The above quote, at approximately the same time as Osborne Reynolds was proposing a random description of turbulent flow, was probably the initialization of a deterministic approach to turbulence, after Poincare’s findings that relatively simple nonlinear dynamical systems may exhibit chaotic random-like behavior that was, in fact, completely deterministic.

Despite the fact that following studies of such systems throughout the first half of the 20th century, it was not until 1963 that American meteorologist Edward Norton Lorenz would propose possible links between “deterministic chaos” and turbulence.

The above quote then present us with the interesting question… Does it really matter if we prefer the statistical view to the nature of a set of deterministic equations or a deterministic view on equations of which only a random in nature solution could be achieved?…

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 flow fields. 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)*.

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.

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.

#### Reynolds Sress Models (RSM)

It was shown how curvature and rotation effects could be somewhat overcome in the framework of “first moment models” such as the v2-f turbulence model by invoking the concept of elliptic relaxation, nevertheless 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).

As we recall, the Reynolds stresses arising in the RANS formulation are modeled entirely in terms of the Boussinesq hypothesis in “first moment models” such as k–ε, k-ω, SA and essentially all those which indirectly invoke a model for the Reynolds stresses.

The earliest steps to use of second-moment closure (also known as Reynolds stress models, RSMs) could be traced back to Rotta’s work at the early 50’s although most of the advancement in the field took hold in mid seventies by Launder et al. and came to light shortly after introduction of the modern two equation eddy-viscosity models.

Such methods provide partial differential equations for each component of the Reynolds stress tensor rather than depending on the Boussinesq hypothesis.

In what follows I shall describe the derivation of RSM in a quite easy to follow manner, beginning with RANS equations that without invoking the Boussinesq hypothesis and the concept of eddy viscosity may be written as:

Supplemented by the continuity equation which for incompressible flow amounts to divergence free average velocity field condition.

In what follows I shall define the Navier-Stokes operator for the i’th component of the velocity:

Remembering that I wish to isolate the Reynolds stresses, the next step in the derivation is to form the following operation:

This operation shall allow me to collect terms which shall finally construct a Reynolds stress transport equation. So I shall first consider the time derivative term and cast it in a Reynolds stress form:

Where I have notated the Reynolds stresses as:

From the advective terms of the NS operators I get the following expression to cast in a Reynolds stress form:

The triple correlation term was simplified using the divergence free condition for fluctuating quantities.

Treating the diffusive terms from the NS operator invoked operation above and casting it in a Reynolds stress form it becomes:

I shall complete the process with the treatment of the terms referring to the pressure gradient in the NS operator to be casted in a form suitable for my future Reynolds stress transport equation:

Now finally I may collect each of terms treated above to for a Reynolds stress transport equation:

Examining the first line it looks as if we have achieved a tensor form transport equation by which we may directly solve for the Reynolds stresses. A a system of six independent equations coupled with the RANS equation.

But this is only the first line. In the second line though, all the terms must be modeled in order to close the RSM formulation.

The first, a triple correlation (part of the advective term above) that after expanding and accounting for symmetries shall add another 10 unknowns.

The second, part of the diffusive terms (I have defined as the dissipation rate tensor in the k–ε derivation) is expanded to produce 6 more unknowns.

The third, is the pressure-gradient term of the derivation above which amounts to a fluctuating velocity-pressure correlation and 6 further unknowns for a total of 22(!) unknowns as opposed to the 6 additional encountered in standard two equations turbulence models.

#### Conclusions the RSM derivation

First of all, the above tensor form equation brings into light the “closure problem” as it is seen that as higher and higher moments of the set of RANS equations may be taken, more unknown terms arise and the number of equations never suffices.

Not only that, many of the unknowns are tightly coupled (a phenomena we encountered somewhat lightly with the dissipation equation of the k–ε model) which results in numerical stiffness (i.e. the equation includes some interactions that can lead to rapid variation in the solution susceptible for divergence).

Finally, despite the fact that second moment closures are claimed to “contain more physics” than their Boussinesq hypothesis first moment closures counterparts, the above derivation casts quite a large shadow over that claim since even if it seemed true when the derivation started, all the the advantageous added physics is somewhat hampered by the amount of ad-hoc modeled terms needed to actually close the equations. Since the physics of the model does not really shine from these statistical terms it is not obvious that the models actually contain as much physics as we would like when taking upon ourselves the penalty in computational resources, well, unless some excruciating physical phenomena in our exploration demands such a sacrifice.

Self paced recording:

https://drive.google.com/file/d/1RsC_jC1BOU-AziqRq7-lbKQ-cRLklHCB/view?usp=drivesdk

Understanding Reynolds Stress Models (RSM) for Turbulence Modeling – Part II shall address a reduced form of RSM along with its valuable features – the Algebraic Reynolds Stress Model (ASM) including some special emphasis on the attractive explicit form (i.e. EASM).

A review of ANSYS Fluent different approaches to turbulence modeling could be found in the presentation below.

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Great article, as usual, but as I’m a CFD beginner I wish to understand that more clearly:

“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.”

What you really mean by saying “reciprocal, circular relation” ?

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