Be aware, total energy is conserved but partial energies (kinetics, internal, …) have a production term due to the conversion of a form of energy into another one.

Thanks Filippo, but the concept of mechanical energy is of kinetic energy plus all forms of potential energy that can be converted to kinetic energy. Final conversion to internal energy (dissipation) is considered as a loss of mechanical energy since this cannot be converted to kinetic energy. The downwards flux of mechanical energy that sustains turbulence near the wall may have several components, but the RANS energy equation says that the divergence of the sum of all components balances the loss to dissipation.

My question remains.

Before considering the RANS equations, do we agree that in the standard equations only the total energy equation is conserved and the dissipation appears in the kinetic energy equation and (opposite sign) in the internal energy equation to take into account the irreversibile process?

Yes. It is the sum of the kinetic energy, potential energies and internal energy that is conserved. It is, however, a bit limited to regard the RANS equation as a kinetic energy equation. It is equally a gravitational potential energy equation and a pressure potential energy equation. It is the sum of the divergences of the fluxes these three that equals the dissipation rate.

Are you considering only the statistical steady case? That is only an energy equilibrium (in statistical sense) between production and dissipation.

Then, you can analyze different form of the Energy equations under Reynolds decomposition.

we consider the work done on an element of fluid near the wall, right?

K. G. Mcnaughton

I assume you are familiar with the Wilcox textbook. Starting from Sec.4.1 he introduced the general term of Production of kinetic energy.

The nomenclature of "shear production" is more common in the meteorology community but it appears as a term in non-divergence form that takes into account how the mean velocity acts on the kinetic energy by means of viscosity. This terms appears relevant in the region close to the wall. In other words, it is the work (irreversible) produced by the mean velocity.

Yes, I consider the steady case because that is the only case analyzed experimentally. The question then concerns `fully developed' turbulence.

I am familiar with the literature, back to Reynolds paper of 1895. Basically, all textbooks repeat what he wrote. You tell me, in your own words, what is the mechanism of shear production.

If the mean flow that `produces' turbulence near a wall, then what sustains the mean flow near the wall? That energy must come from somewhere?

To expand the question, we note that the divergence of the flux of kinetic energy formally leads to two terms in the RANS energy equation, usually called the `production' and `transport' terms. What is the physics underlying this division? Is it the same for all flows and all flow regimes?

This is my opinion:

- looking to the production of turbulent kinetic energy we see the product between the gradient of the mean velocity and the correlation of the fluctuations. Close to the wall, in the TBL, the dU/dy is relevant and positive and, even if the fluctuations near a wall can have low intensity, this term expresses the capability of the mean flow to produce turbulent kinetic energy like a source (it is not in divergence form). Note that I assume the mean flow is substained if energy is provided to the flow system by means of BCs.

- "transport" term is better understood if we consider the statistically unsteady case. This way we have dk/dt+U.grad k = Dk/Dt, that is the lagrangian derivative of the kinetic enrgy. This term is balanced by the other terms.

I don't know if that answers your questions, let me know.

You are stating the tautology that that the RANS production term represents turbulence production. I ask, what is the mechanism by which turbulence is `produced'?

Try this: For statistically-steady, horizontal flow over a flat wall the vertical flux of kinetic energy can be written as, $w E_K$. Take the vertical divergence of this flux: $\frac{\partial w E_K}{\partial z}$, fill in the definition of $E_K$, do the usual Reynolds expansions, take an ensemble average and you have the `production' and `transport' terms of the RANS energy equation. The Navier-Stokes equations are not involved. So what do these names mean?

K. G. Mcnaughton

not only tautology ... you have to consider the two kinetic energy equations, E_bar and k. By comparison, you will see the mechanism in the term that subtracts kinetic energy to the mean flow to put in terms of production for the k energy. This is general but you can write also in terms of the TBL equation for a flat plate.

I cannot say it better than how Pope wrote in Sec. 5.3 and page 126, 127.

I don't understand your statement that the original NSE are not involved.

Just to add that the dynamics of this mechanism can studied by means of the well known simple 1d model of the so called Burgulence.

Of course, that would not represent the TBL but it appears useful.

I should have been more explicit. E_K is the total kinetic energy in my example. It has mean and fluctuating parts, so E_K=w'((\overline{u}+u')^2+v'^2+w'^2)/2. Now continue and you will see that I am right.

The equation for E is actually discussed in Sec.3 in the Pope textbook. Production is generally a denomination for k, not for E.

Thus, I think we agree, E is different. But there is a term in non divergence form also for E.

All the textbooks say much the same thing, and I think their arguments are inadequate. Let’s stay with the basics and start with the total mechanical energy, E_M, in flows at statistically steady state. Here the divergence of the flux of mechanical energy equals the dissipation rate. That is a given, following directly from the definition of mechanical energy and the law of conservation of energy. Turbulent flows are dissipative systems that will run down and stop unless maintained by an external source of mechanical energy.

We can, if we want to, use Reynolds’ rules to write E_M in two parts: a part based on the mean velocities and a turbulence part. We then have two energy equations linked by a common term, positive in one and negative in the other. Conservation applies to the two parts taken together. If we have a physical process called `production’ of TKE in the turbulence part then we must have a source of that energy somewhere in the two equations. If that source is to be the mean flow then it's flux must be included in the linkage term, and there must be an energy loss in the mean equation. But what then maintains the energy of the mean flow?  How does that energy get there? This usually involves turbulent transport from somewhere, and that too must involve the linkage term.

For a wall-bounded shear layer the source of all mechanical energy lies at larger scale, in the free stream or the free atmosphere above the boundary layer. Kinetic energy is introduced at the top as a ZPG BL or ABL grows by entrainment. It is carried downwards by turbulent eddies. The linking term is then a two-way street so that the mean and fluctuating parts of the energy equation are inextricably entwined. The conceptual simplification that Reynolds, in 1895, hoped to achieve by separating out mean and fluctuating parts is not realized.

Given this, my original question stands. What is the mechanism of `shear production'?  We would like to know about flow dynamics, and so about flow instabilities, growth of disturbances, or whatever else is involved.

While I agree (and it is well known) that the Reynolds decomposition never realizes a real separation between mean and fluctuations, I have to say that your statement :

Conservation applies to the two parts taken together. If we have a physical process called `production’ of TKE in the turbulence part then we must have a source of that energy somewhere in the two equations. If that source is to be the mean flow then it's flux must be included in the linkage term, and there must be an energy loss in the mean equation. But what then maintains the energy of the mean flow?  How does that energy get there? This usually involves turbulent transport from somewhere, and that too must involve the linkage term.

is well described by the Eq.(5.131)-(5.132) in the Pope textbook. Then the energy mechanism in the TBL is also described in Sec. 7.3.5.

Let me use a simple model, a control volume of rectangular shape surrounding a flat plate where a BL develops.

Clearly, we have k=0 at inlet, at wall, and on the external region where v'=0.

However, k is not zero at the outlet, therefore the sum of the fluxes has the only contribute at the outlet along with the diffusive (normal direction) flux of k. That must be balanced by something that acts locally, in the interior of the BL. That is described by Eq.(7.177).

Then we can introduce the BL approximation for the mean energy Eq.(5.131) and that provides the full set of equations you addressed in your comment.

Is that in contrast to your requirement of a "linkage" term? Energy for the mean flow is subtained by the BCs (in absence of a mean pressure gradient) and the production of k in the TBL is subtained by the vertical direction term (7.184).

Please, address me what are the equations you find inadequate in the Pope description.

To those who are following this discussion: Filippo and I are having an off-piste discussion by email, trying to find common ground, so to avoid going round in circles in our comments here.

Howdy K. G. Mcnaughton,

"off-piste" is skiing "on snow that has not been compacted into tracks or groomed trails" Hmmm. Actually, on reflection that is very good. Your question struck me in the original as "How do fluids establish shear physically in a boundary layer flow?" and the discussion has been on "Where is shear production treated in a mathematical/computer model of boundary layer flow?" I think you would like both options treated. Now, the "snow" is well compacted in the models and their descriptions, and Filippo Maria Denaro is clearly an expert therein so I'll just read about that above and when you folks come back.

The nature of boundary flows is more in my line. Assuming the physical view is also wanted, I'll try. Your background, papers since 1895, means you know about this, but I like cover the bases and alternate wording often helps me. An atmospheric boundary layer shear flow may be pseudo-steady state during a period of slowly changing pressure fields of the general circulation. The steady state is supported by synoptic scale system pressure gradients that force the flow at the surface. Your specific question: "But what then maintains the energy of the mean flow?" is answered by the general circulation of the atmosphere. Your companion question: "How does that energy get there?" is simply answered by the atmospheric response to the twofold influences of unequal distribution of solar energy with latitude and thermal differences between land masses and oceans. The complex answer takes longer. With either answer, the energy supply for a local boundary layer flow is assured. In the lab or an experiment, the flow is maintained by a fan or other device with an external power source. In computer models I expect it to be in the boundary conditions which have no energy problem.

Air in contact with the ground has zero velocity, the no-slip condition. Air aloft of the boundary layer partakes of the general circulation. The greater momentum of the air aloft mixes downward by molecular diffusion and turbulent diffusion and its energy is partially dissipated to heat by sufficiently random motions in both. Thermally driven vertical mixing and turbulence related to them are also very important, but they have not been noted in earlier posts, so I'll set them aside.

This obvious stuff about which you already know aside, what is a neat answer to: "Does anyone know the mechanism of shear production of turbulence?" Probably not, but we do know about the shear production mechanism and I'll try to sketch that. Shear is a result of momentum exchange between air parcels of different velocities in a boundary layer. The momentum exchange results in an intermediate value of momentum for the mixed parcel. In laminar flow this is accomplished by transverse motions of molecular unrest which perform the diffusion. Turbulence is the same and different. Fluid parcels vary widely in size and may have transverse motions equal to the boundary layer thickness, but their motion does not contain the collisions of molecular unrest to force them in random directions. Now, what we need to express is how turbulence air parcels form and by what forcing do they move laterally to the synoptic pressure gradient?

I'm going to play this straight as I see it, "off-piste." It will be a view against which to review textbook versions: Turbulence consists of fluid structures and flow among them. Numerous structures have been found. I will limit this post to structures with an internal pressure distribution among general gustiness. The internal pressure distribution is critical to my answer to your question. Consider the transition to turbulence on a flat plate. The Reynolds Number at which transition occurs is affected by roughness elements on the plate, it is lower for a rough plate than for a smooth one. Why? Because the flow around the roughness elements does not cling to their surface, but separates. [[new =>]] Flow past the roughness elements entrains molecules from the separated volume because the unrest molecules leave normally but the passing flow has a forward vector component added to its unrest. This produces a lower pressure "in the wake" and the passing flow is deflected toward it. The force producing flow curvature (acceleration) appears as enhanced energy in the flow and reduced energy in the wake, that is, a pressure gradient with higher pressure in the flow and lower pressure in the wake. The "eddies" thus formed constitute larger roughness elements that continue to interact with the flow and a range of structures (eddies) performs asymmetric diffusion and pressure gradient formation that forces the curvature in intermediate scales. This process actually extends to large scales as you have seen when new clouds form and develop curved, even spiral, forms against a slower parcel containing the original cloud. The internal pressure distribution provides sufficient coherence so such parcels that are forced by deflection against slower parcels produce motion in all directions in the boundary layer flow with something like molecular unrest, but be careful with all analogies. However, the coherence does not control the parcel surfaces which interact as boundary layers between relatively moving parcels and form new structures in that process. All the coherent structures move and mix as their momentum forces them thereby smoothing the momentum distribution of the shear flow. They lose energy to random thermal activity when the interactions on surfaces are sufficiently small scale. This augments the energy conversion of forming eddies with internal low pressure balanced with rotational kinetic energy that enhance the energy of the fluid. The eddies decay to an ambient energy sum by means of rotational kinetic energy "filling" the core low pressure. [[also a new view]] Thus, "downwards transfer of mechanical energy from the flow above" is distributed into structures and gusts that contain turbulent kinetic energy and dissipation. Is this helpful?

In the Reynolds decomposition version, the fluctuations of longitudinal momentum are driven laterally by the fluctuations of normal momentum to effect the mixing of momentum between layers by a statistical correlation of those fluctuations (Reynolds stress). It is actually the same story when you stand back a bit. The mixed momentum produces the flow shear, but dissipation is normally parameterized as I recall because the thermal scale is beyond any computer, except the fluid itself, which actually is a pretty fast computer.

Happy Trails, Len

Hi Len,

Welcome to the discussion. Your first point, that the energy and momentum of atmospheric boundary-layer flows is maintained by the general circulation is, of course, correct. So how is it that the textbooks on atmospheric boundary layers (ABLs) do not mention this? Wyngard’s book "Turbulence in the Atmosphere" (2010) is a prominent exemplar. No mention is made of the entrainment of kinetic energy and momentum into ABLs. How can we understand turbulence in the atmospheric surface layer if we don’t start by acknowledging that most of the momentum and energy comes down from above. My original question is a rhetorical one, asked to focus attention on what is curiously overlooked. The idea of `shear production’, which implies a local source of kinetic energy, is part of the problem.

Now to some disagreement. Your idea of momentum exchange is a K-theory explanation: essentially it is that localized `eddies’ with random motions mix momentum down a velocity gradient. I understand convective ABLs rather differently. I think that large eddies transfer momentum down through the bulk of the ABL without the need for a velocity gradient. Momentum is transferred from the free atmosphere to the top of these ABL-spanning large eddies in the entrainment zone, and then directly down to smaller shear eddies near the ground. The required `rigidity’ of the intervening large eddies is provided by pressure forces which allow the large eddies to retain their shapes over time. This `rigidity' counters the tendency of the large eddies to accelerate at the top and slow near the ground. Smaller shear eddies near the ground then transfer the momentum on down towards the ground, but much less efficiently. They produce more waste heat in the surface layer (dissipation) because increased entropy production is the essential cost of the higher organization associated with the shear turbulence. This conceptual model is for convective ABLs; it is clearly not a universal model for all flow regimes in ABLs. Neutral and stable ABLs have different eddy structures and seem to transfer momentum by different mechanisms.

I will leave for later your comments on how to rewrite the textbooks, for fear of distracting from the main points.

Filippo and I have mostly aligned our views. I will discuss that in a separate contribution.

If you focus in atmospheric BL, it is better to address that the energy transfer in quasi-2d turbulence has some differences. External large structures can also get energy from smaller ones in an inverse cascase.

Howdy K. G.,

Thank you for the welcome.

As I noted in the "large eddy Reynolds Number" question: "Afterthought: I offer thoughts in my replies on Q&A that attempt to address the information provided." So, let's try again.

I guess we are expected not to read about ABL turbulence until we have been indoctrinated in atmospheric science more generally, and therefore we are expected to know about the general circulation and the upper air going past our boundary layer location. I am happy to have addressed that omission in textbooks. I agree with you, however, in being irritated by being "expected to know" whatever the author doesn't want to provide. It is a frequent problem in my life since in many situations my path has not included "what any old fool knows."

I do not agree with the K-theory identification of my treatment and do not see it in what I wrote. It is not just diffusion in shear flow with a lot of different sized eddies, like bigger molecules. That view is too simplistic, too kinematic, in my opinion. I tried to avoid the tone of L. F. Richardson's ditty about "Big whirls have little whirls . . ." and replace it with my [new =>] version because I find the whirls good poetry and bad physics, but that is covered by the [new =>] section in my post. My emphasis on interaction in layers between eddies is very different from the whirls image. Also, I do not care about rewriting textbooks, I would just like to comprehend turbulent fluid dynamics as a science and scrape off the engineering conveniences that dominate most exchanges in the field. The fluid knows, but it is very hard to approximate so capable a computer accurately with our lumbering along electronic devices and the equations we can solve either analytically or numerically.

Yes, convective boundary layers are very different, and even mechanical turbulence altered by heating that is insufficient to gather the warm surface layer into a thermal are different from the basic image of mechanical turbulence addressing the method of shear production in turbulence that I offered. I did note: "Thermally driven vertical mixing and turbulence related to them are also very important, but they have not been noted in earlier posts, so I'll set them aside." to excuse that omission. In fact, I find little actual disagreement with my "This process actually extends to large scales as you have seen when new clouds form and develop curved, even spiral, forms against a slower parcel containing the original cloud." although one must admit that that case does require thermals.

"I think that large eddies transfer momentum down through the bulk of the ABL without the need for a velocity gradient." Of course, "Shear is a result of momentum exchange between air parcels of different velocities in a boundary layer. The momentum exchange results in an intermediate value of momentum for the mixed parcel." is my version. I do prefer the image that thermal forcing of convection produces return flow in the form of your "large eddies" to focus on dynamics in preference to kinematics, but the meaning is present in either wording.

My image that the formation of eddies causes a partition of kinetic energy from the flow into reduced pressure in the core of a rotation with enhanced pressure in the fluid is very important! Valid is a separate concern, but the point is that "pressure recovery" and fluid heating occur during the eddy formation, and later the "low" is filled by the kinetic energy of the eddy rotation as the eddy dissipates. I wouldn't mind having seen that in textbooks, at least as a speculation instead of relying on eddies becoming small enough to be, in essence, already dissipation heat.

Oh well, I doubt we could agree on a statement for the press, but the range of ideas here appeals to me.

Happy Trails, Len

The turbulence production mechanism is associated with the interaction of the tensor of turbulent fluctuations (Reynolds tensor) with the tensor of mean velocity gradients (shear). This mechanism extracts energy from large structures (mean fields), and the power developed maintains the turbulent field by feeding the energy cascade. This will transfer this energy through the turbulence spectrum from the large structures (large productive scales) to the smallest structures (dissipating scales or Kolmogorov scales) passing through intermediate scales of inertial transfer whose range widens with increasing Reynolds number. In the absence of a mean velocity gradient (uniform flow for example), the production is zero. In this case the initial turbulence (produced by a grid for example) relaxes towards a homogeneous and isotropic situation and turbulence degenerates by viscous dissipation.

Dear all,

I find interesting the discussion and your idea. However, I think that the discussion requires some math to better detail the meaning of what we are discussing.

I attach a file I wrote where I detail the integral formulation for 1) totale energy and 2) kinetic energy in the NSE. Then, the RANS equations for the 3) mean kinetic energy 4) turbulent kinetic energy.

I hope they are correct, let me know if some term is wrong.

Now, consider a volume (also 2D) of rectangular shape that has dimension 0

Filippo Maria Denaro Thank you for putting into equation, the integral form of the energy balances of the turbulent flow. You write (the last two equations) the balances of the kinetic energy of the mean motion (Km) and the turbulent kinetic energy (k) of the turbulent motion. These equations show that the shear production term of k appears with opposite signs in the two equations, meaning that the turbulence production is fed by the large scales of the mean motion. This is dissipated by viscosity (the last term of the last equation, by definition negative).

However, I don't understand why you put (d/dt=0 for RANS). Suppose you instantly create turbulence in a fluid at rest by homogeneous agitation. In the absence of movement putting d/dt=0 means that this turbulence will remain indefinitely which is physically not consistent. We should have in this case dk/dt=disspation so that turbulence could degenerate.

See also (in French)

Chapter SMTF-Chapitre 2 : Les échelles de la turbulence

Chapter SMTF-Chapitre 3 : Description statistique de la turbulence

Jamel Chahed

For Km this is a consequence of the assumption of a steady field in the Reynolds decomposition. In this sense, any application of the statistical mean produces a steady function.

Maybe you are thinking to the URANS formulation that is very different.

Note that the shear production could change sign depending on the resulting sign of the Reynolds tensor component.

PS: the example of grid turbulence produces zero mean velocity.

Filippo Maria Denaro wrote "For Km this is a consequence of the assumption of a steady ". The mean velocity is not obligatory steady in turbulent flows. The same goes for the Reynolds stress tensor. The atmosphere is for example a turbulent airflow where the mean velocity is not steady. the same goes for the mean and turbulent kinetic energy.

Filippo Maria Denaro wrote "the example of grid turbulence produces zero mean velocity". When we have zero mean velocity, no turbulence production may occur. However, in this case, Homogeneous Turbulence evolves under the effect of eddies stretching toward an isotropic character and degenerates under viscosity dissipation.

See more on:

Chapter SMTF-Chapitre 3 : Description statistique de la turbulence

Jamel Chahed

what are you addressing is the adoption of an ensemble averaging, no longer a statistical time-averaging. Formally, that mean is indeed time-dependent. That would represent the theoretical framework for URANS.

But if we invoke the ergodicity property, all statistical mean must converge to the same solution. Therefore, in energy equilibrium, also the ensemble averaging will produce a steady solution.

A totally different approach is the time filtering, However it is not a statistical operator.

Filippo Maria Denaro wrote "But if we invoke the ergodicity property, all statistical mean must converge to the same solution" Ergodicity means "statistical steady turbulence" which only applies to steady turbulent flows. It suffices to establish the statistical equations of the turbulent flow to realize that the statistical averages of the mean and turbulent fields are not necessarily, steady fields.

See more on

Chapter SMTF-Chapitre 3 : Description statistique de la turbulence

Again, unsteady statistical turbulence requires the adoption of the ensemble averaging. However that would be really produce a time dependent mean only under non-equilibrium turbulence, that is an external source of energy. For example the motion produced by a piston.

Any other approach to unsteady turbulence requires a time-filtering, that is a non statistical operator.

There is an old question I posted on RG about this topic

Filippo Maria Denaro Okay My bad

Not your bad.

Just the mathematical meaning of the operation you want to apply:

f_bar(x,t; delta_t) = 1/delta_t Int[t0, t] f(x, tau) dtau

this mean is a function of time defined by a local time-averaging.

This way you can write the time-dependent equations as you desire.

But they differ from the statistical approach because this mean operator applied on the mean is no longer idempotent and the mean of the fluctuation variable is no longer zero.

You must consider further terms.

The following relatively old paper is an example among many kinds of research focussing on the problem of unsteady flows using both Reynolds Averaged Navier-Stokes equations (RANS) standard approach, and Large Eddy Simulation (LES):

Chang, Y. S., & Scotti, A. (2004). Modeling unsteady turbulent flows over ripples: Reynolds‐averaged Navier‐Stokes equations (RANS) versus large‐eddy simulation (LES). Journal of Geophysical Research: Oceans, 109(C9).

Available on:

Article Modeling unsteady turbulent flows over ripples: Reynolds-ave...

Indeed there is a lot of confusion between the concept of URANS and that of temporal LES.

Note the concept of phase averaging for pulsatile flows, that is shown by Wilcox.

On statistical averaging: Temporal and spatial averaging operators.

In unsteady turbulent flow, the statistical quantities relating to the turbulent motion at a given instant, defined on the time scales of the average motion, are obtained by applying the statistical average operator on scales much smaller than the scales associated with the average fields, which are those of the large flow scales.

Applied to small scales, statistical averaging operators (time averaging for ergodic processes for example) assume that on these small time scales the turbulent signal is instantaneously ergodic. The same goes for the statistical averaging obtained by applying a volume averaging operator which assumes spatial homogeneity. It is assumed that this is the case locally (at small spatial scales) even in non-uniform flow. This is even the fundamental basis of local metrology in turbulent flow where at each step of experimentation the evaluation of the turbulent scales involved is necessary in order to determine suitably the experimentation protocol.

See more on:

Chapter SMTF-Chapitre 3 : Description statistique de la turbulence

Math is math … “assumptions” are some approximation. What the consequences of such approximations?

consider a turbulence without external time dependent forcing. Which is the statistical operator ? The time averaging over a finite period is not statiscal. That is a local time avereging. Tou should verify that the period T is so long to cover the full range. In line of principle for any local time avereging differs from and does not vanish.

Here you can introduce your assumption about a large T and what do you get?

This is the relevant question about the standard assumptions.

just as a joke, assume a function like a sum of sin(kt) components. Now, please, address me a statistical operator that produces a time-dependent mean and zero mean of fluctuatuons.

Jamel Chahed

have a look to this

https://www.researchgate.net/post/URANS-what-is-the-meaning-for-statistically-steady-flows-and-what-compared-to-LES

This is a matter of turbulence scales to consider each time we have to carry out experimental data on statistical quantities in unsteady turbulent flows using intrusive or non-intrusive tools. All published databases on unsteady turbulent flows were carried out on the basis of these theoretical analyses. Otherwise, how they are measured?

And how are they solved numerically? Are you sure to be able to distinguish the unsteady RANS from the LES equations ?

Look at there equations and you see that no averaging/filtering operetor is really applied.

Provide me the example of the sum of sine function.

People just made a traditional “assumtpion” a wrong theoretical approach in using the unsteady RANS.

See for example the following paper which "compares the performance of eight Reynolds-Averaged Navier–Stokes (RANS) two-equation turbulence models and two sub-grid scale (SGS) large eddy simulation (LES) models in the scenario of unsteady flow around a finite circular cylinder at an aspect ratio (AR) of 1.0 and a Reynolds number of Re=20000"

Zhang, D. (2017, October). Comparison of various turbulence models for unsteady flow around a finite circular cylinder at Re= 20000. In Journal of Physics: Conference Series (Vol. 910, No. 1, p. 012027). IOP Publishing.

Available on:

https://iopscience.iop.org/article/10.1088/1742-6596/910/1/012027/pdf

Jamel Chahed

just the fact that the authors denotes RANS and LES as two turbulence model means he did not understand the issue.

If I was the referee of the journal, I would have reject the paper just for this assumption.

RANS and LES produce different variables since are different formulations, not models!

Please, read with attention the discussion I posted and the attached paper in it.

Some Pages extract from Wilcox. There are here (2.9) the source of the theoretical flaw in unsteady RANS.

Then, you can see that URANS and LES equations are totally equivalent, the only difference should appear in the closure model. Each model should take into account for the different variables (statistical, averaged, filtered). That is a further issue.

In general, a time dependent mean produces non vanishing cross term in the convective flux.

Filippo Maria Denaro "just the fact that the authors denotes RANS and LES as two turbulence model means he did not understand the issue". Please note that the authors do not speak of two turbulence models but compare eight Reynolds-Averaged Navier–Stokes (RANS) two-equation turbulence models and two sub-grid scale (SGS) large eddy simulation (LES) models

Of course, two-equation turbulence models generate turbulent viscosity closure for RANS equations. The same goes for LES which needs closures for unknown filtered quantities and this is done using sub-grid models or subgrid viscosity.

Please read before claiming "If I was the referee of the journal, I would have reject the paper just for this assumption" Fortunately you were not. This is a relevant paper, already well-cited despite its relatively short time from publication

Jamel Chahed

were you the referee?

The paper has minus relevance in turbulence field and is based on the wrong assumption that the statistics of a filtered LES field can be rigorously compared to the RANS formulation. From the LES solution you get the statistics of the filtered field. The author of the paper says the grid is fine but do not show if they can realize a solution where the filtering effects can be disregardable.

Have you seen the URANS and LES equations in the paper? Have you seen the real application of some statistical mean operator on each term? They are equal, only the additional closure model term would characterize if one solves either RANS or LES. Furthermore, the LES variable is space filter-dependent by means of the implicit grid and discretization filter shapes, it represents a class of solution, the limit for the spatial filter width being the NSE solution (DNS). That is not the case of URANS solution. HAve you any idea of what solution would be obtained in URANS for the grid size going to zero?

Comparing the two formulations is just of some interest from euristic point of view, it add nothing to the theoretical basis of URANS. It add nothing to the issuse highlighted in the Wilcox textbook.

There many published papers that just repeats standard and wrong way of analysing the formulation.

Thus, before you claim something that is not a mere repetition of the approximate view, already appeared in literature, think the about relevant publications in the field.

Have you read carefully the Spalart paper I addressed? Have you read the discussion with other people? If you have some valid statement to add you are wellcome, otherwise I consider this discussion closed since it is useless.

And I apologize to bother the readers for such discussion about issues that expert in the field know already very well.

Filippo Maria Denaro wrote "There many published papers that just repeats standard and wrong way of analysing the formulation." too easy and pretentious. Then FMD wrote, "Thus, before you claim something that is not a mere repetition of the approximate view, already appeared in literature, think the about relevant publications in the field." For example? A single reference that clearly contradicts what I stated would be sufficient.

PS: Spalart paper does not contradict what I said nor Wilcox. I know his book by heart since I was student.

For more on RANS modeling in single and multiphase flows see this old paper on second-order turbulence closure for turbulent bubbly flows:

Article Eulerian–Eulerian Two-Fluid Model for Turbulent Gas–Liquid B...

As for LES you may consult this old bibliography:

Thesis LES : Simulation des Grandes Echelles de la Turbulence

I expect that there is nothing to add and consider this discussion closed.

K. G. Mcnaughton wrote "... This conceptual model is for convective ABLs; it is clearly not a universal model for all flow regimes in ABLs. Neutral and stable ABLs have different eddy structures and seem to transfer momentum by different mechanisms."

Neutral ABL corresponds to the case of adiabatic ABL. As soon as the ABL is not adiabatic, the temperature gradients will deviate from the reference gradient (adiabatic) and then appear gravity forces associated with the buoyancy which can be expressed according to the Boussinesq hypothesis in the momentum equation. The writing of the equation of the turbulent kinetic energy, with this additional term, makes appear two terms of production of the turbulence: a term (dynamic present in neutral ABL) associated with the fluctuating and mean fields interaction and an additional correlation term of the fluctuating velocity and temperature coming from the buoyancy expressed according to the Boussinesq hypothesis involving the deviation from the adiabatic situation. The ratio between these two terms of production of different physical natures makes it possible to define the Richardson Number in the turbulent energy equation. This adimensional number is fundamental in atmospheric dispersion models to characterize the ABL's stability (or instability). The Richardson number is also interpreted locally as the ratio between the altitude and a length scale (Monin-Obokuv length), which in turn characterizes the stability of the atmosphere at a given altitude.

For more see (in French)

Chapter AD-Chapter 5: Turbulence and Dispersion in the Atmospheric B...

Chapter AD-Chapter 6: Atmospheric Dispersion Modeling

K. G. Mcnaughton In turbulence, shear production refers to the mechanism by which the mean flow gradients contribute to the generation and sustenance of turbulent kinetic energy. It is a significant source of turbulence in shear flows such as those found in boundary layers, jets, and wakes. The mechanism of shear production can be understood through the following steps:

1. Mean Velocity Gradient: In a turbulent flow, there is typically a mean flow with a velocity gradient in the direction perpendicular to the flow. This velocity gradient represents a change in the mean flow velocity with respect to the position in the flow domain.

2. Reynolds Stress: Turbulent flows are characterized by fluctuations in velocity at different scales. These fluctuations, known as turbulent eddies, contribute to the Reynolds stress tensor. The Reynolds stress represents the correlation between the fluctuating velocities in different directions.

3. Production of Turbulent Kinetic Energy: The mean flow velocity gradient interacts with the Reynolds stress to produce turbulent kinetic energy. This interaction occurs through the transfer of momentum from the mean flow to the turbulent eddies. The eddies are stretched and deformed by the mean velocity gradient, leading to an increase in turbulent kinetic energy.

4. Redistribution of Turbulent Kinetic Energy: The increased turbulent kinetic energy generated by shear production is then distributed throughout the flow domain. Turbulent eddies transport and mix this energy, leading to further turbulence and mixing in the flow.

Overall, shear production is a process by which the mean flow velocity gradients drive the production of turbulent kinetic energy, enhancing the turbulent characteristics of the flow. It is an important mechanism in sustaining turbulence and influencing the behavior of various turbulent flows, including boundary layers, jets, and wakes.

It's important to note that the mechanism of shear production is just one aspect of the complex dynamics of turbulence. Turbulent flows involve various other processes such as turbulent diffusion, dissipation, and interactions between different scales of eddies. Understanding and modeling turbulence remains an active area of research in fluid dynamics.

Gopal Sharma : Hi Gopal, your contribution is a good account of what is in many textbooks. Your summary sentence "shear production is a process by which the mean flow velocity gradients drive the production of turbulent kinetic energy, enhancing the turbulent characteristics of the flow. It is an important mechanism in sustaining turbulence and influencing the behavior of various turbulent flows” uses the word “mechanism” twice? What is that mechanism? What kinds of eddies are invoved? What are their dynamics? Where does the mean flow get its energy from?

In convective atmospheric boundary layers, which are my special area of interest, kinetic energy comes from the atmosphere at large, as pointed out by Leonard Hall , and from the sun as gravitational potential energy delivered mostly at the ground. Applying the Reynolds’ decomposition to the (total) flux of kinetic energy, WU^2, ensemble-averaging the result and then taking the divergence we have

\frac{\partial}{\partial z}\; \overline{WU^2}/2= \overline{u'w'}\;\frac{\partial \overline{U}}{\partial z}+\overline{u'^2w'}/2

There are two terms on the right, one usually named `production’ and the other with various names, but often called `turbulent transport’ Together they describe the vertical divergence of the flux of kinetic energy. Can they be interpreted singly? They occur in various ratios in various flows and parts of flows. Is the mechanism underlying the divergence of the kinetic energy flux near the ground the same as in the entrainment zone at the top of the CBL? Eddy transport processes near the ground are dominated by attached shear eddies, while transport in the entrainment zone at the top of the CBL is dominated by the tops of the large eddies that span the CBL. Can `shear production’ have a common mechanism in each -- just scaled differently? I don’t think so.

I feel I am starting to repeat myself, but this is important. Many LES and experimental studies report the vertical divergences of the energy fluxes in turbulent flows such as CBLs, which is to say they report the profiles of the terms in the RANS energy equation, but very few indeed report the energy fluxes themselves. Why this blind spot?

K. G. Mcnaughton The lack of reporting the actual energy fluxes in turbulent flows, such as Convective Boundary Layers (CBLs), despite the availability of profiles of the terms in the Reynolds-Averaged Navier-Stokes (RANS) energy equation, can be attributed to several factors. Here are a few possible reasons for this blind spot:

1. Experimental Challenges: Measuring energy fluxes directly in turbulent flows can be technically demanding and challenging. It often requires specialized instrumentation and techniques that may not be readily available or easy to implement. The complexity of accurately measuring energy fluxes may discourage researchers from reporting them directly.

2. Computational Limitations: Numerical simulations and RANS models provide valuable insights into the energy dynamics of turbulent flows. However, these models often focus on solving the RANS equations, which provide information on the individual terms in the energy equation rather than the fluxes themselves. Computational limitations or the inherent nature of RANS models may make it more feasible to report the profiles of the terms rather than the energy fluxes directly.

3. Emphasis on Phenomenological Understanding: Many studies in turbulent flows aim to understand the physical mechanisms and processes governing the flow behavior rather than solely quantifying energy fluxes. By reporting the profiles of the individual terms in the energy equation, researchers can gain insights into the relative contributions and interactions of different processes, leading to a deeper understanding of the flow physics.

4. Focus on Model Development and Validation: Researchers may prioritize developing and validating turbulence models rather than directly reporting energy fluxes. These models often rely on closures based on the RANS equations, and by comparing the profiles of the terms with experimental data, they can assess the performance and accuracy of the models.

5. Lack of Standardization: There may not be a standardized methodology or consensus on how to accurately measure and report energy fluxes in turbulent flows. This lack of standardization could contribute to the disparity in reporting practices across different studies.

It is important to note that while the direct reporting of energy fluxes may be limited in some studies, the profiles of the terms in the energy equation can still provide valuable insights into the energy dynamics of turbulent flows. Future research efforts may focus on addressing the challenges associated with measuring and reporting energy fluxes directly to further enhance our understanding of turbulent flow behavior.

Gopal Sharma

In answer to your points:

1 I focus on the word “may”. Energy fluxes have been reported at least as far back as the 1970s. Calculating fluxes is inherently easier than calculating their divergences—indeed flux divergence is usually calculated from the difference in a flux at adjacent levels.

2 I focus here on the word `dynamics’. Dynamics is purged from the RANS equations at the point where the ensemble average is taken. We deal with averages over realizations that have no connection in time, hence the averaged quantities have no dynamics. Closure assumptions try to compensate for this loss.

3 I focus on the word phenomenological. According to Wikipedia, "A phenomenological model forgoes any attempt to explain why the variables interact the way they do, and simply attempts to describe the relationship, with the assumption that the relationship extends past the measured values”. RANS models are phenomenological models. surely we can try for some understanding of real causes in flows.

4 I am quite happy that many papers focus on testing existing phenomenological models. That leaves plenty of room for more adventuresome approaches, and for the development of conceptual models designed to provide explanations and and ideas to guide further developments.

5 This is clutching at straws!

In my opinion energy fluxes are rarely reported because existing approaches do not frame important questions concerning them. Why would they if kinetic energy is `produced' locally, and that's the end of it?

K. G. Mcnaughton ; I think, In a turbulent flow, the production of turbulent kinetic energy through shear production is indeed a local process that occurs due to the interaction between mean flow gradients and turbulent eddies. However, the story does not end there. The produced kinetic energy is subsequently subjected to other processes in the turbulent cascade.

Once the kinetic energy is produced locally through shear production, it undergoes a cascade of energy transfer between different scales of turbulence. The energy is transferred from larger eddies to smaller eddies through a series of interactions. This process is often referred to as the energy cascade.

As the kinetic energy cascades down to smaller scales, it encounters a range of turbulent eddies with varying sizes and velocities. The energy transfer occurs through a combination of mechanisms, including vortex stretching, diffusion, and interactions such as eddy collisions and mergers.

At the smallest scales, where the kinetic energy is dissipated, it is converted into thermal energy through viscous dissipation. This dissipation process is responsible for dissipating the turbulent kinetic energy into heat, leading to a decrease in overall turbulence intensity.

Therefore, while shear production locally generates turbulent kinetic energy, the energy is subsequently subjected to the energy cascade, where it undergoes transfer and dissipation processes. This cascade is an essential aspect of turbulent flows and helps to maintain the overall turbulent characteristics throughout the flow domain.

It's important to note that the details of the energy cascade and dissipation can vary depending on the specific flow conditions and turbulence models being used. Studying and understanding these processes are crucial for accurately modeling and predicting turbulent flows in various engineering and scientific applications.

There is no specific physical meaning in the shear production term. Why?

1) the integral equation for the point-wish kinetic energy has only the dissipation term.

2) The shear production term appears only after a certain decomposition is introduced. This decomposition is not unique but changes depending on the adoption of a statistical mean, a time-mean, a local volume-mean, a spatial filtering. For each choice, you have a different meaning of the main variable and the Reynolds stress.

3) In RANS there is no sense to talk about cascade energy since only the zero-Th order statistics can be deduced. All the turbulence scales are involved by the statistical averaging.

in conclusion, before to talk about the meaning and the mechanism that is involved in the shear production term, one has to clearly define what are the mean operato, fluctuations and mean velocity.

RANS Equations are written using a decomposition of the instantaneous quantities into average and fluctuating quantities then applying "statistical averaging". The latter should verify the Reynolds rules otherwise we would speak of something else than RANS equations. The time averaging operator verifies Reynolds rules when the signal is ergodic over a sufficiently long time scale compared to the time scale of turbulent fluctuations and the spatial averaging operator verifies these rules on a volume scale on which local statistical homogeneity could be assumed.

Howdy Gopal Sharma,

Please take a moment to check out my exchange with K. G. Mcnaughton on Page 2 of this discussion. You will find there, finally:

"My image that the formation of eddies causes a partition of kinetic energy from the flow into reduced pressure in the core of a rotation with enhanced pressure in the fluid is very important! Valid is a separate concern, but the point is that "pressure recovery" and fluid heating occur during the eddy formation, and later the "low" is filled by the kinetic energy of the eddy rotation as the eddy dissipates. I wouldn't mind having seen that in textbooks, at least as a speculation instead of relying on eddies becoming small enough to be, in essence, already dissipation heat."

Thank you in advance,

Happy trails, Len

Hi Keith,

Thank you for posting this nice discussion. Since the so called "Shear production term" involves u'w', I want to add a point, regarding the erratic behaviour of the quantity u'w' in atmospheric surface layer flows, where the constant flux layer is never achieved (specially in unstable conditions).

Keith and I wrote a paper on that in Boundary-layer meteorology, (Article An Empirical Scaling Analysis of Heat and Momentum Cospectra...

Cheers,

Subharthi

Subharthi Chowdhuri Thank you for this interesting paper which provides insightful analyses and measurement data on dynamic and thermal turbulence phenomena within the ABL, as well as their consequences on the calculation of the Monin Obukhov length, the estimation of which is of paramount importance in characterizing the stability state of the ABL and in predicting atmospheric pollution dispersion.

Subharthi Chowdhuri

yes, not only in ABL the empirical evaluation of the Reynolds stress components is difficult. I am not an expert of experimental devices but I am not sure anemometry can be able to measure the correct turbulent fluctuations. I tend to think they filter out some wavenumber components. Furthermore, is their dimension adequately small? In simple words, this device would produce something analogous to the results of the LES solutions.

At y+

This discussion seems to be heading towards agreement on a number of issues, and opening up new questions in others.

I think we are all agreed that the RANS equations are well derived and reliable, so far as they go. We also seem to be agreed that they are phenomenological equations, the physics being discarded in the act of Reynolds' averaging. Physical interpretation of their terms must therefore rely on information from outside the RANS framework. A prime symptom of this is that the set of RANS equations is not closed so that direct solutions are not possible unless closure assumptions are added. Their job is to put back in enough of the lost information that reasonable solutions can be calculated for particular situations. Well and good.

To return to our question, we can't address questions like "What is the mechanism of shear production?", without going beyond the RANS formalism. No such extra information has been offered in our discussion. Statements like "the energy comes from the mean flow" lie within the RANS framework and so are without substance. Indeed they are misleading, to they extent that they are taken as statements about physics. It is a pity that this is not more widely recognized.

An interesting byway in the discussion has been the question of what kind of averaging should be used in "Reynolds'" averaging procedure: ensemble, volume, time, line etc.. Many experimentalists, myself included, use fixed instruments, so our measurements are made in time. If this information is to be used to calculate eddy fluxes or to give information on the spatial structures of eddies then area or line averages must be calculated. This is achieved by appeal to the ergodic hypothesis in the form of Taylor's frozen turbulence hypothesis. This remains just an hypothesis despite its roots in the 19th Century. However, technology moves on, and temperatures can now be measured essentially instantaneously along the line described by a thin optical fiber (Yu Cheng et al. 2017; doi:10.1002/2017GL073499). The significance of this has yet to be absorbed into our science of turbulent flows.

Howdy K. G. Mcnaughton,

A Research Spotlight, Two Kinetic Theory Thought Experiments -Channel Flow with Cylinder and Dynamically Induced Asymmetric Diffusion, has been uploaded. It is much better organized and complete, should you be interested. Thank you, I really needed to think it through!!

Happy Trails, Len