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Velocity gradient tensor
Hi everyone,
Has anyone done research into the velocity gradient tensor? I would appreciate a few links of recent research work in both cfd & experimental. I have searched reasonably heavily, but so far, I have not located what I'm looking for. Thanks so much. desA |
Re: Velocity gradient tensor
Can you be more specific? What is it about the velocity gradient tensor that you are looking for?
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Re: Velocity gradient tensor
Thanks for your reply, agg.
agg wrote: Can you be more specific? What is it about the velocity gradient tensor that you are looking for? desA's reply: Simply-put, has anyone investigated the velocity gradient tensor in relation to the turbulence phenomenon? desA |
Re: Velocity gradient tensor
I'm not sure if this answers your question, but as an example the velocity gradient tensor has been used in vortex detection (swirling strength) in turbulent flows. Check out this paper for more information:
"Analysis and interpretation of instantaneous turbulent velocity fields" Adrian, Christensen, Liu Experiments in fluids 29 (2000) 275-290. |
Re: Velocity gradient tensor
What are you talking about man?!
The velocity gradient IS the main source of turbulence. Look at the production term of Reynolds stresses. Half the contraction of that is the production of turbulent kinetic energy. Gradients in velocity feed the turbulence. |
Re: Velocity gradient tensor
Thanks agg & Pennysworth.
Thanks for the link agg. I'd seen a similar reference around 1998 to Liu's work. I'll try & download the reference you've provided. If you perhaps have a copy, I'd be grateful to receive it as my e-mail address. :) Based on some of my current research, I would say that you may both be correct - vortex motion & the turbulence phenomenon seem to be connected with the velocity gradient tensor, in some way. Based on earlier work, I am expecting the velocity gradient tensor to be responsible for a jump mechanism. I'm currently calling this the 'flipper' mechanism. I'd love to see what findings other folks have discovered. desA (diaw) |
Re: Velocity gradient tensor
Hi,
You can have a look at Chong et al. (JFM 357), Ooi et al. (JFM 381) and perhaps some other papers by Cantwell in J. of Fluid Mech. It has been used quite heavily in studies on vortices etc. |
Re: Velocity gradient tensor
What is your e-mail id?
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Re: Velocity gradient tensor
Dear desA,
From what I understand, the gradient of the velocity vector produces (in 3D) a 3x3 matrix of derivatives. These matrix is not a tensor in its own right because is not "objective" or "frame indifferent".. In fluid mechanics, we usually refer to the symmetric part of the velocity gradient, Sij = 0.5 [grad (u) + grad^T(u)].. This one is objective and frame indifferent. I imagine you are referring to Sij. From here, Sij is very much the backbone of the stress behavior in fluid mechanics: The constitutive equation for stress and velocity is made a function of Sij, and/or its invariants. 1 - Newtonian fluid is an example 2 - Generalized Non-newtonian fluids use the "shear strain rate" as the measure of gradient of velocity.. The "shear strain rate" is just the second invariant of Sij for incompressible fluids. 3 - Pennysworth just mentioned that "velocity gradient" is the main source of turbulence production.. Though correct is just part of the story.. The time derivative of a second order tensor, such as u_iu_j, in an Eulerian frame requires two additional terms to satisfy the "frame indifferent" constraint (regardless of turbulence).. For example, the transport equation for the "elastic part of the stress" in a viscoelastic fluid will be something like Tensor2_dot + u.grad(Tensor2) +/- (grad(u) Tensor2 + grad(u)^T Tensor2) = sources the first term is the local time derivative, the second is the advected part and the third is the production of stress due to velocity gradients.. The third exist just to satisfy mathematical constrains, regardless of the meaning of the variable Tensor2 (second order tensor) I find the following links very helpful: http://www.grc.nasa.gov/WWW/K-12/Num...2002211716.pdf In this one, you will find that not all 3x3 matrices are second order tensor, but all second order tensor can be represented by 3x3 matrices.. http://endo.sandia.gov/SEACAS/Docume..._Continuum.pdf Check for the Frame Indifference section.. Good luck, Opaque |
Re: Velocity gradient tensor
Agg wrote: What is your e-mail id?
desA replies: e-mail: momentumwaves@gmail.com Thanks so much agg. desA (diaw) |
Re: Velocity gradient tensor
Thanks very much 'tom' for those references.
desA |
Re: Velocity gradient tensor
Hi opaque. Thanks so much for your very detailed input & the references. The NASA reference has a nice approach, especially with the 'action' of a 2nd order tensor on a vector. The questions around what defines a tensor, & the velocity gradient are interesting, in that there do seem to be a few different approaches in the literature.
opaque wrote: From what I understand, the gradient of the velocity vector produces (in 3D) a 3x3 matrix of derivatives. These matrix is not a tensor in its own right because is not "objective" or "frame indifferent".. desA replies: Definition of a tensor: In terms of the definition of what constitutes a tensor, a useful reference is: Borisenko A.I., Tarapov I.E., 'Vector & Tensor Analysis with Applications", Dover Publications,1968. Mase G.T., Mase G.E., "Continuum Mechanics for Engineers", CRC Press, 2ed, 1999. The dyad product of two vectors provides a tensor (Borisenko, pg 64), based on rotational transformation around the origin of the reference coordinate system. This falls in line with the ?accepted? definition of a tensor. This tensor is rotationally invariant around the origin of the reference frame origin. Moving onto the dyad product of nabla & velocity, we then have a tensor nabla_v. Mase refers to this as the 'vector gradient' (Mase, pg 35). I've also seen references to nabla_v' being the transpose of the 'velocity gradient matrix'. So, nabla_v would appear, by definition, at its most primitive level, to be a tensor. ----------- Invariance & material frame indifference: Mase (pg 195 - ) has a good look at this concept, in terms of the suitability of constitutive equations. They introduce a number of additional invariance concepts, to which not all tensors will conform. Certain combinations of tensors can be formed to provide this additional invariance requirements. This would fall in with your point 3. ----------- I'm currently researching nabla_v & its role, if any, in the (momentum-)wave phenomena I've been observing. There is also an interesting apparent link between vortices & waves/oscillations. Thanks so much for your input. desA |
Re: Velocity gradient tensor
A couple of minor points S_ij is usually the deviatoric stress D_ij is the symmetric part of the velocity gradient - in general they are not the same thing.
For frame indifference (see "Continuum mechanics" by Chadwick) all that is required, for a fluid, is that the stress tensor should be a function of D_ij. The Cayley-Hamilton theorem then gives the general result sigma = -pI + f_0.D + f_1.D^2 where sigma is the stress tensor, p is the pressure and f_0, f_1 are functions of the 3 invariants of D (i.e. tr(D), det(D) and adj(D); tr(D)=0 for an incompressible fluid). This is the general form for the stress tensor for a fluid (usually called a Reiner-Rivlin fluid). with f_1 = 0 we have f_0 = const. Newtonian fluid, f_0 proportional to the square root of adj(D) is the Smagorinsky subgrid turbulence model used in LES. Similar arguments hold for a solid with the velocity gradient replaced by the deformation gradient. The difference between a fluid and a solid is that particle labels in a fluid are interchangeable while in a solid they are not (a consequence of this is Kelvins circulation theorem). |
Re: Velocity gradient tensor
Thanks Tom for your excellent insights. Always very much appreciated. (I was hoping you'd look in)
Mase covers some of this, although in not as much detail, in terms of the constitutive equation for the viscous stress tensor (p 286). Again on p288 as the stress tensor. With Stokes condition & incompressible fluid (isochoric motion) we lose a few more terms, until we finally arrive at the simple form of the incompressible N-S (as we all well know). dv/dt+ v o nabla_v = n*nabla2_v - (1/d)nabla_p + a where: dv/dt partial derivative a = body acceleration d = density (constant property fluid) n = kinematic viscosity Now, the convection term at v.nabla_v is straightforward, but, what of the delta_v (nabla2_v) term? Can we use: nabla2_v = nabla o nabla_v ? ie. v_i,jj = (v_i,j),j If so, then we seem to have two sets of nabla_v terms - one for convection & the other for the diffusion term (Eulerian form). Have I missed something glaringly obvious in my approach? I'd welcome input here. desA (diaw) |
Re: Velocity gradient tensor
"Can we use:
nabla2_v = nabla o nabla_v ? ie. v_i,jj = (v_i,j),j" Strictly speaking the viscous term is div( gradv + (gradv)^T ) which is, in component form ( v_i.j + v_j,i ),j = v_i,jj + v_j,ij For an incompressible fluid v_j,j = 0 so the last term vanishes yielding your formula. Basically, as pointed out in an earlier post, v_i,j is not a tensor since it is not invariant under reflections (i.e. moving from a left handed system to a right handed one). The reason why it's not invariant is best seen by writing v_i,j = (v_i,j + v_j,i)/2 + ( v_i,j - v_j,i )/2 The first term on the right is the symmetric part of the velocity gradient which is invariant under the orthogonal group O(3) while the second is the anti(or skew)-symmetric part which is not invariant with respect to reflections. Basically this last term is related to the rotation of the fluid (vorticity) and so when viewed in a mirror changes sign. Since the stresses must be observer independent they cannot depend upon the anti-symmetric part of the velocity gradient. Strictly speaking div is only defined for Tensors and it is defined via, for some tensor T and any choice of constant vector a, [div(T)].a = div(Ta), which is not the same as your formula! Try appying this formula to T_ij = u_i,j and T_ij = u_i,j + u_j,i - only one of them gives the correct answer. "Continuum Mechanics: Consise theory and problems" by Peter Chadwick is worth looking at. Hope my ramblings aren't too confusing, Tom. |
Re: Velocity gradient tensor
Thanks for your input, Tom. As always, lots of food for thought. While you are on-thread, I'd like to probe a little deeper, if I may.
(Tom): Basically, as pointed out in an earlier post, v_i,j is not a tensor since it is not invariant under reflections (i.e. moving from a left handed system to a right handed one). (desA): Fair-enough. So, in order to strictly be a tensor, it seems that it has to be invariant under *both* rotation & reflection. (I had been considering the rotation definition.) A few thoughts: Would v_i,j be classed as a pseudo-tensor - if it is invariant under basis rotation (I'll re-check the basis rotation)? Although this system is not a tensor, can the principal values & principal directions still be computed (per usual approach)? If so, would there be any significance in the eigenalues becoming complex. What part does nabla_v play, if any, in the creation of jump/shock phenomena in the Euler equations? In other words, what drives the shock - pressure, velocity, or both together? ------------ (Tom): Strictly speaking div is only defined for Tensors and it is defined via, for some tensor T and any choice of constant vector a, [div(T)].a = div(Ta), which is not the same as your formula! Try appying this formula to T_ij = u_i,j and T_ij = u_i,j + u_j,i - only one of them gives the correct answer. (desA): This, I had not come across, or certainly hadn't thought about. I'll work through the implications. Excellent. So, basically, it is not possible to decompose nabla2_v => nabla o nabla_v, since nabla_v is not a tensor. Fair enough. ----------- (Tom): "Continuum Mechanics: Concise theory and problems" by Peter Chadwick is worth looking at. (desA): I see it's a Dover publication. I'll get one on order. Thanks so much for the reference. ---------------- (Tom): Hope my ramblings aren't too confusing, Tom. (desA): I'm always very grateful for your wise input. These e-discussions have been invaluable in shaping a large number of my academic sorties along this research path. I will always be extremely grateful for your kindness. (If anyone rambles on, it's me :) desA |
Re: Velocity gradient tensor
"(desA): Fair-enough. So, in order to strictly be a tensor, it seems that it has to be invariant under *both* rotation & reflection. (I had been considering the rotation definition.) "
Basically Tensors fall into 2 categories (1) those invariant under the rotation group SO(3) and (2) those invariant under reflections and rotations O(3). The decomposition of the velocity gradient into L_ij = u_i,j = (u_i,j+u_j,i)/2 + (u_i,j-u_j,i)/2 = D_ij + W_ij creates a symmetric and skew-symmetric tensor. D_ij is the deformation tensor which is invariant under O(3) while W_ij is the spin tensor which is invariant only under SO(3). Since the physics must be invariant under the larger O(3) group the stress tensor must only depend upon D_ij. As pointed out above by the other "tom" the W_ij term is important in vortex dynamics (as should be obvious from its form). It was incorrect of me to say that L_ij was not a tensor - I meant that physically it's only the symmetric part that is important in the stresses because the Navier-Stokes equations are in covariant form so that invariance under reflections is important. I think your problem is that your contracting the wrong index in your div; e.g. div(T) = T_ij,i not T_ij,j ! Hope this clears things a bit (it's a long time since I studied tensors - close to 20 years). "What part does nabla_v play, if any, in the creation of jump/shock phenomena in the Euler equations? In other words, what drives the shock - pressure, velocity, or both together?" It's the density equation (coupled to the pressure via the equation of state) that causes the shock - this is why incompressible flows do not have shocks. The pressure-density coupling is why the speed of sound, and hence Mach number, is so important in shock formation. The different types of jump condition are discussed in Chadwicks book but only the shock-wave is possible in a compressible fluid. The best place to find out more about shock waves is the book by Courant & Friedrichs or, the cheaper, "High speed flow" by C.J. Chapman. |
Re: Velocity gradient tensor
Thanks again Tom for your input. Things are now a lot more clear.
(Tom): It was incorrect of me to say that L_ij was not a tensor - I meant that physically it's only the symmetric part that is important in the stresses because the Navier-Stokes equations are in covariant form so that invariance under reflections is important. (desA): This makes sense. (Tom): I think your problem is that your contracting the wrong index in your div; e.g. div(T) = T_ij,i not T_ij,j ! Hope this clears things a bit (it's a long time since I studied tensors - close to 20 years). (desA): From direct tensor notation development, I'm getting - substituting for d_ij onto either nabla, or T_ij, it comes out the same way (developed in cartesian tensor form): div(T) = T_ji,j *e_i (vector) I'll get down to settling this one in my head during the next few days. Thanks so much for the pointers. (I'm pretty much hooked on tensors at the moment - they are so useful). (desA): "What part does nabla_v play, if any, in the creation of jump/shock phenomena in the Euler equations? In other words, what drives the shock - pressure, velocity, or both together?" (Tom): It's the density equation (coupled to the pressure via the equation of state) that causes the shock - this is why incompressible flows do not have shocks. The pressure-density coupling is why the speed of sound, and hence Mach number, is so important in shock formation. The different types of jump condition are discussed in Chadwicks book but only the shock-wave is possible in a compressible fluid. The best place to find out more about shock waves is the book by Courant & Friedrichs or, the cheaper, "High speed flow" by C.J. Chapman. (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 convection-type 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. Thanks for the Courant & Friedrichs, Chapman links. I've got access to the first one, if I remember our library correctly. --------- Oscillation-type waveforms: Now, that we have firmly established that incompressible flows cannot evidence pressure-type waves, due to fixed density, would it, in your opinion, be possible for these flows to show oscillation-type solutions? If so, then I may just happen to be in some luck, & my previous year's research has all been worthwhile :) If oscillation solutions are possible, could these be considered as time-decaying wave-forms? These should display resonance-type & vibration-type effects. If it does seem plausible for these wave-forms to be evidenced, then I'm going to clean up my current paper I'm working on & submit it promptly. :) desA |
Re: Velocity gradient tensor (errata)
Correction:
(desA):... If so, then viola - a pressure convection-type wave equation, driven along by -div(v)! |
Re: Velocity gradient tensor
"Now, that we have firmly established that incompressible flows cannot evidence pressure-type waves, due to fixed density, would it, in your opinion, be possible for these flows to show oscillation-type solutions?"
Yes this is well known. A simple example is Taylor-Couette 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. "If oscillation solutions are possible, could these be considered as time-decaying wave-forms? These should display resonance-type & vibration-type effects..." 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): 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 convection-type 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." I think you're referring here to the well-known 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). |
Re: Velocity gradient tensor
Thanks Tom for your input.
(desa): "Now, that we have firmly established that incompressible flows cannot evidence pressure-type waves, due to fixed density, would it, in your opinion, be possible for these flows to show oscillation-type solutions?" (Tom): Yes this is well known. A simple example is Taylor-Couette 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): Taylor-Couette flow bifurcation I have read about. What is 'ABC flow'? Has any major inroads been made in this area for the full Navier-Stokes, for say flow within confined domains, lid-driven 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 time-decaying wave-forms? These should display resonance-type & vibration-type 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 convection-type 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 well-known 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 N-S still considered good-enough 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 time-lag ending up in a 'telegraph-like' 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 micro-fluid & nano-fluid flow regimes? ------------- Thanks again Tom for your exceptionally kind contributions. As always, I now have a mountain of reading ahead of me... :) desA |
Re: Velocity gradient tensor
(desA): Taylor-Couette flow bifurcation I have read about. What is 'ABC flow'? Has any major inroads been made in this area for the full Navier-Stokes, for say flow within confined domains, lid-driven 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 mid-eighties - see papers in JFM by Goldstein, Wu, Smith and there co-workers through the 80's and 90's. "- Are the N-S still considered good-enough 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 time-lag ending up in a 'telegraph-like' 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 Magneto-hydrodynamics 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 Navier-Stokes 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 micro-fluid & nano-fluid flow regimes?" Probably - since you may violate the continuum hypothesis. |
Re: Velocity gradient tensor
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 force-balance, 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 velocity-component 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 large-scale effects can also be seen in these acceleration-fields. desA |
Re: Velocity gradient tensor
(desA): ***What physical reason is given to the occurrence of this Hopf bifurcation? Is there a change in force-balance, 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. |
Re: Velocity gradient tensor
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 pressure-type waves, due to fixed density, would it, in your opinion, be possible for these flows to show oscillation-type solutions?" Italics: (Tom): Yes this is well known. A simple example is Taylor-Couette 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 pressure-type waves, due to fixed density, would it, in your opinion, be possible for these flows to show oscillation-type solutions?" ---------------------- just a suggestion. |
Re: Velocity gradient tensor
Thanks for an excellent suggestion.
Can I ask *how* you enable (format) the various options under the cfd-online editor? This has been the trick that has eluded me until now. :) Once again, thanks for your suggestion. desA |
Re: Velocity gradient tensor
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 force-balance, 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 'viscous-shock line'. The mechanism for viscous shock (vs) is a drawn-out 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 |
Re: Very quick html
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:
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Re: Very quick html
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Re: Very quick html
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 |
Re: Very quick html
I was also <h2> blissfully unaware </h2> of it for long time. It happens though.
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Re: Velocity gradient tensor
(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 'viscous-shock line'. The mechanism for viscous shock (vs) is a drawn-out 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 re-invent 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 Orr-Sommerfeld 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 |
Re: Velocity gradient tensor
(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 'viscous-shock line'. The mechanism for viscous shock (vs) is a drawn-out 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 re-invent the wheel! (desA): I'm afraid I don't quite follow which wheel I'm re-inventing. I'll provide my simple observations. The research I've been following is to essentially understand the momentum equations in their original as-derived form - as force balances on a moving fluid particle, with consequent acceleration. This logic essentially carries the original physical logic of the N-S into their final incompressible formulation. The N-S as derived are a simple non-linear 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 time-evolution snapshot stops changing visually is when we have reached a Eulerian steady-state. At this juncture, a number of standing-wave phenomena (time-variable-fixed-space) can be observed. The fact that modal-type effects can be simply observed in various flow fields, can be explained by a slow & fast decomposition of the N-S, 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 N-S 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 low-speed region where instability just begins. My research is basically exploring the action & mechanism of how the force-fields morph & modify themselves as the flow field develops. In the end, simulations basically boil down to a structure composed of a large-scale array of non-linear 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 Orr-Sommerfeld 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 macro-form, & am not sure how exactly we are able to work up from the local N-S form to these macro-effects with current available mathematical tools. I'm trusting that the large-scale 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 macro-scale 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 re-set the links? Thanks once again for your applied wisdom & critique - it is greatly appreciated. desA |
Re: Velocity gradient tensor
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"? |
Re: Velocity gradient tensor
Thanks Tom for your reply. (I've been at the receiving end of the Asian internet tsunami & have been off-line 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 |
Re: Velocity gradient tensor
(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 "Infinite-dimensional 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 Navier-Stokes 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). |
Re: Velocity gradient tensor
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 |
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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