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December 11, 2006, 08:18 
Re: Velocity gradient tensor

#21 
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Thanks Tom for your input.
(desa): "Now, that we have firmly established that incompressible flows cannot evidence pressuretype waves, due to fixed density, would it, in your opinion, be possible for these flows to show oscillationtype solutions?" (Tom): Yes this is well known. A simple example is TaylorCouette flow when instability is lost through a Hopf bifurcation. Another example is ABC flow which can also loose stability via a Hopf bifurcation. There are lots of examples of this in the literature on weakly nonlinear stability theory. (desA): TaylorCouette flow bifurcation I have read about. What is 'ABC flow'? Has any major inroads been made in this area for the full NavierStokes, for say flow within confined domains, liddriven cavity, flow over obstacles? I've seen a spate of moving wave & solition solutions in the Chaos, fractals & solitions Journal, of late.  (desA): "If oscillation solutions are possible, could these be considered as timedecaying waveforms? These should display resonancetype & vibrationtype effects..." (Tom): Strictly speaking if they are decaying they won't exhibit resonance. Resonance requires mode interaction at a bifurcation point; i.e. a point where stability is lost to two or more modes whose wavenumber and/or frequency are integral multiples. You can get transient growth via this mechanism but it does not result in finite amplitude equilibrium solutions. (desA): What if the amplitude term were to decay in some places, whilst simultaneously growing in other regions  ie. a trading effect. I have a few examples of this mechanism using my 'wave probe' approach. I've been trying to make sense of the findings.  "(desA): I've not worked in the compressible region, but would guess that the variable density will also enter into the Continuity equation, basically turning it into a pressure form. If so, then viola  a pressure convectiontype wave equation, driven along by nabla_v! If so, this makes perfect sense now & I can see in my mind's eye how this all strings together." (Tom): I think you're referring here to the wellknown acoustic wave equation. Lighthill did something like this in the 50's or 60's. For a barotropic fluid P=f(rho) and so rho can be eliminated from the continuity equation to get an evolution equation for P (alternatively you can write rho=g(P) and eliminate P from the momentum equations). (desA): It does sound very similar. I have Lightfoot's book to hand & will check out his points. A very sound source indeed.  A few more questions  thoughts out loud:  What are the current thoughts regarding turbulence mechanisms?  Are the NS still considered goodenough to explain the turbulence phenomenon, or are folks beginning to think about altering them in the hope of obtaining more clarity? I've seen inclusions for timelag ending up in a 'telegraphlike' equation.  Has work been done in understanding the force field/s within fluid domains?  Does academia in general feel that we have theoretical answers for fluid flow phenomena from the end of the incompressible, viscous regime, up to say 90% of the speed of sound?  Are our major hurdles more of a numeric nature, than physical understanding? If numeric, then time will solve this  if physical, what are we to do?  Will we need to develop new equations for microfluid & nanofluid flow regimes?  Thanks again Tom for your exceptionally kind contributions. As always, I now have a mountain of reading ahead of me... desA 

December 12, 2006, 09:40 
Re: Velocity gradient tensor

#22 
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(desA): TaylorCouette flow bifurcation I have read about. What is 'ABC flow'? Has any major inroads been made in this area for the full NavierStokes, for say flow within confined domains, liddriven cavity, flow over obstacles? I've seen a spate of moving wave & solition solutions in the Chaos, fractals & solitions Journal, of late.
ABC flow is an exact solution to the equations of motion involving 3 constants (A, B & C) which is periodic in all spatial directions. It's main interest is that the particle paths are actually chaotic  people who work on the geodynamo study this flow. Correctly performed weakly nonlinear analysis yields an approximation to an nonlinear solution and so it corresponds to an approximation to a solution of the full equations. The existence of such a full solution is guaranteed by the implicit function theorem. This approach can be used to prove existence of certain types of solution to the full equations. In most of the flows you are talking about the problem lies in having an analytic solution to start with and then being able to perform the required stability analysis. In general some computation needs to be performed since even in the simplest cases the eigenvalue problem cannot be solved exactly. However computations for flow past a circular cylinder indicate that there is (as should be expected since there is no other way this can happen) a Hopf bifuraction at a critical Reynolds number which gives rise to a time periodic wake. (desA): What if the amplitude term were to decay in some places, whilst simultaneously growing in other regions  ie. a trading effect. I have a few examples of this mechanism using my 'wave probe' approach. I've been trying to make sense of the findings. Sounds to me like you've got a convective instability; i.e. a moving "packet" of local instability. That is a growing instability that is washed downstream and out of the domain leaving a decaying signal in its wake.  " What are the current thoughts regarding turbulence mechanisms?" After a brief explosion of interest and quite a bit of progress in the 80's and 90's this has gone a bit quiet. This is probably due to the mathematics being rather labourious (and quite a challenge for the novice) and progress in numerical simulations  although the latter hasn't really shown any new insights that were not in the former asymptotic theories. I would say that current thoughts on transition haven't changed much since the mideighties  see papers in JFM by Goldstein, Wu, Smith and there coworkers through the 80's and 90's. " Are the NS still considered goodenough to explain the turbulence phenomenon, or are folks beginning to think about altering them in the hope of obtaining more clarity? I've seen inclusions for timelag ending up in a 'telegraphlike' equation." There's no evidence to say they are not and a lot of evidence to say they are (i.e. the papers by Smith and Goldsein). " Has work been done in understanding the force field/s within fluid domains?" If you mean other forces then look at Magnetohydrodynamics as an example. " Does academia in general feel that we have theoretical answers for fluid flow phenomena from the end of the incompressible, viscous regime, up to say 90% of the speed of sound?" Not really the person to answer this. However whats special about 90% of the speed of sound? There has been considerable work on transonic and hypersonic flow! " Are our major hurdles more of a numeric nature, than physical understanding? If numeric, then time will solve this  if physical, what are we to do?" I think it's a bit of both. Theoretically the NavierStokes equations are just too difficult to deal with analytically and numerically the structural instability of the equations at High Reynolds number makes accurate simulation at anything other than low Reynolds number almost impossible. " Will we need to develop new equations for microfluid & nanofluid flow regimes?" Probably  since you may violate the continuum hypothesis. 

December 13, 2006, 00:58 
Re: Velocity gradient tensor

#23 
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Thanks Tom for a very complete answer. It seems like we still have some way to go in all of this.
(Tom): However computations for flow past a circular cylinder indicate that there is (as should be expected since there is no other way this can happen) a Hopf bifuraction at a critical Reynolds number which gives rise to a time periodic wake. (desA): ***What physical reason is given to the occurrence of this Hopf bifurcation? Is there a change in forcebalance, or net particle acceleration, for instance?*** Why I ask, is that in many of my simulations for flow over a cylinder within a confined domain  at almost incompressible flow, I observe a phenomenon leading from the enclosed object towards the domain wall. From my observations, this represents a velocitycomponent reversal. Directly after this phenomenon, periodic flow emerges. Actually, this phenomenon is also evident in open domains, only one has to look carefully for it. If my observations are in line with what you are 'seeing' mathematically as a Hopf bifurcation/s, then it also occurs before the object  but without periodic flow emergence. I'd be very interested in knowing about flow visualisation work of these phenomena, as it sounds like this may very well be some of what I have been observing. I have developed some neat visualisation tools to show up many of these effects. **** In what way does this Hopf bifurcation differ from a shock line? **** From my work, I have a pretty good idea of what the acceleration field does around these 'lines of interest'  a number of largescale effects can also be seen in these accelerationfields. desA 

December 15, 2006, 07:41 
Re: Velocity gradient tensor

#24 
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(desA): ***What physical reason is given to the occurrence of this Hopf bifurcation? Is there a change in forcebalance, or net particle acceleration, for instance?***
A Hopf bifurcation has very little to do with physics by itself  it is a general statement about the behaviour of differential equations and how the solution properties change at a bifurcation point. Basically if you have a steady suolution u(x,R) which looses stabitity to become a time periodic solution v(x,t,R) at some critical value of R then a Hopf bifurcation must occur at this point; i.e. the linearized (about u) differential operator has a pair of (conjugate) purely imaginary eignevalues. Trying to pin this down further to some physical balance is rather pointless and restrictive  the generality of the Hopf theorem means that it is generally applicable to any problem involving differential equations. 

December 17, 2006, 21:30 
Re: Velocity gradient tensor

#25 
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I like one thing about you very much that you take lot of pains in writing a reply. I wish to suggest something, when you take so much pains it would be better to do a little more and make the reply more readable (its alreay very good way of replying), you can make the other persons comment as italics or bold example 
Bold: (desa): "Now, that we have firmly established that incompressible flows cannot evidence pressuretype waves, due to fixed density, would it, in your opinion, be possible for these flows to show oscillationtype solutions?" Italics: (Tom): Yes this is well known. A simple example is TaylorCouette flow when instability is lost through a Hopf bifurcation. Another example is ABC flow which can also loose stability via a Hopf bifurcation. There are lots of examples of this in the literature on weakly nonlinear stability theory. or something like this (desa): "Now, that we have firmly established that incompressible flows cannot evidence pressuretype waves, due to fixed density, would it, in your opinion, be possible for these flows to show oscillationtype solutions?"  just a suggestion. 

December 18, 2006, 23:42 
Re: Velocity gradient tensor

#26 
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Thanks for an excellent suggestion.
Can I ask *how* you enable (format) the various options under the cfdonline editor? This has been the trick that has eluded me until now. Once again, thanks for your suggestion. desA 

December 19, 2006, 00:40 
Re: Velocity gradient tensor

#27 
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Hi again Tom, I've been away for a few days.
(desA): ***What physical reason is given to the occurrence of this Hopf bifurcation? Is there a change in forcebalance, or net particle acceleration, for instance?*** (Tom):A Hopf bifurcation has very little to do with physics by itself  it is a general statement about the behaviour of differential equations and how the solution properties change at a bifurcation point. Basically if you have a steady suolution u(x,R) which looses stabitity to become a time periodic solution v(x,t,R) at some critical value of R then a Hopf bifurcation must occur at this point; i.e. the linearized (about u) differential operator has a pair of (conjugate) purely imaginary eignevalues. Trying to pin this down further to some physical balance is rather pointless and restrictive  the generality of the Hopf theorem means that it is generally applicable to any problem involving differential equations. (desA):Practically, though, the bifurcation actually has _everything_ to do with the physics represented by the pde. For the momentum equations, the underlying physics is an acceleration field  observation of which will show regions of different activity. The region in 2d space towards the narrow region between enclosed pipe & wall, causes a change in the acceleration field. This is observed as a line where the velocity changes abruptly. It so happens that after this bifurcation that periodic flow is observed. Waves, periodicity & complex eigenvalues essentially predict wave phenomena  physically & mathematically. I refer to these waves as 'momentum waves'. Moving wave & standing wave phenomena are both present in such flow domains. The Hopf bifurcation thus seems to predict what I have termed the 'viscousshock line'. The mechanism for viscous shock (vs) is a drawnout affair consisting of a series of vs lines dividing the flow domain into regions. For an enclosed cylinder, the upper & lower momentum waves basically mesh some distance downstream from the obstruction & sequence themselves (interleave). The aspect to observe in wave & oscillation phenomena, is the phase information  this produces a wealth of hidden information. Observation of the instantaneous acceleration field is also rather revealing. The Reynolds experiment & critical Reynolds number is characterised by a change in vs line structure. It is beginning to appear that this may also the underlying 'magic' to turbulence  wave phenomena. Ironically, the approach I've been working on, also goes some way towards explaining some deterministic chaos effects in a sane way (wip). At times, magic, mathematics & physics are indistinguishable. desA 

December 19, 2006, 01:50 
Re: Very quick html

#28 
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This is very easy. The editor supports html. Anyway here is the summary:
line break is : <<t>br> </t> Bold is : <<t>b> text <<t>/b> italics is: <<t>i> text <<t>/i<t>> underline is: <<t>u> text <<t>/u<t>> Lists: Unordered or Bulleted lists <<t>ul> ... <<t>/ul> delimits list. <<t>li> indicates list items. No closing <<t>/li> is required. For Example: <<t>ul> <<t>li> apples. <<t>li> bananas. <<t>/ul> Which looks like this when viewed through a browser:


December 19, 2006, 01:56 
Re: Very quick html

#29 
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December 19, 2006, 02:49 
Re: Very quick html

#30 
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Hi zxaar,
Thanks so much for that information. I was blissfully unaware of the ability to use html under the current editor. That tutorial was perfect. desA 

December 19, 2006, 03:29 
Re: Very quick html

#31 
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I was also <h2> blissfully unaware </h2> of it for long time. It happens though.


December 20, 2006, 13:05 
Re: Velocity gradient tensor

#32 
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(desA):Practically, though, the bifurcation actually has _everything_ to do with the physics represented by the pde. For the momentum equations, the underlying physics is an acceleration field  observation of which will show regions of different activity. The region in 2d space towards the narrow region between enclosed pipe & wall, causes a change in the acceleration field. This is observed as a line where the velocity changes abruptly. It so happens that after this bifurcation that periodic flow is observed.
Waves, periodicity & complex eigenvalues essentially predict wave phenomena  physically & mathematically. I refer to these waves as 'momentum waves'. Moving wave & standing wave phenomena are both present in such flow domains. The Hopf bifurcation thus seems to predict what I have termed the 'viscousshock line'. The mechanism for viscous shock (vs) is a drawnout affair consisting of a series of vs lines dividing the flow domain into regions. Are you sure about this  part of what you are saying sounds like you are trying to reinvent the wheel! The point I was getting at about the Hopf bifurcation having little to do with physics and that there is no reason in trying to go further is a valid point. Try explaining why the Blasius boundary layer goes unstable when the displacement Reynolds number is around 500 using nothing more than physical balances. In the unlikely event you succeed in this perturb the velocity profile and then determing when this goes unstable? The standard argument involves solving the OrrSommerfeld equation under a parallel flow approximation; i.e. you need o solve an eigenvalue problem! As another test of simple physical reasoning consider Rayleigh's inflection point theorem. Using nothing more than simple force balances determine the physical significance of the inflection point? The simple forces rules out the use of vorticity arguments and variational principles for this so the explanation in Lin's book and that by Kelvin (and also Arnold) are ruled out. Some useful (free to download for now) papers you should probably read Review Review + new results Instability and Separation Some more papers 

December 21, 2006, 00:46 
Re: Velocity gradient tensor

#33 
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(desA):Practically, though, the bifurcation actually has _everything_ to do with the physics represented by the pde. For the momentum equations, the underlying physics is an acceleration field  observation of which will show regions of different activity. The region in 2d space towards the narrow region between enclosed pipe & wall, causes a change in the acceleration field. This is observed as a line where the velocity changes abruptly. It so happens that after this bifurcation that periodic flow is observed.
Waves, periodicity & complex eigenvalues essentially predict wave phenomena  physically & mathematically. I refer to these waves as 'momentum waves'. Moving wave & standing wave phenomena are both present in such flow domains. The Hopf bifurcation thus seems to predict what I have termed the 'viscousshock line'. The mechanism for viscous shock (vs) is a drawnout affair consisting of a series of vs lines dividing the flow domain into regions. (Tom): Are you sure about this  part of what you are saying sounds like you are trying to reinvent the wheel! (desA): I'm afraid I don't quite follow which wheel I'm reinventing. I'll provide my simple observations. The research I've been following is to essentially understand the momentum equations in their original asderived form  as force balances on a moving fluid particle, with consequent acceleration. This logic essentially carries the original physical logic of the NS into their final incompressible formulation. The NS as derived are a simple nonlinear sum of force vectors, equated to resultant particle acceleration (worked back to Lagrangian acceleration). These terms can then be constructed from the emergent velocity solution field, combined & visualised graphically. No assumptions, or restrictions are made, other than the selection of the appropriate solver method. The force/acceleration fields are plotted as the solution evolves in time. In this sense, moving wave activity is nothing other than a time progression of the acceleration field through space. When this timeevolution snapshot stops changing visually is when we have reached a Eulerian steadystate. At this juncture, a number of standingwave phenomena (timevariablefixedspace) can be observed. The fact that modaltype effects can be simply observed in various flow fields, can be explained by a slow & fast decomposition of the NS, along the lines of a typical Reynolds decomposition  with a linear wave (fast) tied to the bulk (slow) field via a momentum closure link (no approximations, or lost terms  wip). In fact, the slow decomposition can also be a fixed reference condition with the NS recast as an excursion from this condition. The tensor form makes this analysis pretty straightforward  philosophically, at least. The 'modal' effects are very interesting visually in that there is a velocity component direction swing across these 'vc' lines. I would love to see links to papers where this kind of visual approach has been used to understanding the physical nature of the lowspeed region where instability just begins. My research is basically exploring the action & mechanism of how the forcefields morph & modify themselves as the flow field develops. In the end, simulations basically boil down to a structure composed of a largescale array of nonlinear fluid elements. The vector field derived at each time evolution point & the macroscopic groups & structures are presently being explored. (Tom): The point I was getting at about the Hopf bifurcation having little to do with physics and that there is no reason in trying to go further is a valid point. Try explaining why the Blasius boundary layer goes unstable when the displacement Reynolds number is around 500 using nothing more than physical balances. In the unlikely event you succeed in this perturb the velocity profile and then determing when this goes unstable? The standard argument involves solving the OrrSommerfeld equation under a parallel flow approximation; i.e. you need o solve an eigenvalue problem! As another test of simple physical reasoning consider Rayleigh's inflection point theorem. Using nothing more than simple force balances determine the physical significance of the inflection point? The simple forces rules out the use of vorticity arguments and variational principles for this so the explanation in Lin's book and that by Kelvin (and also Arnold) are ruled out. (desA): At this juncture, my research is in more of a 'observe & interpret' mode. I firmly believe that we need to study the force/acceleration field effects in a macroform, & am not sure how exactly we are able to work up from the local NS form to these macroeffects with current available mathematical tools. I'm trusting that the largescale array of fluid element approach, coupled with a number of new visualisation tricks, will provide greater insight into the physical mechanisms at work. I can 'see' definite lines in the flow field across which velocity swings occur. This occurs in both FEM & FVM simulations, with different solvers used. I'm truly open to challenge on my approach, & would love to see publications where either a similar approach has been used, or where explanations are given for macroscale field effects. If someone else has done it all before, then I'd certainly like to read about it. (Tom): Some useful (free to download for now) papers you should probably read (desA): Thanks for the links, Tom, but somehow I can't access the papers. I have registered with the Journal. Would you be able to reset the links? Thanks once again for your applied wisdom & critique  it is greatly appreciated. desA 

December 29, 2006, 08:15 
Re: Velocity gradient tensor

#34 
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The links appear to expire after a short period of time. So it's probably best to get the papers using the references.
"Transition to turbulent flow in aerodynamics" R.I.Bowles, 2000. in Phil. Trans. R. Soc. A. Volume 358, Number 1765. The special volume of Phil. Trans. R. Soc. A. on separtion and instability. Volume 358, Number 1777 / December 15, 2000 "Transition of free disturbances in inflectional flow over an isolated surface roughness" By Savin, Smith & Allen. (1999). Proc. R. Soc. A. Volume 455, Number 1982. The other link should still work. Going off some of the description in your reply have you looked at any of the work on "inertial manifolds" and "slow manifolds"? 

December 31, 2006, 01:54 
Re: Velocity gradient tensor

#35 
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Thanks Tom for your reply. (I've been at the receiving end of the Asian internet tsunami & have been offline for a few days).
(Tom):The links appear to expire after a short period of time. So it's probably best to get the papers using the references. "Transition to turbulent flow in aerodynamics" R.I.Bowles, 2000. in Phil. Trans. R. Soc. A. Volume 358, Number 1765. The special volume of Phil. Trans. R. Soc. A. on separtion and instability. Volume 358, Number 1777 / December 15, 2000 "Transition of free disturbances in inflectional flow over an isolated surface roughness" By Savin, Smith & Allen. (1999). Proc. R. Soc. A. Volume 455, Number 1982. The other link should still work. (desA): Thanks for the detailed links. I'll try & capture them via our library. (Tom):Going off some of the description in your reply have you looked at any of the work on "inertial manifolds" and "slow manifolds"? (desA): Thanks so much for that direction  I'd been edging that way. Would you perhaps have any references that could get me up to speed reasonably quickly, in a language that a physicist's brain could follow? I think this direction should allow me to explore the 'slow/fast' phenomena with some decent tools. Thanks again for your very kind contribution. desA 

January 2, 2007, 07:33 
Re: Velocity gradient tensor

#36 
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(desA): Thanks so much for that direction  I'd been edging that way. Would you perhaps have any references that could get me up to speed reasonably quickly, in a language that a physicist's brain could follow? I think this direction should allow me to explore the 'slow/fast' phenomena with some decent tools.
I don't think there are any simple books on this. The simplest book on "inertial manifolds" is probably J.C. Jameson "Infinitedimensional dynamical systems: An introducion to dissipative parabolic PDEs and the theory of global attractors" published by Cambridge university press. For "slow manifolds" you probably need to look for a review article since I don't know of any book. The problem with the NavierStokes equations is proving the existence of these manifolds! In the case of an inertial manifold the existence proof would yield the long time existence theorem for the initial value problem for the NS system (although the converse is not true). 

January 3, 2007, 01:31 
Re: Velocity gradient tensor

#37 
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Thanks very much Tom for your reply.
May I take this opportunity to wish you a prosperous New Year for 2007 & to thank you for your many patient replies in this forum. desA 

September 3, 2015, 05:44 

#38 
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Rajendra Singh
Join Date: Aug 2015
Posts: 11
Rep Power: 3 
Transpose of the velocity gradient tensor, The stress tensor is defined as
tau = viscosity*(l+l^T) where "l" is the velocity gradient tensor "^T" represent the transpose operator. That's the definition of the stress tensor. Simplified expressions like the one you showed (viscosity*velocity_gradient) come in particular cases (e.g. u = [u(y),0,0]), where many terms in the velocity gradient tensor are zero and the tensor "becomes" a scalar. 

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