Hi Vivek,

Good question. Actually, while there are some large vortices which remain due to the stirrer, the process of turbulent dissipation will continue.

There is nothing inherently unstable about a vortex. In fact, vortices are very stable structures, which is why they can persist for long periods (just look at the great red spot on Jupiter). Note as well that the fluid itself isn't rotating or trapped in the vortex, it's passing through, just at different rates, but the structure of the vortex remains.

Turbulence is a characteristic we associate with the fluctuation of fluid velocity vs the mean fluid flow. The mean flow is essentially the stable structures in the flow, although even these may vary periodically.

Anyway, the main thing to keep in mind is that turbulent vortices usually alternate in direction. Behind a cylinder they will appear to roll off the top and the bottom, but the same thing happens on a wall, with eddies shedding in the clockwise and counter-clockwise direction (relative to the plane of the mean free-stream flow and wall normal).

While a vortex is stable, multiple vortices are not. Two vortices curling in the same direction will actually attract each other, while vortices curling in opposite directions will repel.  The breakdown of large eddies into smaller ones occurs as a result of these interactions. As vortices are shed, curling in opposite directions, they push away from each other, but this push isn't uniform along the length of the vortex due to small variations in the vortex. This eventually tears the vortex apart into smaller vortices and the process continues at smaller and smaller scales, just as you described.

In your stirred tank, however, the stirrer is generating more vortices curling in one direction than another. As a result, smaller vortices curling in a common direction will accumulate (because they attract) to create larger ones.

I hope the explanation helps. 

Best regards,

Robin

The size and distribution of eddies will depend upon the forcing function, its energy imparted to the fluid medium. All eddies are an attempt to dissipate energy within the fluid. If the energy is low, then laminar flow will suffice; but, for large energy, exceeding the elastic properties of the fluid, this will cause turbulence. Essentially, the fluid cannot keep pace with the forcing function, thus deteriorating into turbulence.

Hi 

Exactly, size and distribution of eddies depend on forcing function and the energy it imparts to the fluid. In my previous example, rotation of stirrer is the forcing agent.

Now  the claim that the large eddy breaks into smaller ones to dissipate the energy-how is it valid? Because, if you keep the forcing function constant, then number, distribution and size remains constant. Then how the larger eddies transfer the energy to smaller eddies to dissipate the energy by viscous action (Since the all the eddies are getting energy from the forcing function)? 

Please try to analyse this.

Thanks

Vivek

Hi Vivek,

In the case of a stirer, you are right that in the statistical sense the largest eddies will remain the same if you keep the same rotation speed. Yet, that does not mean that large eddy do not transfer kinetic energy to the small ones.

On the contrary, the kinetic energy transfer will balance with your forcing so that there is no accumulation of energy at any given wavelength. Eventually, the dissipation by molecular viscosity at the Kolmogorov scale will also compensate the energy coming from largest scales and will therefore also be equal to you forcing (still in the statistical sense!).

This is exactly what people investigate simulating forced HIT, the mecanisms responsible for this transfer of energy along the different scales of the turbulent cascade.

Best regards,

Alexis

Thank you.

Yes the energy that is being provided by the forcing agent, is equal to the total kinetic energy of rotation of all eddies present in the system plus the heat energy produced due to the viscous dissipation. 

So what i understand is that the forcing agent provides the energy to the bigger eddies and the bigger eddies keep the energy required for its own rotation and transfer the rest of the energy to the rest of the eddies and this way energy transformation of goes on until it reaches  the smallest eddies to be dissipated as heat by viscous action. Is this right?

Look at it.

Thanks

Vivek

Vary the lengh scales (le) in the turbulence model . It may be helpful . higher le for  Large eddies and lower le for smaller eddies. try this ..

Vivek,

You are right, once the transfer of energy reaches the kolmogorov scale, the system is in a steady state in the statistical sense.

Best,

Alexis

Thank you sir.

If the stirring is stopped, then there is no forcing function acting anymore and then the bigger eddies length scale will be decreasing and eventually reach to the kolmogorov microscale and dissipate as internal energy. Right?

Kolmogorov microscale depends on the viscosity and the dissipation rate. How does the microscale depends on the forcing function (or the rpm in this case)?

Thanks a lot sir for responding to my questions.

Thanks

Vivek

@Vivek,

What you ask for is not so clear cut and most must measure to find what the inner scale is; but, I have this paper on estimating the turbulent scales: http://arxiv.org/pdf/1112.6033.pdf

Actually i am trying to understand all these physically.

My question was that does kolmogorov microscale depends on the forcing function or it is a property (viscosity) of fluid?

In my above example., i talked about the stirring in the liquid medium. If the stirring is stopped, the bigger vortices will become weak (its kinetic energy will be decreased) and

gradually it will be obliterated by viscous action once it reaches the microscale. 

I hope this will clarify my doubts.

Please keep the discussion going on.

Thanks

Vivek 

All right, consider a solid, it has some strength to it, Young's modulus, if you were to use some implement and apply enough force to exceed its intrinsic strength, then what would happen? It would crack and break. Now, fluids also crack and break; but, because they're fluid, the medium flows and conforms to the new conditions. 

If you apply enough force to form, say, two eddies, then fine, they'll rotate and dissipate energy along their edges; because, as they rotate, the edges heat up, the properties of the fluid changes and allows for dissipation. Keep in mind, the eddies rotating represent a certain amount of energy that is captured within that rotation and it rotates around and around, 'til it is dissipated; hence, the reason and purpose of eddies.

Now, increase the force and the two eddies suddenly are not enough to dissipate the extra energy, then more eddies appear, smaller eddies will form. Large eddies will rotate so fast, they too will cause yet more eddies to form, which is the same process--to dissipate extra energy.

Like I said, the specifics of how large the eddies, the number of them, the extent to which they cover the fluid space, etc, all these things are, in general, not too easy to answer; because, the process is a statistical process and there simply is no deterministic way of predicting--in all exactness--where, when and what will happen. 

Kolmogorov statistics are just that--statistics. He studied the typical behavior and attempted to find relations between material property and turbulence. You can relate viscosity, heat capacity and thermal conductivity to turbulence, as with other material properties. 

@Vivek

Attached is a paper where scales and other properties of turbulence are derived, including fundamental constants like von Karman's and Kolmogorov's using (stochastic) geometry and Fokker-Planck equations for interacting vortex-dipole ensembles. Actually we now understand turbulence well - in principle. If it comes to applications, eg. in mixer systems, at finite Reynolds numbers, in non-Newtonian fluids, some more thoughts need to be invested, what we did. In all cases the key variables for a statistical vortex  ensemble and its effects on scalar mixing are TKE and enstrophy. In stratified fluids the vortices still interact  with the fluid column as resonator ... You've chosen a complex problem...

well according to me kolmogorov microscale should  not depends on the forcing function instead it should be property of fluid alone. Depending on the viscosity of the medium the size of the smallest eddy could differ so does the size of the larges eddy generated. No matter how much energy is supplied ,  the microscale (length as well as time) is  function of system properties for example kinematic viscosity. However, the fact that kolmogorov microscale are function of turbulence energy dessipation rate assures that the dessipation effect of the medium is accounted. But as the energy is tranferred from one point in the medium to another there are losses that comes from the continuity of the medium itself , which are not accounted in the Kalmogorov scale. Thus in your example of stirrer at constant speed the size of the eddies at microscale is comparable to  the affinity of the medium to respond and absorb the turbulence.

Also i do not agree to you statement that the largest eddy retains the energy required to rotate and pass the rest to smaller ones that would  means you are considering the size of the eddies to remains constants for a fixed rotational speed. Instead it should be treated as probabilistic where the size as well as amount of energy in the eddies varies every instant  and the system should not allow constancy. The primary reason for the aformentioned lines  is the nature does not prefer constancy  and turbulence itself is a probablistic in nature. We cannot fix a scale to the size instead we outline the extremes.

This has been an excellent discussion of the current status of modeling small scale turbulence for which interesting approaches are compared. We are getting beyond "Big Whorls form......"

However for weather scale turbulence, the mesh sizes are far from micro scale and thus ignoring the imbedded vorticity (the spin tensor) within a mesh volume is questionable.  What progress has been made since I last looked at this problem in 1990?.

@Aditha

The Kolmogorov scale is a variable that characterizes quasi-steady states ("Fliessgleichgewichte") in a statistical ensemble-average sense in a fluid. It is defined as \lam = (\nu^3 / \eps)^1/4 wherein \lam =  microscale, \eps = dissipation rate and \nu = molecular viscosity. In the steady state we have \eps = forcing so that \lam depends very well on the forcing. Turbulence is not that easy, and it took about 100 years to arrive were we are today...

In addition of the answer from Alexis Giauque, the viscous dissipation will become in heat into the stirrer. Similar kind of phenomenon appears in centrifugal pumps. When you close the outflow valve, fluid circulates into the pump only, and the power that you give for moving the rotor is only dissipated by viscous and turbulent friction, therefore the fluid temperature rises...

Thanks to all for participating in this discussion. This is having a great opportunity for learning. 

One thing we are mentioning always is that viscous dissipation is the final process whereby the turbulent kinetic energy is dissipated as heat(internal energy of the fluid, temp raises). By the way what does the viscous dissipation means physically?

So what we can say that viscous dissipation is more near the walls because larger no of microscales are present there? That is the reason that epsilon(dissipation rate) is more near the wall compared to the bulk of the medium.

Keep the discussion going on. may be questions are simple, but lot of learning is there.

Regards

Vivek

When you rubs yours hands between themselves, you generates heat. In a similar way, the viscous dissipation is friction between fluid particles that generates heat. The Re number is a ratio between inertial and viscous forces, and Re number for dissipation is of order one (try to compute for have an idea, the length and times scales for dissipative vortices, for a bulk Re of 20,000 in a plane plate flow for example, using similarity laws, e.g. see Tennekes and Lumley's or Pope's books). The vorticity has de capacity to generates a great mixing. Fluid particles that initially were long away between themselves, thanks to vorticity then could interact between themselves.The viscous dissipation occurs in all places into the flow, depending of the geometry and flow but it, generally, is more confined in the boundary layer (there are strong velocity gradients here that produce vorticity...) where production, transport and dissipation of turbulence occurs.  But not at wall, where the velocity is zero and the no-slip condition must be fulfilled.

Thanks Miguel....

Generally viscous dissipation remains confined in the viscous sublayer because the viscosity effect is more prominent here in this region. 

Turbulence causes the mixing in the macroscale, microscopic mixing (molecular level mixing) is not achievable through the turbulence. Because it requires higher intensity of turbulence and large number of microscale vortices, which is very difficult to achieve?

give the counter argument.

Thanks

Vivek

Dear Vivek,

In my knowledge, classical theory of turbulence involves the concept of cascade of energy, implying three scales' ranges, or categories: Integral length scales, Kolmogorov length scales and Taylor microscales. (see e.g. Tennekes and Lumley, 'A First Course in Turbulence',1972). Here, a complete description of these scales are given. Also, maybe the Richardson poem (1922) could be inspiring for you: "Big whorls have little whorls that feed on their velocity, and little whorls have smaller whorls and so on to viscosity - in the molecular sense."

@Vivek & all:

The Kolmogorov range within the turbulence spectrum is more or less frictionless like a good mechanical gear or a clockwork. Dissipation of mech energy into heat happens in grainy 3D elements with "diameters" near the Kolmogorov scale. In these elements scalar mixing is slow. I.e. you get fast mixing by turbulence, but final smoothing of conc contours takes longer. At solid boundaries the energy-containing scale is not free but limited by the distance to the wall. Read my paper attached to my previous note ...

@Bruce:

Thanks for your paper on CAD and ship design. Yes, thanks to integrated circuits meanwhile engineers are far ahead of the state of 1990 and R. F. Richardson. Together with Kolmogorov he remains number one in turbulence. He's had a right nose for science but an additional talent in poetry. I draw my hat!

Dear Helmut,

Yes, a great scientist but, not so good in poetry, isn't?, despite of that, his poem is very know in ' the turbulence's community', it's not little...

Thanks a lot Prof. Miguel Coussirat and Helmut Ziegfeld Baumert for putting some thought provoking comments. I found some mistakes in my understanding. 

Surely there are small vortices in the larger vortices (integral length scale), and smaller will have even smaller ones (Taylor microscale) and finally vortices with length scale equivalent to the Kolmogorov microscale, when dissipation takes place.

Now Kolmogorov microscale large enough compared to the molecular length scale. So the final smoothing of concentration does not takes place by turbulence alone. Molecular diffusion becomes significant in this period to make final (blending) smoothing.

So the siize of those 3D grains depends on the dissipation rate and dynamic viscosity.Dynamic viscosity is the property of fluids, which can't be changed, but, dissipation rate can be changed by means of turbulence in medium to bring the Kolmogorov scale smaller as much as possible near to the molecular scale.

Make an argument.

Thanks

Vivek

I understand that this generalization should not be done. But put some comments so that we can further carry on the discussion.

Vivek, you gave a more or less correct verbal description of turbulence. Note that the closure problem is solved today and new challenges wait for young guys like you. From my point of view it is (1) internal waves and the so-called wave turbulence including mixed aspects which deserve attention, and (2) the functioning of turbulence in non-Newtonian fluids and further (3) all sorts of turbulence in compressible media. Yes, also collapsed turbulence (Kolmogorov scale = energy-containing length scale) is a practically extremely  relevant point, but with löw degree of fundamentality. Well, it's difficult to classify because it is a degenerate case ...

I'm in agreement with Helmut.

From another viewpoint, Computational Fluid Dynamics (CFD) recently gives some lines for understanding turbulence phenomena also. Recent works concerning to Direct Numerical Simulations (DNS) in simple geometries and for moderately-high bulk Re, (aprox 20,000-50,000) give some insight about coherent structures and vorticity distribution, but at the moment, DNS is in its infancy due to, the required computational resources are enormous!!!, e.g., channel flow:

Reτ = ρUτh/μ, y+ = ρ yp uτ /μ, Uτ = (τw/ρ)^1/2τ = Ue (cf/2)^1/2=, h=height/widht of the channel, yp disntance from wall for first grid point and cf computed from 'classsic' formulations for skin friction coeff.

Grid requirement: N ~ (Reτ)9/4 ~ 1x10^7 for Reτ = 800, N=number of nodes

Time step requirement: Δt ~ (Reτ)-1/2 ~ 1x10^-5 for Reτ = 800

Despite of that, the works related with DNS studies are becoming increasingly frequent.

Thanks Professors for having a nice discussion.

I am following a book of Wilcox, a nice book.

Is there any other good and easily understood book.

Thanks

Vivek

Vivek, Wilcox is a great choice. His fine nose and solid physics background led him to a system of eqns which is fundamentally correct (in contrast to k-eps or k-kL etc.).

You may learn a lot phenomenology from Monin & Yaglom, 1st volume. (The second volume, mainly by Akira Yaglom, has only limited results, frankly spoken - correct, but "not even wrong" like Wolfgang Pauli once characterized such texts.) Tennekes' & Lumley's "A first course in turbulence..." is also great. A fine booklet but not necessarily for the beginner is Uriel Frisch's "Turbulence: the legacy of A. N. Kolmogorov". However, you should read the paper I recommended earlier, wherein von Karman's constant is derived as 1/sqrt(2*pi), and wherein the system of k-omega eqns by Wilcox are shown to be fundamentally correct.

Without doubts, Wilcox, and Tennekes & Lumley are nice options....in the second one there are a lot of exercises in order to understand better the topics. Pope's book is also good. There is other, from Davidson, An Introduction for Scientists and Engineers', also so good, where k-eps models are derived in a so pretty way, and one from Durbin and Pettersson 'Statistical Theory and Modeling for Turbulent Flows'. also with exercises.

Miguel,

without doubt Davidson is a great (and voluminous) book, but please accept that k-eps CAN NOT be derived from conservation laws, it only may be "written down".

The only  conserved mech variables in turbulence are turbulent kinetic energy, TKE, and enstrophy (~ r.m.s. vorticity ~ omega). That's "forgotten" by almost ALL engineers, including Rodi, Launders, Spalding and the many others (myself included when I was a beginner). Wilcox is an exception although he did not derive the equations; he also "wrote them down". A derivation from firtst principles you only find there where you also find 1/sqrt(2*pi) for Karman's constant ...

Dear Helmut, I think that Davidson book has a interesting explanation in terms of dimensional analysis, for the k-eps model, but only, for the k eq (as all books treting this topic). All people that also use the epsilon equation for closing the turbulence model, including RSM models know (I think so...) that the transport equation that results for eps, is only derived from dimensional basis without physical ones, as the k equation.... and for sure, I being in the path of beginners in turbulence is true for me... I have engineering formation... and it's very difficult to solve this 'strong restriction' . I'm an 'user' of turbulence models and at the moment I'm exploring the SAS option in SST k-om in order to model cavitating flows in turbomachinery (complex flow/geometry) because LES or other options, are not feasible for such flows. Best regards.

k-om is the physically correct choice, and SST k-om is nice in particular as it is said to apply to high AND low-Re flows (I have not tested it. It is an industry code.) It would be worth to check whether the many coefficient functions in SST k-om CONVERGE to my theoretical values for Re ==> \ifnty. Do you have a student who might check that, with active support from my side?

The formation of smaller vortices always takes place in regions where there is a shearing of the fluid and where the shear layer is unstable w.r.t. vortex formation (this is not the case on length scales smaller than the Kolmogorov length).

In your case the stirrer creates a shear layer, in which large vortices are formed. In the shear layers surrounding those vortices, smaller vortices are formed and so on to the Kolomogorov length scale (actually the smallest vortices are several times bigger than this length).

So energy is transferred from the stirrer to the the shear layer and from there to the smallest vortices, which dissipate into heat.

@Michael: Not even wrong!

>>>actually the smallest vortices are several times bigger than...

An important practical problem is given if these two characteristic scales approach each other. If they become identical then "turbulence as described in the books" (the Kolmogorov cascade) ceases and it remains a chaotic motion with strongly reduced mixing efficiency. From an outside view it is difficult to differentiate between the two states motion. It is actually the spectrum which makes the difference, and the motion of vortex dipoles controls the mixing. This is or some relevance in non-Newtonian fluids ...

@Helmut: I agree. My answer assumed (tacitly, and maybe wrongly) that the stirrer is operated at sufficiently high Reynolds number, so that a fully developed turbulent spectrum can form, and the fluid is Newtonian. Stirring at low Re number obviously creates different phenomena and non-Newtonian fluids offer another broad range of possibilities ...

@MIchael: In a non-Newtonian fluid you may start stirring with high Re and ending at a  Re plateau well below the initial Re value. And vice versa!

Thanks all for sharing your expertise. I just got to know so many things.

Shear layer is created in the region where there  is  large velocity gradient, region surrounding the large vortices. Like in the case of the stirring, large vorttices are formed in the region between stirrer shaft and and the container wall. Shear layer is present at the outside of the large vortices.

I hope this is correct.  

I am now slightly deviating from this discussion towards the multiphase flows.

When particle moves through the fluid medium, one time scale is used to characterize its motion is the particle's velocity relaxation time scale, tau= mp/6*pi*eta*R, R is the radius of the particle, mp is the mass of particle.

My question is what is physically meant by relaxation time scale?

When bubbles flows in a liquid in the turbulent flows, bubbles try to accumulate inside the vortices, whereas particles  in the gas flows, tend to accumulate in the shear zone.

Why this happens?

Please got through above points and put some light on this.

Thanks

Vivek

Your first sentence is mainly correct. You should, however note that large vortices may be part of the main flow field (not all vortices are created by shear layers).

The particle relaxation time is a time by which the particle velocity (if different from the local flow velocity) relaxes toward the flow velocity by friction.

Inside strong vortices the pressure is lower than outside. So there is a buoancy like force, heavy particles are transported out, while light particles (bubbles) move in.

Thanks for your reply..

Whether the particle velocity will be picking up to the fluid velocity depends on the several factors, like mass of particle, size, density. Sometime particle is so big that it will never flow in the same velocity as that of fluid. What i mean is that when we throw two material in the river, we see that one material flowing nearly with the velocity same as that of fluid whereas other material lagging far behind. 

So the relaxation time for the former particle is finite but for the latter particle is infinite?

In case of latter part of your answer, it is true that pressure inside the vortices are lower than in the shear layer. Buoancy depends on the size of the particle and the density of the fluid. If the bubble size and the particle size are same, buoyancy force acting on them will be same. Whatever pressure difference you are talking about is in the continuous phase. How this pressure difference is related to the buoyancy? 

Please look at this and put some comments.

Thanks

Vivek

This discussion is related to explaining the apparent reduction in turbulent intensity of dilute polymer solutions for drag and pressure drop reduction.  I've always regarded the polar moment of inertia of the polymer as affecting the generation of turbulent vorticity but never thought of it as multiphase flow. 

@Bruce & Vivek:

Interesting points, whereas I think that the debate initiated by Vivek was so far interesting and  fair but now, with Vivek's last question, tends to leave a simple academic Q&A game, moving towards a serious engineering study of some length and effort, which may possibly exceed the limits of an open-source exchange of views.

The relaxation time of larger particles is larger, true, but also a very large particle will eventually move at the fluid velocity (assuming it to be spatially and temporally constant), since there is a finite friction force which becomes only zero if the relative velocity is zero.

Regarding your second question: the pressure forces are the same, but in rotational flow you also have the centrifugal forces, which acts on mass. This is large for the particle and small for the bubble.

I think after some time the drag force (which depends on the relative velocity) will become become constant and there will be some finite relative velocity , which means particle velocity will never equal to the fluid velocity.

What i am thinking in the case of second question that in the vortex flow, velocity is higher inside the vortex and lower in the outside shear layer, now flow inside the vortex is essentially is a laminar flow, if we apply the Bernoulli equation we can say that pressure inside the vortex is lower than outside. May be we can say that for bubble, this pressure gradient is higher than the centrifugal force, forcing the bubble to be inside the vortex, whereas for the case of particle centrifugal force dominates over the pressure gradient forcing it out of the vortex.

Look at this.

 Thanks

Vivek

regarding your first assumption: according to Newton's second law a net force will accelerate the particle, so the constant drag force should accelerate it. The drag force only becomes zero if particle and fluid velocity are equal. If you take gravitation into account, then indeed the particle will acquire a constant velocity downwards, this effect is used to measure viscosity.

Ya that means for some particles, particles will never reach the fluid velocity, which essentially means that for those particles relaxation time is infinity.

Hi all,

I am going in to the multiphase flow again. 

I was going through a book of multiphase flow written by Brennen.

When particles mode through a  fluid phase, then a force called the induced inertia or added mass force act on the particle. So how is this force originated? Is this force same as that of buoyancy force? 

Please keep the discussion going on.

Thanks

Vivek

THe size of turbulence also depends on size of inception points  and surface roughness of stirrer.

Just read p. 62-63 in Brennen's book. The induced inertia occurs in unsteady particle movements even in potential flow.

Thanks

I understand.

I am going to the different topics now.

In the LES (large eddy simulation), filtering operation is done using the filtering kernel.

Can you please anyone explain this operation and this kernel in detail?

Thanks

Vivek

hello, just an observation...the energy cascade can be also reversed in some type of turbulent flows...the merging of vortical structures in quasi 2D flows is an example.

furthermore, turbulence can be inhibited by rotation ..

I would rather comment on relaxation time of particles. The fact that the particle velocity of particles in a fluid with gravitation do not agree with the fluid velocity does not mean that the relaxation time in infinite. The relaxation time is just defined without external forces on the particle.

Thank you Professor.

Yes when there is no external fields like gravitational field, then relaxation time does not work.

But in absence of external field also, say i am giving an example. Say a large body is thrown into the river along with a small body. Relative velocity between fluid and the small body becomes zero very quickly compared to the larger body, which takes larger time to reach the fluid velocity. Even i think it may not reach the fluid velocity, though i am not very sure.

Look at the question on LES also.

Thanks

Vivek

Dear Vivek

filtering in LES can be subdivided in two categories: implicit and explicit

implicit: the equations are only formally filtered, a real filter operation does not exist and is not coded but both the computational grid and the numerical discretization act implicitly as filtering.

explicit: the equations are filtered and the filtering operation is really coded, often acting on the convective term.

Note that the dynamic SGS modelling requires a test-filtering, an explicit filtering operation that is coded both for implicit and explicit-based filtering.

You make a very good point: if small eddies of the size of polymers chains are perturbed, why should cascade at larger scales be changed? This notion of cascade is actually quite vague and has not been tested precisely, it is unclear how it represents processes taking place actually or is just an image. Another possibility for explaining dissipation in turbulence is by the occurrence of finite time singularity in solutions of Incompressible Euler equations, a deep undecided questions.  This finite time singularity and the ensuing dissipation could be perturbed by polymer additive. My personal opinion is that people have a more critical look at those questions than they usually do. 

Yves Pomeau

My answer has more journalistic form, but in my opinion the theory of turbulence is in a very awkward position. The general idea of this theory was expressed by the "order of observers", i.e. by researchers who proposed this descriptive idea, where the terms "coherent structures" or "cascade of vortices" play the basic role (I agree with Yves Pomeau that this formulation is quite vague).

Maybe this attitude could be an useful strartpoint  for the further efforts, but it was suddenly overwhelmed by the swift development of CFD. In consequence the "order of calculators" appeared and so many (too many!!!) scientists try to simulate the turbulent flows making ud]se of (almost always commercial) computer software.

I will risk some negative comments, but in my opinion an especially negative role (remebering about positives) is played here by the "k-epsilon" model. It is an extremally speculative construction, but since it gives so attractive results (as claim so many authors of papers) - practically nobody tries to take up more theoretical considerations  on turbulence.

But let's go back to your topic. Generally we must distinguish two different families of vortices - let's say "advective" (generated by the flow conditions and the system geometry) and "fluctuative" (which really are the essence of turbulence).

Your stirrer will obviously create one main "advective" vortex and probably some smaller (depending on the chamber shape)  and you can simulate them easily using existing possibilities (CFD), but what about these fluctuations and transport of energy between vortices?

The concept offered by "observers"  doesn't explain this problem, as you have already stated in your question. Unfortunately, existing models of turbulence (especially LES, together with all variants) is also helpless. In my opinion we should come back to the theoretical roots of hydrodynamics. Who will be the lucky finder?

Cordial greetings - Jerzy M. Sawicki

@Jerzy:

I basically agree with most of your points, but k-eps is less negative if we put it in the wide turbulence-closure landscape of k-kL, k-om, k-eps ... models and their so many relatives and modifications.

I made an attack to solve the turbulence problem and could finally succeed (attachment), only BECAUSE I once did start with k-eps! I learned that the common structure of ALL the phenomenological closures is identical with a system of Fokker-Planck equations (FPE) with sources and sinks. FPE is nothing else than a generalized form of the dynamic equation of heat conduction (or Brownian motion in Einstein's sense) with spatially variable conductivity and with sources and sinks. Key point: It is a CONSERVATION LAW for the variables under consideration. Major question then: Is eps a conserved quantity? Clearly not, and also not the other combinations - with one exception: k-omega. Only energy (TKE) and enstrophy (omega, the rms measure of vorticity) are conserved in Eulerian turbulence. This gives you from a purely geometrical (stochastic-dynamic) theory without any empirical param the values of the three major universal constants of turblent motion. A most beautiful result (at least for me, if you allow) is von Karman as 1/sqrt(2*pi) = 0.399. We all know the international standard value based on measurements is 0.40. More complex expressions are found for Kolmogorov 1 and 2, in good agreement with the observations.

Now, what was the basic trick to get this? I have seen these analogies of semi-empirical closures with FPE and knew that FPE is a particle-based picture. But how connect particles with turbulence? Well, I knew that vortex tubes in Eulerian turbulence are closed lines and that dipoles of those tubes need to move in space, and that they may collide etc. At the end it was the change of view point, from the (hopeless) continuous picture of Navier-Stokes to the singular solutions of Euler taken as particles.

Hi, Helmut! Yes, I agree that you have mentioned many very important aspects of the turbulence closure models and that so many very useful things can be done. I only wanted to underline (in this "journalistic" form, i.e. with exaggeration), that we are forgetting about the basic "philosophical" layer of this problem - viz. the turbulence is generated even when nothing changes in time in the system, that this phenomenon is a kind of  self-inducing fluctuations. And from the mathematical point of view we are not able to get the unsteady solution of a differential equation for the steady boundary conditions. But a very time-consuming discussion should be necessary in this case (with a proper amount of bear) - sincerely - JMSawicki.

Hi, Helmut! Yes, I agree that you have mentioned many very important aspects of the turbulence closure models and that so many very useful things can be done. I only wanted to underline (in this "journalistic" form, i.e. with exaggeration), that we are forgetting about the basic "philosophical" layer of this problem - viz. the turbulence is generated even when nothing changes in time in the system, that this phenomenon is a kind of  self-inducing fluctuations. And from the mathematical point of view we are not able to get the unsteady solution of a differential equation for the steady boundary conditions. But a very time-consuming discussion should be necessary in this case (with a proper amount of bear) - sincerely - JMSawicki.

Jerzy, I absolutely agree, the laminar-turbulence transition is still poorely understood (or even not understood at all), an open wound, and at least for me remaining a conundrum. This contrasts with developed turbulence, thanks to intermediate steps ahead like k-eps and Mellor-Yamada and others, in particular Wilcox who was introducing the basic features of k-om (at Boeing, if I am not in error).

Regarding math of Navier-Stokes (NS), Ed Feireisl and Antonin Novotny worked hard about singular limits and explained me in Prague (with a certain beverage) that the key is the qualitative change of the boundary conditions if you approach hyperbolic Euler from parabolic NS by reducing viscosity. This I understood as main challenge for mathematicians, and even the Clay Prize (armored with ~10^6 US$) didn't help solve the problem - so far!

Best wishes to Gdansk, the destination of my first bicycle tour abroad from Berlin, summer 1965 ...

Dear Helmut,

I used several RANS models: the Spalart Allmaras model, both k-eps and k-om 'families', the v2f from Durbin, and also some versions of RSM in order to model several configurations of isolated and impinging jets systems (heat transfer), and isolated airfoils (e.g., NACA0009, 0015, 066), blade's cascades, Venturis, injectors (for comparing mean and fluctuating velocity fields, Cl and Cd coefficients, and void fraction in cavitating flow problems). After a lot of work, only in some cases the SST version shown a little better results in mean values of velocity and other mean quantities (e.g. lift/drag coefficients or Nusselt number), in others the performance was similar as other RAS (or RANS) models. In spite of that the results for mean values are encouraging, the fluctuating quantities are in general poorly fitted. I did my bet for RANS models because the LES option is not affordable for industrial flows. My main goal is to know the sensitivity of the several parameters that these models have (parameters in the production, destruction, other terms...), in order to apply them to modeling complex flow/heat transfer problems. I think that despite that many people work in this subject, the problem is open yet. Maybe I can get an interested student in order to check your proposal. Other problem is that here there is not beer, there is wine!

Regards,

Miguel

Wine and turbulence!!! I am going to Mendoza on foot, but for the time beeing - away on holidays with my grandchildren - greetings - Jurek.

@Miguel:

The problem is open in so far, and I explicitely agree with your points, that the theory I propose is only valid under idealized conditions. E.g. ideal boundary layers, ideal honogeneous turbulent box, ideal linear stratification and infinite Reynolds number etc. One may compare this with Maxwell's equations which predicted the exitence of electromagnetic waves in vacuum, without which we today would have neither smartphones nor twitter not intergalactic (potential) communication etc. If you look at the details you will find easily that Maxwell is NOT enough for electronics. One needs MUCH more. But without Maxwell those applications would not be understood and could not effectively be modified (only by trial and error) etc. etc. Similarly the laws of mechanics: They deal with idealizations like POINT masses (!) for things like Sun or Earth ... With the new theory we today really understand turbulence; it has opened  a road towards various applications. I am just working about some of them but will not count chickens before they are hatched. One most simple but maybe laborious point would be to check a number of k-om versions (a list will possibly always be incomplete...) in their limit case of Re ==> \infty and compare them with my theory. It would be pure paperwork: collect all available papers about k-om, bring them in a unique format and compare the coefficient functions for high Re, no experiments needed, even no programming, only LATEX maybe needed to publish the results.

With respect to wine, I prefer Riesling from Saar/Mosel and light versions of Late Burgunder. But I admit that my experiences with red wines from Latino America is too limited to judge. So I am afraid that we need to change our methodology: practice rather than theor. hydrodynamics, an international effort together with you and Jerzy to find out major features of turbulence in certain test liquids -- before and after digestion ... ?

Dear Helmut (and Jerzi)

It is very interesting your viewpoint... and it is encouraging your opinion in order to follow in my path. Of course the door of possible collaboration is open... and of course, after the digestion. Best regards...

Different size boif pertubation points give rise to diferent size of turbulence

Hi Vivek:

You do not need to invoke Kolmgorov's theory of isotropic, homogeneous turbulence to characterize creation of small scale- which is only true for 3D flow invoking vortex stretching. Another approach would be to look at inhomogeneous flow based on enstrophy transport equation, as given in:

Diffusion in inhomogeneous flows: Unique equilibrium state in an internal flow: Tapan K. Sengupta, Himanshu Singh, Swagata Bhaumik, Rajarshi R. Chowdhury, Computers and Fluids, 88 (2013) 440–451

This is explained for 2D flow also, which does not have vortex stretching.

Hope it helps.

Hi all after a long time I am here.

Thanks a lot  for having nice discussion so far. 

First coming to the Dr T. K.Sengupta's suggestions. Is it possible to approximate the turbulent flow in the 2D, because turbulent flow is inherently 3D with vortex stretching present.

Thanks

Vivek

Some turbulent flows have the chance to be considere 2D in statistical sense. That depends often on the geometry and is at the basis of the RANS simulations.

Geophysical flows are also suitable to be characterized in 2D for the large dominant eddies that are several order of magnitude greater than eddies in the third dimension.

However, DNS and LES formulations are 3D according to the physics of turbulence

Thanks Maria for explaining. Some trivial queries are their.

what is wavelengths of fluctuations in turbulent flows?

Is is same as the turbulent length scales?

Thanks

Vivek

Hi Vivek,

Good question. Actually, while there are some large vortices which remain due to the stirrer, the process of turbulent dissipation will continue.

There is nothing inherently unstable about a vortex. In fact, vortices are very stable structures, which is why they can persist for long periods (just look at the great red spot on Jupiter). Note as well that the fluid itself isn't rotating or trapped in the vortex, it's passing through, just at different rates, but the structure of the vortex remains.

Turbulence is a characteristic we associate with the fluctuation of fluid velocity vs the mean fluid flow. The mean flow is essentially the stable structures in the flow, although even these may vary periodically.

Anyway, the main thing to keep in mind is that turbulent vortices usually alternate in direction. Behind a cylinder they will appear to roll off the top and the bottom, but the same thing happens on a wall, with eddies shedding in the clockwise and counter-clockwise direction (relative to the plane of the mean free-stream flow and wall normal).

While a vortex is stable, multiple vortices are not. Two vortices curling in the same direction will actually attract each other, while vortices curling in opposite directions will repel.  The breakdown of large eddies into smaller ones occurs as a result of these interactions. As vortices are shed, curling in opposite directions, they push away from each other, but this push isn't uniform along the length of the vortex due to small variations in the vortex. This eventually tears the vortex apart into smaller vortices and the process continues at smaller and smaller scales, just as you described.

In your stirred tank, however, the stirrer is generating more vortices curling in one direction than another. As a result, smaller vortices curling in a common direction will accumulate (because they attract) to create larger ones.

I hope the explanation helps. 

Best regards,

Robin