Avaraging Procedure

For an arbitrary quantity  the decomposition into a mean and fluctuating part can be written as

The term on the left-hand side is the instantaneous value, the first term on the right hand side is the mean value and the second term on the right-hand side is the fluctuating value. The mean value can be obtained using an ensemble averaging procedure

where N is the number of ensembles and W denotes a weighting function. For steady-state flows, the mean value can also be obtained using a time averaging procedure

Setting the weighting factor in Eqs. (2) and (3) averaging procedure, used mainly for incompressible flows. In this case Eq. (1) is written as to unity, results in the conventional Reynolds

where the bar denotes the Reynolds average. If the averaging is applied to the continuity equation for an incompressible flow field it becomes

For the same flow field, the momentum equations are

An additional term is introduced into the momentum equations, the last term in Eq. (6), and it is called the Reynolds stresses. This term cannot be expressed in terms of the averaged quantities and has to be modelled.

If the density is varying and the decompositions is applied, the continuity equation becomes (7) Note that an additional term appears, the last term in the continuity equation.

Density-weighted Averaging or Favre Averaging To avoid this additional term, a density weighted averaging procedure, called Favre averaging, is used by setting the weighting factor in Eqs. (2) and (3) to

With this decomposition into a mean and fluctuating part, Eq. (1) can be written as: where the tilde denotes the Favre average.

The Favre average is applied to the continuity equation and it becomes

By applying the density weighted average, the problem with an additional term, as in Eq. (7) is avoided. This is the reason for using Favre averaging on density varying flows.

The Favre averaged momentum equations are where the Reynolds stresses are still present and have to be modelled.

But applying the Favre averaging procedure has not increased the number of the terms that have to be modelled. In a density varying flow field, such as a reacting flow field, all the governing equations are averaged using the Favre averaging procedure.

Variable Density Turbulent Flows

many problems from industry or aeronautics involve effects of density variations. These variations in density may have various origins: thermal dilatation, compression, supersonic regime, mixing of fluids, chemical reactions, etc.

Some variable density flows can also be produced by the mixing of miscible fluids but with differing densities. Chemical reaction, like combustion, induces complex phenomena coupled with turbulence. Most of the modeling methods used in practice still rely on a direct extension of existing methods for incompressible flows. The effects of compressibility are however complex and may require us to reconsider most of closure hypotheses.

Favre Averaging

Transport equations

The Turbulent Field Equations

Kinetic Energy Equation

This equation presents a new term for dilatation production and also an additional pressure term due to the fact that the Favre fluctuation is not zero in the mean. We shall also note the occurrence of a pressure-dilatation correlation term

Reynolds stress transport modeling in the framework of mass weighted averaging

Dissipation rate equation

Turbulent heat flux equations

DNS-LES-DES

The detached eddy simulation (DES) model is based on a modified version of the Spalart-Allmaras model and can be considered a more practical alternative to LES for predicting the flow around high- Reynolds-number, high-lift airfoils.

The DES approach combines an unsteady RANS version of the Spalart-Allmaras model with a filtered version of the same model to create two separate regions inside the flow domain: one that is LES-based and another that is close to the wall where the modeling is dominated by the RANS-based approach.

The LES region is normally associated with the high-Re core turbulent region where large turbulence scales play a dominant role. In this region, the DES model recovers the pure LES model based on a one-equation sub-grid model. Close to the wall, where viscous effects prevail, the standard RANS model is recovered.

The application of DES, however, may still require significant CPU resources and therefore, as a general guideline, it is recommended that the conventional turbulence models employing the Reynolds-averaged approach be used for practical calculations.

The DES model belongs to the class of models usually referred to as an LES/RANS coupling modeling approach. The main idea of this approach is to combine RANS modeling with LES for applications in which classical LES is not affordable (e.g., high-Re external aerodynamics simulations).

The DES model is based on the one-equation Spalart-Allmaras model. The standard Spalart-Allmaras model uses the distance to the closest wall as the definition for the length scale d, which plays a major role in determining the level of production and destruction of turbulent viscosity.

The DES model replaces d everywhere with a new length scale ~ d, defined as