3} (μ_{t} \nabla \cdot \tilde{v}) I

(1386)

where $S$ is the strain rate tensor given by Eqn. (1130) and computed from the resolved velocity field $\tilde{v}$ .

The subgrid scale turbulent viscosity $μ_{t}$ must be described by a subgrid scale model that accounts for the effects of small eddies on the resolved flow. Currently, three subgrid scale models are available in Simcenter STAR-CCM+:

Smagorinsky Subgrid Scale model
Dynamic Smagorinsky Subgrid Scale model
WALE Subgrid Scale model