#### The v2-f Turbulence Model

In the v2−f model, the variable v^2, and its source term f , as variables in addition to the k and ε (turbulence kinetic energy and turbulence dissipation) parameters of the k−ε eddy-viscosity turbulence model.

The model hence solves for three transport equations for the turbulence kinetic energy, turbulence dissipation and the normal velocity squared, while a fourth elliptic relaxation equation is solved for the source term. The reason for choosing v^2, as explained above is its similarity to the second moment closure of the wall normal Reynolds stress in the near wall region.

The derivation of the elliptic relaxation equation is quite complex originating from the Pressure-Poisson equation with the *rapid* and *slow parts *of the pressure laplacian and involves Green’s function as solution for a modified Helmholtz equation – there is no way I blog such an exhausting derivation… 😉

The model formulation becomes:

With pressure-strain term defined:

and the relaxation equation is solved for the source term f of the normal velocity:

where the turbulent time and length scales are determined as:

Subsequently to performing the surgical identification of the different terms in the transport and elliptic relaxation equations, it should be remembered that we are still left out with some added constants to be calibrated.

In turbulence modeling calibration of the model is at least as important as the derivation of the model itself. Calibration is achieved with the help of experimental and numerical results of the type of ﬂow that should be modeled. The calibration process is also the first step in which the range of validity of the model would be revealed to close inspection and not just postulated from physical reasoning.

For the v2-f turbulence model the calibrated closure constants are:

However, as much the physical reasoning behind the model is sound, the original formulation, is found to be very sensitive to boundary conditions at the wall, a fact which hampers its computational use severely.

The ζ – f turbulence model

To alleviate the stiffness of the v2-f formulation D.R. Laurence et al. devised a model of which stiffness at the wall is much less severe. The formulation is achieved by transformation of the v2 equation to a ζ=v^2/k and another in the elliptic operator for the source function f.

The transformation renders the ζ as not directly dependent on the turbulence dissipation ε and the complete formulation takes the form:

now the boundary conditions for the source function and for ζ go to zero at the wall which makes it possible to solve the system uncoupled. Actually, the stiffness of the original formulation could be avoided if the equations for v2 and f where to be sovled simultaneously, but most codes (commercial or in-house) use segregated solvers.

The v2-f model is still inferior to RSM for highly 3D, swirling flows with strong secondary circulation as it holds only one attractive feature of RSM (e.g. energy blocking) but recent advancements as the incorporation of the model in Wall-Modeled Large-Eddy Simulation (WMLES) as a hybrid RANS-LES approach may be quite an interesting utilization of the v2-f model.

#### Incorporation of v2-f turbulence model in hybrid RANS-LES simulations

Today’s industry need for rapid answers dictates CFD simulations to be mainly conducted by RANS simulations whose strength has proven itself for wall bounded attached flows due to calibration according to the law-of-the-wall. However, for free shear flows, especially those featuring a high level of unsteadiness and massive separation RANS has shown poor performance following its inherent limitations.

*The turbulent boundary-layer and the “law of the wall”*

In Large-Eddy Simultion (LES) the large energetic scales are resolved while the effect of the small unresolved scales is modeled using a subgrid-scale (SGS) model and tuned for the generally universal character of these scales. LES has severe limitations in the near wall regions, as the computational effort required to reliably model the innermost portion of the boundary layer (sometimes constituting more than 90% of the mesh) where turbulence length scale becomes very small is far from the resources available to the industry. Anecdotally, best estimates speculate that a full LES simulation for a complete airborne vehicle at a reasonably high Reynolds number will not be possible until approximately 2050…

*LES simulation of isotropic turbulence*

On the other hand, for free shear flows of which the large eddies are at the order of magnitude as the shear layer, LES may provide extremely reliable information as it’s much easier to resolve the large turbulence eddies in a fair computational effort.

As such, researchers have shifted much of the attention and effort to hybrid formulations incorporating RANS and LES in certain ways. In most hybrid RANS-LES methods RANS is applied for a portion of the boundary layer and large eddies are resolved away from these regions by an LES.

One of the most popular hybrid RANS-LES models is Detached Eddy Simulation (DES) devised originally by Philippe Spalart. The term DES is based on the Idea of covering the boundary layer by RANS model and switching the model to LES mode in detached regions thereby cutting the computational cost significantly yet still offering some of the advantages of an LES method in separated regions.

Although the Spalart-Allmaras (SA) Turbulence model has been widely used for DES its near-wall damping, a result of direct construction of the eddy-viscosity transport equation, does not distinguish between velocity components. As explained in the above paragraphs the v2-f formulation models the suppression of wall normal velocity ﬂuctuation caused by non-local pressure-strain eﬀects. This anisotropy has been shown to improve prediction of separation and reattachment.

Such a hybrid RANS-LES methodology, devised by K. Sharif (NASA). As CDES∆ k^3/2/ε the model switches to RANS v2-f turbulence model.

In LES mode 1-transport equation for the turbulence kinetic energy suffices (since the length scale is grid-dependent), but three equations (v2-f and ε) besides the turbulence kintic energy are required for the RANS mode. In order to achieve that, the v2-f model is reduced to 1-eq SGS in the LES mode. This is done by modification of the coefficient in the elliptic relaxation equation for the source term f, such that v^2 is equal to 2/3k in the LES mode if isotropic turbulence is assumed.

*DES hybrid RANS-LES formulation based on v2-f turbulence model *