CFD-FVM-05: Large Eddy Simulation and Direct Numerical Simulation

Large eddy simulation

The larger eddies need to be computed for each problem with a time-dependent simulation. The universal behaviour of the smaller eddies, on the other hand, should hopefully be easier to capture with a compact model. This is the essence of the large eddy simulation (LES).

Spatial filtering of unsteady Navier - Stokes equations

Filtering functions

May be expressed as

where is a filtering function. is the filtered function, is the original (unfiltered) function, and is the filter cutoff length.

Note: The overbar indicates spatial filtering, not time-averaging. The integration is not carried out in time but in three-dimensional space. The filtering is a linear operation.

The commonest forms of the filtering function in three-dimensional LES computations are

• Top-hat or box filter
  

• Gaussian filter
  
  typical value for parameter is 6

• Spectral filter
  

The top-hat filter is used in finite volume implementations of LES. The Gaussian and spectral cutoff filters are preferred in the research literature. The spectral method cannot be used in general-purpose CFD

For FVM, the cutoff width is often taken to be the cube root of the grid cell volume:

Filtered unsteady Navier - Stokes equations

The unsteady Navier–Stokes equations for a fluid with constant viscosity are as follows:

To solve , we write

The first term on the right hand side can be calculated from the filtered – and – fields and the second term is replaced by a model. Then we have LES momentum equations:

The last term can be considered as a divergence of a set of stresses :

where sub-grid-scale stresses .

A decomposition of a flow variable for LES is

where is the filtered field and is the unresolved part.

Based upon, the first term of the rhs of the can be written as

Then

Thus, we find that the SGS stresses contain three groups of contributions:

• Leonard stresses :
• cross-stresses :
• LES Reynolds stresses :

The Leonard stresses are solely due to effects at resolved scale. The cross-stresses are due to interactions between the SGS eddies and the resolved flow. Finally, the LES Reynolds stresses are caused by convective momentum transfer due to interactions of SGS eddies and are modelled with a so-called SGS turbulence model.

Smagorinksy - Lilly SGS model

In Smagorinsky’s SGS model the local SGS stresses are taken to be proportional to the local rate of strain of the resolved flow :

Meinke and Krause (in Peyret and Krause, 2000) showed that the whole stress is modelled as a single entity by means of a single SGS turbulence model:

Using the cutoff width as the length scale and the as the velocity scale, the SGS viscosity is evaluated as follows:

where is a constant. Successful LES turbulence modelling might require case-by-case adjustment of or a more sophisticated approach.

Higher-order SGS models

An alternative strategy to case-by-case tuning of the constant is to use the ideas of RANS turbulence modelling to make an allowance for transport effects. Replacing the velocity scale by the square root of the SGS turbulent kinetic energy , we have

To account for the effects of convection, diffusion, production and destruction on the SGS velocity scale we solve a transport equation to determine the distribution of :

where and is a constant. This is the LES equivalent of a one-equation RANS turbulence model.

Advanced SGS models

The Smagorinsky model is purely dissipative: the direction of energy flow is exclusively from eddies at the resolved scales towards the sub-grid scales. Leslie and Quarini (1979) have shown that the gross energy flow in this direction is actually larger and offset by 30% backscatter – energy transfer in reverse direction from SGS eddies to larger, resolved scales.

Based on the application of two filtering operations, Bardina et al. (1980) proposed

where is an adjustable constant. Above model improves the correlation between the LES and DNS results, but may generate negative viscosities. To overcome this problem, a damping term is added:

The dynamic SGS model (Germano et al., 1991) defines the difference of the SGS stresses for
two different filtering operations with two cutoff widths and :

The SGS stresses are modelled using Smagorinsky’s model assuming that the constant is the same for both filtering operations. It can be shown that this yields:

with and

The angular brackets indicate an averaging procedure.

Initial and boundary conditions for LES

Initial conditions

The initial conditions for LES are usually obtained from DNS.

Solid walls

Fine grids with near-wall grid points, graded non-uniform grids, wall functions.

Inflow boundaries

Inflow boundary conditions are very challenging because they may be the convected downstream.

Outflow boundaries

The familiar zero gradient boundary condition is used for the mean flow, and the fluctuating properties are extrapolated by means of a so-called convective boundary condition:

Periodic boundary conditions

The distance between the two periodic boundaries must be such that two-point correlations are zero for all points on a pair of periodic boundaries. This means that the distance should be chosen to be at least twice the size of the largest eddies so that the effect of one boundary on the other is minimal

LES applications in flows with complex geometry

For non-uniform grids, should be corrected to take into account grid anisotropy in the three dimensions:

where the grid-anisotropy factors are given by and .

General comments on performance of LES

• Post-processing of LES results yields information relating to the mean flow and statistics of the resolved fluctuations.
• The ability to obtain fluctuating pressure fields from LES output has also led to aeroacoustic applications for the prediction of noise from jets and other high-speed flows.
• Flow instabilities have serious consequences for combustion, and the information generated by LES calculations is uniquely applicable to the development of this technology.

Direct Numerical Simulation

The potential benefits of DNSs:

• Precise details of turbulence parameters, their transport and budgets at any point in the flow can be calculated with DNS.
• Instantaneous results can be generated that are not measurable with instrumentation, and instantaneous turbulence structures can be visualised and probed
• Advanced experimental techniques can be tested and evaluated in DNS
  flow fields.
• Fundamental turbulence research on virtual flow fields that cannot occur in reality

The issues being tackled in the DNS research literature:

• Spatial discretisation
  • spectral element methods
  • Higher-order finite difference methods
• Spatial resoluti
• Temporal discretisation
• Temporal resolution
• Initial and boundary conditions

Summary

(worth reading)