Dear Leonard Hall

Let me start discussing first the incompressible flow model, that is the case where the velocity field is divergence-free. What you call "pressure" has no physical meaning in energetic terms since has no equation of state. On the other hand, if we use the vorticity equation, no pressure gradient appears in the vorticity dynamics. Thus, curvature of a votical structure (that is an eddy) depends only on the vorticity dynamic (convective transport, stretching term, diffusion). The kinetic energy equation is derived from the momentum equation and the only term containing the "pressure" is the Div(vp) term, p (Div v) being zero.

The second discussion appears in case of a compressible flow model, this framework changes, you have an equation of state for the pressure and the dilatation term in the kinetic energy equation appears. On the other hand, the vorticity equation has a barometric term.

In conclusion, I think you are addressing the second case.

Howdy Filippo Maria Denaro,

Thank your for your contribution. I'm very pleased to have your response. We will need the usual mathematically oriented models in the long run and having them "up front" from an expert is great.

Incompressible flow is a convenient fiction for mathematicians and engineers, very valuable in its own space, but it is useless and misleading when one seeks to understand the dynamics of actual fluids.

To clarify, I am not addressing the equations in either case. My objective it to understand "how" with respect to the nature of energy transfer from flow kinetic energy to heat. I have proposed a mechanism as a substitute for an energy cascade to "tiny eddies that are already thermal in effect" which is my idea of the cascade answer. Incidentally, the inertial subrange is an expression of the geometry of eddy sizes, and may not be that valuable as evidence of how the energy cascades.

I look forward to your comments on my view in my question notes since your expertise treats the phenomena from a different perspective.

Happy Trails, Len

P. S. My exchange with K. G. Mcnaughton on page 2 of his question What is the mechanism of `Shear Production’ in turbulence? may help. lfh

Hi Len

without any use of mathematics, I can provide an example. Immagine turbulence for inviscid flows. If your hypothesis would be correct, dissipation of kinetic energy would appear. Actually you would see an energy cascade extending up to infinite wavenumbers. That means the pressure is not able to dissipate mechanical energy.

Howdy Filippo Maria,

Excellent! Another important element of the discussion right up front!

Inviscid flow is also a convenient fiction for mathematicians and engineers. One cannot accept the inviscid "convenience" after reading about the Kinetic Theory of gases and liquids in several volumes from Jeans' The Dynamical Theory of Gases to Chapman and Cowling's The Mathematical Theory of Non-uniform Gases. Real fluids mix at molecular scale and inviscid is impossible. This is also obvious because sets of molecules cannot support infinite wavenumbers and inviscid theory produces them. I'm afraid I cannot imagine turbulence for inviscid flows.

As for pressure "dissipating" kinetic energy: the process is that the formation of the pressure gradient in a curved flow "distributes" energy which was formally kinetic into random molecular activity. Imagine an adiabatic, inviscid reduction of a volume of air to 1/16 of that volume. It gets so hot it ignites a spray of diesel fuel. The distributed energy appears as increased pressure, for example, pressure recovery behind a cylinder.

This is a different way to view fluids in motion, is it not at least a direction worthy of exploration? Can it be just wrong?

Happy Trails, Len

Hi Len,

yes, inviscid fluid model is indeed an approximation of the real turbulence. Kologorov discussed about that in its theory.

I used it because such approximation has no effect on the role of the pressure.

However, I see you prefere to see the the dynamics of real flow in euristic way.

At this point, I need to introduce some math to discusse the role of the pressure into the kinetic energy equation. I am talking about the general equation, that is I do not introduce any decomposition in mean and fluctuations parts.

The pressure enters by means of the isotropic part P of the stress tensor.

One writes the isotropic part (Newton model):

P =[-p+(lambda+2mu/3) Div v]

From this expression, you see that your idea can be right if we consider the action due to P, not to the thermodynamics pressure p.

But, if the flow is divergence-free there is no action of the two viscosity coefficients. What is more, for compressible flows we should debate about the validity of the Stokes hypothesis (lambda+2mu/3)=0. Actually, this aspect is debated in case of strong compression (shock wave).

If we focus only on the role of the pressure p = rho*R*T (EOS) into the kinetic energy equation, one gets

R*Div (rho*v*T) - rho*R*T (Div v)

and there is only a convective transport of internal energy in the first term.

Where the dissipation appear?

Howdy Filippo Maria,

"From this expression, you see that your idea can be right if we consider the action due to P, not to the thermodynamics pressure p.

But, if the flow is divergence-free there is no action of the two viscosity coefficients. . . ."

Perhaps it would be relevant to consider a flow that is not divergence-free like a fluid would use, after all a young mountain stream does things about stones in a rapids that we cannot calculate exactly, and the gentle breeze that brings you the scent of a rose through the rose bushes is also hard to calculate. Affirming the value of mathematics as far as it goes, lets look at a volume of a fluid for a moment, say air to accommodate an atmospheric scientist.

I accept kinetic theory and therefore the volume is filled with molecules in motion for me. We may measure temperature, pressure, bulk motion, etc. in the volume, each of which may be related to the molecular unrest which may include a common vector component for all molecules due to motion of the volume. When acceleration of the bulk motion occurs it is because of a pressure gradient! This was a key realization back in Oct. '82 when I was wading through the convenient fiction of potential flow and its associated speculations. My idea is that the partitioning of the energy forcing the acceleration, say a flow curvature, separates it into pressure increase and pressure decrease with the latter somewhat balanced by the kinetic energy of the curving flow. This is non-viscous pressure recovery and transformation of flow kinetic energy to random molecular motions. If the equations are complete, it will be there.

A helium balloon moves forward in a closed vehicle during acceleration of the vehicle because the air in the vehicle also must be accelerated by a forward directed pressure gradient in the vector sum of gravitation and acceleration. When a flow volume is slowed into a curve it is forced by an adverse pressure gradient. Now, if one recognizes lower pressure inside the curve (vehicle windshield) one must see higher pressure beyond it (vehicle back window). If the equations in use still do not cover this dynamic case, there remains work to do - if the idea is as valid as I think it is.

And yes, my question is about "how it happens," not how can we calculate and use it without understanding the process in detail. I seek an heuristic view to see in a mountain stream eddy beyond a stone.

Happy Trails, Len

Dear Len

any intuition in science requires to be translated in a physical and mathematical model.

Since you are approaching your idea within the kinetic theory and you see the fluid as an ensemble of molecules, you are out of the limit of validity of the continuum model. Thus, there is no longer a macroscopi concept for density function, pressure function, velocity function, temperature function and there is no gradient operator to work with. Your model considers elastic/anelastic interactions between molecules, based on simple Newton law. This model is indeed used, for example in terms of lattice Boltzmann.

In this framework, no macroscopic viscosity is introduced and dissipation of energy is differently described.

I am not an expert of molecular theory, I stay within the continuum model of a fluid.

Regards

Filippo

Howdy Filippo,

Well written and clear; I agree. Thank you for helping from the continuum model perspective. I still seek an answer to the original question in any formal treatment that can handle it, of course.

My original approach was a thought experiment to explain the presence of a vortex in the wake of a bluff body. I visualized asymmetric diffusion caused by the common vector added to molecules sharing bulk motion past the bluff versus simple molecular unrest in the wake of the bluff. In the corner molecules would depart into the flow, but "replacement" would occur beyond the corner. It is written up well enough in my presentation Structure Formation in Turbulent Flows should that be of interest, but I prefer further discussion here to sending contributors elsewhere.

Again, thank you for your valuable contributions.

Happy Trails, Len

Len,

I cannot add much more into this discussion. My opinion, staying within the continuum hypothesis, is that the only way pressure can dissipate kinetic energy is if the second viscosity coefficient does not vanish and there is a change in the volume of fluid.

That means removing the Stokes hypothesis.

On the other side, from the microscopic point of view, your question is not in my specific field of expertize. I think that in such framework, ”pressure dissipation“ is something due to an anelastic interaction.

Best Regards

Filippo

Howdy Filippo,

Very interesting, and potentially very important observations, thank you!

I've briefly refreshed my memory on the Stokes hypothesis and I intend to look further. It may be the assumption that separates the continuum model and the molecular unrest model inappropriately in their mathematical forms. That reminds me of a comment by George Bernard Shaw “The British and the Americans are two great peoples divided by a common tongue,” but then I associate things that way. I started from the question "Why does the flow curve behind an obstruction" and couldn't live with the available answers.

My concept of Dynamically Induced Asymmetric Diffusion (DIAD) sketched earlier does transfer molecules between volumes mixing their properties. Very interesting. Incidentally, pressure recovery may be accurate but confusing by placing the wrong emphasis on what is actually governing the result.

Happy Trails, Len

Hello Len,

Stokes hypothesis is largely used in fluid dynamics, however some debate about the validity appears in case of strong compression.

I would'nt focus too much on structures due to geometry of a body, vortical structures appear also in free turbulence.

Thus, your question is open to 1) macroscopic level: action of the second viscosity coefficient in the Newtonian flow model 2) microscopic level: anelastic interation between molecules whereas energy is not reversible.

Howdy Filippo,

Actually, in my efforts to write this up I have considered the asymmetric diffusion to occur wherever flow passes anything including other flow elements. The bluff body version is simple to visualize, but one finds strong interactions between a rising thermal and a flow with curling structures in the newly impacting fluid. In the K. G. Mcnaughton question on Shear Production I wrote: "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."

I seem to be questioning the Stokes hypothesis quite generally. The process I imagine is so clear in my mind that I had better step back from it and take a deeper look into the theory and physics of bulk viscosity. You are enriching my awareness, which is "why" discussion in my book.

Happy Trails, Len

Howdy Folks,

I have looked into the Stokes hypothesis (lambda+2mu/3)=0, first in Wikipedia for convenience and little satisfaction. I then turned to G. K. Batchelor for science and Frank M. White for engineering views. Sent by reference to Sir Horace Lamb's Hydrodynamics (6th Edition 1932) as the originator of volume viscosity (1st Edition 1879), I found in Art. 325 and 326 that the Stokes hypothesis equation is implied by the definitions of p in Art. 325. So far I have found no physical justification for it despite its place as a principal component of the Navier-Stokes Equations.

After 40 years of intermittent thought and presentation, this issue of Stokes hypothesis is very encouraging to me as a route to acceptance of asymmetric diffusion in turbulence. If it is real, asymmetric diffusion will matter and may have been worth examination and tracer experiment to a career.

Perhaps the question has become: "Does shear viscosity effecting asymmetric diffusion lead to bulk viscosity dissipation in turbulent flow?"

Happy Trails, Len

Howdy Folks,

Did I just explain "how" the Stokes hypothesis is implemented in a turbulent flow? That is, shear viscosity transfers kinetic energy by asymmetric diffusion to appear as bulk viscosity energy dissipation. Hmmm, interesting.

Happy Trails, Len

Len, the "bulk viscosity" is generally disregarded in turbulence. To be its value relevant, we need to consider strong density variations such as that in shock waves.

Howdy Filippo,

Yes, I have read that also. Since you have reviewed my Structure Formation in Turbulent Flow it is apparent that you are not convinced of the significance of asymmetric diffusion. Whether it exists so slightly it may be disregarded is unknown to me, but you have seen how I think it happens.

Also, just for the record, I would still like an answer to my original question of "how" flow kinetic energy is dissipated.

Happy Trails, Len

Hello Len,

1) The conventional form of the diffusive flux of momentum is 2*mu*Grad v. This way, the macroscopic diffusion is modelled as an isotropic homogeneos term while the continuum hypothesis allows to model v as the macroscopic mean velocity of a small volume of molecules.

What you denote as "asymmetric diffusion" (I assume diffusion of momentum) should be expressed by means of a tensor form of the molecular diffusion. If you would introduce an "anisotropic" diffusion, you should also re-define what is the macroscopic velocity in the continuum model.

2) The dissipation of kinetic energy appears owing to the action of the molecular diffusion and the velocity gradient. That is an angular deformation of a volume of fluid. The dissipation term into the kinetic energy equation is always positive since it writes mu*(Grad v:Grad v). And the same term, with opposite sign, appears in the internal energy equations. Note that the action of dissipation is distributed in spectral space as mu*k2, that is higher frequencies (small turbulent structures) contributes to the dissipation. In your idea, an isotropic viscosity would act as mu(k) but would'nt that alter the classic idea of inertial cascade?

Howdy Filippo,

"but would'nt that alter the classic idea of inertial cascade?" YES! at last! I consider L. F. Richardson's little ditty okay poetry and poor physics . I checked years back and found that filling a volume with eddies of decreasing size yields the slope of the inertial subrange geometrically. That eddies of a range of sizes are produced in turbulence is a trivial observation in clouds and rivers. That the energy is passed down the size scale per the poem violates the appearance of many a cloud. I do not buy it, cute but not nature. The math works, but does the flow know about it?

However, my problem was why flow curves in the wake of an obstacle when I had rejected the absurd idea of infinite inertial resistance that was left after removing viscosity and compressibility. Batchelor spends pages on the defense of deleting physics from fluid dynamics as I recall. Dynamically Induced Asymmetric Diffusion (DIAD) struck me one fine October day up at Spring Hollow in Logan Canyon as how the low in the wake of a body is formed and the rest of the story follows. I may need to word it better, and I shall read your suggestions, always welcome, to see if I am able to find a way to communicate better, but the DIAD exists in some measure. Lord Kelvin was right and wrong with his ~"ideas are good, numbers are better"~ observation: both are necessary. After all, physics did not have it "all sewn up" yet like he is reported to have thought.

I think to "catch my drift" we must note that I have focused on diffusion of molecules rather than diffusion of either momentum or energy. The latter may be found in the consequences of diffusion of molecules, but we must stay with the image of these individuals and question whether they are sufficient in number to consider their effect. I agree with your observations 1) above with that modification.

I commented enthusiastically on 2) above. I should be more gentle, and I do understand the continuum version you describe: "The dissipation of kinetic energy appears owing to the action of the molecular diffusion and the velocity gradient. That is an angular deformation of a volume of fluid." It seems to me that there must be an additional paragraph that explains how the action of the molecular diffusion randomizes flow kinetic energy. That the volume suffers angular deformation does not seem to be enough to qualify as random, which must be required of dissipation. How is the common momentum of the diffused molecules lost to dissipation, since it would be passed on during scattering and remain in the fluid. That the shear is smoothed is not the same as the momentum becoming randomly directed. I'm afraid I still like the sense of my DIAD - no surprise there.

Happy Trails, Len

There are some issue I am not sure to understand:

1) atmospheric turbulence is essentially quasi-2D and the energy cascade has a different dynamics, smaller turbulent structures merges into larger structures. This is clearly different form the idea of inertial cascade in 3D turbulence.

2) What "diffusion of molecules" exactly means? Could you better detail?

I accept a random path of molecules interacting each other, exchanging their momentum by contact. The concept of macroscopic viscosity is just a way to extract a function of the temperature from the kinetic theory, that is also to define separately the macroscopic convection and diffusion of momentum.

Thus, I am not able to understand if you want do describe your idea within either the microscopic or macroscopic framework.

At a microscopic level, all we see is discrete, it is not really rigorous to define something that is an "eddy" until we come to a macroscopic level.

Howdy Filippo,

I actually seek to conceive the nature of Nature in turbulent flow, and in that quest the limits of macro- versus micro- are misleading. They may be used in an effort to understand, but must not prevent it. My objective is comprehension not practical use of results, which also is a valid human activity. Granted, the thin sheet of the atmosphere may be treated as 2D for some purposes. The examples of the curvature formations I have noted in clouds are not 2D - the turbulence therein is biased by gravitation, buoyancy, precipitation drag, etc. but volumes of turbulent atmosphere have similar extent in three directions - 3D.

Okay, let's consider a small event in an effort to communicate: please explain to me why flow turns into the wake of a body in a channel flow experiment. I have offered an explanation of the production of the force necessary to accelerate it into the wake region. If my objection is valid that inviscid and incompressible assumptions work by mistake because of the implied infinite inertial resistance to any other path, what is left to cause the flow curvature?

And thank you for your continuing effort, I appreciate it. I am certified capable of learning in several ways.

Happy Trails, Len

Dear Len

clearly, owing to a different background we have also some difference in our vision of the basic problem. I am used to associate a physical intuition to a proper model that guides me by means of math.

I have to address my opinion about:

- turbulence in atmosphere: yes we see a 3D dynamics in clouds but they are not on large inertial length, what we see is in the small range of scales. In more physical words, our life experience about turbulence is in the viscous sublayer, somehow a range of low Re numbers scales. Thus, curvatures in clouds is differently originated from curvature on large atmospheric structures.

- curvature: why do you associate curvature of the flow to the turbulence? Curvature appears since from laminar flow regime, just consider the laminar vortex shedding in a 2D case. Thus, the nature of that has nothing to do with turbulence. Again, what has that to do with the concept of pressure dissipation? In laminar regime we have no extension of an inertial range, viscosity is relevant everywhere. On the other hand an ideal inviscid flow around a bluff body would never generate an eddy but has curvature to follow the geometry of the body. The energy of the flow is totally reversible (total pressure is constant), no dissipation appearing.

To help us using a more common language about curvature and vortical structure, maybe you could find interesting this hystorical paper:

Article Hussain, F.: On the identification of a vortex. JFM 285, 69-94

Howdy Filippo,

I agree with your observations in "- turbulence in atmosphere:" However, while "Thus, curvatures in clouds is differently originated from curvature on large atmospheric structures." that observation is not important to the complex motions in clouds. Granted they have limited wavenumber range compared to a fully developed turbulence with an inertial subrange, but are they not still turbulent motions in a flow? Certainly the waves and synoptic systems of high and low pressure are effectively 2D, but when an upward flow with tens of meters scale penetrates a flow aloft and the flow around it is very similar to that of a cylinder in a laboratory, I see a similarity.

You have made a very valuable observation: "clearly, owing to a different background we have also some difference in our vision of the basic problem. I am used to associate a physical intuition to a proper model that guides me by means of math." And I am seeking a movie-in-my-mind that has understanding the process(es) of dissipation included with the images. Imagine a toy balloon increasing in size along with a person inflating it as an example. It would include the physiology of exhalation into the balloon, tension in the balloon surface, etc., etc.

Oops, had to start the wife's garden tiller and now the internet repairman is on his way. I shall read the article you provided and respond later since it is getting late over there. Until tomorrow . . .

Happy Trails, Len

Howdy Filippo,

Your observation: "I am used to associate a physical intuition to a proper model that guides me by means of math." is critical. It is an excellent method when the model and model equations are correct and complete, and I use it regularly as well.

Since the early '70's when I got working on inertia in general and flow in particular I have been of the opinion that the mathematics of turbulent fluid dynamics is inadequate. The convenient fictions to which I have referred were unsatisfactory. In '82 a key issue of flow curvature into the wake of objects answered "what" was happening there and a pressure gradient acceleration into the wake answered "how" (or why) it was happening for me. A dynamically induced asymmetric diffusion was the process conceived to express "how" the pressure gradient was produced and I formulated it with figures and calculations. My intuition from natural flows observed for 37 years by that time and practically memorizing Van Dyke's Album of Fluid Motion, while seeking to comprehend the process illustrated in each photo, were my primary references. I also read extensively. Unfortunately for us I have not been able to formulate my "physical intuition" as a "proper model" the math of which may be used as a guide. I find what is available without my DIAD insight to be inadequate as a guide for "reading the physics in the equations." I shall try to do better; I consider your opinion worth the effort to communicate my opinion clearly.

I have read the article you suggested and it was a good review of decades-old reading. Also during the last year I have read about 60 articles on reverse flow vortex separators and at least 25 articles on coherent structures dated 1967 - 2022 in pursuit of a better answer to a reverse flow vortex separator question here on RG Q&A. I found in your suggested article "Among the four definitions, only the lambda sub 2 - definition is found to represent the topology and geometry of vortex cores correctly for the large variety of flows considered in this paper. The lambda sub 2 - definition corresponds to the pressure minimum in a plane,. . ." This use corresponds to my intent when I write "pressure." That I then see it as a measure of molecular unrest in a specific case and associate a measure of internal energy and temperature with that same molecular unrest has caused us difficulty. For me the internal energy is enhanced by asymmetric diffusion driven by the flow kinetic energy. The latter also forms a lower pressure region inside a curve which may develop into a vortex.

"- curvature: why do you associate curvature of the flow to the turbulence?" (Atmospheric science (PhD) has many equations and models for some of the following intuition.) To my mind flow curvature requires a force to accelerate directional change. I see that force as a pressure gradient. The sloshing about of a turbulent fluid must have a full array of local pressure gradients to force the accelerations therein as well as momentum of fluid between accelerations. "Curvature" emphasis is a convenient visual aid to indicate the need for these local pressure gradients since most everyone is familiar with tornadoes. I see these local pressure gradients formed from kinetic energy of the flow. This is obvious in the curved flow at a stagnation point and I have formulated a diffusion diode to produce it in a wake. The curvature of the flow in a Hele-Shaw wake of a cylinder is not turbulent, but the low inside the curve must be there: the flow is not forced to remain in the streamlines we observe in the laboratory by the bulk of the fluid being immovable. On the other hand, I think your method has been betrayed by inadequate theory and its mathematics guidance: "In laminar regime we have no extension of an inertial range, viscosity is relevant everywhere. On the other hand an ideal inviscid flow around a bluff body would never generate an eddy but has curvature to follow the geometry of the body." I do not agree, and I have been unable to communicate a valid basis for that disagreement so far. I shall work on it. I hope my answers above help somewhat.

Happy Trails, Len

Fluid viscosity is main cause of dissipation of kinetic energy into heat in turbulent flow.

Howdy Mohsen Jahanmiri,

Thank you for including this observation. The earlier discussion has focused on what I mean in my idea, so having this standard answer is valuable.

The question word is "how." That viscosity causes momentum exchanges and resistance to flow is true and heating occurs in the fluid, but it is not like friction heating because there is no "rubbing" between layers to produce heat that way. What is the physical mechanism by which fluid viscosity causes flow kinetic energy to increase internal energy, i.e., how did it get randomized?

Happy Trails, Len

Dear Len

to be honest, I cannot add more to the discussion without a physical and mathematical model that translate your physical idea.

In your words I see a mixing of concepts linked to both the miscoscopic and macroscopic models.

First, I find hard to understant what you mean for "asymmetric diffusion" of molecules. At small scales, a discrete world appear, we can use the Newton law f=ma descrbing the interactions. It seems you are describing the fact that the random movement of molecules must be described in macroscopic way by a different concept of a continuous viscosity function in tensor form.

I insist with the fact that curvature of a flow (that is in macroscopic view the curvature of a streamline ) has nothing to do with turbulence. It is in the nature of the flow to occupy all the available flow because the pressure gradient acts ( v.grad p) and produces an acceleration that do that.

As an example, think of a laminar flow (low Re number) in an open channel where the flow goes behind a backward facing step. You get a curvature in the flow due to the pressure balance over a volume of fluid that is accelerated and curves. There is no turbulence and no range of structures, only one recirculation region behind the step.

What is more, this happens even if you let tend the viscosity (macroscopic) to vanish. The streamline curves. Therefore, your "asymmetric diffusion" is nothing but a pressure-based force in a macroscopic view. But I don't see that as a source of entropy production that transform the kinetic energy in heat. At least in a macroscopic view. That would require a process wherein you modify the meaning of the pressure. I don't know how, but your model should provide a "pressure" field in which you have a mechanism where the total pressure is no longer constant for inviscid flows. In other words, your pressure is no longer represented by an isotropic tensor.

I hope other people could contribute to clarify my doubts.

Best regards

Filippo

Howdy Filippo,

I also hope other people will contribute and that a better expression of my conception will emerge from discussion. My view is a mixed view of micro- and macro- worlds which I am trying to express, and it contains enough individuality so that points get overlooked and focus becomes too strong on specific points. I did write that asymmetric diffusion occurs in the wake to produce a pressure gradient there while stagnation pressure curves flow on the upstream side of an obstruction. Also, I did try to separate curvature from turbulence as a more general case, right down to Hele-Shaw flow. My presentation is too much "of my mind movie following the process" instead of clear exposition.

Your contributions to this point are the first anyone has offered on any of my thoughts, and I am very thankful. Your clarification of micro- / macro- elements in my thoughts and the revelation that Stokes hypothesis is closely related to what I am trying to clarify is invaluable. We may hope that someone, maybe even one of us, will find the connection like Max Planck's insight that enabled calculation of the black-body emission spectrum without the extremes that lay in the earlier equations. Certainly we must accept both kinetic theory and continuum fluid treatments and find a larger model/math to encompass both.

Under it all is the fact that fluids can do it. Fluids are hybrid analog-digital computers with analog speed and direction between discrete encounters of molecules, with many-on-one collisions to explain extremes. Our modern computers are oxen-drawn lumber wagons on a muddy road in a forest with crescent moon illumination compared to the stunning speed and precision of molecular unrest in a fluid. We need to work on it.

Asymmetric diffusion:

Hello Len,

let see if other researchers want to contribute.

In the meanwhile, my idea of the kinetic theory and the source of the macroscopic concept of viscosity (and diffusion of momentum) is described in this textbook

https://ia802601.us.archive.org/34/items/introductiontoth000487mbp/introductiontoth000487mbp.pdf

Howdy Filippo,

Good plan. I'll check out the textbook. The figure in my last post illustrates my meaning, but that it describes the nature of Nature in this case is still open.

Happy Trails, Len

Howdy Filippo,

Actually, I have Sir James Jeans "Introduction to the Kinetic Theory of Gases" on my shelf and I had read it long ago. This is good. I'll try to express my ideas in ways aligned with the manner of exposition by Jeans. I prefer bound volumes for reading, but it is much more convenient to search the computer version that I downloaded.

Happy Trails, Len

Howdy Filippo,

Thank your for your patience. I have finished the Sir James Jeans book. It has enriched my understanding of my thought experiment leading to asymmetric diffusion, and also clarified how to report what I have thought. I shall work on it a while to incorporate those clarifications. We'll find I have realized that my figure a few posts back illustrates both diffusion at the corner and viscosity further along which also drags fluid away from a low pressure at the corner. More later.

Meanwhile, as a thought, our agreement that we are viewing the events from macro- and micro- worlds and the twain do not meet has an option now. I realized that Brownian movements do exist in both worlds because molecular unrest drives the Brownian particles and continuum drag retards them. From the book Page 222: "The foregoing ideas are applicable to the Brownian movements, although in a somewhat modified form. Each Brownian particle will perform a sort of random walk, but as its mass is much greater than that of the particles it encounters, persistence of velocities becomes all important.

"Following Einstein* and Smoluchowski, we may treat all collisions with other molecules as forming a statistical group, and regard the Brownian particle as ploughing its way through a viscous fluid. . . ."

Happy Trails, Len

Dear Len

now think about the fact the in the macroscopic continuum model the viscosity is a function of the temperature.

The equation of state says the T=p/R*rho, this you can approach the link between macroscopic viscosity and pressure.

Howdy Filippo,

Your comment is in line with my past thoughts. I seek here to clarify my meaning with more careful use of terms and with quotations from Sir James Jeans' An Introduction to the Kinetic Theory of Gases

https://ia802601.us.archive.org/34/items/introductiontoth000487mbp/introductiontoth000487mbp.pdf

The online version is a first edition (1940) acquired on 17 March 1941 by the Kansas City. Mo. Public Library, while my bound volume is a second printing dated 1948. The book is the same age as myself, but old guys can get it right. The book is an excellent presentation passing over the ideas and math in stages of increasing depth with clear descriptions of the molecular unrest involved. (This was called the Spiral Curriculum, on another thread, apparently a modern advance in the theory of education.)

Now, my question notes read: "I think that the process of flow curvature formation at any scale in a fluid requires a pressure gradient across the curvature. The result outside the curve is increased internal thermal energy there. During subsequent decay of the curved flow, the curving kinetic energy fills the low pressure inside the curvature." So, what does this mean? The Hele-Shaw Revisited figure attached here and the Asymmetric Diffusion Illustration posted earlier are both in Structure Formation in Turbulent Flows and I recommend having them handy because I believe some of this is new, and old answers will cause confusion. Bold type is used for words defined in the excerpts from Sir Jeans book following my opinion of what this all means.

The Model

This was an attempt to address the problem for me that flow curves into the wake of objects when I saw previous explanations as wrong. Incompressibility or inviscid fluid treatments set all the streamlines that occur as rigidly defined by an otherwise immovable fluid, or of effectively infinite inertial mass, and flow is constrained thereby. Newton's first law was overwritten kinematically because the flow could do no other against the rigid flow of the theory. This is illustrated in the Hele-Shaw figure as a pressure force directed from the environment toward the object that forces the flow to curve into the wake. Given Newton's 1st law describing flow overshoot unless accelerated laterally, and given Newton's 2nd law requiring a force of acceleration, I posited a low pressure at the surface to provide a pressure gradient force for acceleration, as shown in the figure.

In an effort to understand how the low pressure could form I formulated an approach that became called Dynamically Induced Asymmetric Diffusion (DIAD), a diffusion diode. The Asymmetric Diffusion Illustration figure is a schematic view of the difference between viscosity in shear flow and diffusion plus viscosity in a flow past an object (note improved wording here.) The illustration expresses that diffusion of molecules into the wake adjacent to the flow blockage is prevented by the addition of mass-velocity to thermal-velocity of the molecules. Diffusion out of the wake is not similarly affected because mass-velocity is zero there: there is a net loss of molecules from the wake and a net increase of molecules in the passing flow. Comparison with the vertical pressure gradient in the atmosphere produced the equation for a molecular number density gradient that would result in an acceleration of 10 times the acceleration of gravity (10g). Beyond a volume so affected adjacent to the flow blockage, viscosity is the more valid process, which will increase the wake fluid mass-velocity and retard the through flow mass-velocity. Further changes will be that of flow toward the low in the wake of the flow blockage, flow away from the corner of the flow blockage due to viscosity affected mass-motions, and reaction to the slowed mass-velocity of the passing flow above the obstruction, plus the fact that the pressure gradient resulting from the low pressure at the flow obstruction will accelerate fluid in its presence. Now, this is at best a snapshot to identify the important issues; a full transient treatment from flow initiation to large eddies would be quite difficult.

My question notes show similarity to proofing one's own work: one tends to see the expected text and misses the actual omissions or errors. Given this new description of my "model" of the cause of flow curvature into a wake, I shall attempt to clarify the notes. Firstly, molecular unrest of the Kinetic Theory of Gases and Liquids is behind my thoughts. Secondly, as you will find in quotations following, the kinetic energy of the molecules is why we can measure or derive pressure, temperature, and internal energy in a volume of fluid. Thirdly, my model sets up a pressure gradient by changing the distribution of molecules. Since number density affects internal kinetic energy, I see its increase in the passing flow and its decrease in the wake as a measure of that change and cause of the pressure gradient. (Improvements will be welcome.) Now, curvature of flow in a fluid requires a pressure gradient, whether it is due to stagnation or to "low" formation, and folks familiar with tornadoes understand what "low" means. I swept ahead, and was appropriately asked about it, when I saw the variation of pressure indicated by curvatures of all scales in a flow as illustration of energy partitioning into molecular unrest in the higher pressure volumes in keeping with my model. I speculated that the kinetic energy of rotation around the inner low was roughly equal to the thermal energy "deficit" of the low and they would dissipate together. Thus, mass-motion kinetic energy in a flow becomes distributed as thermal energy in higher pressure outside curving flow, which is seen as dissipation in continuum models.

Well, that's it for the moment. Filippo Maria Denaro has provided in these exchanges the first feedback I have on this topic. The view above is improved, and I hope for an even better model to come.

Quotations from Sir Jeans' book.

Pages noted with quotations apply to either edition.

It was a pleasure to run through the history in the Introduction. I had remembered Lucretius' observation that a fish required a particulate fluid in order to swim. It would be prevented from moving were the water connected as a continuum. One is reminded of Chuang Tzu: "The Dexterous Butcher" who lived in tune with the Tao, and whose knife never needed to be sharpened because he passed it in interstices and never actually cut with it. I had not remembered the moist shirt example included by Sir Jeans which also is excellent to set the stage for Kinetic Theory.

Page 11b – 12a. pressure -. . . the pressure of a gas gives a measure of the total kinetic energy of motion of its molecules, . . . and Page 19b. . . . pressure is equal to two-thirds of the kinetic energy per unit volume.” and Page 72b. Art. 50. This equation shews that the pressure may be regarded as made up of two contributions, one coming from the kinetic energy of motion of the molecules, and the other from the potential energy of their intermolecular forces, including of course the forces of collision.

The essence of the kinetic theory view of pressure is that it stresses this first contribution; the second figures only as a small correcting term. . . .

Page 14b. temperature - . . . absolute temperature of a gas must be measured by the kinetic energy of motion of its molecules . . . Page 28a. This shews that E/T must be a constant, and gives us our first insight into the physical meaning of absolute temperature ; we see that it is simply proportional to the average kinetic energy of motion of a molecule.” and Page 29b. If we warm up the gas inside an enclosure which is kept at

constant pressure, both p and v must remain constant, so that,

from relation (19), NRT remains constant. . . . when we light a fire in a room, we do not increase the heat-content of the air of the room; this remains unchanged. By increasing the pressure, we drive some molecules out of the room, and the original amount of heat, now being distributed over a smaller number of molecules, gives more energy to each.

Page 15b. evaporation cooling - . . . since only the quick moving molecules escape, the average kinetic energy of the molecules which remain in the liquid is continually being lowered. Thus, as some molecules of the liquid evaporate, the remainder of the liquid decreases in temperature.”

Page 157a. conduction - As the result of a long chain of collisions, energy may be transported from a region where the molecules have much energy to one where they have but little energy : studying such a chain of collisions we have in effect been studying the conduction of heat in a gas. viscosity - If we examine the transport of momentum we shall find that we have been studying the viscosity of a gas. . . . diffusion - Finally if we examine the transfer of the molecules themselves we study diffusion.

Page 204b. Diffusion, it will be seen, is a transport of a quality, while viscosity and heat-conduction are transports of quantities. The difference rests ultimately upon the circumstance that qualities remain unaltered by collisions, whereas quantities do not. [[lfh note: a molecule remains that molecule during diffusion. ]]

Page 113: We now see that the velocity of any molecule may be regarded as made up of

(i) a velocity u0, v0, w0 which is the same for all molecules, being the velocity of the centre of gravity of the gas, and

(ii) a velocity U, V, W relative to the centre of gravity of the gas, which of course is different for each molecule.

We may speak of the former velocity as the "mass-velocity" of the gas, and of the latter velocity as the "thermal" velocity of the molecule, since it arises from the thermal agitation of the gas.

Page 117b. . . . In other words, all directions are equally likely, as indeed we should expect. For when the gas has no mass-motion there is no reason why thermal velocities should favour one direction more than another. When the gas has a mass-motion, the principle of relativity takes charge of the situation, and shews that the distribution of thermal velocities must be the same as when the gas has no mass-motion. . . .

Page 119b. Mass-Motion and Molecular-Motion

92. We have seen that in the most general "steady state" possible, the motion consists of a thermal motion compounded with a mass-motion. The mass-motion has velocity components u0, v0, w0, while the thermal motion has velocity components u - u0 , v - v0, w - w0 , which we have denoted (p. 1 12) by U, V, W.

Page 120a. This shews that the total energy of a gas may be regarded as the sum of the energies of its mass-motion and its thermal motion.

Let us suppose that a vessel containing gas, which has so far been moving with a velocity of which the components are u0, v0, w0, is suddenly brought to a standstill. This will of course destroy the steady state of the gas, but after a sufficient time the gas will assume a new and different steady state. The mass-velocity of this steady state will obviously be nil, and the energy wholly molecular. The individual molecules have not been acted upon by any external forces except in their impacts with the containing vessel, and these leave their energy unchanged. The new molecular energy is therefore equal to the former total energy. . . . In the language of the older physics, one would say that by suddenly stopping the forward motion of the gas the kinetic energy of this motion had been transformed into heat. In the language of the kinetic theory we say that the total kinetic energy has been redistributed, so as now to be wholly molecular.

Page 121a. . . . The conception of the perfect non-viscous fluid postulated by hydrodynamics is an abstract ideal which is logically inconsistent with the molecular constitution of matter postulated by the kinetic theory. Indeed in a later part of the book we shall find that molecular structure is inconsistent with non-viscosity; and shall be able to shew that the actual viscosity of gases is simply and fully accounted for by their molecular structure. If we introduce viscosity terms into the hydrodynamical discussion, the energy of the original motion becomes "dissipated" by viscosity. On the kinetic theory view, this energy has been converted into molecular motion. In fact the kinetic theory enables us to trace as molecular motion energy which other theories are content to regard as lost from sight.

Page 161a. Now in Art. 92, it was found that the total energy of the gas

could be regarded as the sum of the energies of the molecular and mass motions; indeed, the sum of the energies of the molecular-motions of the two molecules now under discussion is easily seen to be m(u2 + v2 + w2) + mu02. The first term is equal to the energy of the molecular-motion of the two molecules at the start; the second term represents an increase which must be regarded as gained at the expense of the mass-motion of the gas. Thus we see that the phenomenon of viscosity in gases consists essentially in the degradation of the energy of mass-motion into energy of molecular-motion; this explains why it is accompanied by a rise of temperature in the gas.

Happy Trails, Len

Dear Shakil,

Your explanation is quite thorough and justified. I may add that the theory through which big eddies breaks into small one is called cascade theory.

Howdy Rana Hamza Shakil and Mohsen Jahanmiri,

Your mutual "it is known that" is thorough and justified as history. It is good to have it so clearly stated, and I appreciate your contribution to provide readers with a larger picture.

However, inviscid flow is used there as if valid, just for a start! It is unnatural. From Sir James Jeans book noted above: "Indeed in a later part of the book we shall find that molecular structure is inconsistent with non-viscosity; and shall be able to shew that the actual viscosity of gases is simply and fully accounted for by their molecular structure."

My observation about a young mountain stream being able to solve the problem of a few stones in a rapids makes it much more capable than history's best efforts because a fluid is that capable an hybrid analog-digital computer. Think about it as it may open avenues for yourselves.

The historical methods have been good enough for engineering, etc., but I am seeking to understand turbulence as a natural phenomena. What are you doing? Let's explore together and contribute observations that extend or refute efforts to include kinetic theory. I do not claim to be right, but just grinding out another decimal place with refined history may not be the future of turbulence research and understanding.

Happy Trails, Len

Howdy Folks,

A point of concern about my diffusion diode concept and figures, included in earlier posts, is that they show only one molecular exchange incident. "So what (?)" comes to mind quite easily. That numerous additional exchanges would occur is implicit, but something stronger would be comforting. A supplemental view has come about because Filippo Maria Denaro suggested I read Sir James Jeans book, An Introduction to the Kinetic Theory of Gases. That volume contained a treatment of Brownian Motion that struck home. My refreshed awareness that in Brownian Motion the impulse is delivered to the particle by numerous molecules finally produced this needed diffusion diode supplement in my thoughts.

We note that the impulse of numerous molecular collisions with the particle cause its response, but we also recognize that in any particle-sized volume of the fluid such sets of molecules must exist even without a particle response to indicate them. If we go to the figures and discussion of the diffusion diode and replace the single molecule exchange shown as an illustration with a group of molecules that could drive a Brownian particle out of the wake, that large number of molecules would leave the wake if no particle were present to impact, and we have the stronger affect needed to support believability.

Also noted by Filippo Maria Denaro in our earlier posts, the macro- vs. micro- approaches to fluid dynamics are separated by a great difference in scales and mathematical treatments. That Brownian Motion is a natural process that lies in both scales was observed. The insight here is even better, for a Brownian Motion impulse operating in the absence of a particle may be the more effective process of asymmetric diffusion that justifies a deeper exploration. That such impulses would occur is clear from successful Brownian Motion theory.

It's just a thought "fresh off the mind." The ResearchGate purpose includes exchanging such ideas early to enable development in discussion, so here it is: discussion certainly has helped me.

Happy Trails, Len

Howdy Folks,

A second point of concern about my diffusion diode concept and figures, included in earlier posts, is that the path lengths over which the asymmetric diffusion is shown seem too short to matter. Again, "So what (?)" comes to mind quite easily. Even with the "numerous molecules" of the impulse generators of Brownian particle motion noted in the previous post, one worries about their penetration distance into the flow away from the wake. Not to worry.

If, and it does, the loss of a few molecules from the corner of the bluff body illustrated produces a lower pressure there, flow toward that volume will be forced from the higher pressure in the wake further from the corner. That acceleration will lead to flow, the flow will overshoot the low pressure, it will surge into the passing flow and it will enhance the process of low formation at the corner: a rotating wake flow structure begins.

Now, look at the Hele-Shaw streamlines and see a small low form from the asymmetric diffusion as passing flow overshoots the cylinder surface in accord with Newton's 1st Law. This low will result in an adverse pressure gradient beyond it that will accelerate fluid upstream, a well known mechanism of flow separation and turbulence inducement. That viscous mixing prevents appearance of separation does not prevent the low pressure or the normal pressure gradient that causes the streamline to curve around the cylinder.

Commercial: "If you look for the bad in people expecting to find it, you surely will." Abraham Lincoln. That is also true of finding fault with new approaches to understanding. How much thought and effort was expended between "I wonder if we could model a particle as a tight string" to M-Theory?

Happy Trails, Len

Howdy Folks,

A more complete answer to How is flow kinetic energy dissipated in turbulence? has come to mind. I’m working on equations, but comprehending the concepts is more important when inviting discussion. This is a work in progress.

Overview: Given the need in a flow to curve into a wake that is created by the physical environment or a structure in the flow, the process of asymmetric diffusion that produces the required pressure gradient occurs anywhere it is natural in the flow, not just at a surface. That is, the diffusion diode is posited as the mechanism producing the local pressure gradients that curve the flow everywhere into the wake of obstructions. Thus, for example, in the Hele-Shaw laboratory flow all departures from Newton’s 1st Law motion are forced by Newton’s 2nd Law accelerations due to local pressure distributions where they are needed.

More generally, a fluid adjusts its flow past an obstruction at molecular unrest speed, roughly the speed of sound in the fluid. At low flow velocity the adjustment everywhere is sufficient to produce smooth streamlines that are mimicked by the false streamlines of potential flow. As higher velocities of the flow occur, there will be a velocity at which the diffusion diode curvature-forcing will no longer be sufficient and the flow will separate with reverse flow resulting. This flow separation changes the flow regime to turbulent flow locally. At a bluff body the separation occurs with any flow velocity; over a curved surface the transition to turbulence is estimated by the Reynolds Number, Re; on a flat surface uneven features of the flow, say those driving Brownian Motion particles or even waves in the flow, will be the location of flow separation and turbulence. The actual Reynolds Number at transition to turbulence will be affected by the details of the surface roughness and the smoothness of the incident flow in the latter cases.

This natural process is inefficient due to partial randomization of the momentum transferred while forming the local pressure gradients that curve the flow. That is the process of flow kinetic energy loss to thermal motions in the fluid that is called “dissipation” in continuum fluid theories. The momentum of the flow volume receiving the wake molecule(s) is adjusted to a reduced streamwise value, and adjustment of the energy in the event passes flow kinetic energy into thermal-motion energy. Curvature of the passing flow has slowed it as well as deflecting it laterally, therefore the flow kinetic energy involved has been randomized to that degree. In continuum fluid theories the term “pressure recovery” is used for conversion of flow kinetic energy to pressure in a wake as the streamwise mass-velocity is reduced; I see that as another view of the mass-velocity kinetic energy being passed to thermal-motion energy. Both processes transfer mass-motion kinetic energy into thermal-motion energy, and therefore describe “how flow kinetic energy is “dissipated.” That my question should have been “ in turbulence also.” means I have learned again.

The molecular exchanges treated are all symmetric in molecule count except the loss from zero mass-motion fluid beyond an obstruction. This increases the molecular number density of the flow and reduces it in the wake. Viscosity enhances this loss by accelerating fluid away from the obstruction, which fluid is replaced from the wake. Actual reverse flow may occur and lead to a well-defined rotation in the wake of a bluff body.

Numbers: For air at standard temperature and pressure (Pages in Sir James Jeans The Dynamical Theory of Gases, 4th Ed.):

• For instance, the density of a gas may generally be supposed homogeneous throughout a cube of edge equal to one millimetre, while such a cube is large compared with the scale of molecular structure, containing, as has already been mentioned, about 2.75 x 1016molecules in the case of a gas under normal conditions of pressure and temperature. (Page 15a.)
• The average distance apart of adjacent molecules at atmospheric pressure will clearly be about (2.705 x 1019)-(1/3)cms., or 3.33 x 10-7 cms. Or 3.33 x 10-6 mm. (Page 8b.)
• Regarding the molecule of hydrogen as a sphere of radius 1.36 x 10-8 cms. the number of collisions per cubic centimetre of hydrogen at 0° C. is found to be about 2.037 x 1029 per second. Each collision is the termination of two free paths, hence the number of free paths described in the gas just considered is about 407 x 1029 per second. Hence on division we see that the mean length of these free paths is 1.125 x 10-5 cms. Or 1.125 x 10-4 mm. (Page 10a.)
• Thus the values we have obtained for the mean free path are approximately true for all gases so long as the molecules are supposed uniformly to be spheres of radius 1*36 x 10-8 cms. (Page 10a)
• Thus only one molecule in 148 describes a path as great as 5 mean free paths, only one in 22,027 a path as great as 10 mean free paths,
• One molecule in 109 difference between moving and stationary fluid volumes over 10 mean free paths yields a 10g pressure gradient in air to curve the passing flow into the wake of the obstacle. (Slide 9 NMSU Seminar.)
• This computes to 109 / 22,027 = 45,398.8 molecules per cubic mm that have path-lengths of 10 mean free paths from collisions which occur at 2.037 x 1026 per cubic millimetre per second.

Testing: Evaluation of asymmetric diffusion would involve calculating the distribution of pressures in the curving flow that are required to produce actual streamlines and compare them to the pressure distribution that could be produced by the diffusion diode process. To approach this calculation, one could choose a separation surface between Newton’s 1st Law mass-motion fluid and stationary fluid behind a sufficient flow obstruction. For fluid volumes of 2.75 x 1016 molecules (about 1mm edge cube) the obstruction may be a curved surface or surface roughness and the volumes need not be cubical. That is, one may consider a wedge between the 1st Law mass-motion and a curved surface along with a similar reflected volume in the passing fluid.

For each molecule in the wake one may expect a molecule in the passing flow with opposite thermal-velocity that can be associated with it. If this pair of molecules exchange locations by 10 mean free paths of thermal-motion from 5 mean free path depths in each volume, the result will be the Blocked Flow Case diagrammed in the Asymmetric Diffusion Illustration.gif figure provided on my June 2 post.

The scale of that diagram is given by 10 mean free paths between the lines of molecules, so the square corner could be a roughness element of less than 20 mean free paths or 2.25 x 10-3 mm. The molecule from the wake will be accelerated in the flow by momentum exchange; the molecule from the flow will accelerate wake fluid away from the flow obstruction. When one considers the number of molecules that could take part in such an exchange per millisecond, the prospects of effectiveness of asymmetric diffusion become more believable.

Departure from Newton’s 1st law would occur over the stream width beyond an object and pressure adjustment would proceed at molecular unrest distribution velocity. There is no need for deep penetration from the surface since local adjustment of curvature obtains. It is posited that the flow reaches a steady state of asymmetric diffusion sufficient to maintain the local pressure variation curving the flow up to flow rates that result in separation. TO TEST: Calculate Re for separation in several flows based on a diffusion diode approach.

At least, this is a more complete and understandable presentation of Dynamically Induced Asymmetric Diffusion, although still a work in progress.

Happy Trails, Len

Howdy Folks,

Many thanks to a couple readers. It helps to have someone to whom I am writing. I hope this development bears valuable fruit. It rings true for me, and this venue is as good as any to report its growth, like news from the front: thought unfolding online. The sense of how a fluid adjusts to a flow obstruction is becoming better in each step. It may even be right!

Consider a curved streamsurface in a flow and small volumes near the convex and concave sides of the streamsurface as sources of 3-D radial free path molecules. The “impact” of such molecules on the concave side of the streamsurface would be stronger because their paths would be on average more perpendicular to the streamsurface. The geometry of the free paths would thereby yield asymmetric diffusion across the curved surface as in earlier posts, and the affect would be proportional to curvature. Newton’s 1st Law separation affects also would act at the streamsurface, like along a solid body, enhancing the asymmetry.

This curvature asymmetric diffusion would occur with either stagnation or wake flow curvature. At a stagnation point, the stagnation pressure is due to transfer of mass-motion kinetic energy to thermal-motions of the molecules during deceleration of the flow. The concave flow deflected by stagnation could involve concave flow asymmetric diffusion that would support the stagnation pressure generation. Where the streamsurface becomes convex around the obstruction, the flow on the concave side of the streamsurface would again experience curvature asymmetric diffusion. In either stagnation or wake curvature, asymmetric diffusion supports or enhances the pressure gradient needed to cause flow curvature in accord with Newton’s 2nd Law of Motion.

Curvature asymmetric diffusion that occurs across all streamsurfaces in a flow is the clarification of: There is no need for deep penetration from the surface since local adjustment of curvature obtains. from the previous post.

Happy Trails, Len

Howdy Folks,

It’s time for a thought experiment that clarifies the activity of molecular unrest in a specific case of channel flow with a transverse cylinder, as I see it. Molecular unrest has no grand ambitions, like flow, inertia, force: the molecules engaging in thermal-motion just bounce around. When mass-motion occurs it is because reflection of molecules moving that direction has been reduced. Please! refrain from reading Nature from continuum fluid mechanics equations and continuum concepts like pressure gradient during this thought experiment! As noted earlier, this is a work in progress, attempting to state my ideas with more clarity. Equations and predictions are in the plan. Discussion is welcome.

Setting: We take a channel of rectangular cross section with a transverse cylinder as our setting. The channel is of sufficient width (lateral) and height (vertical) so that channel-surface boundary layers are not important to the experiment. The channel also is long enough to prevent end effects with an entry engineered to have minimal flow disturbances and a similarly engineered exit with an exhaust device to produce flow in the channel. The fluid is air at standard temperature and pressure (0°C and 1 atmosphere). This fluid involves about 2.75 x 1016 molecules per mm3 that collide at a rate of 2.037 x 1026 per second. Molecular unrest described by these values produces an hybrid digital/analog computer of astounding resolution and speed. That is how a light breeze can calculate the solution to a few intervening rose leaves flawlessly while delivering the scent of a rose, while the approximations that are forced on humans by the limits of mathematics and computing facilities cannot. The analog motions from discrete encounters of molecules in molecular unrest are the computing algorithm. We begin with the air in equilibrium, uniform thermal-motion of the molecules everywhere. Molecular motions between collisions obey Newton’s 1st Law of motion; accelerations of molecules at collisions obey Newton’s 2ndLaw; and colliding molecules obey Newton’s 3rd Law. This thought experiment does not include interactions involving the measure of action given by Planck’s constant, although they are important in other fluid and flow states.

Start-up: When started, the exhaust device removes molecules from a volume of air immediately upstream. Molecular unrest in the channel is unaffected except for that adjacent upstream region that interacts with the volume of reduced number density. Molecules in the interacting region have increased freedom of movement into the depleted volume and the thermal-motion of molecular unrest is thereby redistributed into thermal-motion and mass-motion. Redistribution is effected like the maintenance of equilibrium without any demon to sort the molecules. The thermal motions of all molecules remain individual, while the mass-motion component from increased freedom to move downstream is common. The reduced number density due to mass-motion from the upstream interacting region effects exchanges with its upstream region in the same way, and the volume of reduced number density proceeds upstream at a rate governed by the molecular unrest. The exhaust device continues to remove fluid to produce a steady mass-motion through it.

It is imperative that one comprehend that molecules in mass-motion have a common motion included in the now-biased molecular unrest. We separate for communication the common mass-motion from the biased molecular unrest and obtain thermal-motion as unbiased, but the fluid molecules just bounce around.

At the Cylinder: The portion of the transverse cylinder of interest in this thought experiment is a length of its center long enough to treat the flow as two-dimensional there but far from the channel walls. Description focus is on the upper half of the cylinder with obvious treatment of the lower half. The center of the reduced number density volume moving upstream is assumed to have a plane surface larger than the cylinder portion of interest. The cylinder will require adjustment of the downstream mass-motion, which is modeled to have been constant and parallel to the channel axis to this location. As the reduced number density volume approaches the cylinder, enhanced depletion of the intervening volume will result in mass-motion along the cylinder surface to the lowered number density volume, because the cylinder supplies no air. While the reduced density volume passes over the cylinder, redistribution of thermal-motion of molecular unrest to mass-motion will increase because the cross-section through which it acts becomes smaller. The adjustment is simply due to molecules moving against reduced opposition, as in the first instance of exhaust device startup. Molecular unrest fills the need, and thermal-motion is redistributed into mass-motion to do that.

At the top of the cylinder, the mass-motion is parallel to the channel axis and at its maximum speed. Upstream of the diameter, molecular unrest responds from the volume available, and mass-motion along the cylinder surface results in addition to axis-parallel response. After the reduced number density volume has passed beyond the cylinder, molecular unrest adjusts the molecular number density near the cylinder to establish a steady state. At the upstream cylinder centerline, the air is stopped. The stoppage produces a molecular number density variation decreasing outward from a maximum at the centerline. This maximum results from mass-motion molecules penetrating the volume, which molecules are redistributed to thermal-motion. Further redistribution of mass-motion to thermal-motion produces gradients of molecular number density that result in a steady state. One may see the results in flow visualization illustrations, for example the Hele-Shaw flow illustrated earlier. To avoid “known dynamics” it is important that we see this flow result as thermal-motion redistribution and number density variation that effect the mass-motion of molecules through the collisions of molecular unrest.

Entry: Upstream from the cylinder to the entry the process will continue to produce a mass-motion equal to the downstream rate. A steady state will develop.

Notes: We set aside possible oscillation of a reduced number density volume as not informative in our thought experiment. The experiment objective is comprehension of the behavior of the air when passing the cylinder. Also, the above treatment is an effort to scrape off all of the “it is known that” confidences from inadequate theories and equations from the past until their value in a new awareness may be determined. Again, Please! forget about reading Nature from continuum fluid mechanics equations and continuum concepts like pressure gradient during this thought experiment!

Three Cases: The mass-motion and redistribution between mass-motion and thermal-motion downstream along the cylinder surface are quite different from the upstream surface of the cylinder. We shall consider three cases: initial response to tangential mass-motion, steady state curvature of mass-motion in the wake of the cylinder, and mass-motion separation from the cylinder.

Initial response to tangential mass-motion. Consider the moment that the reduced molecular number density volume has begun to pass the top of the cylinder. There is a volume between the downstream cylinder surface and the tangent plane to the top of the cylinder that we must explore. Given the distribution of molecular unrest in this case, what is the number density distribution in that volume? Quantitatively, it will be affected by the rate of molecule removal by the exhaust device against the entry restriction because the adjustment by molecular unrest has a limited rate. We shall assume only a moderate variation here and seek a qualitative number density distribution which may be scaled when comprehended. Channel axis-parallel mass-motion results above the tangent plane, while less oriented mass-motion into the reduced number density volume continues below the tangent plane. The latter will include mass-motion along the surface to compensate for the no air-supply constraint of the cylinder, while the mass-motion will become channel axis-parallel far away from the cylinder. In the specified volume, a transition will result from removal of molecules by the latter flow that are not replaced by the channel axis-parallel flow above the tangent surface. Exchange of molecules across the plane tangent to the top of the cylinder involves the thermal-motion of molecules in each region, and the mass-motion components of molecular velocities. The latter will produce a downstream mass-motion response to collisions in the volume. The movement of molecules through the tangent plane into the passing mass-motion volume will cause redistribution of axial mass-motion to thermal-motion there as these molecules are included in the flow. Additional surfaces considered from the cylinder to the channel top will experience relative mass-motion and thermal-motion differences across them. The mass-motion decreases with distance from the cylinder and therefore the thermal-motion in the mass-motion volumes will increase. Molecular exchanges across a surface will result in mass-velocity toward the cylinder.

Steady state curvature of mass-motion in the wake of the cylinder. We begin by considering a series of surfaces from the cylinder surface out to effectively channel axis-parallel mass-motion of the fluid. The upstream curvature of these surfaces is due to molecular number density variation due to redistribution of mass-motion and thermal-motion as the mass-motion is reduced then increased to move around the cylinder and over the top of the cylinder. All this redistribution of molecular unrest motions is quite “easy to see” and needs no further comment here.

The process of curvature downstream of the top of the cylinder is the mystery. It is a steady state, and each molecule is involved in motion of zero curvature at each instant between collisions. How is the mass-motion caused to curve? There is nothing available but molecular unrest, so the answer must be asymmetry in thermal-motions that leads to curving of the mass-motion. This asymmetry lies in variation of the number density of thermal-motion molecules of the fluid in this thought experiment. The variation near the cylinder is caused by expansion of the cross-section of the channel downstream of the top of the cylinder. Mass-motion is redistributed to thermal-motion and the mass-motion components of those thermal-motions move them into the increasing volume near the surface of the channel where lesser molecular number density produces fewer collisions. This process will also be active away from the surface as molecular unrest responds to the resulting reduced molecular number density it produces. The process is like the axial mass-motion due to removal of molecules by the exhaust device at startup. As in the first response case the additional surfaces considered from the cylinder to the channel top will also have mass-motion that decreases with distance from the cylinder and therefore the thermal-motion in the mass-motion volumes will increase with distance. Molecular exchanges across a surface there will result in mass-velocity toward the cylinder. That the fluid is slowed as well as turned means the molecular number density is increasing to the downstream value of uniform mass-velocity in the channel.

Mass-motion separated from the cylinder. There will be a mass-motion as the exhaust is increased at which the process of redistribution of mass-motion to thermal-motion to fill the expanding cross section of the flow will be inadequate. The mass motion will depart from the surface and upstream flow along the surface will occur. At this event, the flow forms structures that grow, shrink, and interact. To an extent the activity of these structures is a large-scale version of molecular unrest and many efforts are underway to solve the problems they provide. This variety of forms redistributes mass-motion to thermal-motion which process is called dissipation in continuum theories of turbulence. They are turbulence, and they effect transitions of mass-motion to thermal-motion in the molecules of molecular unrest of which they are made over a full range of scales, like the redistributions that occur along the channel in this thought experiment.

Observations: This thought experiment is the result of decades of observation, reading, and reflection. It has as a motivation an explanation of curvature of flow into the wake of objects when one realizes that incompressible and inviscid are not natural conditions. It is a concerted effort to clarify molecular unrest as it applies to a channel with a transverse cylinder. It is related to the possibility of the “diffusion diode” in earlier posts, but valid is more important than theory support. Discussion, clarification, and improvement will be welcome.

Happy Trails, Len

Howdy Filippo Maria Denaro ,

You might find the thought experiment in my previous post interesting. It is a consequence of your suggestion that I read Sir James Jeans An Introduction to the Kinetic Theory of Gases and my choice to read his The Dynamical Theory of Gases as well. Writing the thought experiment triggered greater insight into processes because I had to address problems with answers that ring true. It also produced a richer view of a laminar flow past a cylinder with believable cause of curvature. I really appreciate our discussion that has led to it.

Happy Trails, Len

Howdy Folks,

An assertion in my thought experiment seems too convenient to be valid: How is the mass-motion caused to curve? There is nothing available but molecular unrest, so the answer must be asymmetry in thermal-motions that leads to curving of the mass-motion. The idea that the fluid would have a distribution of molecular number density just right to effect curvature of the mass-motion looks like wishful thinking or erroneous intuition at best.

The molecules of molecular unrest have a mass-motion component in their thermal-motions when there is freedom to move in a direction as a result of reduced molecular number density there. The steady state produced by the exhaust device against entry restriction results in equal numbers of molecules passing each plane normal to the axis of the channel in equal times. When the channel is restricted by the cylinder, the mass-motion is curved by molecular number density gradients so it passes through the available cross-section. These gradients are number density variations that strengthen to a value sufficient to prevent further increase by equilibrium between number density and mass-motion penetration.

Molecular unrest molecules are independent of one another, therefore molecular unrest is irrotational: curl v = 0. One may find a potential, phi, and the velocities locally would be response to gradients of that potential, v = - grad phi. This view breaks down near the cylinder surface because fluid does not slip along the surface. There is even evidence that molecules do not reflect off the surface at an expected angle, but adhere and then escape from the surface at any angle. This no-slip condition adjacent to mass-motion is not irrotational. The molecular unrest has external constraints on its freedom there.

But, the state of downstream mass-motion away from both, no-slip at the cylinder surface, and adjustment to channel expansion there, is different. In the "outer flow," with care, we may use streamlines illustrating Potential Flow Theory, or a Hele-Shaw flow visualization, to illustrate mass-motion. The similarity exists because molecular unrest is irrotational. The cause of curvature is in the thermal-motion molecular number density gradients that produce the mass-motion response, after all. I think we may consider this an answer to “How?” or “Why” mass-motion curvature occurs downstream of the cylinder.

Were surfaces “perfect reflectors” of molecules during collisions, then molecular unrest would exist right up to the surface. However, intermolecular forces and adhesion to surfaces violate that condition, and molecular unrest breaks down. The surface affected molecules bias the interactions and shear is rotational. This condition produces a boundary layer along the surface. In the cylinder case this boundary layer exists in addition to the cross-section variations of contraction and expansion discussed earlier.

Happy Trails, Len

Howdy Folks,

Oops! Erratum: Molecular unrest molecules are independent of one another, therefore molecular unrest is irrotational*: curl v = 0.

*Molecular unrest molecules are independent of one another, therefore thermal-motion of molecular unrest is irrotational. The mass-motion may include shear with viscosity and whether irrotational needs to be evaluated.

This is better, sorry, the original did not set well.

Happy Trails, Len

Howdy Folks,

The thought experiment detail of the surface boundary layer on the downstream surface of the cylinder needs examination and explanation. Mass-motion decays there because of the no-slip stoppage and resulting momentum transfer to the surface. The surface drag due to the no-slip condition is a tangential force producing rotational shear in the fluid. This is very different from the redistribution of mass-motion thermal-motion at the stagnation point of the cylinder. It also is different from the free shear far from the no-slip condition, where the fluid possesses a constant mass-motion plus thermal-motion magnitude that is redistributed between mass-motion and thermal-motion as molecular unrest adjusts to the cylinder constriction.

Why is shear subject to rotation, producing rotational or irrotational curvature? What holds molecular unrest molecules in a curve? The answer must be a Low: a High would not have focus to produce the curve around a center. Low and High represent volumes of reduced and increased molecular number density, relative to the region number density. A mass-motion to a low number density location carries individuals along. “In Baghdad, if you are near the bus door at a stop, that is your stop!” (Comment by a colleague from India in 1975.)

In the surface boundary layer, a Low and its main-motion curvature are rotational: they possess vorticity. Latent rotational curvature, present as vorticity in rotational shear produced by surface drag, is natural to shear; rotational curvature will appear if a seed is available, like surface roughness or the Low described here. Viscosity is also natural to shear; it mixes and maintains shear, thereby disturbing growth of Lows. If of sufficient strength, a Low does not simply fill by radial mass-motion. Instead, the Low strengthens because molecular unrest molecules from the faster layer of the shear miss it downstream and molecules from the slower layer of the shear miss it upstream! Their contribution is tangential, not radial. The molecules that would fill the Low from the same layer are subject to interference by collisions with the faster and slower molecules from the other layers. Rotational curvature forming in shear is not a problem, even vortices, they are just the nature of Nature there. A similar phenomenon also occurs when draining a water reservoir, the vortex that forms at the outlet is not a problem, it is due to vorticity in the water because the Earth is rotating, which vorticity must be drained with the water.

In my opinion, this extended set of long posts answers the question of "How is flow kinetic energy dissipated in turbulence?" Thank you for your patience and assistance.

Happy Trails, Len

Howdy Folks,

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