The Reynolds number of the largest eddies in a turbulent flow can be calculated using the following equation:

Re = UL/ν

where Re is the Reynolds number, U is the characteristic velocity of the flow, L is the characteristic length scale of the flow, and ν is the kinematic viscosity of the fluid. The characteristic length scale can be taken as the size of the eddies, and the characteristic velocity can be taken as the velocity scale of the flow at that size.

Howdy Robert Demyanovich,

I have been uncomfortable about Reynolds Number for decades. On the face of it, the value is intended to inform us about the relative significance of inertial forces and viscous forces with respect to the initiation of turbulence in a flow. The former supports turbulence, the latter damps turbulence, but one finds different velocity and length scales are used for each flow configuration. There is valuable consistency in the measure it provides and it is a useful guide if one has data to justify the critical Reynolds Number in each flow case.

However, If you wish "to determine the Reynolds number exactly" you must fail because the "largest eddies in a turbulent flow" are not all the same size in any real case, and to be exact for longer than an instant that variation must be addressed. What you use as "the fluctuating velocity" is critical, so as if you were in charge, you must determine the inertial forces involved for the time scale of your system and how they could trigger turbulence, then explain how you determined them and then work with them. Note that the presence of eddies of any size indicates your flow is already turbulent and I remain uncomfortable in using Reynolds Number "after the fact" in this way.

One is limited to some sort of compromise as to values to use for either U or L. In the current state of our knowledge of turbulent fluid dynamics your best bet is to use an engineering value such as any you have cited or has been offered by Chathumal Chandrasiri in his post. To explain your choice is a valid option that will justify it to anyone who is appropriately bewildered by the complexity of turbulent flows.

Beware of correlations when you are seeking to associate natural processes with data. Correlations are convenient and can be useful within the bounds of their data base, but they may be independent of the physical dynamics of the flow. I think the values of U and L you use must be shown to have triggered the turbulence upstream that you now seek to characterize with a Reynolds Number for the largest eddies. All in all, you have a very interesting (complex) problem and how to bound that complexity is the nature of mathematical physics and engineering. Good Luck.

Afterthought: I offer thoughts in my replies on Q&A that attempt to address the information provided. However, given the value of the Reynolds Number as a measure of whether a laminar flow will become turbulent, the very idea of "the Reynolds Number of the largest eddies" seems to be absurd as a measure of natural fluid dynamics and it is a bit confusing to me, sorry. . .

Happy Trails, Len

Hi Robert, your question is an interesting one to think about. Your paper is probably written by now, but I have a very different perspective to that offered by Leonard Hall.

Start with dimensional analysis and with the proposition that the mean statistical properties of some flows can be fully characterized by a length (say BL depth, pipe diameter, nozzle diameter for a round jet), a velocity (say velocity at the top of a BL, mean flow velocity in a pipe, nozzle velocity in a round jet) and the viscosity of the fluid. We can form a Reynolds number in each case as UL/v. A given value of the Reynolds number will then determine all the mean statistical properties of the fully-developed flow in each case. We needn’t be concerned that the flow itself is chaotic, so that the eddies themselves have variable shapes and sizes; we just concern ourselves with the ensemble mean values of the various statistical descriptors of the flow. The properties of the eddies are important in determining actual relationships, but we don’t need to know about them to know that relationships will exist.

A consequence of all this is that all other mean lengths and velocities will be specified once the Reynolds number is specified. This means, in your example, that the Kolmogorov microscale will bear a fixed  relationship to any other characteristic length of the system. It also means that any characteristic velocity can be used to define a Reynolds number that will be proportional Uv/L, and so equally useful in any analysis.

So why does your large-eddy fluctuating velocity work less well than the mean flow velocity? Perhaps the clue is that a Reynolds number defined using mean flow velocity gives only a very good, but not a perfect correlation. Measurements and sampling are never perfect, so the poorer correlation with the fluctuation velocity likely reflects increased error in measuring that scale.

Howdy K. G. Mcnaughton,

Your discussion of the value of the symbol ratio, UL/v, is well presented and I affirm that the symbol ratio UL/v may be well used as you describe. However, while a rose by any other name may smell as sweet, its character as a flower is not changed thereby and vice versa for an angry skunk one calls a rose.

On the other hand, it is misleading to use the valuable label defined for a specific purpose, namely a measure of transition to turbulence, Reynolds Number, as a catch-all for application of the values used in the symbol ratio to other purposes. That was my point, and I really hope that everyone appreciates the value of accurate communication that is addressed here.

Call UL/v Fred or Mehitable if you wish when you use it for other purposes than as a measure of transition to turbulence, even when using it for the character of the resulting turbulence, but do not call it Reynolds Number. Turbulent fluid dynamics is so difficult a field that confusing the issue with mixed meanings for terms is not wise, however convenient.

Happy Trails, Len