Large eddy simulation (LES) is classified as a space filtering method in CFD. LES directly computes the large-scale turbulent structures which are responsible for the transfer of energy and momentum in a flow while modelling the smaller scale of dissipative and more isotropic structures. In order to distinguish between the large scales and small scales, a filter function is used in LES. A filter function dictates which eddies are large by introducing a length scale, usually denoted as∆in LES, the characteristic filter cutoff width of the simulation [10]. All eddies larger than ∆are resolved directly, while those smaller than∆are approximated.
3.2.1 Filtering of Navier-Stokes equations
In LES, the flow parametersφis separated into a filtered, resolved partφ¯and a sub-filter, unresolved part,φ0,
φ= ¯φ+φ0 (25)
Define a spatial filtering operation by means of a filter functionG(X, X0,∆)as follows:
φ(X, t)¯ ≡ Z ∞
−∞
Z ∞
−∞
Z ∞
−∞
G(X, X0,∆)φ(X0, t)dx01dx02dx03 (26) Whereφ(X, t)¯ = filtered function,φ(X, t)= original (unfiltered) function and∆= filter
cutoff width
Here, the overbar indicates spatial filtering, not time-averaging. Equation (26) shows that filtering is an integration,just like time-averaging in the development of the RANS equations, only in the LES the integration is not carried out in time but in three-dimensional space [1].
As mentioned, the filter function dictates the large and small eddies in the flow. This is done by the localized function G(X, X0,∆). This function determines the size of the small scales
G(X, X0,∆) =
(1/∆3 kX−X0k ≤∆/2
0 kX−X0k>∆/2 (27)
Various filtering methods exist, the top-hat filter is common in LES. The function repre-sents Eq. (27). The top-hat filter is used in finite volume implementations of LES. The cutoff width is intended as an indicative measure of the size of eddies that are retained in the computations and the eddies that are rejected. In CFD computations with the finite volume method it is pointless to select a cutoff width that is smaller than the grid size. The most common selection is to take the cutoff width to be of the same order as the grid size. In three-dimensional computations with grid cells of different length∆x, width∆y and height∆z the cutoff width is often taken to be the cube root of the grid cell volume,
∆ = p3
∆x∆y∆z (28)
Now, the unsteady Navier-Stokes equations for a fluid with constant viscosity µ are as follows: If the flow is also incompressible we havediv(u) = 0, and hence the viscous momentum source termsSu,Sv andSw are zero.Filtering of above equations,
∂ρ
∂t + div(ρ¯u) = 0 (33)
∂(ρu)¯ This equation set should be solved to yield the filtered velocity field u,¯ ¯v and w¯ and filtered pressure field p. We need to compute convective terms of the form¯ div(ρφu) on the left hand side, but we only have available the filtered velocity field u,¯ v,¯ w¯ and pressure fieldp¯[1]. To make some progress we write,
div(ρφu) = div( ¯φ¯u) + (div(ρφu)−div( ¯φ¯u))
The first term on the right hand side can be calculated from the filtered φ¯and u¯ fields and the second term is replaced by a model. Substitution into above equation and some rearrangement yields the LES momentum equations:
∂(ρ¯u) In these equations, the first two terms on the left hand side denote the rate of change and convective fluxes of filtered x−,y−and z−momentum. And third and forth terms on the right hand side denote the gradients in thex−,y−andz−directions and diffusive fluxes of filtered x−, y−and z−momentum. The last terms are caused by the filtered operation. They can be considered as a divergence of a set of stresses τij. In suffix notation thei−component of these terms can be written as follows:
div(ρuiu−divρu¯iu) =¯ ∂(ρuiu−ρu¯iu)¯
W here τij =ρuiu−ρu¯iu¯ =ρuiuj−ρu¯iu¯j (41) The termτij is known as the subgrid scale (SGS) Reynolds Stress. Physically, the right hand side of Eq. (41) represents the large scale momentum flux due to turbulence motion.
The nature of these contributions can be determined with the aid of a decomposition of a flow variableφ(x, t) as the sum of (i) the filtered functionφ(x, t)¯ and (ii)φ0(x, t).
φ(x, t) =φ(x, t) +¯ φ0(x, t) (42) Using this decomposition in Eq. (41) we can write the SGS stresses as follows:
τij =ρuiuj−ρu¯iu¯j = (ρu¯iu¯j−ρu¯iu¯j) +ρu¯iu0j+ρu0iu¯j+ρu0iu0j (43)
thus, we find that the SGS stresses contain three groups of contributions:
• Leonard stresses Lij: Lij =ρu¯iu¯j−ρu¯iu¯j which represent the interaction between two resolved scale eddies to produce small scale turbulence.
• cross-stressesCij: Cij =ρu¯iu0j+ρu0iu¯jare the cross-stress terms that describe the interaction between resolved eddies and small-scale eddies.
• LES Reynolds stresses Rij: Rij = ρu0iu0j is the subgrid scale stress that represents the interactions between two small scale eddies
3.2.2 Smagorinksy-Lilly SGS model
To approximate the SGS Reynolds stress, a SGS model can be employed. The most commonly used SGS models in LES is the Smagorinsky-Lilly model. In a flow, it is the shear stress and the viscosity of the flow that cause the chaotic and random nature of the fluid motion. Thus, in the Smagorinsky-Lilly model, the effects of turbulence are represented by the eddy viscosity based on the well known Boussinesq hypothesis.
The Boussinesq hypothesis relates the Reynolds stress to the velocity gradients and the turbulent viscosity of the flow [8]. It is therefore assumed that the SGS Reynolds stress Rij is proportional to the modulus of the strain rate tensor of the resolved flow S¯ij = 12(∂u¯i/∂xj+∂u¯j/∂xi) where µSGS is the SGS eddy viscosity. Leonard stresses and cross-stresses are lumped together with the LES reynolds stresses in the current versions of the finite volume
method. The whole stress τij is modeled as a single entity by means of a single SGS The Smagorinksy-Lilly SGS model builds on Prandtl’s mixing length model and assumes that we can define a kinematic SGS viscosityνSGS, which can be described in terms of the one length scale and one velocity scale and is related SGS viscosity by νSGS =µSGS/ρ.
Since the size of the SGS eddies is determined by the details of the filtering function, the obvious choice for the length scale is the filter cutoff width∆. The velocity scale is expressed as the product of the length scale∆and the average strain rate of the resolved flow ∆× kSk, where¯ kSk¯ =
where CSGS is the Smagorinsky constant that changes depending on the type of flow.
For isotropic turbulent flow, theCSGS value is usually around 0.17 to 0.21. Basically, the Smagorinsky SGS model simulates the energy transfer between the large and the subgrid-scale eddies. Energy is transferred from the large to the small subgrid-scales but backscatter sometimes occurs where flow becomes highly anisotropic, usually near to the wall [4]. To account for backscattering, the length scale of the flow can be modified using Van Driest damping, and suggested that CSGS = 0.1 is most appropriate for this type of internal flow calculation.
The Smagorinksy model has been successfully applied to various flows as it is relatively stable and demands less computational resources among the SGS models. But some disadvantages of the model have been reported,
• Too dissipative in laminar regions.
• Requires special near wall treatment and laminar turbulent transition.
• CSGS is not uniquely defined.
• Backscatter of flow is not properly modelled
Germano and Lilly conceived a procedure in which the Smagorinsky model constant, CSGS, is dynamically computed based on the information provided by the resolved scales of motion. Hence, the dynamic SGS model has been introduced. This model employs a
similar concept as the Smagorinsky model, with the Smagorinsky constantCSGSreplaced by the dynamic parameter Cdym [8]. The parameter Cdym is computed locally as a function of time and space, which automatically eliminates the problem of using constant CSGS. In the dynamic SGS model, another filter is introduced which takes into account of the energy transfer in the dissipation range. Performing the double filtering allows the subgrid coefficient to be calculated locally based on the energy drain in the smallest scales. Generally, the dynamic model predicted better agreement with experimental work in region of transition flow and the near wall region.
Some advantages of the dynamic model over the Smagorinsky-Lilly models are,
• Dynamic SGS automatically uses a smaller model parameter in isotropic flows.
• Near the wall, the model parameters need to be reduced; the dynamic SGS model adapts these parameters accordingly.