Why do we use friction Reynolds number in turbulence?

Moreza7 · January 27, 2021, 15:40

Hello,

I was just reading some turbulent flow papers and most of them used friction Reynolds number:
$Re_{\tau}=\frac{u_{\tau}h}{\nu}$
where:
$u_{\tau}=\sqrt{\frac{\tau_{w}}{\rho}}$
Why do they use friction Reynolds number? Does it have any advantages or special concept?

FMDenaro · January 27, 2021, 17:44

Quote:

Originally Posted by Moreza7

Hello,

I was just reading some turbulent flow papers and most of them used friction Reynolds number:
$Re_{\tau}=\frac{u_{\tau}h}{\nu}$
where:
$u_{\tau}=\sqrt{\frac{\tau_{w}}{\rho}}$
Why do they use friction Reynolds number? Does it have any advantages or special concept?

That is quite a traditional choice in wall bounded turbulence (see Lumley). Instead of using the bulk velocity as characteristic velocity in the Re number, the wall-based velocity is used. The idea is to have a non-dimensional velocity of O(1) in the viscous sub-layer.

sbaffini · January 27, 2021, 17:58

Also, most near wall profiles nicely collapse on each other when nondimensionalized with the wall units

Moreza7 · January 27, 2021, 19:52

Quote:

Originally Posted by FMDenaro

That is quite a traditional choice in wall bounded turbulence (see Lumley). Instead of using the bulk velocity as characteristic velocity in the Re number, the wall-based velocity is used. The idea is to have a non-dimensional velocity of O(1) in the viscous sub-layer.

Thank you for your reply.

I know the equation is like the equation of y+, but what's the relation between friction Reynolds number and having non-dimensional velocity in viscous sub-layer?

Also, there's another problem. When we are going to simulate a flow by Bulk Velocity Reynolds number, we can easily set the Reynolds number before the simulation. But while the friction Reynolds number depends on the shear stress on the wall, it can only be calculated after the simulation. Then how can we say that we're gonna simulate a flow with a specific friction Reynolds number when we don't have shear stress at the wall before the simulation?!

Best Regards

sbaffini · January 28, 2021, 05:07

Quote:

Originally Posted by Moreza7

Thank you for your reply.

I know the equation is like the equation of y+, but what's the relation between friction Reynolds number and having non-dimensional velocity in viscous sub-layer?

Also, there's another problem. When we are going to simulate a flow by Bulk Velocity Reynolds number, we can easily set the Reynolds number before the simulation. But while the friction Reynolds number depends on the shear stress on the wall, it can only be calculated after the simulation. Then how can we say that we're gonna simulate a flow with a specific friction Reynolds number when we don't have shear stress at the wall before the simulation?!

Best Regards

y+ and the friction Reynolds number are related by the fact that both are based on the same velocity scale (and viscosity of course). So that:

$y^+ = \frac{y}{L} Re_{\tau}$

with L being the length scale in Re_tau.

Now, there are cases where, indeed, you can fix the friction Reynolds number and let the bulk flow be what it has to be. This happens, for example, in fully developed flows for channels and pipes, where you can directly link the driving pressure gradient to the friction Reynolds number.

In all the other cases, either you know the corresponding bulk Reynolds number in advance or you can set up a sort of PID controller to drive the flow where you want.

But note that for the fully developed flows I mentioned, when not performed with spectral methods, the driving mechanism of the flow is always the pressure gradient and not the bulk flow, so it tipically is the other way around, it is the pressure gradient that is iterated/controlled to achieve a given mass flow rate.

FMDenaro · January 28, 2021, 06:02

Quote:

Originally Posted by Moreza7

Thank you for your reply.

I know the equation is like the equation of y+, but what's the relation between friction Reynolds number and having non-dimensional velocity in viscous sub-layer?

Also, there's another problem. When we are going to simulate a flow by Bulk Velocity Reynolds number, we can easily set the Reynolds number before the simulation. But while the friction Reynolds number depends on the shear stress on the wall, it can only be calculated after the simulation. Then how can we say that we're gonna simulate a flow with a specific friction Reynolds number when we don't have shear stress at the wall before the simulation?!

Best Regards

If you use u_tau for the nondimensional form of the equations, you get into the non-dimensional momentum equation where you have directly the Re_tau value to insert. This way, you have then a specific non-dimensional pressure gradient that drives the flow. No need to evaluate the u_tau value at least if the dimensional values are not required.

gnwt4a · January 28, 2021, 06:59

for higher Res and beyond the buffer layer the mean shear stresses in the fluid are carried by the reynolds stresses while the viscous ones are negligible. therefore it has been assumed that the mean shear stress at the wall is a proper scale for turbulence of bounded flows. however, it seems that dns flows driven by a constant flow rate (ie constant bulk reynolds number) tend to reach steady state faster that those driven by a constant pressure drop. this ambiguity is probably related to that lack of robust scaling velocity in turbulent flows.
--

leobisco · November 15, 2023, 07:58

Quote:

Originally Posted by sbaffini

But note that for the fully developed flows I mentioned, when not performed with spectral methods, the driving mechanism of the flow is always the pressure gradient and not the bulk flow, so it tipically is the other way around, it is the pressure gradient that is iterated/controlled to achieve a given mass flow rate.

So, given that defintion of the problem, i have a certain value of mass flow rate, Bulk velocity and Bulk Reynolds number.

Which is the associated value of friction Re? Can i know it a priori?

I have this doubt since i have found several turbulent channel DNS and each one is associated with a certain friction Re. In the simulations pressure gradient is iterated to achieve a given mass flow rate, how do the authors know the proper value of mass flow rate to set? I.E. which is the mass flow rate associated with the desired value of friciton Re?

LuckyTran · November 15, 2023, 10:09

You have the friction factor as a function of bulk Reynolds number which relates the pressure gradient to the mass flow. So it doesn't matter whether you iterate pressure gradient or calculate the mass flow for a fixed pressure gradient.

$\frac{\partial P}{\partial x} = f \frac{1}{2} \rho {U_b}^2$

$U_b = \frac{\dot{m}}{\rho A_c}$

From that you can also show that the friction factor also yields the ratio of the bulk Reynolds number to friction Reynolds number for fully developed flows.

January 27, 2021, 15:40	Why do we use friction Reynolds number in turbulence?	#1
Moreza7 Senior Member Join Date: Jan 2018 Posts: 121 Rep Power: 8	Hello, I was just reading some turbulent flow papers and most of them used friction Reynolds number: $Re_{\tau}=\frac{u_{\tau}h}{\nu}$ where: $u_{\tau}=\sqrt{\frac{\tau_{w}}{\rho}}$ Why do they use friction Reynolds number? Does it have any advantages or special concept?

January 27, 2021, 17:58		#3
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	Also, most near wall profiles nicely collapse on each other when nondimensionalized with the wall units Moreza7, aero_head and ParsaT like this.

January 28, 2021, 06:59		#7
gnwt4a Member EM Join Date: Sep 2019 Posts: 59 Rep Power: 7	for higher Res and beyond the buffer layer the mean shear stresses in the fluid are carried by the reynolds stresses while the viscous ones are negligible. therefore it has been assumed that the mean shear stress at the wall is a proper scale for turbulence of bounded flows. however, it seems that dns flows driven by a constant flow rate (ie constant bulk reynolds number) tend to reach steady state faster that those driven by a constant pressure drop. this ambiguity is probably related to that lack of robust scaling velocity in turbulent flows. -- Moreza7 likes this.

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November 15, 2023, 10:09		#9
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,753 Rep Power: 66	You have the friction factor as a function of bulk Reynolds number which relates the pressure gradient to the mass flow. So it doesn't matter whether you iterate pressure gradient or calculate the mass flow for a fixed pressure gradient. $\frac{\partial P}{\partial x} = f \frac{1}{2} \rho {U_b}^2$ $U_b = \frac{\dot{m}}{\rho A_c}$ From that you can also show that the friction factor also yields the ratio of the bulk Reynolds number to friction Reynolds number for fully developed flows.