[truffaldino]

Recently one person asked me why transitional (laminar to turbulent) Reynolds numbers are big (10^3 to 10^5) while all coefficients in Navier-Stokes equation are of order of 1?

The person is not an expert in fluid mechanics and asked for a reasonably simpe explanation.

Is there one?

Truffaldino

[FMDenaro]

Originally Posted by truffaldino

Recently one person asked me why transitional (laminar to turbulent) Reynolds numbers are big (10^3 to 10^5) while all coefficients in Navier-Stokes equation are of order of 1?

The person is not an expert in fluid mechanics and asked for a reasonably simpe explanation.

Is there one?

Truffaldino

Hello,
I can not focus on the meaning of this question.... The Re number depends on the choice of the velocity and lenght, therefore is not a unique definition...for example in channel flows you can have a fully developed turbulence at very low Re(_tau) number. On a flat plate you have flow increasing Re_x number.
A laminar Poiseulle solution is valid for any high Re number, the problem is that physically it is not stable for high Re.
The answer is more related to a stability analysis (linear and non linear) rahter than to the coefficients of the equations...

[truffaldino]

To me this question looks quite reasonable.

Imagine a person who takes a look at NS equation for the first time: This is an equation where all coefficients are 1 and it has one parameter nu which is nu=Re^-1 in the front of Laplacian. All the length and speeds in this system of scales are order of 1.

Then person finds out that the transition happens when the parameter is almost zero nu=10^-3, 10^-5. Equation itself does not contain Re_theta and other things obtained in process of its solution.

So, the person must suspect that there is a simple and fundamental reason why does transition happen when this parameter is almost zero.

[FMDenaro]

Originally Posted by truffaldino

To me this question looks quite reasonable.

Imagine a person who takes a look at NS equation for the first time: This is a equation where all coefficients are 1 and it has one parameter v which is v=Re^-1 in the front of Laplacian. All the length and speeds in this system of scales are order of 1.

Then person finds out that the transition happens when the parameter is almost zero v=10^-3, 10^-5. Equation itself does not contain Re_theta and other things obtained in process of its solution.

So, the person must suspect that there is a simple and fundamental reason why does transition happen when this parameter is almost zero.

if you consider the fact that the for very small viscosity the NS equations are actually perturbed non-linear equation, it is well known the fact that the solution for hyperbolic non-linear equations can be not-unique.
Transition is just a bifurcation of the solution when you get conditions when the non-linear convective parte is more relavant than the diffusive linear part.

The Re number is someway the parameter that guides small perturbation to be suppressed by the diffusive effects or to be amplified by the non-linear convective effects.

[truffaldino]

Originally Posted by FMDenaro

if you consider the fact that the for very small viscosity the NS equations are actually perturbed non-linear equation, it is well known the fact that the solution for hyperbolic non-linear equations can be not-unique.
Transition is just a bifurcation of the solution when you get conditions when the non-linear convective parte is more relavant than the diffusive linear part.

The Re number is someway the parameter that guides small perturbation to be suppressed by the diffusive effects or to be amplified by the non-linear convective effects.

NS without perturbation is Euler equation. So, according your explanation, Euler should be ill-posed in 3D. I remember I heard something about that. Is it well known fact?

And another question: why does diffusive part become more relevant than convective at such a small parameter?

[FMDenaro]

O(1) Div (vv) + O(1) Grad p and as counterpart O(1/Re) Lap v so that for Re>>1 you can see the perturbed Euler equation

[truffaldino]

Yes, all this is evident and well known:

The question is what about well-posedness and ill-posedness of Euler and why does dissipative term becomes more releveant at such a small perturbation.

[FMDenaro]

Originally Posted by truffaldino

Yes, all this is evident and well known:

The question is what about well-posedness and ill-posedness of Euler and why does dissipative term becomes more releveant at such a small perturbation.

Again, analysing critical values for the Re number is a topic of the stability analysis and depends on the type of flow. The answer is not simple...
Basic tools for the stability can be found in classical fluid mechanics books but many issues are still open questions and you can find many papers published in scientific journals such as JFM, PoF, etc

[truffaldino]

I was asking this question since friend of mine, who is not a specialist in fluid mechanics, sent me a note written by his former supervisor who is a specialist in quantum field theory (qft). Using qft techniques he somehow shows that critical Re must be high, and his demonstration is quite general (i.e. suitable, in principle, for any flow).

To me use of qft technique for such a purpose looks quite excessive and I thought that there must exist much simpler explanation.

From cfd literature I red I remember that people uses long analysis (eg orr-sommerfeld etc) and it is not universal. But I have never seen any simple general explanation.

[FMDenaro]

This aspect is very complex and can not be determined by a general observation of the Re number.
I suggest a reading of Chap. 11 of Kundu.

[Martin Hegedus]

To some extent, one can view the transitional Re as a multiple of the characteristic length of a vortex. I guess. I'm just making this up. In other words, it takes lots of vortiticy for transition to occur.

[Martin Hegedus]

Or, another way to look at this. The length value that is chosen for the Re is sort of arbitrary. So, if one chooses a length such that Re=1, what does that length represent?

[truffaldino]

Originally Posted by Martin Hegedus

Or, another way to look at this. The length value that is chosen for the Re is sort of arbitrary. So, if one chooses a length such that Re=1, what does that length represent?

The idea to use vortex sizes and dimensional considerations to explain (qualitatively) big values of critical Re seems to be reasonable.

[truffaldino]

Here is an idea: why not to estimate the total time of decay of biggest eddies through the cascade till the Kolmogorov scale and then multiply this time by freestream velocity: This will give the qualitative estimate of the transition length and the transition Re.

[FMDenaro]

Originally Posted by truffaldino

Here is an idea: why not to estimate the total time of decay of biggest eddies through the cascade till the Kolmogorov scale and then multiply this time by freestream velocity: This will give the qualitative estimate of the transition length and the transition Re.

I do not understand that... biggest eddies in the inertial cascade do not decay (they live in a inviscid-like situation) but are only transported and stretched.
The free-stream velocity, the lenght of biggest eddy define the globa Re number...

[truffaldino]

Ok, this is a wrong idea, but why can't we show that critical Re should be big (not to find it, but just demonstrate that it is big) in Kolmogorov's style.

Say, to estimate order of total dissipated energy and find order of Re when it is possible, or some other very rough considerations.

[Martin Hegedus]

Warning, I know very little about this, but I did some googling and found the Orr-Sommerfeld equations.

http://en.wikipedia.org/wiki/Orr%E2%...rfeld_equation

[Martin Hegedus]

Another link

http://web.mit.edu/1.63/www/Lec-note.../5-4OS-eqn.pdf

[Martin Hegedus]

OK, I gather Orr-Sommerfeld has already been brought up on this site.

[truffaldino]

Yes Orr-Sommerfeld is a very well-known stability analysys based on linearization. It is aimed to find value of critical Re.

But the question is not finding Re, but just to qualitatively demonstrate that it should be big, without solving NS equation, just by some general consideration in Kolmogorov's style.

Say, e.g. to estimate order of total dissipated energy and find order of Re when it is possible, or some other very rough considerations.