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Why CD is used for DES?

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Old   July 2, 2007, 12:07
Default Why CD is used for DES?
  #1
Natasha
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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?
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Old   July 3, 2007, 09:53
Default Re: Why CD is used for DES?
  #2
Tom
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"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.
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Old   July 3, 2007, 10:10
Default Re: Why CD is used for DES?
  #3
momentum_waves
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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
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Old   July 3, 2007, 14:06
Default Re: Why CD is used for DES?
  #4
jojo
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Diffusion --> the shortest wavelengths are the most damped. Dispersion --> the shortest wavelengths are the most phase-shifted.
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Old   July 4, 2007, 05:08
Default Re: Why CD is used for DES?
  #5
Tom
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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"

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Old   July 4, 2007, 23:14
Default Re: Why CD is used for DES?
  #6
momentum_waves
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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

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Old   July 5, 2007, 04:50
Default Re: Why CD is used for DES?
  #7
Tom
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`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.

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Old   July 5, 2007, 05:07
Default Re: Why CD is used for DES?
  #8
momentum_waves
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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
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Old   July 5, 2007, 05:35
Default Re: Why CD is used for DES?
  #9
Tom
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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.
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Old   July 5, 2007, 08:03
Default Re: Why CD is used for DES?
  #10
momentum_waves
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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
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Old   July 5, 2007, 09:37
Default Re: Why CD is used for DES?
  #11
Tom
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"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.

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Old   July 5, 2007, 14:13
Default Re: Why CD is used for DES?
  #12
momentum_waves
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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

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Old   July 6, 2007, 06:10
Default Re: Why CD is used for DES?
  #13
Tom
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"... 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.
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Old   July 6, 2007, 07:51
Default Re: Why CD is used for DES?
  #14
momentum_waves
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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.

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Old   July 6, 2007, 08:38
Default Re: Why CD is used for DES?
  #15
Tom
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"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.
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Old   July 6, 2007, 11:22
Default Re: Why CD is used for DES?
  #16
momentum_waves
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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
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Old   July 9, 2007, 04:49
Default Re: Why CD is used for DES?
  #17
Tom
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"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.
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Old   July 9, 2007, 05:28
Default Re: Why CD is used for DES?
  #18
momentum_waves
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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
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Old   July 9, 2007, 06:21
Default Re: Why CD is used for DES?
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Tom
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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).
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Old   July 9, 2007, 06:41
Default Re: Why CD is used for DES?
  #20
momentum_waves
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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
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