The following from my blog gives a thorough and concise heuristic description of what is arguably the single most important hypothesis underpinning most turbulence models for engineering along with its caveats.

### Motivation

Turbulence modelling is considered by many as witchcraft, by others as the art of producing physics out of chaos, “the last unsolved problem” of classic physics.

A full description of the phenomena is entangled in a seemingly simple set equation, the Navier-Stokes equations, their nature is such that analytic solutions to even the most simple turbulent flows can not be obtained and resorting to numerical solutions seems like the only hope.

But the resourcefulness of the plea to a direct numerical description of the equations is a mixed blessing as it seems the availability of such a description is directly matched to the power of a dimensionless number reflecting on how well momentum is diffused relative to the flow velocity (in the cross-stream direction) and on the thickness of a boundary layer relative to the body – The *Reynolds Number.
*It is found that the computational effort in Direct Numerical Simulation (DNS) of the Navier-Stokes equations rises as Reynolds number in the power of 9/4 which renders such calculations as prohibitive for most engineering applications of practical interest and it shall remain so for the foreseeable future, its use confined to simple geometries and a limited range of Reynolds numbers in the aim of supplying significant insight into turbulence physics that can not be attained in the laboratory.

*Turbulent Boundary Layer (P. Schlatter and D. Henningson of KTH)*

Having said all that, engineering applications could not have been left out and simplified methodologies to capture flow features of interest were developed their complexity and range of applicability dictated by the simplifying assumption, a direct consequence of computational effort limitations and generally predicted by *“Moore’s Law”. *

**Moore’s Law applied to CFD –
Taken from Prof. Phil Roe (Univ. of Michigan) with his celebrated lecture on the history of the development of CFD (a must watch…🤓)**

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

Most of nowadays CFD simulations are conducted with the Reynolds Averaging approach. Reynolds Averaged Simulation (RANS), the “working horse” of industrial CFD 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. This is of course an obvious consequence to the fact that taking these higher moments is simply a mathematical endeavor and has no physical contribution what so ever.

*Reynolds-stress tensor*

Levels of modeling 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 for the turbulent length scale (or time scale), which (from a pure dimensional analysis perspective) suffice to define an eddy-viscosity (analog to its kinetic gasses theory derived counterpart, albeit flow dependent instead of flow property), then by invoking the “Boussinesq Hypothesis”, relate the Reynolds stresses to the mean strain rate. In this sense 2-equations models can be viewed as “closed” because as they possess sufficient equations for constructing the eddy viscosity with no direct use for experimental results.

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 such as Spalart-Allmaras (SA) turbulence model) these models possess sufficient equations for constructing the eddy viscosity with no **direct** use for experimental results.

Now there is a caveat to the theoretical underpinning of the Boussinesq hypothesis that is of a more mathematical nature and is seldom recognized in the engineering literature. The source of the Reynolds stresses is actually in the averaging of the nonlinear advective terms of the Navier-Stokes equation. The Boussinesq hypothesis on the other hand, leads to replacing these with linear diffusive terms. There is a certain natural balance between nonlinear advection and linear diffusion (dissipation). Adding this specific diffusive term, which actually in many situations is quite large, results in equations that are more dissipative than should. This is not to bad for converging numerical simulations, but it hinders the possibility of tracking evolving flows and recognizing correct bifurcations (…and it’s bad physics 😊). Actually one of the reasons (among many…) of why predicting transition with 2-eq RANS models is impossible without an ad-hoc additive.

Moreover, 2-equations models 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, today industry need for rapid answers dictates CFD simulations to be mainly conducted by 2-equations models whose 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.

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

*Near wall cell size calculation*

*The above “Near wall cell size calculation” explanatory video *

**The “Boussinesq Hypothesis”**

**The “Boussinesq Hypothesis”**

The Boussinesq Hypothesis stands in the basics of eddy-viscosity related turbulence modeling. The linear Boussinesq hypothesis major claim is that the principal axis of the Reynolds stresses coincide with those of the average strain

Now if I shall define a traceless tensor as :

Then this is the anisotropic stress tensor and under the *linear Boussinesq hypothesis* it could be written as:

This is generally a linear constitutive law between the stress and strain tensors in a direct analogy to the constitutive relationship for Newtonian fluids:

where Rn is the viscosity stress tensor. This relationship has empirically been proven regarding the kinetic theory of gasses as the reference to a first order approximation of the velocity gradient to be simplified according to the upcoming description.

#### approximation of Newtonian viscosity

A hypothesis in the kinetic theory of (rare) gasses is that molecules passing through y=0 are holding their characteristic momentum from the velocity layer they where coming from:

In the molecular level a decomposition is proposed (note: does it not remind the Reynolds decomposition?…):

u=U+u”

(While U is defined by U(y) and u” is molecular random movement).

The sudden flux of every property through y=0 is proportional to the normal to plane velocity normal to plane. Concerning the description above it is v”. Hence the sudden change in momentum through a differential element dS may be described as:

After conducting an ensemble average this becomes:

As by definition the stress acting on y=0 may be described as:

As it’s accustomed to break the stress into pressure and viscous stresses in the following manipulation:

This brings us to:

Just look at the wonderful resemblance to Reynolds stress… 🙂 one only has to exchange u”, v” (a random molecular motion), with u’, v’ (turbulent perturbation in the Reynolds decomposition).

#### The kinetic theory of gasses – standing on the shoulders of giants

Now we are following reasoning from the kinetic theory of gasses. According to that we regard an average number of molecules moving through a unit area in the y direction.

For ideal gas the molecular velocity is following the maxwellian distribution, such that all directions are equally possible. The average molecular velocity shall be the thermal velocity :

On average half of the molecules follow to the positive side and the others to the negative. if we take the vertical velocity these becomes:

Now we integrate on a hemisphere:

This means that the total molecules on the route for the positive direction:

Now back to our previous description:

In their way from P to Q each molecule is ” typical of where they come from”, hence each molecule from P carries about a negative momentum:

This means that the total momentum flux from to the negative side (to first Taylor expansion approximation):

On the same grounds, the total momentum flux from to the positive side (to first Taylor expansion approximation):

Summing both sides it becomes:

Now we may write:

Where:

The assumptions that guarantees that a first Taylor expansion shall be valid require:

#### The analogy of the boussinesq hypothesis to the derived consequences from the theory of kinetic gasses

The Boussinesq hypothesis is based on the same principles only Boussinseq “out of the box thinking” led him to the following postulates:

molecule————————->Fluid parcel

Mean free path —————>Mixing length

It is very straightforward to write the following, derived directly from the above:

and by that:

To enrich the validity of the hypothesis, two derived assumptions should be valid:

- every fluid parcel is characterized by a Lagrangian length scale which randomly changes such that the average is lmix – indeed to every fluid parcel following a Lagrangian path one could assume characteristic enough to derive lmix.
- The problem relies in the fact that lmix might not be smaller than variation in the average flow properties. This is due to the spectral gap problem which is not evident in the molecular counterpart.

#### Shortcoming of the Boussinseq hypothesis

- It is possible to define lmix, but it is a property of the flow rather than the fluid (such as the case in the kinetic theory of gasses) thus universality may not be expected.
- Scale separation does not exist due to the
*spectral gap problem*, a problem avoided in its molecular counterpart. - Failure to predict flows with sudden and abrupt changes in the strain of the averaged flow. In the Taker-Reynolds experiment on an almost isotropic flow a rate of strain flux is applied on a unified averaged flow (U,-ay,az), where a is the a constant rate of strain. following some distance the strain is abruptly stopped.

While the experiment shows a gradual return to isotropy, the Boussinesq hypothesis predicts a sudden return with the exact moment of the abrupt strain stopped.

Moreover, with gradual increase in strain flux from zero the Boussinesq hypothesis predicts a sudden jump in the anisotropy.

These two failure modes presented are due to the inability of the Boussinesq hypothesis to account for history changes which implies a serious cause and effect failure (a result can not occur before or exactly with its cause). - The failure to give a reliable prediction to swirling flows, slows over curved surfaces separations etc…

The Boussinesq hypothesis ties between the average velocity tensor of the flow and the Reynolds stresses in a linear relation. therefore even in the equations for the kinetic energy enters the influence of the strain tensor which is the symmetric part of the velocity tensor after a decomposition to a symmetric an antisymmetric part.

The antisymmetric part is the rotation tensor defined as:

And it doesn’t appear in the equation for the kinetic energy nor in the Boussinesq hypothesis. As a consequence the behavior of the Reynolds stresses doesn’t take into account instances of rotation combined with a high gradient in the flow strain of the average flow, cases of separation or cases flow above a highly curved geometries, cases of a rotating system and the appearance of a centrifugal force which brings forward the non-gallilian nature of RANS.

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