diaw: I'm trying to understand why, in the pipe case, the solutions are isolated. This can certainly be said for traditional development of say gravity waves, where the perturbation is introduced, 3 sets of equations are developed - one for bulk flow, one for deviatoric & the final - for the non-linear linkages. The approach then makes simplifying assumptions which essentially reduce the non-linear linkage equation to zero - leaving two independent solutions operating in the same space - or drops the bulk solution completely - as in the case of gravity waves.

This is a standard property of nonlinear equations - solutions can exist which are not bifurcations from a common "base state". You really need to read about what was done to obtain these solutions and not look at a "popular" account which does not tell you all the facts. You probably also need some some understanding of nonlinear stability theory/bifurcation analysis to appreciate what the results actually mean.

There are some nice colour pictures of these solutions in Wedin, H. & Kerswell, R.R. ``Exact coherent structures in pipe flow: travelling wave solutions'' J. Fluid Mech. 508, 333-371, 2004.

diaw: Granted, I am certainly referring to transient wave-forms in the context of a transient wave-field. given a consistent boundary disturbance, would there be no possibility of non-linear 'normal modes' forming? I think that perhaps the difference in viewpoints may be one of considering a reasonably consistent source of input 'noise' as a boundary-condition, or of flow-generated noise/perturbations. I would love to know if sound measurements of the onset of instability have been conducted - I'm sure they have. Does flow in a pipe make a noise? In some cases, it certainly can.

Most experimental fluid experiments use sound waves to excite the flow and induce transition. Turbulence does produce noise - just go outside on a windy day. There is a considerable amount of work on this - numerous papers by Lighthill, Crighton and Howe (Howe's also written a book vortex sound). To study this type of problems you must abandon the incompressible assumption and work with a fuller set of equations since the solenoidal condition filters all acoustic waves from the equations.

There is no such thing as a nonlinear normal mode (by definition!) and even if there was there is no need to invoke it to describe the problem - the linear "branch cut" argument in the complex plane gives the solution for small amplitude disturbances (just as in the gravity wave case).

diaw: What happens when we 'pull' flow out of a pipe - by setting a slight suction on the outlet boundary? (This one can be rather fun to observe, although some solvers will definitely not want to oblige.)

In an incompressible fluid cavitation occurs (you are taking out more fluid than you are putting in) and the solution to the problem (as posed with noslip bcs) has failed to exist - hence the solvers that fail are the ones that are correct! For the solution to make sense you need to augment the statement of the problem with suitably altered boundary conditions.

diaw:What would a pressure-wave look like in a pipe?

As I've already stated there is no pressure wave in the incompressible limit.

Thanks Tom for those comments. A fair bit to digest.

diaw: What happens when we 'pull' flow out of a pipe - by setting a slight suction on the outlet boundary? (This one can be rather fun to observe, although some solvers will definitely not want to oblige.)

Tom:

In an incompressible fluid cavitation occurs (you are taking out more fluid than you are putting in) and the solution to the problem (as posed with noslip bcs) has failed to exist - hence the solvers that fail are the ones that are correct! For the solution to make sense you need to augment the statement of the problem with suitably altered boundary conditions.

diaw:

I agree with cavitation occuring. I have a different perspective, however, on the role of the 'squishy' element & du/dx = - dv/dy & its role in allowing cavitation-type events & mass (momentum) re-distribution to occur. I guess this will go back to the p=const vs div(V)=0 debate.

diaw:What would a pressure-wave look like in a pipe?

Tom:

As I've already stated there is no pressure wave in the incompressible limit.

diaw:

Perhaps again we disagree in terms of the role of du/dx = - dv/dy?

--------- Concepts of singularities in flow fields:

Could I ask what you views are on the possibilty of singularities existing in fluid flow fields? If so, what physical form would these take?

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Physical approach (a full circle):

When all is said & done, what I always fall back on, in terms of trying to understand the physical nature of flows is that, in general we have flows that meet the density=const requirement (as per my reference to Gas Dynamics), & that to all intents & purposes this can be at reasonably modest velocities. We do most definitely observe both wave activity & bulk activity co-existing together in such flows.

When we started earlier discussions, we discussed gravity waves evident at the surface & I tried to draw down the pressure effects to an imaginary plane below the surface. A simple force balance would show that the gravity waves must be balanced by upthrusts from below - otherwise need to bridge from some distant river boundary, or obstacle - surface tension would merely convey the support from the distant boundary. Thus, to maintain equilibrium in the flow field at both surface & below, pressure fields must be active. In other words, see a pressure on the surface, you will be sure to see a pressure distribution below it on the cutplane, in some form. A cfd simulation will show pressure distributions on this cutplane that reasonably closely match the gravity-wave phenomena on the surface. This is basic physics, as I see it. It must hold.

Flows in musical instruments rely on pressure waves to do their job. Change the boundary condition, & we get a different wave structure & different sound. We have different harmonics etc. What happens when we move the fingers? Pressure waves operate. Does air move along the musical instrument (low bulk speed + co-existent wave)?

There are so many examples around us of fluid phenomena that operate at very low velocities that exhibit wave & bulk phenomena. As far as engineers & physicists seem to be concerned, at these velocities (M<0.3 ~ 100 m/s!, for gases), the flows can be considered as density=constant. This makes no assumptions about the activity of du/dx + dv/dy = 0. It seems as though the mathematical & physics interpretations may be fundamentally at odds - even, though, in the end, we arrive at the same form of the continuity equation. The fact that infinite communication velocities cannot occur in practice, shows that something is incorrect in the physical outworking of the div(V)=0 approach. This worries me.

Perhaps we have created restrictions on our perceptions that are preventing us from seeing a larger picture spreading from low-speed to high speeds, where both waves (transient, persistent, non-linear) & bulk flows are active. Could it be that we are perhaps not modeling nature in a completely physically meaningful way?

In the end, if our equations do not allow us to arrive at answers that match the physics we observe around us, then we have really done something wrong.

In my understanding of density=consant, with a 'squishy' (deformable) element, I'm still not convinced that pressure waves cannot move back & forth until standing-wave structures set themselves in place. I know that we have discussed the implications of the mathematical interpretation of these flows, but to my observeration, they don't ring true.

In my view, we really need a physical paradigm shift to take us away from this viscous, low-speed, 'incompressible' trap. If not wave+bulk field interaction, then perhaps we can develop an alternative understanding - but, it should not resort to the idea of turbulence being a statistical concept. This cannot be correct - nature is not that way - it may perhaps have much more to do with how we interpret nature. Nature is simple - yet complex. When we begin to become overly complex in our understanding then I vote that it may be time to re-think the physics.

When we can explain & model the Reynold's experiment in a pipe - then we may be in the right ballpark.

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The elegance of the dual-nature of the N-S - the 'hidden nature' & a co-existent non-linear models, with presure inter-linking may go some way to answering some of these questions.

I will read every link you have so kindly provided - this is a blessing indeed. I also plan to devote my time to gaining a complete understanding of the bridging paradigm we so desperately need. I plan to build the numerical models in a few different environments to test out the concepts. Lets see where it all goes.

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Tom, thanks so much for your extremely kind debate & for making me a lot wiser than when we first began. You have been wonderfully patient, kind & generous - thank you. You are indeed a master craftsman.

Des Aubery...

Wave-solutions & div(v) re-visited

During the last few days, after the previous discussions, I settled a number of research firmly issues in my mind. I would draw your attention to an excellent book, "The Physics of Vibrations & Waves", Pain H.J., 6th Ed, Wiley, 2005

(1) Incompressibility & dilation

For a constant mass,

Compressibility: B = - dP/(dV/V)

Incompressible => B -> infinity; => dilation = (dV/V) -> 0

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(2) Deformable fluid element

Relationship between longitudinal compression & lateral distortion Proceed with derivation as per 'solid' constant mass element. With a few observations, the div(v) = du/dx + dv/dy = 0 emerges.

Proceeding along the development of elastic waves in solids is extremely informative.

Conclusion: The deformable 'incompressible' fluid element meets both dilation = 0, div(V)=0 requirements. This element supports wave activity.

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(3) Momentum (velocity) & pressure waves in N-S

The full non-linear form of the incompressible N-S equations admit a solution of the form:

() = ()o.e^(-at).e^i(kx + ly -b.wt)

where: a ~ visc*K^2 ; b ~ fn(()o.e^(-at).e^i()) (non-linear)

At this point, no simplifying assumptions are made - no linearisation etc.

The main topic of interest to note is the term e^(-kvisc*K^2.t), which, in most practical terms (kvisc~855e-9) would tend to e^(~0)~1. This admits the wave solution in any time window within reason. It is impractical to consider the limit t-> infinity, as this would mean that all experiments would take longer than the average lifetime. For all practical intents & purposes, as suitable time for observation should be selected as the 'steady-limit'.

The 'b' finding implies that 'b' itself admits wave-solutions, & thus the timing within the solution constantly moves around as time advances. The e^(0)~1 decay term, to all intents & purposes, admits a 'long-term' standing-wave solution. I have observed these effects at scales, in many models, which are definitely not in the region of numeric-roundoff-error.

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With this latter research clarification, I am now comfortable with the 'transient wave-solution' to the N-S & understand the role it plays. It appears that the problem may not purely lie in numeric roundoff, but in the N-S solutions themselves.

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Can I respectfully ask advice on suitable Journals to consider when publishing my research findings?

Thanks again, diaw... (Des Aubery)

"Can I respectfully ask advice on suitable Journals to consider when publishing my research findings?"

If you really believe you have something new and interesting to say about the equations of motion then you should consider either the Journal of fluid mechanics, Proceedings of the Royal society of London A, European Journal of Mechanics B/fluids, physics of fluids or possibly Theroetical and computational fluid dynamics.

However I would suggest that, if you submit to one of these journals, you prepare yourself for some harsh critism/rejection (JFM rejects around 55% of submitted papers as policy!). It is always worth the extra effort to publish in a respected journal though.

Tom:

"Can I respectfully ask advice on suitable Journals to consider when publishing my research findings?"

If you really believe you have something new and interesting to say about the equations of motion then you should consider either the Journal of fluid mechanics, Proceedings of the Royal society of London A, European Journal of Mechanics B/fluids, physics of fluids or possibly Theroetical and computational fluid dynamics.

However I would suggest that, if you submit to one of these journals, you prepare yourself for some harsh critism/rejection (JFM rejects around 55% of submitted papers as policy!). It is always worth the extra effort to publish in a respected journal though.

diaw:

Thanks so much Tom, for the Journal list. Those are indeed very well respected Institutions. I would certainly expect some very harsh criticism/rejection - this would most certainly be in order. I believe that this would be well worth the effort.

In some ways, your gentle wise comments during the course of this debate have been preparing my mental resolve. At each point, it has sent me more & more deeply into the books to try & explain what I have seen. This has forced me to go back to basics in some areas & explain my case more thoroughly. No doubt, the review team of some of these Journals will put my research through the distilling fires - but, so, let it be.

Once again, thank you so much for your debate & wisdom. It has been suncerely appreciated.

Des Aubery...