[rdemyan]

I am confused about how to calculate the Reynolds Number (Re) of the largest eddies in a turbulent flow. It seems that many discussions I read (university course notes, for example) state that the characteristic velocity of the largest scale eddies is the fluctuating velocity and it should be used to calculate the Re for the largest eddies. However, course notes by Moshe Matalon at the University of Illinois at Urbana-Champaign state on page 4:
"A characteristic of a turbulent flow is the existence of a wide range of length scales, or eddy sizes. The largest size L, is the macroscale or the flow scale. Eddies of size L are characterized by the mean flow velocity U. The Reynolds number of these eddies Re = UL/v is large so that the effect of viscosity is negligibly small."

https://cefrc.princeton.edu/sites/g/...Lecture-14.pdf

I am trying to determine the Reynolds number exactly for the purpose of correlating to the range of scales for my system (the ratio of my microscale to the large eddy scale). For example the ratio of the Kolmogorov microscale to the largest eddy is equal to the Re to the -0.75 power.

It turns out that in my system the mean flow velocity seems to provide a good Re correlation (R2 = 0.91). If I use the fluctuating velocity, I get a somewhat poorer correlation (R2 = 0.87). But if you look at the charts, there is no question that the first chart (R2 = 0.91) looks like a better fit, despite the second chart having an R2 of 0.87. My concern is how to explain this in a paper, since it seems that most believe the fluctuating velocity and not mean flow velocity should be used to calculate the large eddy Re.

[LuckyTran]

Generally, people use the mean flow velocity to calculate the Reynolds number because there's no way general way to know the fluctuating velocity level a priori and the mean velocity represents the total available kinetic energy anyway.

The Re^(3/4) is a scaling law. You cannot show separation of scales using 1 scenario. You must test/simulate a broad range of Reynolds numbers and only then you can show that the separation between the largest and smallest eddies goes like Re^(3/4)

The mean velocity and fluctuating velocity are related via the turbulence intensity. If turbulence intensity is constant at all conditions then it wouldn't matter which velocity you use, one is proportional to the other. However, turbulence intensity is weakly dependent on Reynolds number. Hence, it is known that the Re^(3/4) scaling is not perfect because if it was (then turbulence intensity wouldn't depend on Reynolds number at all).

If you consider the example of calculating "Reynolds number of a pipe," you take the pipe diameter and average velocity. Nobody tries to calculate this Reynolds number using the fluctuating velocity. Of course "Reynolds number of a pipe" isn't exactly the same thing as "Reynolds number of Kolmogorov scale" or "Reynolds number of Taylor microscale." The point of the Re^(3/4) scaling is that the smallest scales become smaller at a specific rate, not that you can exactly predict how small the smallest scales will be without knowing anything about the state of the turbulence.

[rdemyan]

I'm specifically interested in the Reynolds number for the largest eddy. I merely gave the Kolmogorov scaling as an example of what I am trying to do. I'm trying to scale my microscale to the largest eddy.

The scaling law for the Kolmogorov microscale can be derived via equation.



where is the Kolmogorov microscale, is the largest eddy, u is the fluctuating velocity and is the viscosity. Also, the energy dissipation rate, , is given by:



The use of in the above derivation would indicate that fluctuating velocity was used and not a mean velocity.

[LuckyTran]

You should be careful that the greek lambda is notation usually used for the taylor microscale. Some texts will call this the "integral scale.". It is definitely not the largest scale so this notation is not the favorite.

Still, these are scaling laws. The ratio of length scales is proportional to (~) Re, not exactly equal (=). And since u'/U=I, it shouldn't matter if you use u' or U in Re since they are both of the same order of magnitude.

When Reynolds number is calculated using the fluctuating velocity, it is usually explicitly noted as a turbulent Reynolds number. I will add that if you see "characteristic velocity of a flow" then it refers to the mean velocity, whereas characteristic velocity of the largest eddy (i.e. the large eddies) should be the fluctuating velocity. There are multiple characteristic velocities and you (or the writer) should be careful which characteristic velocity is being referred to. However, the characteristic velocity of the flow and largest eddy are almost the same thing (they differ by the turbulence intensity).

Still I don't know how you're curve-fitting a power law to a single data point and getting a correlation coefficient. Maybe some extra presentation might give some hints as to what you are trying to do and where you are having trouble.

[FMDenaro]

If you look at the next pages of the note, the (auto)correlations can be of some help when you think that they are strictly linked to the spectral energy distribution.
When you look at a spectra in a classic turbulence, you see a peak of energy corresponding to a certain frequency. You can get the length from such frequency that is the characteristic lenght of the most energetic eddy of size L.

Your question can be now posed like: what is the velocity used for computing the spectra, mean or fluctuation? How do you think they differ ?

[rdemyan]

Originally Posted by LuckyTran

You should be careful that the greek lambda is notation usually used for the taylor microscale. Some texts will call this the "integral scale.". It is definitely not the largest scale so this notation is not the favorite.

Still, these are scaling laws. The ratio of length scales is proportional to (~) Re, not exactly equal (=). And since u'/U=I, it shouldn't matter if you use u' or U in Re since they are both of the same order of magnitude.

When Reynolds number is calculated using the fluctuating velocity, it is usually explicitly labeled turbulent Reynolds number. I will add that if you read "characteristic velocity of a flow" then it refers to the mean velocity. Whereas characteristic velocity of the largest eddy (i.e. the large eddies) should be the fluctuating velocity. There are multiple characteristic velocities and you (or the writer) should be careful which characteristic velocity is being referred to. However, the characteristic velocity of the flow and largest eddy are almost the same thing (they differ by the turbulence intensity).

Still I don't know how you're curve-fitting a power law to a single data point and getting a correlation coefficient. Maybe some extra presentation might give some hints as to what you are trying to do and where you are having trouble.

Regarding lambda, I agree. I don't know how to create a capital lambda in LATEX which is what I wanted to do. But definitely I meant for lambda to be the large eddy size for this discussion.

From my experimental data, I have been able to come up with a microscale size (for my system) as a function of energy dissipation rate. I have 103 data points total. In my original post, I was talking about graphing the microscale sizes divided by the large eddy scale sizes as a function of the Reynolds number. After trying various Reynolds number definitions, I have found that the mean velocity is producing a better correlation than if I use the fluctuating velocity.

You seem to be agreeing with the notion that the fluctuating velocity should be used for the largest eddy. It turns out that in the system I am studying, there are no boundaries (well, the liquid is bounded on all sides by air, but there are no solid boundaries). If the largest eddy were the same size as the largest size scale of the system, would it then make more sense to use the mean velocity for the largest scale eddy as opposed to the fluctuating velocity. However, most definitely the turbulence and hence the largest scale eddy is caused by the fluctuating velocity.

Perhaps this is what Matalon means. If the size scale of the largest eddy is the same as the largest size (macroscale) or flow scale, then the mean flow velocity could be used. Presumably then, if the largest eddy size scale is less that the flow scale, it might be better to use the fluctuating velocity although Matalon does not state this.

Maybe the problem is that I don't understand the difference between the largest eddies related to the geometry and the largest eddies ad meant to be an integral scale. I suppose the latter means the size scale for most of the "production" energy. Perhaps then if the integral scale equals the largest eddy based on geometry, then the use of the mean flow for both works fine. I don't know.

[rdemyan]

Originally Posted by FMDenaro

If you look at the next pages of the note, the (auto)correlations can be of some help when you think that they are strictly linked to the spectral energy distribution.
When you look at a spectra in a classic turbulence, you see a peak of energy corresponding to a certain frequency. You can get the length from such frequency that is the characteristic lenght of the most energetic eddy of size L.

Your question can be now posed like: what is the velocity used for computing the spectra, mean or fluctuation? How do you think they differ ?

I don't understand how people come up with the spectra for the eddies. I mean in a turbulent flow you can come up with a good estimate for the largest eddy; you can calculate the Taylor microscale and especially the Kolmogorov microscale. But how do you come up with the turbulence kinetic energy for other scales. Do you just assume they all exist and try to calculate the amounts with references points of the largest and Kolmogorov (Taylor) microscales?

[FMDenaro]

Originally Posted by rdemyan

I don't understand how people come up with the spectra for the eddies. I mean in a turbulent flow you can come up with a good estimate for the largest eddy; you can calculate the Taylor microscale and especially the Kolmogorov microscale. But how do you come up with the turbulence kinetic energy for other scales. Do you just assume they all exist and try to calculate the amounts with references points of the largest and Kolmogorov (Taylor) microscales?

I mean that you can "read" the eddy size distribution along the spatial frequency axis. You have k=2*pi*n/L, being L the periodicity length and n the wavenumber. Therefore (1/k) is the characteristic lenght of the eddy having a certain density spectral energy E(k). The integral lenght can be read this way, just as (1/k) where the peak of energy appears.

Similarily, the Taylor microscale can be see as the (1/k) lenght where you see the onset of the dissipation range, that is the deviation from the inertial range.


Clearly, in a complex real flow, such a theoretical framework has some issues...

[sbaffini]

Originally Posted by rdemyan

I am confused about how to calculate the Reynolds Number (Re) of the largest eddies in a turbulent flow. It seems that many discussions I read (university course notes, for example) state that the characteristic velocity of the largest scale eddies is the fluctuating velocity and it should be used to calculate the Re for the largest eddies. However, course notes by Moshe Matalon at the University of Illinois at Urbana-Champaign state on page 4:
"A characteristic of a turbulent flow is the existence of a wide range of length scales, or eddy sizes. The largest size L, is the macroscale or the flow scale. Eddies of size L are characterized by the mean flow velocity U. The Reynolds number of these eddies Re = UL/v is large so that the effect of viscosity is negligibly small."

https://cefrc.princeton.edu/sites/g/...Lecture-14.pdf

I am trying to determine the Reynolds number exactly for the purpose of correlating to the range of scales for my system (the ratio of my microscale to the large eddy scale). For example the ratio of the Kolmogorov microscale to the largest eddy is equal to the Re to the -0.75 power.

It turns out that in my system the mean flow velocity seems to provide a good Re correlation (R2 = 0.91). If I use the fluctuating velocity, I get a somewhat poorer correlation (R2 = 0.87). But if you look at the charts, there is no question that the first chart (R2 = 0.91) looks like a better fit, despite the second chart having an R2 of 0.87. My concern is how to explain this in a paper, since it seems that most believe the fluctuating velocity and not mean flow velocity should be used to calculate the large eddy Re.

I don't understand to which charts you are referring but, within the validity of the Kolmogorov picture, energy enters at the largest scales. That is, it excludes cases where energy enters at a scale smaller than the largest L. Also, those estimates are up to a multiplicative factor, and not meant to give the exact values of anything, only their scaling.

From the practical side, it really matters what you are trying to do. The Kolmogorov theory is too generic (and with flaws) to give you accurate answers, if that is what you want. There certainly are more specific correlations and studies for a given flow that you can use.