Understanding the Boussinesq Hypothesis and the Eddy-Viscosity Concept

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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”. 

length-scalr1.pngMoore’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 StressReynolds-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.

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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.

IMG_0646
The turbulent boundary-layer and the “law of the wall”

y+_Calculation
Near wall cell size calculation

The above “Near wall cell size calculation” explanatory video 

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

tau1-e1502087910611S
Now if I shall define a traceless tensor as :Tרשבקךקדד

Then this is the anisotropic stress tensor and under the linear Boussinesq hypothesis it could be written as:
Linear Bousinesq hypothesisThis is generally a linear constitutive law between the stress and strain tensors in a direct analogy to the constitutive relationship for Newtonian fluids:
newtonian fluids constituvr lawwhere 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.

Mapping between Microscopic and Macroscopic Realms

The two levels of descriptions, the microscopic-molecular and that of the macroscoping-continuum assumption are totally autonomous, but also, they have a different range of applicability, manifested in their different vocabularies. For example, we are not allowed to talk about the temperature of an atom, or about the pressure exerted on it. This is described by the concept of emergence, when an entity is observed to have properties its parts do not have on their own. These properties or behaviors emerge only when the parts interact in a wider whole. Note that it doesn’t mean that an emergent description can contradict what the laws of physics allow to happen on a lower level description, it is actually fully constrained by what the lower-level description allows, even if we do not have enough information or resources to show that it does.
Moreover, It is especially straight forward for the case of fluid dynamics, to show that the macroscopic description entailed by it can be directly obtained from the microscopic description. In other words, there is an explicit mapping from the world of molecules to that of fluids. The macroscopic description serves as an effective theory for the microscopic theory, even though historically speaking, long before we knew it was made of molecules, we’ve invoked the vocabulary describing air’s pressure and velocity.
On the other hand, when mapping a state from the macroscopic realm to one state in the microscopic, we find that there are many different states in the microscopic theory that may describe the same state in the Macroscopic theory.

As an example for this explicit mapping from the microscopic to the macroscopic realm, we may take the example of the macroscopic property viscosity:

We regard an average number of molecules moving through an (unit) area in a specific direction.

For ideal gas the molecular velocity is following the maxwellian distribution (note: this is a manifestation many different states in the molecular theory get mapped to the same state in the fluid one), such that all directions are equally possible, and the average molecular velocity is the thermal velocity:

Vth

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:

V_Vertical

Now we integrate on a hemisphere:

V_th_Integreating

And get that the total molecules on the route for the positive direction:

number of molecules

We look at a typical molecules and the route they make without colliding:

molecules and momentum

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

mOMENTUM- NEGATIVE

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

minus

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

positive momentum

Summing both sides it becomes:

final stress

Now we may write:

txy

Where:

myu

And we’ve mapped viscosity!!

Scale Separation

The continuum assumption underlying fluid dynamics would not be valid if the effects of particular molecules were important individually, rather than only in aggregate. For this to not be the case, and for the fluid dynamics description to be valid and autonomous to the “black box” of the microscopic realm we need to assure Scale Separation.

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

  • Kn
  • Lmpf criteria

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:

tauxy

and by that:

txy fluid

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…

In Sum (and an important remark)…

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:
Anti symetric
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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