[arungovindneelan]

Reynolds number is an important parameter determines the nature of the flow. No physics, even turbulence is not perfectly connected to the present formula of Reynolds number. In pipe flows, at Re = 2300 fluid will lose laminarly is not always correct. There are lot of experimental evidence shows that we can push this limit much higher. I don't like the derivation of the Reynolds number given in the following links:

https://www.grc.nasa.gov/WWW/BGH/reynolds.html

link 2

We know for most of the viscous cases. There are other assumtions in the derivation but I don't want to complecate it much now.

This may be fine in the 19th century because of the lack of sophisticated experimental techniques. We are following this in the 21st century even after the discovery of PIV and CFD.

I believe we are using one parameter (Re) not completely relevant to the flow. This could be one of the reasons why we are unable to come up with any good theories in fluid flows especially turbulence and transition.

Please share some better derivation of Reynolds number.

Thank you.

[FMDenaro]

Have a look to this discussion
https://www.researchgate.net/post/Th...eynolds_number

[sbaffini]

The link provided by Filippo already contains some meaningful explanations. Let me just say that:

1) There is no such thing as Re number derivation

2) What you have in the links you posted is just some nice slide for the masses to grasp the main concept in the Re number

3) The only relevant subject here is Buckingam's PI theorem. If you know and understand it you will see how and where Re comes out. And you should if you are into experimental fluid dynamics.

4) If you end up with a single Re number when non-dimensionalizing your problem, it is just the tautological outcome of your assumptions. If you are careful enough to also properly non-dimensionalize the geometry and the boundary conditions, you will see that every single detail of the problem can introduce its own Re number. A single Re number is really meaningful when you only have 1 velocity and 1 length, and even in that case there might be differences (i.e., uniform flow around a cylinder vs a square cylinder, where maybe the boundary curvature might be a parameter to define additional Reynolds numbers)

[FMDenaro]

I totally agree with Paolo.

The fundamental understanding of the Re number idea is that is a ratio between inertial (convective) and diffusive fluxes of the momentum but the resulting meaning can be translated in terms of a ratio of characteristic velocities, lenghts, times, viscosities and so on.

And do not forget the relevance also in numerical simulation as the Re number is in the y+, in the cell Re distribution.


By a general point of view, the Re number cannot say, bay alone, much about the real physics of the flow regime. For example, in pipe flow the exact steady laminar solution is valid for any value of the Re number. The problem is in the stability of such solution in term of the Re number and a clear answer for that has to be found... Re=2300 is an old estimation, I agree that recent studied confirmed that a laminar state exists at higher values.

[arungovindneelan]

Originally Posted by sbaffini

The link provided by Filippo already contains some meaningful explanations. Let me just say that:

3) The only relevant subject here is Buckingam's PI theorem. If you know and understand it you will see how and where Re comes out. And you should if you are into experimental fluid dynamics.

I like Buckingam's PI theorem for simple linear cases. It is based on matching the dimension of different parameter involved in the problem. this kind of analysis tries to make non-dimensional number based on parameters which drive the flow. Though this analysis is beautiful, it is may cause some of the following problems because it cares about the only dimension. For example
not .

Newton discovered differential calculus because he realized the velocity of falling apple is not constant. If he applied v=h/t it will give average velocity, not the real velocity.

Similarly, I believe dimensional analysis can give an overall picture (macro) but may miss some complete physics.

Overall I believe the dimensional analysis is macro-scale analysis (algebraic) but we need some expression that valid form all scales using the differential model.

[arungovindneelan]

Originally Posted by FMDenaro

I totally agree with Paolo.

The fundamental understanding of the Re number idea is that is a ratio between inertial (convective) and diffusive fluxes of the momentum but the resulting meaning can be translated in terms of a ratio of characteristic velocities, lenghts, times, viscosities and so on.

And do not forget the relevance also in numerical simulation as the Re number is in the y+, in the cell Re distribution.

Since Reynold number explicitly coming out when we non-dimensionalizing the NS equation, it is an interesting parameter.

[arungovindneelan]

Originally Posted by FMDenaro

I totally agree with Paolo.

The fundamental understanding of the Re number idea is that is a ratio between inertial (convective) and diffusive fluxes of the momentum but the resulting meaning can be translated in terms of a ratio of characteristic velocities, lenghts, times, viscosities and so on.

And do not forget the relevance also in numerical simulation as the Re number is in the y+, in the cell Re distribution.

Since Reynold number explicitly coming out when we non-dimensionalizing the NS equation, it is an interesting parameter. Some papers and books qualitatively said that when the increase in Reynolds number viscous force may not enough to hold fluid so the flow separation occurs. If Reynolds number cannot quantify both the forces correctly how can it correctly predict flow separation?

I personally don't believe the cell-Reynolds number should be less than 2 for NS equation. This is a safe limit and thump-rule, I haven't seen any proof for this. If you come across any proof please share. Anyway, it is good because having something which is safe is better than having nothing.

But there is a proof for this for linear convection-diffusion equation using positivity analysis (some peoples call this maximum principle analysis) and it is quite good for linear problems solved using linear numerical schemes.

I like to know whether some proof for Kolmogorov length scale other than dimensional and order matching which I studied in turbulence course. I like to read some solid physical explanation or experimental proof.

Thank you.

[FMDenaro]

Originally Posted by arungovindneelan

Since Reynold number explicitly coming out when we non-dimensionalizing the NS equation, it is an interesting parameter. Some papers and books qualitatively said that when the increase in Reynolds number viscous force may not enough to hold fluid so the flow separation occurs. If Reynolds number cannot quantify both the forces correctly how can it correctly predict flow separation?

I personally don't believe the cell-Reynolds number should be less than 2 for NS equation. This is a safe limit and thump-rule, I haven't seen any proof for this. If you come across any proof please share. Anyway, it is good because having something which is safe is better than having nothing.

But there is a proof for this for linear convection-diffusion equation using positivity analysis (some peoples call this maximum principle analysis) and it is quite good for linear schemes.

There is no relation between Re number and separation in the flow. The real problem in the Re number is the velocity and lenght are not linked in a unique choice. This problem appears clearly in turbulent flows. Consider a typical energy spectrum. If you consider the x-axis, that is a wavelenght-based representation (L^-1), you can easily convert that in the local Re axis. Theferefore you see that the turbulence contains a wide range of characteristic scales, each one representing a differend Re number until to reach the laminar condition in the range between the Taylor and the Kolmogorov scales.


The cell Re number =2 comes from the positivity of the coefficients of the second order central discretization in the linear convective/diffusive equation. In a more general framework, Reh=O(1) everywhere is a contraint to have a real DNS.

[arungovindneelan]

Originally Posted by arungovindneelan

Since Reynold number explicitly coming out when we non-dimensionalizing the NS equation, it is an interesting parameter.

Originally Posted by FMDenaro

Have a look to this discussion
https://www.researchgate.net/post/Th...eynolds_number

But in the discussion, most of them agree it is a ratio of inertial and viscous force. If Re cannot quantify both the terms correctly, how can we believe the analysis based on Reynolds number!

[FMDenaro]

Originally Posted by arungovindneelan

But in the discussion, most of them agree it is a ratio of inertial and viscous force. If Re cannot quantify both the terms correctly, how can we believe the analysis based on Reynolds number!

No, it is a ratio betwee the convective flux rho*V^2 and the diffusive flux mu*Delta V / Delta L. Therefore, you see that such ratio depends on the local evaluation of the characteristic lenght and velocity. Only if you have a steady laminar flow you can consider that the flow has only one characteristic lenght and velocity. In real flows you have a wide range of scales and the ratio assumes different values. And yes, the analysis based only on the Reynolds number is misleading as you can read in the discussion on RG.

[FMDenaro]

Have a look also to this discussion
https://www.researchgate.net/post/Wh...ence_in_fluids

[arungovindneelan]

Originally Posted by FMDenaro

There is no relation between Re number and separation in the flow. The real problem in the Re number is the velocity and lenght are not linked in a unique choice.

I main factors affecting flow separations are , sign of second gradient of velocity, pressure gradient etc. If Reynolds number were calculated based on the above formulation. It will be one of the parameters to quantify the flow separation with other two non-dimensional number quantify (pressure gradient - viscous) and(inertia-pressure gradient).

Unfortunately, some researchers quantified it as a weak parameter (next to above-mentioned quantity) probably because of poor Reynolds number formula.

It is difficult to calculate the Reynolds number based inertial and viscous effect. But using accurate unsteady simulation, we can quantify those non-dimensional number for each cell and try to come with some empirical relation by performing the simulation on different test cases.

Have you come across any papers reported this kind of analysis?

Do you think this kind of analysis makes sense and may come with a better understanding of flow separation and turbulence?

[arungovindneelan]

I strongly agree with your point that the Reynolds number cannot be based on a single scale. In addition to that Reynolds, number formula is not agreeing with the ratio of two forces. This could be one of the reasons why most of the flow field is not related to the readymade formula of Reynolds number based on a single scale but still, we can find a good number of papers using this concept in reputed journals!

[sbaffini]

I think you are making confusion, at least in the way you are exposing your doubts, between the meaning and role of one or more non-dimensional parameters and the full equations and the underlying dynamics.

Using non-dimensional numbers is not expected to reveal the full dynamics of a set of equations or the relative physical problem. That is, the numerical values of the non-dimensional parameters of a problem, OBVIOUSLY, do not carry the whole information of the problem.

You insist in using derivatives and a certain Re number definition, but that approach is far from correct in deriving non-dimensional numbers. Sure, you can interpret such numbers in some intuitive ways, but that only comes afterward.

Not only the Re number doesn't come from a definition, but using a derivative in it is even farther from its origin, which has nothing to do with equations (less than ever derivatives). That's why I insist in citing the Buckingam's theorem, which is the only relevant source here. Long story short, you can come up with Re number while doing experiments for fluids and knowing nothing about their math.

The whole stuff is more or less as follows:

1. You have a physical problem you want to investigate and you identify ALL the physical parameters it might depend on. Note how the due diligence is up to you, to collect all the relevant parameters.

2. You identify, by PI theorem, the non-dimensional parameters of the problem at hand. They are less than the original physical parameters the problem depens on.

3. PI theorem tells you that, in non-dimensional terms, all the problems with the same non-dimensional parameters will have the same solution.

Let's make an example. In the steady pipe flow, if you identify a single length and a single velocity (or pressure gradient) as physical parameters, you end up with a single Re number. So far so good. But transition actually has to do with disturbance amplification. Is it a velocity or length (roughness) disturbance? Of what type? This requires additional information to be properly formalized and produce the relative non-dimensional numbers. Most times this information is only very partial (can you really describe perfectly the roughness?), and so the outcome is only an approximation. The both of us could perform the same transition experiment in a pipe using a common method to quantify roughness and using it as only source of disturbance only to later find out that one of us or both overlooked something because our results for the transition Re number differ.

[arungovindneelan]

Originally Posted by sbaffini

I think you are making confusion, at least in the way you are exposing your doubts, between the meaning and role of one or more non-dimensional parameters and the full equations and the underlying dynamics.

Using non-dimensional numbers is not expected to reveal the full dynamics of a set of equations or the relative physical problem. That is, the numerical values of the non-dimensional parameters of a problem, OBVIOUSLY, do not carry the whole information of the problem.

Thanks for the elaborate answer.


My intention was to tell some of the drawbacks of such analysis and some of the ways to overcome that. It will be hard for more experienced guys to digest such claims and I guess you are experimental+CFD person. Since we didn't archive much using the present formulation of some of the non-dimensional number, we are forced to think about the problem in the present formulation.

These are my view, this may be correct or wrong. I will explore this if I get an opportunity in future. This cannot be carried by a single researcher. We have to redo the study from simple test cases without worrying about publication. That is quite hard but achievable using machine learning.

[sbaffini]

To be sure we understand each other, I'm not saying that what you want to explore is wrong or old, whatever it is, just that it probably is something different from the way Re is defined and should be interpreted.

The way you are presenting the matter is like if you said: "we are using the wrong food to cure the cancer". And I am replying with: "while a possible cure for cancer, if ever found, might be something taken trough the mouth, we do not typically use that term to refer to a cancer cure" and "go ahead with your cancer cure search but we use the word food to refer to something else".

It's just a matter of using the correct terminology so that we understand each other.

[FMDenaro]

Originally Posted by arungovindneelan

I strongly agree with your point that the Reynolds number cannot be based on a single scale. In addition to that Reynolds, number formula is not agreeing with the ratio of two forces. This could be one of the reasons why most of the flow field is not related to the readymade formula of Reynolds number based on a single scale but still, we can find a good number of papers using this concept in reputed journals!

Who told you that the Re number must be based only on the ratio of two forces? The history of this number is in the experiment of O. Reynolds that managed only a group formed by velocity, diameter and viscosity of the fluid. Then, the non-dimensional form of the NSE show a relevant task:
you can get a non dimensional form using any velocity and lenght you want but the physical relevance and meaning of the non dimensional numbers is obtained only if you use the characteristic parameters such that ALL the terms in the non dimensional equation are of O(1). This is the correct idea of the non-dimensional numbers.

But you need again to think about real problem where L and V are actually a range of characteristic scales, as is typical in turbulence.
You can also define the Reynolds number introducing the local gradient


Re = V^2 *(1/|Grad V|)/ni


but that does not change much, it is also a ratio between times, velocities, viscosities...



Thus, to understand you better, please detail your idea of a correction on the number.

[arungovindneelan]

Originally Posted by FMDenaro

Who told you that the Re number must be based only on the ratio of two forces?
Thus, to understand you better, please detail your idea of a correction on the number.

Please check the following link in the wiki:

This I have taken a screenshot from the standard fluid dynamics book (please check the attachment): "Fluid mechanics" by Kundu, Cohen, Dowling
5th edition 146th page.

I believe you have your own right not to accept it as long as you can defend your view that is different scales are not accounted in this formula. My objective of this question is to learn a better derivation of Reynalds number. I have pointed out the limitations of the existing formula and assumptions made. My claim is: it didn't quantify inertial and viscous force correctly even at the inlet conditions.

Let assume we have some solution of flow past cylinder using PIV or CFD.
Using numerical methods we can calculate inertial, pressure and viscous forces over the whole domain and analyse the whole domain (instead of single non-dimensional number framed using inlet) using three non-dimensional number formed using viscous, inertial and pressure force. We need numerical methods because internal, viscous force involve gradient terms and we need integration to calculate the pressure force over the domain or cell.

The reason why I used non-dimensional number over the whole domain instead of single non-dimensional number at the inlet are:

1) In the following paper, the authors mentioned that they failed to get the result when they directly applied neural network on images of simulation with inlet and outlet conditions.

2) They got a good result when they consider the influence of boundary using one non-convectional input to the neural network that accounts for boundary points.

3) Even the convolutional neural network failed to give a good result just by considering inlet conditions. Please note that in fluid dynamics generally, we analyse the flow just by considering inlet conditions. Some papers use momentum thickness or boundary layer thickness for the Reynolds number that makes more sense. But I hope, I have given enough reason why we need a better formulation for Re number if it is based on the ratio of two forces.

In our study, we should account for boundary conditions. Since we have the solution, we can study the reasons for flow separation or transition using the new non-dimensional number formed using the above approach.

[arungovindneelan]

Originally Posted by sbaffini

To be sure we understand each other, I'm not saying that what you want to explore is wrong or old, whatever it is, just that it probably is something different from the way Re is defined and should be interpreted.

Generally, most of the researchers never listen to any points I make which is aginst the classical theories and thanks for listening this .

I have also mentioned the ways and reasons to utilize the new formulation of the non-dimensional number using numerical methods. If you wish you can suggest the limitations or problems with the new analysing procedures.

[FMDenaro]

"Using numerical methods we can calculate inertial, pressure and viscous forces over the whole domain and analyse the whole domain (instead of single non-dimensional number framed using inlet) using three non-dimensional number formed using viscous, inertial and pressure force. We need numerical methods because internal, viscous force involve gradient terms and we need integration to calculate the pressure force over the domain or cell. "


To be honest that appears tautological. If you compute a full numerical solution (but be careful, you need to solve a DNS!) from the dimensional form of the equation you can then evaluate each single term of the momentum (and energy) and build the non-dimensional numbers in any way you want. That is common for example when we evaluate the three cell Reynolds numbers or the y+ law from the solution. But ho would you proceed to solve the non-dimensional form when you need to set the value of Re as input?

I am not able to understand your final goal, you get a field distribution of local Re numbers, that's ok, and then?