Stability theory - at once straightforward to understand but often difficult in practice.

All flows have a basic state which, for an isothermal flow, would correspond to sufficiently low values of the Reynolds number, Re. If this flow is perturbed by a very small amount, then the disturbance may be decomposed into a set of different spatial modes each with its own decay rate. This is the realm of linearised stability theory.

There will be a critical value of Re above which at least one of these modes will grow, enter the nonlinear regime and form a new flow pattern. This is often a steady flow but sometimes unsteady. There will always be some symmetry of the basic state which has been broken. There could also be a competition between more than one growing mode and the eventual flow may depend on the form of the initial disturbance.

Now sometimes it is possible for large disturbances to form a new flow even when Re is less than the above critical value - such cases are called subcritical instabilities, as opposed to supercritical instabilities.

At even higher values of Re even this new flow pattern will destabilise. Much now depends on the type of problem - the next pattern could be a new steady pattern, or it could be an unsteady flow. There is sometimes a sequence of destabilisations as Re increases further. This is the joy and the frustration of fluid mechanics, for there are many different routes to turbulence and many researchers have devoted their careers to studying these matters.

To answer your question, in the above scenario it is the Reynolds number whose value determines what sort of flow or range of flows are possible. In convection problems it may be the Rayleigh or Grashof numbers. In rotating systems it might be the Taylor number. All of these nondimensional parameters express a balance of forces. In the case of the Rayleigh number, its value tells one how strong buoyancy is compared with viscous damping.

Thanks Andrew,

Your answe very valuable for me...

Dear Nazaruddin Sinaga read this paper you can easily find the parameters to characterize this stability and instability of flow:

Article Stability and Instability Criteria for Idealized Precipitati...

Thanks Muzammal Hameed Tariq

To distinguish between the different states of fluid flow one must consider how the fluid reacts to a disturbance in the initial state.These disturbances will relate to the initial properties of the system, such as velocity, pressure, and density. James Clerk Maxwell expressed the qualitative concept of stable and unstable flow nicely when one said

"when an infinitely small variation of the present state will alter only by an infinitely small quantity the state at some future time, the condition of the system, whether at rest or in motion, is said to be stable but when an infinitely small variation in the present state may bring about a finite difference in the state of the system in a finite time, the system is said to be unstable."

That means that for a stable flow, any infinitely small variation, which is considered a disturbance, will not have any noticeable affect on the initial state of the system and will eventually die down in time. For a fluid flow to be considered stable it must be stable with respect to every possible disturbance.

On the other hand, for an unstable flow, any variations will have some noticeable affect on the state of the system which would then cause the disturbance to grow in amplitude in such a way that the system progressively departs from the initial state and never returns to it. This means that there is at least one mode of disturbance with respect to which the flow is unstable, and the disturbance will therefore distort the existing force equilibrium.

Laboratory experiments are a very useful way of gaining information about a given flow without having to use more complex mathematical techniques. Sometimes physically seeing the change in the flow over time is just as useful as a numerical approach and any findings from these experiments can be related back to the underlying theory. Experimental analysis is also useful because it allows one to vary the governing parameters very easily and their effects will be visible.

When dealing with more complicated mathematical theories such as Bifurcation theory and Weakly nonlinear theory, numerically solving such problems becomes very difficult and time consuming but with the help of computers this process becomes much easier and quicker. Since the 1980s computational analysis has become more and more useful, the improvement of algorithms which can solve the governing equations, such as the Naiver–Stokes equation, means that they can be integrated more accurately for various types of flow.

Thanks Ahmed Abdelhameed

Your reply is very helpful for me.

Regards,

Nazar

If it is a convection problem, the critical Rayleigh number characterize the transition from the steady to the unsteady state. But if it is a dynamic system, Critical Reynolds number is the responsible of this characterization. Also, it depends on the nature of the shape containing the fluid and the inner bodies.

Thank you Basma Souayeh

Is there a parameter that we can use to characterize this stability value?

Hi Nazaruddin,

I have studied these issues for my Ph.D. You can consult the two papers by Fortin and myself. You can use the diagnostics for flow stability used on those papers.

They are theoretically sound and at the same time not very complicated to code.

Steven Orszag (MIT) tried using stability analysis to predict turbulent flow in 1971. He calculated Re=5772 which is clearly higher than what's seen experimentally. The paper from JFM might still be interesting to you. Article Accurate Solution of the Orr-Sommerfeld Stability Equation

Thanks Mohamed Jardak

Would you send me the link for the two papers, please?

Thank you very much.

Nazaruddin,

Sorry for the delay. I have been quite busy as of late. I added the paper that I told you about.

Ok Mohamed Jardak, no problem.

Thank you very much....

For Eric Smith, The value Re=5772 given by Orszag is valid for the value of alpha = 1.02 only. This is very important.

Dear Sirs,

I don't want to be rude, but Reynolds number doesn't define anything. With any given Reynolds number (=flow velocity) the flow can be stable/laminar or turbulent.

And even the acceleration doesn't cause turbulence, it's well known that increasing velocity (=acceleration) actually laminarize the flow. So what causes it?

According to my lab & nature observations, the start of "Turbulence" can be clearly defined through Froude number. If flow velocity exceedes Fr= SQRT(3) and then decelerates, the flow becomes turbulent.

This idea provided me accurate results for energy losses in Hydraulic jump.

https://www.researchgate.net/project/Turbulence-22

The Reynolds number like the Froude number and maybe other numbers are just the result of non-dimensionalization of the Navier–Stokes equations.

They are both bifurcation parameters serving to characterise the state of the flow, seen as a dynamical system, during its transition from stable to chaotic. Maybe the Froude number is better suited for a certain type of flows more than others, but saying the Reynolds number doesn't define anything and with any given Reynold number the flow can be stable/laminar or turbulent is inaccurate.

Hi Jouni Jokela

I always like your reply, it is very interesting for us, especially for your lab observations that the start of "turbulence" can be clearly defined through Froude number .

My opinion are as follows:

1) Yes, I agree with you that the velocity (solely) can make the flow laminar or turbulent. That's the same case with the viscosity and characteristic length of Reynolds number.

2) I do not agree if you say that Reynolds number doesn't define anything, at least we may regards it as the ratio of inertia force and viscous force.

3) I agree that increasing velocity (=acceleration) actually laminarize the flow.

Could you tell us why does the acceleration and deceleration have a different effect on the turbulence of the flow?

Thank you very much....

Nazaruddin Sinaga

You ask, If I can Tell why Acceleration and Deceleration have different effect on the turbulence.

Well I think this is quite simple actually. I have approached this through the Hydraulic jump in Open channel flow, and afterwards noticed that this works also in pipe flow etc.

First; As we know, Viscous forces are able to keep the flow together and keep it Laminar. And as long we remain on Laminar regime, the flow is reversible. So actually the fluid can also considered as a deformating solid. Ie.metal wire or Paperclip. You can bend it with force, but if you reverse these forces, it can be returned to it's original shape. Just if you bend it enough many times, then it breaks and you can't reverse it anymore. You have two pieces. The Paperclip has become "turbulent". The Elastic forces are not transferred to it's separated parts.

This is THE problem of Navier Stokes equations, and thats why they cant have continuous smooth solution. Their Premises are wrong. The fluid is splitted in parts like solids when it become turbulence. This means there are independent fluid objects ("drops", but their size can be anything) which are not transferring viscous forces between them. Instead of that It's friction and collision what is happening. This causes the chaos. But as the flow is also splitted in parts, all these "drops" produces their own surface, which consumes energy. And this energy is lost. It can't be transferred to any reasonable movement and when the flow laminarizes again, it simply turns to heat. So this provides the method to test this idea; Turbulence starts when energy losses begins.

I could proceed with this in high details how the idea works also in inviscid fluids, as the mains aspect is Surface tension... But I need to skip that, as I dont have the time.

So back to the acceleration - deceleration... It's easy to notice that in Open channel flow Y1 flow depth is 50% of Y2 flow depth @ Fr=SQRT(3), and this is exactly the point where turbulence begins. So to create a Turbulence, the flow must be accelerated above Fr=SQRT(3) velocity, and then it must be decelerated, which causes the flow to break in parts.

You find my papers through my profile under my project Turbulence, but I want to link you at least to this key LAB observation, that there is no energy loss in hydraulic jump if Fr< SQRT(3)

Data Figure 8 from HYD-399 USBR

This data is very old, and there is no previsously known equation which fits to this data. But It was actually very easy to me to find this equations, I just accepted the previously ignored roots of the flow equations, though they seemed to provide "nonsense" results, and interpreted these results differently with these ideas.

And this simple task provided a perfect fit to this data.

So, Actually I have solved the "Navier Stokes Existency & Smoothness Problem", and at the moment I just wait that someone figures that out. And I am not in a hurry, and I dont even feel any need to fight me through some proper publication or so. Some day, some one, will be able to follow these thoughts, but even that doesnt matter, as our thoughts of nature doesnt really change anything....

https://www.researchgate.net/project/Turbulence-22

Dear Jouni Jokela

Thank you very much for your answer....

Hi Mohamed Jardak

I think you know very well about the topics. Could you give your comment about Jouni Jokela 's last response?

Thank you very much.

Nazaruddin Sinaga

Well thanks. I just returned while I noticed that the explanation of why acceleration laminarized the flow remained missing.

I mean after flow has become Turbulent it's "drops" must settle down and unite back to each other. Normally this happens after the energy is dissipated. I mean the start of the energy losses with splitting and surface creating is not the main cause of energyloss, but it make the difference, as the broken fluid also has much more freedoms to move, and thus also the amount of viscous losses grows exponentially.

But anyway, this settling down takes quite some time, and acceleration actually cant do so much to this, so how can it "laminarize" the flow then?

This is because these "drops" are pulled long and they become fibre-alike. This increases their surface amount, and also the neighbour-fibres velocities are unified, which finally accelerates the settling down process too.

But still the Turbulence disturbance basically travel over this acceleration. I've played with this issue on natural stream as shown in this video, where I shoot vortex-rings over such an acceleration to kill the laminar jump.

https://www.youtube.com/watch?v=GPMMWQ0EEII

As can be noticed small disturbances doesn't survive the acceleration, but these bigger eddies can go through it.

Ok Jouni Jokela

I've got your message, and I will learn more in depth about it.

Thanks friend...

Have a nice time....

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