"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.

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

" 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).

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

I'm thoroughly enjoying this phase of my research.

mw...

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...

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.

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...

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

"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.

"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...

"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).

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

mw...

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...

"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).

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.)

"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).

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.

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...

"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.

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