Turbulence

[Sriman Dutta]

Hi everybody,

Turbulence is a still a complex unsolved topic in mathematics. According to Wikipedia, turbulence is one of those unexplained problems in physics, "Is it possible to make a theoretical model to describe the behavior of a turbulent flow — in particular, its internal structures?".

Turbulence includes complex vortices of fluid flow. Analyzing it properly is still not possible. Is there a way out? Can we create a new theory of turbulence?

Now, my real question is: Laminar flow is thought to be composed of multiple plates that flows along with a flow velocity v in the direction of flow. This hypothesis is of Couette flow. Since turbulence involves a lot of chaotic processes, can we regard it as the result of disintegration of the so-called parallel hypothesized plates in a laminar flow?

 

Please excuse me if my topic of discussion seems impractical.

[studiot]

Depends what you mean by an unsolved problem.

 

There are well established mathematical methods for handling turbulent flow.

 

They are obviously not the same as for laminar flow or there would be no point in making the distinction.

 

We do, however, understand less about the mechanism of turbulent flow than laminar flow. But it is not chaotic (in the mathematical sense), that is something else again.

 

So let's try to narrow the focus of your question down a bit?

[Bignose]

I highly suggest you red a copy of Stephen Pope's Turbulent Flows. It is a good introduction to a very complex subject.

[Sriman Dutta]

Depends what you mean by an unsolved problem.

 

There are well established mathematical methods for handling turbulent flow.

 

They are obviously not the same as for laminar flow or there would be no point in making the distinction.

 

We do, however, understand less about the mechanism of turbulent flow than laminar flow. But it is not chaotic (in the mathematical sense), that is something else again.

 

So let's try to narrow the focus of your question down a bit?

I would like to know the mathematical methods you are talking about.

[studiot]

I would like to know the mathematical methods you are talking about.

 

First you need to be clear about the difference between

 

Steady flow

Uniform flow

Laminar Flow

Turbulent flow

Vortex flow

Rotational flow

Irrotational flow.

Brownian motion

 

The main difference between laminar and turbulent flow is that the transport or transfer coefficients for laminar flow are molecular coefficients and are properties of the fluid, whereas for turbulent flow they are known as eddy coefficients and are properties if the flow.

Flow is never fully laminar or fully turbulent, the total transport properties are the sums of the molecular and eddy coefficients and the ratio of these coefficients determine whether we call the flow turbulent or laminar.

[studiot]

I would like to know the mathematical methods you are talking about.

 

 

Have you lost interest in turbulence?

[sethoflagos]

Hi everybody,

Turbulence is a still a complex unsolved topic in mathematics. According to Wikipedia, turbulence is one of those unexplained problems in physics,......

Which one?

 

Depending on context, there's 70 odd distinct terms in the Navier-Stokes equations alone (to which must be added the continuity equation and applicable equations of state plus an infinity of variations in boundary conditions......)

 

In short, an infinity of mathematical challenges, a few of which have (near) exact analytic solutions, and many more of which are amenable to numerical methods.

 

It so happens, that for low Reynolds numbers and steady boundary conditions in simple geometries, all but two terms tend to zero after some settling time. So the mathematics of laminar flow tends to be steady state and simply analytic.

 

As Reynolds numbers increase, new terms gain significance; the mathematical complexity increases exponentially (for want of a better term); and almost invariably, the partial time derivatives remain significantly and indefinitely non-zero. This, we call turbulence. But it covers a myriad of flow regimes from the stirring of a coffee cup to the expansion of the universe. And yes, even the flap of a Brazilian butterfly's wing.

 

There is no sense in which the richness of turbulence somehow emerges from the Couette scenario. Better to wonder that somewhere within the infinite ocean of possible fluid flow regimes, there is one tiny island where 80 or so parameters are practically zero, and the regime is sufficiently simple that we can actually start to get our heads around it.

[Tim88]

Hi everybody,

Turbulence is a still a complex unsolved topic in mathematics. According to Wikipedia, turbulence is one of those unexplained problems in physics, "Is it possible to make a theoretical model to describe the behavior of a turbulent flow — in particular, its internal structures?".

Turbulence includes complex vortices of fluid flow. Analyzing it properly is still not possible. Is there a way out? Can we create a new theory of turbulence?

Now, my real question is: Laminar flow is thought to be composed of multiple plates that flows along with a flow velocity v in the direction of flow. This hypothesis is of Couette flow. Since turbulence involves a lot of chaotic processes, can we regard it as the result of disintegration of the so-called parallel hypothesized plates in a laminar flow?

 

Please excuse me if my topic of discussion seems impractical.

 

Laminar flow can also consist of closed line rotating eddies (often confusingly also called vortexes!), which are obviously not "parallel plates" (and no plates are hypothesized anyway). Probably you confused a practical application of laminar flow with the general theory of laminar flow.

 

But your understanding is roughly correct, turbulence is characterized by chaotic processes caused by instabilities that break up the flow lines, and the local flow changes as function of time.

[studiot]

 

Tim88

But your understanding is roughly correct, turbulence is characterized by chaotic processes caused by instabilities that break up the flow lines, and the local flow changes as function of time.

 

I would be happier if you would substitute the word random for chaotic. The eddy coefficients (they may be variables as Sethoflagos pointed out) are defined in terms of a random variable, not a chaotic one.

 

Chaotic flow regimes exhibit the bifurcation characteristic of certain forms of chaos.

 

You also need to be careful not to allow your second statement about local flow changing as a function of time to confuse the difference between steady and unsteady flow. Turbulent flow can be steady.

[Tim88]

 

I would be happier if you would substitute the word random for chaotic. The eddy coefficients (they may be variables as Sethoflagos pointed out) are defined in terms of a random variable, not a chaotic one.

 

[..]

 

I would be happier if you don't disagree with the OP's standard usage of terms; it's simply unhelpful - https://en.wikipedia.org/wiki/Turbulence.

[studiot]

 

I would be happier if you don't disagree with the OP's standard usage of terms; it's simply unhelpful - https://en.wikipedia.org/wiki/Turbulence.

 

Here is a properly peer reviewed scientific article concerninig the term chaos (and bifurcation) in fluid flow - open channel flow to be precise.

 

If you look at the bottom you will see links to turbulent flow in channels.

 

Turbulent flow is not chaotic flow.

 

http://scitation.aip.org/content/aip/journal/pof2/6/6/10.1063/1.868206

[Tim88]

 

Here is a properly peer reviewed scientific article concerninig the term chaos (and bifurcation) in fluid flow - open channel flow to be precise.

 

If you look at the bottom you will see links to turbulent flow in channels.

 

Turbulent flow is not chaotic flow.

 

http://scitation.aip.org/content/aip/journal/pof2/6/6/10.1063/1.868206

 

Neat.

"The process by which a laminar viscous flow undergoes transition to turbulence through diverse routes

[...]

(I) laminar steady-state flow

[..]

(VI) aperiodic chaotic state."

 

Thus the author identifies fully developed turbulent flow as an "aperiodic chaotic state", just as I briefly explained to Sriman.

 

Maybe you got confused by the fact that not all "chaotic" flow is turbulent. Once more, that's of little relevance to the topic of turbulence, so that more than an incidental comment about that interesting detail is simply unhelpful here.

 

If Sriman has further questions or comments, I will clarify more of course.

[Sriman Dutta]

You all mean that laminar flow can be considered as rotational by defining vorticity. So, turbulence is something that is too chaotic to be described by Couette flow.

[sethoflagos]

You all mean that laminar flow can be considered as rotational by defining vorticity. So, turbulence is something that is too chaotic to be described by Couette flow.

 

Couette flow as a concept is a 100% shear-driven flow regime. Many turbulent flow cases are more than adequately described by assuming an inviscid fluid - one where there is no viscous shear. So how can turbulence ever be described by Couette (which is just a simple case of laminar flow), or vice versa, when the dominant physical mechanism of each case is so insignificant in the other that it can be (and is) ignored in most practical applications?

 

Vorticity is also, I feel, an inappropriate guide. Any pure laminar flow regime passing through a right angle tube bend will develop a pair of symmetrical 'D'-shaped vortices if only by the consideration that the streamlines must negotiate the change in direction somehow.

 

It is difficult to see how any attempt at understanding the different fluid flow regimes can progress unless we start with their common ground, which is the transformation of internal energy to kinetic energy and back. This inevitably calls into play the governing thermodynamics which can begin to shed some light on the development of high entropy flow regimes.

 

One reason laminar flow is a poor choice of starting point is that the equations that are normally used to describe it do not explicity indicate the energy sources and destinations. That the driving force derives (as typically presented) from the expansion of an 'incompressible' fluid is just one of a number of confusing paradoxes here.

[Sriman Dutta]

I see. Thanks for your help sethoflagos.

My last question - can the Navier-Stokes equation ( that describes the flow of a compressible fluid) be modified to get a general equation for the system of turbulence? Or, do we need to interpret turbulence as a flow that uses more energy than required by laminar flow for covering a given length of pipe ? (considering that we have turbulence within the pipe)

 

Sorry if my questions are a bit impractical. I am a novice in this area.

[sethoflagos]

I see. Thanks for your help sethoflagos.

My last question - can the Navier-Stokes equation ( that describes the flow of a compressible fluid) be modified to get a general equation for the system of turbulence? Or, do we need to interpret turbulence as a flow that uses more energy than required by laminar flow for covering a given length of pipe ? (considering that we have turbulence within the pipe)

 

Sorry if my questions are a bit impractical. I am a novice in this area.

 

The Navier-Stokes equations ARE the general equations of turbulent systems. (Noting that turbulence is the general state of a dynamic universe - orderly, low entropy laminar flow something of a peculiarity existing on the boundary of dynamic and static structures).

 

Anything else is an approximation. In chemical engineering we tend towards semi-empirical equations, usually based on overall energy balance considerations ranging from the (near) isentropic - such as gas expansion through a turbine, to (near) isothermal - such as flow through a long pipeline. Deep, deep down, there is some rooting in Navier-Stokes, but this is not obvious at first glance.

 

You might gain some insight into the subject by studying Richardson and Kolmogorov's theories of energy cascades from large inertia dominated fluid parcels down through the Kolmogorov scales to the microscales where viscous shear is once again dominant and where the system waste heat (in the thermodynamic sense) is finally dissipated. It's more of a statistical, fractal geometry approach than a purely analytic one. But since the purely analytic solution will be a while getting into print, it's as good as anything for providing a perspective.

[Sriman Dutta]

Kolmogorov suggests that the turbulence is caused due to movements of Eddie currents on a very small scale, the Kolmogorov scale. These eddies react vigorously to generate a chaotic flow. Am I correct?

[sethoflagos]

Kolmogorov suggests that the turbulence is caused due to movements of Eddie currents on a very small scale, the Kolmogorov scale. These eddies react vigorously to generate a chaotic flow. Am I correct?

 

Eddies are a result rather than a cause. The root cause tends to be two fluid parcels attempting to occupy the same space. The parcels tend to break up into rotating parcels of ever decreasing size ('shrapnel'), and the transient pressure wave from the 'collision' changes the upstream conditions that initiated that particular conflict, spawning different conflicts with different geometries.

 

In my world (there are always different ways of viewing these complex phenomena) I view turbulence as being characterised by these unpredictable (I use neither 'random' nor 'chaotic' in this context) pressure transients from downstream events impacting upstream flow patterns thus creating sustained flow instability with a great variety of structures on many scales.

 

In contrast, in laminar flow, such peturbations are damped out almost immediately by viscous diffusion due to their low energy content.

Edited October 15, 2016 by sethoflagos

[studiot]

 

Eddies are a result rather than a cause. The root cause tends to be two fluid parcels attempting to occupy the same space. The parcels tend to break up into rotating parcels of ever decreasing size ('shrapnel'), and the transient pressure wave from the 'collision' changes the upstream conditions that initiated that particular conflict, spawning different conflicts with different geometries.

 

In my world (there are always different ways of viewing these complex phenomena) I view turbulence as being characterised by these unpredictable (I use neither 'random' nor 'chaotic' in this context) pressure transients from downstream events impacting upstream flow patterns thus creating sustained flow instability with a great variety of structures on many scales.

 

In contrast, in laminar flow, such peturbations are damped out almost immediately by viscous diffusion due to their low energy content.

 

I particularly liked this description, as well as tour previous post. +1

 

But here is a question for consideration.

 

Is the fluid in an eddy flowing ? After all it is going nowhere.

Edited October 15, 2016 by studiot

[sethoflagos]

 

I particularly liked this description, as well as tour previous post. +1

 

But here is a question for consideration.

 

Is the fluid in an eddy flowing ? After all it is going nowhere.

 

Thanks!

 

I think I would always describe an eddy as flowing, even if it were basically static and self-contained. like swirling tea in a cup after it had been stirred. That would be an example of a circulating flow. Not necessarily going anywhere, but its still converting momentum to heat via rotational flow which is basically an eddy's job description.

 

Many of course are typically carried along at moreorless the fluid bulk velocity, so these have both rotational and linear flow components

Edited October 16, 2016 by sethoflagos

[Sriman Dutta]

I understand that in turbulent flow, the momentum is converted into thermal energy due to vigorous movements (at the Kolmogorov scale) . But this thermal energy can raise the fluid's temperature. The increase in temperature ( if sufficient) can make the fluid vaporize. So there happens to be a loss in volume of fluid. ( it can happen only if the bp of the fluid is quite low/ the atmospheric pressure is low) . So doesn't this change the physical systems of the fluid, especially at the microscopic scale?

[sethoflagos]

It's easy enough to do a few calculations of typical kinetic to internal (thermal) energy equivalences. Typically the temperature changes are small. Turbulent flow in long pipelines is usually treated as an isothermal (constant temperature) process with acceptable accuracy.

 

Moreover, since a fluid often sources its kinetic energy by conversion of some of its internal energy, with an associated cooling effect, reversing that conversion will only restore the original temperature. The First Law tells us this.

 

A problem that is more frequently encountered is when the pressure transients discussed previously start dipping below the liquid bubble point. This leads to localised bubble formation followed by extremely rapid collapse - cavitation. A major part of the hydraulic design of pumped circuits focuses on ensuring that this doesn't occur in the eye of the pumps (the pressure low point in the circuit) as heavy cavitation can chew through a pump impeller in no time.

Edited October 16, 2016 by sethoflagos

[Sriman Dutta]

Cagitation can be dealt with by the Rayleigh-Plesset equation.

[sethoflagos]

Cagitation can be dealt with by the Rayleigh-Plesset equation.

 

Like Couette flow (and indeed, all other simply defined flow regimes) the Rayleigh-Plesset equation is derived from Navier-Stokes. Or rather one set of Navier-Stokes for each of the two different phases, with the introduction of surface tension as a coupled boundary condition linking the two.

 

While there is a certain unity to the Navier-Stokes equations in terms of keeping faith with the First Law, the Second Law tends to be spooned into it via a characteristic pressure-density relationship as essentially a boundary condition moreorless at the discretion of the investigator (with sound justification, one hopes).

 

Perhaps one day, the Second Law may become more integral in the formulation of the Navier-Stokes, and the complexity of turbulent regimes be more clearly defined as an equilibrium distribution of energy in accordance with QED, I wonder which will be solved first.