# Understanding the v2f Model – Part I

Computational Fluid Dynamics (CFD) in the industry had gone through tremendous advancements in the past 50 years and cientists and engineers have developed models of increasing fidelity.
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).

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 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.

RANS differential equation closure models do however contain many assumptions along the way for achieving the final form of the transport equations and as such are calibrated to work well only according to well-known features of the applications they are designed to solve. Nonetheless although their inherent limitations. Nonetheless the modeling methodology strength has proven itself for wall bounded attached flows at high Reynolds number (thin boundary layers) due to calibration according to the “law-of-the-wall”.

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).