There are several possible implications in analsying the Reynolds number, depending on what velocity and lenght we adopt, it can have a very different resulting values for the same flow problem.

You can analyse this parameter also in terms of a ratio between characteristic time or lengt scales of the flow (and, of course, velocity scale).

Finally, it is worthwhile to address the cell Reynolds number that is often used in CFD, in such case the characteristic lenght is the computational mesh size.

Reynolds number express the effect of Inertia forces on viscous forces or reverse of it. This definition is the basic physical form of Reynolds number.

The Reynolds number express a balance between the inertial forces to the viscous forces. That is valid for ducts or closed systems only. It is a relation associated to the role of particle velocities.

For fluids with free surface, it is important the relatión between inertial forces and gravitational waves (Froude number). For free surface problems, the Froude Number is a relatión between particle velocities and celerity of waves in fluids. The viscous forces are of smaller importance for free surface fluids.

Dear Filippo, Taha and Carbonel

Thanks for your replies,

Yes I have already known about it. But I'd like ones to think if there are any other meanings of Re number.

In my mind, I think the Reynolds number is the ratio of "macro activity/parameter" and " micro activity/parameter". For instance, the inertia force, for me, is a macro activity while the viscous force is a micro activity.

As an example, for a laminar internal pipe flow, what about if another meaning of Re is the ratio of the average flow velocity (V), and the "micro velocity", defined as kinematic viscosity/pipe diameter?

That's just as an example.

Thank you

" As an example, for a laminar internal pipe flow, what about if another meaning of Re is the ratio of the average flow velocity (V), and the "micro velocity", defined as kinematic viscosity/pipe diameter?"

That is what I addressed above, you can see the Re number as ratio of two times, velocities or lenghts.

However, "macro" or "micro" depends on the choice of the characteristic lenght. For example you can set L as the integral lenght scale and getting a value about the ratio of inertia at that lenght scale. But if you use as characteristic lenght a different scale, for example the Kolmogorov lenght scale, you get the relevance of the inertia at a microscale. Different values for Re but same flow problems!

You are simply changing the lenght scale of the observer

Thanks Filippo Maria Denaro

I just think like that but not so deep. Could you give us an example how to view a laminar and turbulence flow with those micro parameter, please?

Dear İbrahim Halil Yılmaz

Thanks for your nice explanation.

Did you mean that the Reynolds number is a parameter of stability? And it is a relative condition of global and local attractor?

Nazaruddin Sinaga

A simple example is given in the case of a turbulent flow, for example think about the plane channel flow. You can evaluate the Reynolds number using the centerline average velocity and the half-heigth and get a certain values. But often the chosen velocity in such problem is the wall stress-based Utau and you get a lower value for Re. And finally, you could decide to evaluate the Reynolds using the Kolmogorov lenght eta, this way the Reynolds is of unity order of magnitude. But the flow problem is the same! You are only changing the magnitude of the scale of the observer. A small observer of lenght eta will see the flow around as it was laminar....

Dear Filippo Maria Denaro

So, do you mean that the same flow could be viewed as turbulent (by using centreline velocity which produce high Reynolds number) and laminar ( by using Utau and Kolmogorov length eta)? Please correct me if I am wrong.....

Thanks İbrahim Halil Yılmaz

Do you agree with Filippo Maria Denaro explanation?

Hi Filippo Maria Denaro , could you give your comment, please?

The hystorical experiment of Reynolds was based on pipe flow where a geometrical lenght was clearly present. But turbulence exists in many other flow problem wherein a geometrical lenght does not clearly exist. Just consider the standard case of homogeneous turbulence, the Reynolds is evaluated on the basis of the Taylor micro-scale that is a flow-dependent lenght. The same happens for two jets evolving in a mixing layer

If you see an energy spectra, the wavelength axis can be simply transformed in a local Reynolds number so that you can see the energy content at a certain value of the local Reynolds.

Reynolds number (Re) is given as ratio of inertia force to viscous force. The equation is given as, Re = rho v Lc/mu, where rho refers to fluid density in kg/m3, v refers to mean velocity of fluid in m/s, Lc is characteristic length given by 4 Ac/Pw, where Ac is cross sectional area, in m2, Pw is wetted perimeter, in m and mu refers to coefficient of viscosity, kg/m-s or N-s/m2.

Very important question in fluid mechanics . Re no determines the laminar , transition and turbulent flow depending upon his value . Fanning 's Friction factor can be also calculated for a flow inside the pipe . f = 16 /Re .

The Reynolds ( Re ) number is a quantity(unit-less) which engineers use to estimate if a fluid flow is laminar or turbulent. This is important, because increased mixing and shearing occur in turbulent flow. This results increase viscous losses which affects the efficiency of hydraulic machines. laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion; turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities. Actually, the transition between laminar and turbulent flow occurs not at a specific value of the Reynolds number but in a range usually beginning between 1,000 to 2,000 and extending upward to between 3,000 and 5,000.

Thanks Ahmed Abdelhameed

What seems to me not so clear from this discussion is that the lenght in the definition of the Re number does not necessarily is a geometrical dimension ...

Dear/Filippo Maria

Yes, it is not necessarily therefore in case of flow through boundary layer boundary layer thickness is taken as characteristic length in calculating Reynolds number. Also, Reynolds number based on momentum thickness ( ˆ Ue*theta/new), dimensionless ,however there is a significant difference between Reynolds number in internal and external flow

Regards

I believe the explanation Dr. Sinaga is requesting is the following. In Statistical Mechanics it is studied that, in ideal conditions, the viscosity of a gas is given by (see, for example , F. Reif, "Fundamentals of Statistical and Thermal Physics" , 2009, Waveland Press, page 475, Eq. (12.3.7))

\mu=n m v l /3 (1)

where n=number of molecules per unit volume, m=mass of one molecule,, v=average velocity of molecules,, l=mean free path between molecule collisions. Since density=n m, then the Reynolds number is the ratio

Re= 3 V L/v l (2)

being V and L some macroscopic characteristic velocity and length of the flow. This ratio of macroscopic V L and microscopic v l is the true meaning of the Reynolds number. In not-so-ideal conditions, replace the 3 of Eq. (2) by a most appropriate number, but the meaning is still the same.

Best regards.

Yes Javier Garcia Javier Garcia, you know what I was thinking. However, I have another argument about why I asked about the physical meaning of the Reynolds number. I will tell you later. In my mind, as on Osborne Reynolds experiments, the macro or geometrical parameters, such as pipe diameter and average velocity, is the parameter used to characterize the flow regimes. Maybe Prof. Filippo Maria Denaro Filippo Maria Denaro does not agree with this.

Dear Prof. Filippo Maria Denaro , is there the same physical meaning of Reynolds number when we used the "original parameters" (the macro parameters) and the "nonoriginal" parameters (the micro parameters)?

Please advise me.

Than you very much

Dear Tapan K. Sengupta

Could you share us the Laufer (1954) report, please?

Thank you

This is a very interesting statement:

"Even in the pipe flow of Reynolds, it is incorrect to presume that diameter is the natural length scale."

Ok no problem. I've got it...

Nazaruddin Sinaga

I cannot add more than I wrote above. If you consider a pipe seems easy to chose the geometrical length as the diameter, but what about ducts of different geometries? And if I would simply rewrite the Reynolds number using V*tr instead of a lengh L how do you compute the value? The definition is always valid.

Filippo Maria Denaro

For all ducts of different geometries, we normally use hydraulic diameters.

The idea that I expressed, relating macroscales L,V with molecular scales l, v, can be extended to any macroscopic scale, as long as the hypothesis of continuum holds. Therefore, if k denotes the particular scale that we are considering, be it integral or Taylor or any other, the ratio L(k) V(k)/l v still makes sense and expresses the local Reynolds number for that particular k, until the concept of continuum no longer holds.

Best regards.

From the discussion that we have done, I can draw two conclusions on the physical meaning of the Reynolds number as follows:

1. The Reynolds number is the ratio between the inertial force to the viscous force, whatever the geometry of the flow and the regimes

2. The Reynolds number is a measure of flow instability, where the flow becomes more unstable with the higher Reynolds number.

Do you agree?

Thanks Tapan K. Sengupta

Could you give us more physical meaning of Reynolds number, please?

I hope you may tell us with a simple and clear words.

Thank you

I think that it is a measure to classify any type of flow( either laminar or turbulent or transition between them) where a big difference in the flow behavior could be observed at each flow type.

Dear Khalid S Shibib

That's the the point of what we discussing here. Maybe it is true, for very low or very high Reynolds number, far from transition region.....

A nice explanation Tapan K. Sengupta

Thank you very much.

Dear researchers,

Have a nice weekend with your family....

This is proportional to the ratio of inertial forces and viscous forces (forces resistant to change and heavy and gluey forces) in a fluid flow. The Reynolds number is used to study fluids as they flow. The Reynolds number determines whether a fluid flow is laminar or turbulent.

The ratio of momentum flow through a tube of unit cross sectional area to representative of shear stress due to action of viscosity. Hence this is a measure of relative magnitude of inertia force to viscous force occuring in flow. The higher the Reynolds number, the greater is the relative contribution of inertia effect. The lower the Reynolds number greater is the relative magnitude of viscous stresses.

Reynolds number indicate type flow.

It is laminar if the value is less than 2000. It is turbulent if the value is more than 4000. Between these two value the flow is transient.

It is a ratio of inertia forces to the viscous forces.

Well, imagine a fluid passes through a pipe, a container , or whatever. The motion ((momentum)) of this fluid is the inertia force.

The force that tries to stop that motion is the viscous force.

I hope it is clear for you now.

Dear all distinguished scholars, thank you very much for your participation in the discussion.

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Best regards.

In a study, I actually noticed that for a viscous fluid, it is not meaningful to say that higher Reynold number implies lower viscous force associated with the viscosity of the fluid. But, very concise to remark that increasing Reynold number implies higher inertia force when the viscous force is fixed. This is the major reason why larger values of Reynold number denotes turbulence.