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Understanding The v2-f Turbulence Model


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Computational Fluid Dynamics (CFD) in the industry had gone through tremendous advancements in the past 50 years. Scientists and engineers have developed models of many levels of fidelity for flowfields. On the ladder of CFD one may find many stages. Lifting-Surface Methods that model only the camber lines of lifting surfaces, not the thickness, vortex wakes that must of course be paneled. Linear Panel Methods that solve either the incompressible potential-flow equation or one of the versions applicable to compressible flow with small disturbances. Nonlinear Potential Methods where the velocity is represented as the gradient of a potential, as it is in incompressible potential flow, nonlinearity through effectively incorporating an entropic relation for the density as a function of the local Mach number. Euler Methods, solving the Navier-Stokes equations with the viscus and heat-conduction terms omitted. Coupled Viscous/Inviscid Methods solving the boundary-layer equations in the inner near wall region and matched to an outer region inviscid flow calculations.

One huge leap forward was achieved through the ability to simulate Navier-Stokes Methods Such as Reynolds-Averged Navier-Stokes (RANS).

v2-fAnsys Fluent steady-stae turbulent models

RANS is based on the Reynolds decomposition according to which a flow variable is decomposed into mean and fluctuating quantities. When the decomposition is applied to Navier-Stokes equation an extra term known as the Reynolds Stress Tensor arises and a modelling methodology is needed to close the equations. The “closure problem” is apparent as higher and higher moments of the set of equations may be taken, more unknown terms arise and the number of equations never suffices.

Reynolds Stress
Reynolds-Stress Tensor

Levels of RANS turbulence modelling are related to the number of differential equations added to Reynolds Averaged Navier-Stokes equations in order to “close” them.

0-equation (algebraic) models are the simplest form of turbulence models, a turbulence length scale is specified in advance through experimenting. 0-equations models are very limited in applications as they fail to take into account history effects, assuming turbulence is dissipated where it’s generated, a direct consequence of their algebraic nature.
1-equation and 2-equations models, incorporate a differential transport equation for the turbulent velocity scale (or the related the turbulent kinetic energy) and in the case of 2-equation models another transport equation for the length scale, subsequently invoking the “Boussinesq Hypothesis” relating an eddy-viscosity analog to its kinetic gasses theory derived counterpart (albeit flow dependent and not a flow property) and relating it to the Reynolds stress through the mean strain.
In this sense 2-equation models can be viewed as “closed” because unlike 0-equation and 1-equation models (with exception maybe of 1-equations transport for the eddy viscosity itself) these models possess sufficient equations for constructing the eddy viscosity with no direct use for experimental results.

A drawback evident in almost all eddy-viscosity models is the inability to inherently account for rotation and curvature. This drawback is resulted from relating the Reynolds stress to the mean flow strain and in fact is the major difference between such a modeling approach and a full Reynolds-stress model (RSM). The RSM approach accounts for the important effect of the transport of the principal turbulent shear-stress. On the other hand, RSM simulations are not computationally cost-effective, in as much that one does get an improved physical fidelity that is worth the time and computational resources consumed, not only that, they often do not converge.

The Elliptic nature of pressure and near-wall effects

For incompressible NSE the pressure the pressure in a fluid is by nature elliptic. What this means is that the effect of pressure at one point will affect the entire flowfield instantaneously.
This sentence, albeit presents a simplistic view on the nature of pressure, is misleading in more than one way:

  • First, Real fluids such as gasses are actually highly compressible (regardless of the Mach number). incompressibility is somewhat of an approximation even for liquids. It is true that even for gases (at low Mach numbers), a flow can act as if it were incompressible, in that we can make very accurate predictions using equations subsequently to approximating the density as constant. Nonetheless, even for low Mach numbers, when pressure differences, and density differences are all small, the density differences are of the same order of magnitude as the pressure differences. The reason we may neglect density changes and not pressure changes is due to the density’s role in NSE (and continuity) equations. As pressure differences in NSE (appears under gradient) and small velocity differences have a huge impact on the flow, a small difference in density affects the flow much less such that even in the presence of large velocity disturbances it is justified to use the incompressibility approximation as long as the velocity is much less than the speed of sound.
  • Second, writing that the effect of pressure at one point will affect the entire flowfield instantaneously, might suggest a one-way causation, such that pressure gradient causes acceleration and by that induces velocity (Newton’s second law). Although this is not false, it’s incomplete though. NSE dependent variables such as pressure and velocity hold a reciprocal, circular relation. So as the pressure gradient causes the acceleration, the acceleration sustains the pressure gradient.


Given the simplistic view given above, the importance high fidelity modeling of near-wall effects seems quite clear. Not to be vague, I shall add that the definition of what is exactly this “near wall” is not as important but it stands for the region in proximity to a solid boundary where the assumption of eddy viscosity modeling of homogeneous turbulence to simplify the pressure-strain redistribution tensor doesn’t hold.

Now to continue with my reasoning for relating the above to curvature effects, I shall address yet another issue relating to boundary-layer pressure effects. In first-order boundary-layer theory  is customary to ignore the pressure gradient normal to the surface by assuming that the pressure gradient normal to the wall is zero. Nevertheless it is important to remember that a flat wall is a prerequisite for such an assumption but it’s extremely inaccurate if the wall has pronounced curvature.
So consistent with the local mean velocity and streamline curvature, there will always be a normal pressure gradient within the boundary layer when relating to practical engineering applications.

The concept of elliptic relaxation

In the framework of 2-equation eddy viscosity models such as the k-ε Turbulence Model it is possible to bypass modeling near wall behavior by employing the law of the wall and providing velocity “boundary conditions” away from solid boundaries (what is termed “wall-functions”). In order to integrate the equations through the viscous/laminar sublayer a “Low Reynolds” approach must be employed. This is achieved as additional highly non-linear damping functions are needed to be added to low-Reynolds formulations (low as in entering the viscous/laminar sublayer) to be able to integrate through the laminar sublayer (y+<5). This again produces numerical stiffness and in case is problematic to handle in view of linear numerical algorithms and in any case it does not break the assumption of homogeneity as the wall-normal velocity, a key contributor to mixing is severely damped in the near wall region.
As explained above, especially for pronounced curvature, pressure effects in the wall normal direction render the homogeneity assumption quite inaccurate as the near wall area not homogeneous in this sense and one shall expect the wall normal velocity gradient to be far from constant.

In order to overcome this drawback the elliptic relaxation concept was devised (P. Durbin). Following the above explanation and taking into account the mechanism by which RSM damping occurs, through inviscid blocking of the energy redistribution by the pressure fluctuations, the main idea is to construct an approximation two-point correlation (which is non-existent standard eddy viscosity formulations as they are 1-point closures)  in the integral equation of the pressure redistribution. Then, the redistribution term is defined by a relaxation equation of an elliptic nature.
As the complete formulation shall appear in the following paragraph It’s interesting to note that the elliptic nature is utilized in the k-ω turbulence model only by inspecting the ω-equation in the near wall region when combined with the specified ω values at the wall :

The implication of such behavior in the case of the k-ω turbulence model is the straightforward integration through the laminar sublayer without additional numerically destabilizing damping functions or two more transport equation (which shall generally cause stabilization issues due to reciprocity between the variables).

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.

law of the wall.pngThe 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, which means that turbulent content is such the LES mode may and shall be selected, while for 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 

The end

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