Viscous Reynolds ([math]Re_{\tau}[/math]) estimation from streak size in TCF

francisdv · May 11, 2018, 15:44

Hey everyone,

I have found many solutions on this forum by reading, but the time to ask a question has come. I am a student posed with an assignment in which we have to perform several turbulent channel flow LES simulations.

One of the questions in the assignment is the following:"Using the solution from the

mesh, visualise contours of velocity magnitude in a cutting plane located at y = 0.1. What type of structures are visible? Use the typical dimensions of these structures to estimate the $Re_{\tau}$ of the computed flow. In your answer, explain your approach, providing a diagram if necessary."

I have included a picture of the specified cutting plane. It shows the high and low speed streaks near the wall. I suppose this implies the large scale motions that are typically 2-3 times $\delta$ , but that is also not related to the viscous scales. I don't see how I can relate the size of these streaks with anything on the viscous scales, either $u_{\tau}$ or $\delta_{\nu}$ . I have searched many articles and papers but to no avail.

I realise this might be a too specific subject but I wanted to give it a try. Any help would be much appreciated.

FMDenaro · May 12, 2018, 03:39

To tell the truth, that the first time I see such an exercise for a student...

However, start from the exact definition of Re_tau, it requires the half-channel height, the viscosity of fluid and a characteristic velocity that is referrred as u_tau. By definition rho*u_tau^2=tau_wall. You see that an avaluation of the stress at the wall is required, it is given by mu*(du/dy)|wall

Therefore, I suppose (but I could be wrong) that you have to compute an averaged tau_wall using a finite difference approximation for du/dy. On the wall you have zero velocity for the no-slip condition and the velocity values at the plane y+=0.1 can be used for the discrete derivative. Then perform the average in the plane.

Using the characteristic lenght of the flow structures (however, the velocity magnitude does not detect vortical structures) will define a different Reynolds number.

francisdv · May 16, 2018, 08:27

First of all I would like to thank you FMDenaro

So after your response I emailed my professor the two suggestions, about the (2-3 [ $\delta$ ]) streak size and the discrete approximation of $\tau_{wall}$ . He answered with that he can not give away answers and that I am close and should keep looking.

That let me to conclude that the discrete approximation is not the solution he wants or he would have told me I suppose. So I kept looking and found somewhere in the slides some information about experimental results on streaks. They state the following for low speed streaks:

z+ 100 units apart
x+ 1000 units length
y+ 5-10 units before ejection and sweep events

I figured that this might provide a connection as the "+ units" are the relation between the geometrical lengths (x,y,z) and the viscous length scale ( $\delta_{\nu}$ ).

$Re_{\tau}=h/\delta_{\nu}$ with h half channel width (=1), this becomes a matter of extracting the $\delta_\nu$ from the size in the plots of the streak results and the + units. For z and y this results in a $Re_{\tau}=h/\delta_{\nu}$ of approximately 300 and 50-100, which seems low. However by using streak length x+=1000, and saying the streak is on average 2.5 $\delta$ long. Then the following holds:

length of streak [in + units] = $x+ = 1000 = x/\delta_{\nu}$
with the length of LSM streak [geometrical units] = 2.5 $\delta$ , with $\delta$ in this case equal to half channel width (h, which is equal to 1), it would also be possible to also just measure the length of lowspeed streaks which should lead to an average size of 2.5

$\delta_\nu=2.5/1000$

which leads to:

$Re_{\tau}=\delta/\delta_{\nu}=1/0.0025=400$

this seems a feasible result

FMDenaro · May 17, 2018, 04:07

Quote:

Originally Posted by francisdv

First of all I would like to thank you FMDenaro

So after your response I emailed my professor the two suggestions, about the (2-3 [ $\delta$ ]) streak size and the discrete approximation of $\tau_{wall}$ . He answered with that he can not give away answers and that I am close and should keep looking.

That let me to conclude that the discrete approximation is not the solution he wants or he would have told me I suppose. So I kept looking and found somewhere in the slides some information about experimental results on streaks. They state the following for low speed streaks:

z+ 100 units apart
x+ 1000 units length
y+ 5-10 units before ejection and sweep events

I figured that this might provide a connection as the "+ units" are the relation between the geometrical lengths (x,y,z) and the viscous length scale ( $\delta_{\nu}$ ).

$Re_{\tau}=h/\delta_{\nu}$ with h half channel width (=1), this becomes a matter of extracting the $\delta_\nu$ from the size in the plots of the streak results and the + units. For z and y this results in a $Re_{\tau}=h/\delta_{\nu}$ of approximately 3 and 50, which seems extremely low. However by using streak length x+=1000, and saying the streak is on average 2.5 $\delta$ long. Then the following holds:

length of streak [in + units] = $x+ = 1000 = x/\delta_{\nu}$
with the length of LSM streak [geometrical units] = 2.5 $\delta$ , with $\delta$ in this case equal to half channel width (h, which is equal to 1), it would also be possible to also just measure the length of lowspeed streaks which should lead to an average size of 2.5

$\delta_\nu=2.5/1000$

which leads to:

$Re_{\tau}=\delta/\delta_{\nu}=1/0.0025=400$

this seems a feasible result

In general the Reynolds number can be seen as a ratio of two lenghs (actually also two times), one lenght being the channel half-heigh and the second one being the lenght characteristic of the u_tau velocity. That is you could see u_tau=delta_tau/time_tau.

Now, I can understand y+ as the distance from wall in wall-units (it is nothing else that the local Reynolds number) but what do you mean for x+ and z+? In a channel flow we have periodicity both in streamwise and spanwise directions so that we can define dx+ and dz+ values (computational step sizes in wall units).
In conclusion, y+ is just a position you fix along the wall-normal direction while dx+ and dz+ depends on the computational grid sizes.

francisdv · May 17, 2018, 14:24

I have been going through the article from which the professor presents a figure with the text about the average size in + units. [The structure of turbulent boundary layers, Kline et al., 1967] Unfortunately nowhere in the text are the x+ or z+ units used.

As you mentioned, the viscous Reynolds number is a ratio of scales, channel over viscous length scale. And as is made clear by the article by Kline, the + units are always variables that are non-dimensionalised with the viscous length scale and or velocity. This is where I see the connection, because I suppose that this is the approach that my professor used to define x+ and z+

For y+ this leads to the known definition of y / delta_tau
For x+ I think this implies x / delta_tau
For z+ equivalently z / delta_tau

This then relates to the assignment as the length of the streaks (X = stream direction) can be easily measured in the XZ plane (parallel to the wall), as an x-length. With the given x+ length, delta_tau can be obtained. Which in turn gives the ratio of length scales that is the viscous Reynolds number (with delta known).

full disclosure, I am just a student doing his first course on LES but they require us to get tits deep in the subject with a minimum of information provided. Therefore I am just trying to defend what I am doing, to figure out if any of it makes sense.

I deerly appreciate all the help so far

FMDenaro · May 17, 2018, 14:50

Quote:

Originally Posted by francisdv

I have been going through the article from which the professor presents a figure with the text about the average size in + units. [The structure of turbulent boundary layers, Kline et al., 1967] Unfortunately nowhere in the text are the x+ or z+ units used.

As you mentioned, the viscous Reynolds number is a ratio of scales, channel over viscous length scale. And as is made clear by the article by Kline, the + units are always variables that are non-dimensionalised with the viscous length scale and or velocity. This is where I see the connection, because I suppose that this is the approach that my professor used to define x+ and z+

For y+ this leads to the known definition of y / delta_tau
For x+ I think this implies x / delta_tau
For z+ equivalently z / delta_tau

This then relates to the assignment as the length of the streaks (X = stream direction) can be easily measured in the XZ plane (parallel to the wall), as an x-length. With the given x+ length, delta_tau can be obtained. Which in turn gives the ratio of length scales that is the viscous Reynolds number (with delta known).

full disclosure, I am just a student doing his first course on LES but they require us to get tits deep in the subject with a minimum of information provided. Therefore I am just trying to defend what I am doing, to figure out if any of it makes sense.

I deerly appreciate all the help so far

As you are a student, your effort to understand these turbulence and LES basic topics is appreciable.

I don't know if your channel flow problem is the classical plane channel with bi-periodic conditions in x and z but in this case you should be aware about the meaning of the x+ and z+ coordinates you wrote.
According to the paper (http://citeseerx.ist.psu.edu/viewdoc...=rep1&type=pdf) you will find only y+, the distance from the wall. Thus, defining hypothetically also x+ and z+ would define variable distances from walls normal to x and z directions, respectively. But the characteristic velocities would be now referred as to tangential to these walls.

Conversely, in numerical computation we define dx+ and dz+, the grid sizes along x and z, respectively, using the same wall units defined by y+. Therefore you can define the total spanwise and streamwise lenghts Lx+ and Lz+. For example see https://ntrs.nasa.gov/archive/nasa/c...0000039436.pdf
This way, you can measure the lenghts of the structures in fractions of Lx+ and Lz+.

In conclusion, in defending your conclusions, be careful in defining correctly the lenghts and avoiding misleading

francisdv · May 18, 2018, 09:53

Quote:

Originally Posted by FMDenaro

As you are a student, your effort to understand these turbulence and LES basic topics is appreciable.

I don't know if your channel flow problem is the classical plane channel with bi-periodic conditions in x and z but in this case you should be aware about the meaning of the x+ and z+ coordinates you wrote.
According to the paper (http://citeseerx.ist.psu.edu/viewdoc...=rep1&type=pdf) you will find only y+, the distance from the wall. Thus, defining hypothetically also x+ and z+ would define variable distances from walls normal to x and z directions, respectively. But the characteristic velocities would be now referred as to tangential to these walls.

Conversely, in numerical computation we define dx+ and dz+, the grid sizes along x and z, respectively, using the same wall units defined by y+. Therefore you can define the total spanwise and streamwise lenghts Lx+ and Lz+. For example see https://ntrs.nasa.gov/archive/nasa/c...0000039436.pdf
This way, you can measure the lenghts of the structures in fractions of Lx+ and Lz+.

In conclusion, in defending your conclusions, be careful in defining correctly the lenghts and avoiding misleading

Ah yes, I see what you are saying, we do are using double periodic boundary conditions (this doesn't influence the size measurement as it are coherent structures). I also see that x+ and z+ might be confusing according to the naming convention apparently used, as both would indeed NOT be related to the grid scale in my case. It would indeed be smarter to name it as Lx+ and Lz+, I see that in the NASA article they refer to this as "wall units in x and z direction". I will put it in my assignment this way, be it the way the professor intended or not, at least I learned a lot on the way.

Many thanks for all the help FMDenaro

francisdv · May 22, 2018, 10:01

A quick and final note:

in the following paper (https://aip.scitation.org/doi/pdf/10.1063/1.2717527), the characteristic spacing of 100 viscous wall units in spanwise direction for the low speed streaks, is mentioned to be "one of the more reliable physical constants in the study of turbulence".

For this reason i revised my earlier calculations and found a mistake in the spacing estimation used earlier for the spanwise distance. I used 30 instead of 0.3 for the Z spacing in z+ = z / delta_tau. Using the correct value will result in a Re_tau of approximately 300. Also a more accurate estimation would result in an average spacing of 0.25 and a Re_tau of 400. Which is equal to the value that i was expecting from other reference data.

FMDenaro · May 22, 2018, 11:26

Quote:

Originally Posted by francisdv

A quick and final note:

in the following paper (https://aip.scitation.org/doi/pdf/10.1063/1.2717527), the characteristic spacing of 100 viscous wall units in spanwise direction for the low speed streaks, is mentioned to be "one of the more reliable physical constants in the study of turbulence".

For this reason i revised my earlier calculations and found a mistake in the spacing estimation used earlier for the spanwise distance. I used 30 instead of 0.3 for the Z spacing in z+ = z / delta_tau. Using the correct value will result in a Re_tau of approximately 300. Also a more accurate estimation would result in an average spacing of 0.25 and a Re_tau of 400. Which is equal to the value that i was expecting from other reference data.

Good, I hope this is the answer for your problem. What about the y+ value of the plane you used?

Just consider some observations:
1) "structure" is a term more related to a qualitative than a quantitative entity. There are many proposal in literature to identify vortical structures and each one can give different results. Generally, the magnitude of the velocity is not a way to identify a vortical structure.
2) Owing to the approximate way to measure these "structures", I don't think that would be meaningful try to give a specific value for the Re number in between 300-400. You should consider the order of magnitude you get.

francisdv · May 22, 2018, 17:05

Quote:

Originally Posted by FMDenaro

Good, I hope this is the answer for your problem. What about the y+ value of the plane you used?

Just consider some observations:
1) "structure" is a term more related to a qualitative than a quantitative entity. There are many proposal in literature to identify vortical structures and each one can give different results. Generally, the magnitude of the velocity is not a way to identify a vortical structure.
2) Owing to the approximate way to measure these "structures", I don't think that would be meaningful try to give a specific value for the Re number in between 300-400. You should consider the order of magnitude you get.

considering:

intro) For the y+ value i cannot provide an explanation unfortunately. If indeed the y+ value for the streaks that i posted in the picture is 5 to 10, then the viscous reynolds number that is calculated is too small.

1) I fully agree with the qualitative nature of coherent "structures". To visualise the vortex structures that arise (such as horseshoe vortices) we also use the Q and lambda2 criteria which are more common. As I understand it the velocity magnitude is in this case representative because the low speed streaks are the result of the induced velocity (against the mean stream) by the vortex packets near the wall. (fig 12: https://aip.scitation.org/doi/pdf/10.1063/1.2717527)

2) In papers the usual viscous reynolds numbers i came across for TCF was in the range of 100 - 700. So i wasn't quite sure what kind of accuracy i should be aiming for.

FMDenaro · May 22, 2018, 17:17

You wrote that the cutting plane is at y=0.1, I suppose a non dimensional value. Therefore, y+=0.1*Re_tau. Assuming your estimation of Re_tau=O(10^2) you get y+=O(10). Your plane is far from the viscous sublayer.

francisdv · July 30, 2018, 14:24

hey people,

I recently got the results for my assignment and got full marks, so i suppose the approach with the wall units was the one the professor was hinting at.

In the report I handed in I wrote:

"These streaks are coherent structures for wall bounded turbulent flow and have typical properties. These streaks are alternating low and high speed and exist close to the wall. The low speed streaks are related to quasi streamwise vortices. It was experimentally observed that these low speed streaks are spaced approximately 100 viscous wall units apart by Kline, and later quoted by Adrian to be "one of the more reliable physical constants in the study ofturbulence".[Kline 1967, Adrian 2007] Using this information it is possible to relate the streak observations to the viscous scales."

which i followed with the z+ = z / delta_tau

leading to an estimation of delta_tau and thereby also Re_tau

I figured it would be nice to give this final update on the result.
I hope some day this thread might help someone else.

aarohimudholkar · February 5, 2023, 09:57

Hi, I am doing a course project and wanted to know what Re_tau was, how it was measured and how can it be compared to Re_global or Re_theta.

LuckyTran · February 5, 2023, 21:44

The normal Reynolds number uses the average channel velocity. Re_tau is the friction Reynolds number using the friction velocity.

$Re_{\tau}=\frac{u_{\tau}h}{\nu}$
with
$u_{\tau}=\sqrt{\frac{\tau_{w}}{\rho}}$

For pipes the length scale is typically your pipe diameter and half-height for channels. These Reynolds numbers are typically used in the context of fully developed internal flows.

Re_theta is momentum thickness Reynolds number for developing boundary layers (using the freestream velocity and momentum thickness of the boundary layer as the length scale) and should be discussed in its own appropriate context.

May 16, 2018, 08:27		#3
francisdv New Member Francis De Voogt Join Date: Oct 2017 Posts: 7 Rep Power: 8	First of all I would like to thank you FMDenaro So after your response I emailed my professor the two suggestions, about the (2-3 [ $\delta$ ]) streak size and the discrete approximation of $\tau_{wall}$ . He answered with that he can not give away answers and that I am close and should keep looking. That let me to conclude that the discrete approximation is not the solution he wants or he would have told me I suppose. So I kept looking and found somewhere in the slides some information about experimental results on streaks. They state the following for low speed streaks: z+ 100 units apart x+ 1000 units length y+ 5-10 units before ejection and sweep events I figured that this might provide a connection as the "+ units" are the relation between the geometrical lengths (x,y,z) and the viscous length scale ( $\delta_{\nu}$ ). $Re_{\tau}=h/\delta_{\nu}$ with h half channel width (=1), this becomes a matter of extracting the $\delta_\nu$ from the size in the plots of the streak results and the + units. For z and y this results in a $Re_{\tau}=h/\delta_{\nu}$ of approximately 300 and 50-100, which seems low. However by using streak length x+=1000, and saying the streak is on average 2.5 $\delta$ long. Then the following holds: length of streak [in + units] = $x+ = 1000 = x/\delta_{\nu}$ with the length of LSM streak [geometrical units] = 2.5 $\delta$ , with $\delta$ in this case equal to half channel width (h, which is equal to 1), it would also be possible to also just measure the length of lowspeed streaks which should lead to an average size of 2.5 $\delta_\nu=2.5/1000$ which leads to: $Re_{\tau}=\delta/\delta_{\nu}=1/0.0025=400$ this seems a feasible result Last edited by francisdv; May 22, 2018 at 10:05.

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May 12, 2018, 03:39		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,779 Rep Power: 71	To tell the truth, that the first time I see such an exercise for a student... However, start from the exact definition of Re_tau, it requires the half-channel height, the viscosity of fluid and a characteristic velocity that is referrred as u_tau. By definition rhou_tau^2=tau_wall. You see that an avaluation of the stress at the wall is required, it is given by mu(du/dy)\|wall Therefore, I suppose (but I could be wrong) that you have to compute an averaged tau_wall using a finite difference approximation for du/dy. On the wall you have zero velocity for the no-slip condition and the velocity values at the plane y+=0.1 can be used for the discrete derivative. Then perform the average in the plane. Using the characteristic lenght of the flow structures (however, the velocity magnitude does not detect vortical structures) will define a different Reynolds number.

May 17, 2018, 14:24		#5
francisdv New Member Francis De Voogt Join Date: Oct 2017 Posts: 7 Rep Power: 8	I have been going through the article from which the professor presents a figure with the text about the average size in + units. [The structure of turbulent boundary layers, Kline et al., 1967] Unfortunately nowhere in the text are the x+ or z+ units used. As you mentioned, the viscous Reynolds number is a ratio of scales, channel over viscous length scale. And as is made clear by the article by Kline, the + units are always variables that are non-dimensionalised with the viscous length scale and or velocity. This is where I see the connection, because I suppose that this is the approach that my professor used to define x+ and z+ For y+ this leads to the known definition of y / delta_tau For x+ I think this implies x / delta_tau For z+ equivalently z / delta_tau This then relates to the assignment as the length of the streaks (X = stream direction) can be easily measured in the XZ plane (parallel to the wall), as an x-length. With the given x+ length, delta_tau can be obtained. Which in turn gives the ratio of length scales that is the viscous Reynolds number (with delta known). full disclosure, I am just a student doing his first course on LES but they require us to get tits deep in the subject with a minimum of information provided. Therefore I am just trying to defend what I am doing, to figure out if any of it makes sense. I deerly appreciate all the help so far

May 22, 2018, 10:01		#8
francisdv New Member Francis De Voogt Join Date: Oct 2017 Posts: 7 Rep Power: 8	A quick and final note: in the following paper (https://aip.scitation.org/doi/pdf/10.1063/1.2717527), the characteristic spacing of 100 viscous wall units in spanwise direction for the low speed streaks, is mentioned to be "one of the more reliable physical constants in the study of turbulence". For this reason i revised my earlier calculations and found a mistake in the spacing estimation used earlier for the spanwise distance. I used 30 instead of 0.3 for the Z spacing in z+ = z / delta_tau. Using the correct value will result in a Re_tau of approximately 300. Also a more accurate estimation would result in an average spacing of 0.25 and a Re_tau of 400. Which is equal to the value that i was expecting from other reference data.

May 22, 2018, 17:17		#11
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,779 Rep Power: 71	You wrote that the cutting plane is at y=0.1, I suppose a non dimensional value. Therefore, y+=0.1*Re_tau. Assuming your estimation of Re_tau=O(10^2) you get y+=O(10). Your plane is far from the viscous sublayer.

July 30, 2018, 14:24		#12
francisdv New Member Francis De Voogt Join Date: Oct 2017 Posts: 7 Rep Power: 8	hey people, I recently got the results for my assignment and got full marks, so i suppose the approach with the wall units was the one the professor was hinting at. In the report I handed in I wrote: "These streaks are coherent structures for wall bounded turbulent flow and have typical properties. These streaks are alternating low and high speed and exist close to the wall. The low speed streaks are related to quasi streamwise vortices. It was experimentally observed that these low speed streaks are spaced approximately 100 viscous wall units apart by Kline, and later quoted by Adrian to be "one of the more reliable physical constants in the study ofturbulence".[Kline 1967, Adrian 2007] Using this information it is possible to relate the streak observations to the viscous scales." which i followed with the z+ = z / delta_tau leading to an estimation of delta_tau and thereby also Re_tau I figured it would be nice to give this final update on the result. I hope some day this thread might help someone else.

February 5, 2023, 09:57		#13
aarohimudholkar New Member Aarohi Join Date: Feb 2023 Posts: 2 Rep Power: 0	Hi, I am doing a course project and wanted to know what Re_tau was, how it was measured and how can it be compared to Re_global or Re_theta.

February 5, 2023, 21:44		#14
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,677 Rep Power: 66	The normal Reynolds number uses the average channel velocity. Re_tau is the friction Reynolds number using the friction velocity. $Re_{\tau}=\frac{u_{\tau}h}{\nu}$ with $u_{\tau}=\sqrt{\frac{\tau_{w}}{\rho}}$ For pipes the length scale is typically your pipe diameter and half-height for channels. These Reynolds numbers are typically used in the context of fully developed internal flows. Re_theta is momentum thickness Reynolds number for developing boundary layers (using the freestream velocity and momentum thickness of the boundary layer as the length scale) and should be discussed in its own appropriate context.