Why CD is used for DES?
 

Hi, we know that Central difference is very diffusive for Pe>2, still we are using it for DES? I cudn't get it. have anybody suggestion?

Re: Why CD is used for DES?
 

"I cudn't get it. have anybody suggestion?"

Yes - read a book on numerical methods. The comment

"we know that Central difference is very diffusive for Pe>2,.."

is just wrong - the lead error in CD schemes is dispersive NOT diffusive. If CDS was very diffusive it would not be being used in LES. Upwind shemes on the otherhand are very diffusive which is why they aren't used in LES with explicit SGS terms. This is also why CD schemes are used in DNS.

Re: Why CD is used for DES?
 

Tom wrote:

the lead error in CD schemes is dispersive NOT diffusive. If CDS was very diffusive it would not be being used in LES. Upwind shemes on the otherhand are very diffusive which is why they aren't used in LES with explicit SGS terms. This is also why CD schemes are used in DNS.

Would you be so kind as to expand a little more on the difference between dispersive schemes and diffusive schemes, as well as the observable physical differences.

Many thanks, mw

Re: Why CD is used for DES?
 

Diffusion --> the shortest wavelengths are the most damped. Dispersion --> the shortest wavelengths are the most phase-shifted.

Re: Why CD is used for DES?
 

Basically dispersive schemes, for a linear equation, allow the Fourier components to drift apart with speeds related to the wavenumber. In a diffusive scheme these Fourier coefficients will be damped with time at a rate related to the wavenumber.

for the simple advection equation u_t + u_x = 0, whose exact solution is f(x-t) where f(x) is the initial condition this is manifest as :

diffusive scheme => solution decays with time but the main shape of f should be preserved.

dispersive sheme => solution spreads out and becomes oscillatory.

Try solving this equation with some different numerical schemes to see these effect.

Tom.

PS. "momentum waves" you may want to ask yourself "do waves have momentum"

Re: Why CD is used for DES?
 

Thanks, Tom, for your excellent review of the dispersion versus diffusion mechanism in a linear equation. It is most helpful.

Tom wrote:

PS. "momentum waves" you may want to ask yourself "do waves have momentum"

mw's reply:

Ah, an excellent question indeed.

The 'momentum_waves' moniker is actually short for 'momentum-driven waves in incompressible N-S fluid flows'. Essentially, this waveform evidences itself as an oscillation carried along on an underlying carrier advection flow. An exact solution for the N-S under certain momentum-driven conditions evidences itself as a 'carrier + periodic' solution - in 2D. This is not a pressure-driven waveform.

Under this solution, the diffusion-effect in terms of time-driven decay of the oscillation form is a function of wavenumber squared & time. Simultaneously, a dispersion effect appears to be present which is linearly-related to wavenumber.

Current numeric simulations have validated the periodic solution. The exact solution is also well referenced back to known literature. Yes, incompressible fluid flows can & do express waveforms.

--------

To answer the original question - "do waves have momentum".

An oscillatory momentum waveform contains an instantaneous fluctuating velocity that varies with both time & position. Multiply this by the local mass under consideration & an instantaneous fluctuating momentum must, by definition, exist. If the advection carrier velocity remains constant, then the instantaneous momentum of the flow will vary from position to position.

On the other hand, if the diffusion damping effect were almost zero, the oscillation would have an approximately zero average momentum for each full period. For momentum waves, the diffusion-damping effect is generally not zero, albeit small.

mw

Re: Why CD is used for DES?
 

`To answer the original question - "do waves have momentum"`

Actually the answer is NO they do not have momentum - this is true for waves in all of physics including quantum mechanics.

The 1981 paper "On the 'wave momentum' myth" by McIntyre (JFM 106) discusses this. From the abstract

Controversies over 'the momentum' of waves have repeatedly wasted the time of physicists for over half a century. The persistence of the controversies is surprising, since regardless of whether classical or quantum dynamics is used the facts of the matter are simple and unequivocal, are well checked by laboratory experiment, are clearly explained in several published papers, and on the theoretical side can easily be verified by straightforward calculations. They are illustrated here by some simple, classical examples involving acoustic and gravity waves.

The Andrews and McIntyre papers from (1978), also in JFM, on the "General Lagrangian Mean" discuss the correct concepts of psuedo-momentum/energy, etc.

Re: Why CD is used for DES?
 

So, by all account, then, breaking waves do not transfer, or re-distribute momentum?

A good physics movie should settle that one pretty quickly. :)

mw

Re: Why CD is used for DES?
 

There is a difference between a "momentum flux" due to the wave and wave-momentum! They have different units for a start!

Read McIntyre's paper - it's quite a good review of the subject and isn't very mathematical.

Re: Why CD is used for DES?
 

Thanks Tom. Perhaps we are dealing with semantics more than physics at this point. Consider a tsunami for instance - an earthquake occurs, wave propagates with local shm (almost) motions of local particles which return to previous positions, without a net motion, wave-energy continues to propagate until a distant shore is reached, local fluid suck-back occurs, suddenly wave breaks & spills water onto the land, some water eventually drains back.

Does momentum transfer take place? If not, then how does the water end up on the land? What is the mechanism? What we do know is that a seismic event occurs, a wave is generated & propagates until it eventually breaks. The energy is transferred from source, to sink - local momentum transfer occurs at the end. Does the water from the original seismic event reach you? Not a chance.

I'd like to read the paper - thanks so much for the reference. Does anyone have a copy of this paper, or know where it can be readily located? Our library rarely goes back to that vintage, unfortunately.

---------

A few miscellaneous thoughts:

What would you consider period solutions of the transient N-S to represent? Do fluctuating velocity components about a mean bulk flow (slow-fast) have instantaneous momentum? With a gravity plug wave moving down a channel, what is the local instantaneous momentum history along the channel? What part is bulk flow & what part is fluctuating (wave) component? What about the many smaller surface waves often overlooked on the surface moving up & down - is there local momentum transfer - is there local average momentum transfer?

The Shapiro series (ancient) of fluid videos are fascinating - especially the wave flow in channels presentation.

mw

Re: Why CD is used for DES?
 

"Thanks Tom. Perhaps we are dealing with semantics more than physics at this point."

McIntyre, and I agree with him, would argue that we dealing with the "the correct physical interpretation".

"I'd like to read the paper - thanks so much for the reference. Does anyone have a copy of this paper, or know where it can be readily located? Our library rarely goes back to that vintage, unfortunately"

Try JFM online, that's where I lifted the abstract (haven't got a copy myself - I just remeber reading it quite a few years ago).

Otherwise why not request a copy from Michael - he may still have some offprints.

"What would you consider period solutions of the transient N-S to represent?"

A periodic solution by definition can't be transient.

"Do fluctuating velocity components about a mean bulk flow (slow-fast) have instantaneous momentum?"

No it doesn't - the momentum is the sum of the two terms and it is technically incorrect to call either of the bits instaneously momentum. Think of a small fluid parcel, it only sees the sum and has no concept of some arbitrary split you may introduce in your mathematical formulation.

Re: Why CD is used for DES?
 

Thanks Tom. I'll contact Michael & see if I can locate a copy. Thanks so much for that.

mw wrote:

"What would you consider period solutions of the transient N-S to represent?"

Tom's reply:

A periodic solution by definition can't be transient.

... but, a periodic solution with a decaying multiplier, added to a carrier velocity can indeed be transient.

Tom, I've a copy to hand of Philip Drazin & Norman Riley's latest work "The Navier-Stokes Equations - A classification of flows & exact solutions" (2006).

Working through this excellent book I've been able to basically prove out my original theories regarding wavelike solutions of unsteady flows. He briefly refers to similar work - the prototypical studies being unsteady flow in channels with various manipulations of the boundary conditions. He refers to a method in 2D using a stream-function approach. I've extended the theory further into U,V flows & onwards into 3D, subject to various conditions on wavenumber & so forth. This has basically anchored a lot of my previous research onto a solid footing.

The numeric simulations of the same setup as for the exact solutions produce the desired effects. It is very interesting. I'm working on various variants of this at the moment.

-------- mw wrote:

"Do fluctuating velocity components about a mean bulk flow (slow-fast) have instantaneous momentum?"

Tom's reply:

No it doesn't - the momentum is the sum of the two terms and it is technically incorrect to call either of the bits instaneously momentum. Think of a small fluid parcel, it only sees the sum and has no concept of some arbitrary split you may introduce in your mathematical formulation.

My reason for performing the split is that I'm working on a slow-fast mechanism where the oscillations basically are carried along by the bulk flow. Naturally, the momentum at any point would be the sum of the two components - to be sure. In certain cases, the bulk flow can be set out & then the only velocity under consideration is the fluctuating component.

The name of the game here is essentially the balancing act performed between the convection & diffusion terms - a little similar to the soliton convection-dispersion balancing act.

---------

While we're chatting, I'd like to ask another two questions if I may be so bold:

A velocity field exists in an enclosed flow domain eg. flow over a backward-facing step under transient flow. Is the momentum constant at all points in the flow domain in time-space?

How would you define 'wave momentum' versus 'momentum flux'?

Thanks so much for your kind response.

mw :)

Re: Why CD is used for DES?
 

"... but, a periodic solution with a decaying multiplier, added to a carrier velocity can indeed be transient."

If you mean periodic in space then this is true (under the unphysical boundary conditions of periodicity). In the general IBVP the transient will not be of the form you describe. Solve the problem of an infinite flat plate at y=0 below a fluid at rest. Then set the plate in motion with velocity sin(w.t) - The final state is the classical Stoke's layer solution. However the transient is rather complex - in a more general problem it will be even worse!

"In certain cases, the bulk flow can be set out & then the only velocity under consideration is the fluctuating component."

Except one or two cases which have exact solutions, usually due to the advection terms being trivial, this is never (exactly) the case. If you have a single "wave" component E=exp(ik(x-ct)) then the nonlinear terms will produce higher harmonics. In particular the quadratic terms will produce terms proportional to the mean and second harmonic E^2 ( these in turn interact to produce a secular term due to the fact that E=E^2.E^* -* denotes conjugation).

There are three points

(1) the mean flow is altered by the wave which in turn effects the wave!

(2) You are going to have to do something with the secular terms to remove "resonances" and hence render the approximation uniformly valid.

(3) This type of procedure almost never produces a convergent approximation; i.e. the radius of convergence is zero.

I don't know if this in Norman & Philip's book but you should look at Norman's work on acoustic/steady streaming.

"A velocity field exists in an enclosed flow domain eg. flow over a backward-facing step under transient flow. Is the momentum constant at all points in the flow domain in time-space?"

Well the flow isn't really enclosed (you need inflow and outflow conditions - i think you mean it's an internal flow)

If the momentum was constant at points in space and time then it wouldn't be transient (it would be a steady state - I'm assuming you're allowing for a different constant at each point in space, otherwise its not a solution to the problem).

"How would you define 'wave momentum' versus 'momentum flux'?"

Well I don't define "wave momentum". psuedo-momentum is defined in the paper by Andrews & McIntyre which I mentioned in an earlier post (can't remember its exact definition). It's most useful property is that, for small amplitude oscillations it is equal to the Stoke's drift (i.e. the drift of fluid particles in oscillatory flow - this is why particles below a water wave do not travel in closed circles as predicited by linear theory). The Stoke's drift/psuedo-momentum is the driving term in Norman's work on acoustic streaming and also gives rise to Langmuir circulations in the ocean (via the Craik-Leibovich equations which have an elegant derivation from the General Lagrangian Mean).

Actually you could add the lack of closure of the General Lagrangian Mean equations to my above list of problems in decomposing the velocity into mean and fluctuating parts. The closure of the Reynold's averaged eqautions is another.

Re: Why CD is used for DES?
 

Thanks Tom for your excellent comments & review. This is extremely helpful.

Drazin & Riley's book (2006) is a review of the current state of known exact solutions to the N-S. The work I've found very useful is that performed by Hui (1987), in 2D.

"In certain cases, the bulk flow can be set out & then the only velocity under consideration is the fluctuating component."

Does the velocity to which the fluctuating is referenced, have to be variable? What if a relative N-S equation were constructed relative to some known reference velocity? Would this make the job a little easier? No messy bulk-fluctuation coupling problems.

I'll definitely look into Andrews & McIntyre's work. Thank you for that.

Re: Why CD is used for DES?
 

"In certain cases, the bulk flow can be set out & then the only velocity under consideration is the fluctuating component."

No (also remember that the equations are Galilean invariant). However a fluctuation is usually defined as having zero mean so if you expand the a variable as

u = U + M + u'

where U is the mean and u' is the fluctuating part with mean(u')=0 then M is the mean induced motion due to the fact that mean(u'u') is nonzero; i.e. if u' is not exactly zero then U is not really the mean. Even if U is constant there is no reason to believe that M will be - and it is crucial that it is not since, for example in the steady streaming and Langmuir cases, it generates secondary motions in the fluid via a vortex force contribution to the averaged equations. The fact that M is nozero is also why boundary layer streaks occur via oblique mode interactions.

When |u'| is (very) small M can be determined from u' iteratively (although as I've said the resulting series will not be convergent). However if |u'| is not small then it is impossible to disentangle U and M which is also why there must be a 2-way coupling in general and no "slaving" principle.

Re: Why CD is used for DES?
 

Thanks very much, Tom.

Hui works with the case of steady channel flow with, for instance, the upper plate being subjected to a stretching motion, while Cox has transpiration across one boundary, with the other impermeable.

A quote from Riley - "In both of these situations it is shown that time-periodic limit-cycle solutions emerge following Hopf bifurcations."

The equations are developed with steady channel flow U=const. The fluctuating component is periodic about this steady U value.

Essentially the steam function becomes:

S = S(U) + fn(t)*sin(k.x-w.t)

Extraction of u,v is straightforward.

Now, when the same situation is simulated with a moving top plate, the periodic motion is observed in the lower 3/4 of the flow domain, whilst the upper section closest to the top plate shows the effects of the plate-fluid interface - folds & so forth seem apparent.

--------

I've since extended the theory into U,V steady & onto 3D (wip) with some restrictions. I've managed to isolate plug waves (?surge?) & a few other interesting things so far.

I'd value your insights into this approach.

mw

Re: Why CD is used for DES?
 

"I've since extended the theory into U,V steady & onto 3D (wip) with some restrictions. I've managed to isolate plug waves (?surge?) & a few other interesting things so far."

Sounds like a standard bifurcation analysis - however I'd need to read the paper to make a better judgement (e.g. U=const will in general not satisfy the no-slip boundary conditions - so there's something lacking in the above explanation?). The best advice I have is look up the original papers (if Cox is the one I think it is then you are refering to his PhD work with Drazin - your library may be able to borrow a copy).

It the analysis of the Hopf bifurcation is exact, as is suggested by your message, then this is a rather atypical problem since the stability of the resulting limit cycle is usually determined by weakly nonlinear stability theory which only constructs the periodic solution near to the bifurcation point.

Re: Why CD is used for DES?
 

Thanks Tom for your very constructive comments.

What I will say a this point is that the U=const analysis is helpful in gaining an understanding the portion of the (wide) duct flow, away from the oscillating surface. Near the surface itself, U is far from constant.

The extended theory with U,V=const, plus external acceleration term allows a better understanding of the physics.

In my research, I use a simulation platform to visualise the flow-field & then work back-forth between theory & simulation to extract more understanding. Visualisation plays a major part in this. Thus far, the bifurcations show up very clearly under flow-field interrogation. P4 tri elements, with high-order gaussian quadrature, on a variational scheme, is currently being used.

A question:

How would you relate bifurcation theory to the study of oscillations?

Many thanks for your kind input.

mw

Re: Why CD is used for DES?
 

Bifurcation theory is about the formation (branching) of new solutions from others as parameters are varied. For example if a problem u_t = F(u,R) has a steady solution u=U for all it is possible for this solution to become unstable as R is varied. If at some value of R=R_c the linearized problem has a pair of purely imaginary eigenvalues then there is the possibility of a Hopf-bifurcation which will give rise to a new family of periodic solutions near to R=R_c - a solution with |R-R_c| << 1 can be developed to find this solution and to determine whether it is stable or not. Numerical path tracing algorithms can then be used to trace this new solution over a wider range of values of R-R_c and to then determine further bifurcations of this solution.

For a Hopf bifurcation

u ~ U + Ae^{iwt}Q + ...

with A = O( (R-R_c)^{1/2} ); A actually satifies the equation

A_t = a.A + b.|A|^2A + O(A^5),

for some complex constants a & b. Note that Q is, to lead order, the eigenvector corresponding to the purely imaginary eigenvalues. By determining a & b you can determine whether the bifurcating solution is stable (supercritical) or unstable (subcritical).

Re: Why CD is used for DES?
 

Thanks very much Tom for the review.

How would this concept be used for instance on the Euler equations? Would it show up the characteristic lines, for instance?

Do you perhaps have any solid references you could recommend on bifurcation theory?

I've been working through Crawford. Next in line are O'Neill (1966), Panfilov (2001), Arnold - bifurcation, catastrophe (busy translating his later versions from Russian language - albeit rather slowly).

I'm particularly interested in working in tensor form, if possible. This is where O'Neill seems to be useful. I'd also be interested in a reference which could overlay symmetry onto tensor forms without me having to go all the way through Lie algebra. In my mind's eye, I see everything evidencing itself at tensor level - but this is yet to be proved conclusively, although some rules are emerging.

Once again, thank you for all your extremely kind comments.

mw

Re: Why CD is used for DES?
 

"How would this concept be used for instance on the Euler equations? Would it show up the characteristic lines, for instance?"

Limit cycles don't exist for the Euler equations - the same method of analysis applies but the result is a time-reversible Hopf bifurcation; i.e. the amplitude A cannot be determined uniquely but takes its value from the initial condition.

"Do you perhaps have any solid references you could recommend on bifurcation theory?"

"Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations" by Glendinning

is a good introduction.

A more relevant book for PDEs is "Elementary Stability and Bifurcation Theory" by Ioss and Joseph. This book is a bit harder!

For symmetry problems there's the two volume set by Golubitsky & Schaeffer but I wouldn't recommend them (over zealous use of there ideas - and seriously incorrect in at least one of there results in vol2).

General bifurcation theory is usually in operator form (i.e. functional analysis) which is more general than tensors; i.e the form of the equation I wrote for the amplitude function A is completely independent of the physical/mathematical problem under consideration. Most symmetry based bifurcation theory is concerned with compact Lie groups and so the general theory isn't needed.

Re: Why CD is used for DES?
 

Thanks, once again Tom - this is very valuable information for my work.

Ioss and Joseph. This book is a bit harder!

I have a copy to hand of this book. A 'bit harder' is an understatement. This one was planned to be at the end of the list. :)

mw

Re: Why CD is used for DES?
 

" have a copy to hand of this book. A 'bit harder' is an understatement. This one was planned to be at the end of the list. :)"

Yes but once you understand the (slightly more than) basics it's very good! They need to loose elementary from the title.

Glendinnings book is extremely good though - the only reason I haven't got a copy is that I had to donate the copy I reviewed for a journal to the library.

Note that Glendinnings book is really about the basics as applied odes - in many applications pdes can be treated as infinite dimensional odes. Once you understand this book Ioss & Joseph should be less of a challenge!

It might be worth looking at "Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics" by Verhulst (he's also written some other quite good books).

Re: Why CD is used for DES?
 

Thanks very much, Tom. I'll order both Glendinnings & Verhulst.

I'm thoroughly enjoying this phase of my research.

mw...

Bifurcation visualisation techniques
 

Tom, would you know of any visualisation techniques which have been used to explore bifurcation processes in 2D flow simulations, & in physical experimental studies?

For instance, how would one search for bifurcations - lines, fronts, surfaces & so forth?

(Btw, Glendinnin is on order, Verhulst has been located. Many thanks.)

mw...

Re: Bifurcation visualisation techniques
 

The usual method is to reduce the problem to a single scalar quantity and plot its varaition as the control parameter is varied (preferably a quantity that is constant for the background flow irrespective of the control parameter - changes are then bifurcations).

Another method is sample a point as a function of time and plot a phase diagram; e.g. plot u against v at a particular as time evolves or sample, for example, u at 2 points and plot these against each other. A simple closed loop is a periodic function - if, as a control parameter is varied, the loop circles twice before closing a period doubling has occurred, etc. (if you pick three points you can do plot the 3d curves such as seen in the classic Lorenz attractor/Rossler band when the solutions are chaotic).

Careful choice is important (and can be a bit hit and miss) since plotting the wrong quantity can give misleading impressions such as missing the first period doubling or thinking you've found one when you haven't. Looking at the time series can help here.

If you've got a copy of "wave interactions and fluid flow" by Alex Craik then there's some (turns out to be incorrectdue to numerical problems - rounding errors if I recall) pictures of this type on p.267. I wouldn't recommend buying the book if you don't already have it - it's a bit too sparse for learning from.

As a more complex example you could calculate some global integral such E(t)=int(u^2+v^2)/2 and plot this against dE/dt.

There are many things you can try. In this setting bifurcations are viewed as changes in the topology of the phase portrait.

Hope this is helpful.

Re: Bifurcation visualisation techniques
 

Thanks very much, Tom.

The usual method is to reduce the problem to a single scalar quantity and plot its varaition as the control parameter is varied (preferably a quantity that is constant for the background flow irrespective of the control parameter - changes are then bifurcations).

This approach makes a lot of sense.

Another method is sample a point as a function of time and plot a phase diagram; e.g. plot u against v at a particular as time evolves or sample, for example, u at 2 points and plot these against each other. A simple closed loop is a periodic function - if, as a control parameter is varied, the loop circles twice before closing a period doubling has occurred, etc. (if you pick three points you can do plot the 3d curves such as seen in the classic Lorenz attractor/Rossler band when the solutions are chaotic).

I've been working with this approach on lid-driven cavity flow for the various modes, plotting u-v on a spreadsheet, for instance. Some very interesting results are emerging, & some quite exceptional symmetry. I began with Stokes & then moved onto N-S flows. When the system began to head towards instability (numeric possibly, since it was strongly dependent on mesh size) this could be seen in the u-v pattern, with waves & distortions. Very interesting.

I'll build this up a bit further & couple it with my current simulations - it should be interesting.

One thing I've found to be of interest is the (u/v) or (v/u) ratio in a 2D flow field - especially if combined with scale-leveling approaches. This offers some insights into regions of the flow field which seem to react on a large scale, rather than purely on a local scale.

Thanks so much for your kind contribution.

mw...

Bifurcation in space
 

Following on from the bifurcation visualisation concepts mentioned above, I have a question:

What happens, if, when u & v phase is plotted in 2D space, a series of pitchfork bifurcations show up in the flow domain?

I have observed this phenomenon many, many times in my research - using both FVM & FEM solvers. What does this represent? Would this be a bifurcation in the spacial variables?

Many thanks for your kind contribution.

mw

Re: Bifurcation in space
 

"What happens, if, when u & v phase is plotted in 2D space, a series of pitchfork bifurcations show up in the flow domain?"

The phase is a bad variable - it has jumps in it!

"I have observed this phenomenon many, many times in my research - using both FVM & FEM solvers. What does this represent? Would this be a bifurcation in the spacial variables?"

In this type of problem you cannot talk about "bifurcation in the spacial variables" - it has no meaning! The pitch fork and Hopf bifurcations are global instabilities (the spectral problem is elliptic) and so these bifurcations affect all points within the domain.

Re: Bifurcation in space
 

"What happens, if, when u & v phase is plotted in 2D space, a series of pitchfork bifurcations show up in the flow domain?"

The phase is a bad variable - it has jumps in it!

What about tan, atan2, for instance?

Where I'm going with this, is that for a vector field, with amplitude & direction (phase), a sudden jump in the direction component, if interpreted correctly, should have a physical meaning.

For instance, this problem crops up quite often in some fairly well-known chaos problems eg. Henon-Heiles. When reduced to a vector & direction interpretation, the direction is observed to vary abruptly at certain points in the acceleration field. No wonder the underlying 'velocity' field jumps around like a march hare.

For resonance-type situations, the rapid velocity rise, coupled with a swing of phase by 180' is considered significant.

The pitchfork bifurcation I've been observing occurs in reasonably low-speed flows down a long tube, for instance & builds up over a period of time - ending up with a similar effect to what could be expected of a vibrating plate, although not in perfect rectangles. Just behind a cylinder, for instance, a strong line can be observed in the v-velocity field across which the velocity changes direction.

These pitchfork-type phenomena crop up in flow down a tube typically, when observed on v-flow, for instance. They generally occur in unsteady acceleration fields fairly routinely.

In a meandering flow within a tube, these pitchforks also occur at significant, repeatable positions in the flow field & are seen to begin to bunch up at Reynolds number increases.

mw...

Re: Bifurcation in space
 

"The pitchfork bifurcation I've been observing occurs in reasonably low-speed flows down a long tube, for instance & builds up over a period of time - ending up with a similar effect to what could be expected of a vibrating plate, although not in perfect rectangles. Just behind a cylinder, for instance, a strong line can be observed in the v-velocity field across which the velocity changes direction."

This is not what a pitchfork bifurcation does. The pitchfork bifurcation is described by the equation

A_t = aA - l.A.|A|^2 + O(A^5)

where a, and l are real (l is a constant and a varies linearly with the control parameter). It does not slowly build up over time - it spontaneously happens as a crosses from negative to positive (for a supercritical bifurcation with l>0).

The other point is the phase must jump no matter how you calculate it (if it didn't then there would be no need for branch cuts/Riemann surfaces in complex analysis).

Re: Bifurcation in space
 

Thanks very much, Tom. I'll get into Glendinning when it arrives - hopefully in the next few days).

mw...

Re: Bifurcation in space
 

Glendinning's book arrived today. Thanks for the excellent reference.

I spent the weekend writing up my latest findings on what I've been loosely calling "momentum-driven wave phenomena in incompressible N-S fluid flows". I've been able to write the bulk + oscillation equations in a vector-tensor notation which captures 3d reasonably elegantly.

There are a number of 'nice' features that seem to have popped out along the way - it's been absolutely fascinating:

1. In 2d flows, Hui's solution can be deduced with ease;

2. 3d flows now have related wave-number/velocity relationships.

3. The oscillations turn out to be 'linear' - would you believe it?

4. The crucial 'linearisation' piece to the jigsaw puzzle turns out to be a transformation of div(v)=0 under the oscillation transform. This was found out in reverse-engineering mode & ties up nicely with Hui's base assumptions (Riley).

5. The theory can be deduced from the N-S equations without too much pain by performing a simple bulk+fluctuation split, with the bulk flow being carefully selected.

6. The nabla-v' tensor for this oscillation transform turns out to have zero eigenvalues.

7. The existence region for these momentum-waves is now clearly defined by a geometric surface, inside which they exist, outside which they do not exist. The surface is my old 'friend' I understood to be a singularity surface many moons ago.

I'd love to know if you have ever come across this type of phenomena described in known literature - perhaps in other terminology - for low-speed flows. The way the theory has finally fallen into place so elegantly has shocked me somewhat, but, I guess it was really tying up a number of loose threads. I plan to now work through Glendinning & Iooss to assist me to clean up the theory.

Your comments would be greatly appreciated. My underlying end-game is that of tying wave-phenomena to the turbulence phenomenon - I envisage turbulence to a breaking wave effect, in my mind's eye.

mw...

Re: Bifurcation in space
 

"3. The oscillations turn out to be 'linear' - would you believe it?"

Most exact solutions to the equations have this property (it's why you can write them down). However this is only a handful of solutions and most solutions do not have this property.

The Rossby-Hauwitz waves also have this property (i.e. they are exact through a "miraculous" cancelation of terms in the nonlinearity).

"I'd love to know if you have ever come across this type of phenomena described in known literature - perhaps in other terminology - for low-speed flows. The way the theory has finally fallen into place so elegantly has shocked me somewhat, but, I guess it was really tying up a number of loose threads. I plan to now work through Glendinning & Iooss to assist me to clean up the theory."

Transformations which turn the nonlinear equations into linear ones (such as the Hopf-Cole transformation) is related to the existence of an infinite dimensional Lie-algebra (which the Navier-Stokes equations do not have). This is the origin of the inverse scattering transformation. In the case of Burgers equation the infinite dimensional Lie algebra is related to the solutions of the heat equation (the algegra is spanned by f.d_u where f satisfies the heat equation). The "straightening out" of this vector gives rise the the Hopf-Cole transformation ( f.d_u -> d_w ).

The wave-breaking description of turbulence dates back some 30years to Landahl who was inspired by Whitham's wave-action principle. As I recall Landahl's work was seriously (and quite rightly) criticized by the great Keith Stewartsom. I think the main objection was that the "waves" didn't correspond to causal solutions of the IBVP (although I could be wrong).

Re: Bifurcation in space
 

Thanks so much, Tom, for your insightful comments. I will read up on all the authors mentioned & see what insights I can gain from this.

I must say that I was amazed in reading through Riley's latest book as to how many simplifying assumptions are made along the way in developing 'exact solutions'. Practically, I would imagine that many of these solutions most likely only apply in a part of the flow field, closest to the boundary, or in sections well away from many boundaries. In my research, I've been careful to keep to only two major assumptions: incompressible flows, momentum-driven flows.

The wave-breaking description of turbulence dates back some 30years to Landahl who was inspired by Whitham's wave-action principle. As I recall Landahl's work was seriously (and quite rightly) criticized by the great Keith Stewartsom. I think the main objection was that the "waves" didn't correspond to causal solutions of the IBVP (although I could be wrong).

Could you perhaps elaborate a little more on the 'causal solutions of the IBVP' concept - perhaps with a few references, if possible? I have Whitham's book on order, after having first seen the Russian translation (1977). I had planned to read it a little later.

I have been trying to think of ways that these wave-like solutions could be proven experimentally, under controlled conditions. I'm not sure this would be a trivial task.

(BTW, Glendenning's book is a real find, I must say. I find his style very straightforward & easy to absorb. Thanks again for the reference.)

Re: Bifurcation in space
 

"Could you perhaps elaborate a little more on the 'causal solutions of the IBVP' concept"

This is nothing more than a requirement that the "signalling problem" is correctly specified; i.e. an observer should not be able to see the outcome of an event before it has happened. In (linear) initial boundary value problems this is enforced by the requirement (assuming a Fourier transform in time) that the path of the Fourier inversion curve is suitably defined in the complex frequency plane - this is analogous to the constant in the Bromwich inversion formula in the Laplace transform; i.e. if a flow is impulsively started into motion at t=0 then the derived solution using transform techniques should not give disturbances in t<0.

This also applies to boundary conditions in space where a perturbation at some point x=0 should not effect the point at infinity (i.e. modes growing with x need to be discarded).

These issues are usually covered in books on pdes (especially those ones concerned with analytic solutions of said equations by integral transforms).

Actually when Burger originally derived the "Burger's equation" he had the idea of wave-breaking in turbulence in mind - although this is different from Landahl's modulation (Benjamin-Feir) instability.

"(BTW, Glendenning's book is a real find, I must say. I find his style very straightforward & easy to absorb. Thanks again for the reference.)"

I think it's the best introductory book on the subject - it's much better than than the descriptive methods in Thompson and Stewart which hides what's going on by not delving into the mathematics (and as a result can be quite misleading).

Re: IBVP concept - Burgers
 

The 'IBVP concept' as explained makes perfect sense. The communication limit is something I've been keenly aware of during the numeric simulations. I developed some theory on this & basically ensured that the simulation communication timescale did not exceed the governing physical limit (perceived, or real).

Actually when Burger originally derived the "Burger's equation" he had the idea of wave-breaking in turbulence in mind - although this is different from Landahl's modulation (Benjamin-Feir) instability.

An interesting review of Burger's equation. I would imagine that he would have been up against an interesting dilemma on how to introduce the continuity equation into his '1D momentum equation' - where do du/dx & d2u/dx2 disappear to? I've found that velocity jumps help in this regard - many would disagree, I guess. In 2D & 3D, the continuity equation has 'flex' if the element is allowed to deform under zero dilation - push-pull-pull, push-push-pull of the sides etc. (I'll not resurrect our differences on this topic :) (Also not so simple to model on a fixed numeric mesh).

I would say that the simplest way to understand the 'momentum-driven wave' concept is that of an oscillation that is carried along on a 'carrier' flow. The oscillation dynamics should not violate the correct information transfer limits under those circumstances. The theory also seems to predict the relationship between forcing (boundary) velocity & linear wave direction - it drops naturally out of the transformed continuity equation, & consequent cancellation of the non-linear oscillation term - oddly-enough.

Bifurcation & real-valued solutions
 

I've been working formally through the early stages of Bifurcation Theory & have come across an interesting perspective which I think is fatally flawed...

... the insistence on real solutions.

In some of the early examples, if the solutions were allowed to include complex-valued solutions, then the 'magic' of disappearing & emerging solutions may disappear to some extent.

I've also seen this approach in some books on chaos. I'm convinced that if the complex domain were tracked properly, that a lot of this 'magic' would become a lot more explainable. Allow 'ghost solutions' to enter the problem space & deal with their results.

Very interesting reading.

mw...

Re: Bifurcation & real-valued solutions
 

"I've been working formally through the early stages of Bifurcation Theory & have come across an interesting perspective which I think is fatally flawed...

... the insistence on real solutions. "

No it's right - a solution to the equations must be real. The fact that there are complex solutions is inherent in bifurcation theory - if you track a complex solution as a parameter is varied it may intersect the real space (at which point you have a bifurcation). However the solution has no physical meaning until this happens.

Consider the supercriical pitchfork bifurcation

A_t = R.A - A^3.

The steady state solutions are A=sqrt(R) which are complex for R<0 and real for R>0. As R is varied from R<0 to R>0 these complex solutions move along the imaginary (A) axis towards the origin where they collide before moving off along the real (A) axis. The collision point is the bifurcation point where the two new (real) solutions are formed and become "observable" in the space of real functions (if the velocity field were really complex it would have 6 components rather than the observable 3). There are undoubtedly complex solutions to the Navier-Stokes equations which never intersect the real plane and are thus of no physical relevance.

As an aside: Milton van Dyke looked at computer extended series solutions where the Reynolds number was allowed to be complex. These types of series actually allow you to calculate flows with negative Reynolds number which is a nonsensical solution. However the extension to the complex plane allows the series expansion to have a larger radius of convergence and hence approximate solutions at higher (positive) Reynolds number.

Understanding what is happening in the complex plane is important but you must remember that the physical solution must be realizable.

Re: Bifurcation & real-valued solutions
 

Thanks Tom for the clear explanation.

What's been going through my mind for some time is the inter-relationship between the complex part of say:

A=Ar+i*Ac

and the derivative of exp[i.(...)]

or, cos(), sin(), hyperbolic trig relationships & so forth.

Allowing the complex solutions to express themselves could be an elegant way of managing certain split solutions.

For instance, if we have a tensor as part of the system of equations we're wanting to solve, & this tensor can express principle values that are complex. If the principle values are to span the range min..max, then it should imply that other terms within the equation must also have complex portions. (Based on assumption that principle values must span min...max even for non-symmetric tensors - subject to debate).

For the requirement that only real solutions are allowed to express themselves, then these complex parts (ghost equations) must necessarily add up to zero. This in essence supplies an additional equation within the solution space.

eg. v.nabla_v

if nabla_v* => principle gradients.

If nabla_v* has complex values, then so much 'v' in order for there to be real solution components.

Anyway, these are preliminary musings on my behalf. I hope I've made sense.

mw