Good estimates for the friction velocity?

kinematic_presser · November 27, 2022, 06:42

Hello guys,
I am making a mesh for an airofoil and I need to figure out how close to the wall I am going to put the first grid point for suitability for LES. Of course the usual recommendation is to use a y+ of about 1, but I don't know how to translate a y+ of 1 into physical (y) units. We model the scenario with a flat plate of length

. y is the coordinate normal to the plate and x is the coordinate along the plate, with x=y=0 on the left edge of the plate. The plate is immersed in a velocity field having far-field velocity

. The Reynolds number for this scenario is defined as $Re=\frac{UL}{\nu}$ and is this case is quite large (approx 600 000) so we consider the flow to be turbulent.

We know,
$y^+=\frac{yu_*(x)}{\nu}$

Where
$u_*(x)=\sqrt{\nu~(\partial_y u)(x,0)}$
Is the friction velocity.

So in order to estimate the value of y that corresponds to y+=1 , I need to have a good estimate for the friction velocity. I searched online but the only thing I found was this video (https://www.youtube.com/watch?v=SpkYBNdSOhQ) at 8:27, but he gives no reference.

Anybody know a good source?

FMDenaro · November 27, 2022, 06:51

Quote:

Originally Posted by kinematic_presser

Hello guys,
I am making a mesh for an airofoil and I need to figure out how close to the wall I am going to put the first grid point for suitability for LES. Of course the usual recommendation is to use a y+ of about 1, but I don't know how to translate a y+ of 1 into physical (y) units. We model the scenario with a flat plate of length

. y is the coordinate normal to the plate and x is the coordinate along the plate, with x=y=0 on the left edge of the plate. The plate is immersed in a velocity field having far-field velocity

. The Reynolds number for this scenario is defined as $Re=\frac{UL}{\nu}$ and is this case is quite large (approx 600 000) so we consider the flow to be turbulent.

We know,
$y^+=\frac{yu_*(x)}{\nu}$

Where
$u_*(x)=\sqrt{\nu~(\partial_y u)(x,0)}$
Is the friction velocity.

So in order to estimate the value of y that corresponds to y+=1 , I need to have a good estimate for the friction velocity. I searched online but the only thing I found was this video (https://www.youtube.com/watch?v=SpkYBNdSOhQ) at 8:27, but he gives no reference.

Anybody know a good source?

As you wrote, the friction velocity will be a result of the solution, thus you have to solve, compute, evaluate your y+ and if it is large refine the grid and solve again.
In a channel flow you can see an expression based on the pressure gradient but that is not exactly your case. However, you can consider that for channel flows the bulk velocity-based Re number is O(10) times the Re_tau.

kinematic_presser · November 27, 2022, 08:25

But surely some decent estimate must be already known? My scenario is the standard flat-plate setup for boundary layer theory. How was the expression in the linked video obtained?

EDIT: Yes, known estimates exist. Scroll down here:

https://www.cadence.com/ko_KR/home/t...cs/y-plus.html

kinematic_presser · November 27, 2022, 08:31

To answer my own question: A good estimate is known. Based on Fluid Mechanics by Frank M White,

$u_*(x)\approx\sqrt{ 0.027~\left(\frac{Ux}{\nu}\right)^{-1/7}\cdot \frac{1}{2}\nu U^2}$

Which is obtained by combining the approximate expression (ignoring turbulent fluctuations) for the wall shear stress " $\tau_{\text{wall}}$ " $=\mu~(\partial_y u)(x,0)$ (though this is a very bad choice of notation) with the definition of the friction velocity

FMDenaro · November 27, 2022, 08:57

Quote:

Originally Posted by kinematic_presser

To answer my own question: A good estimate is known. Based on Fluid Mechanics by Frank M White,

$u_*(x)\approx\sqrt{ 0.027~\left(\frac{Ux}{\nu}\right)^{-1/7}\cdot \frac{1}{2}\nu U^2}$

Which is obtained by combining the approximate expression (ignoring turbulent fluctuations) for the wall shear stress " $\tau_{\text{wall}}$ " $=\mu~(\partial_y u)(x,0)$ (though this is a very bad choice of notation) with the definition of the friction velocity

The scaling (1/7) is from the polynomial assumption of the mean steady velocity. In the viscous sub-layer you have u+=y+ -> du+/dy+=1 then you can estimate du/dy|wall

kinematic_presser · November 27, 2022, 11:12

How can we estimate du/dy from du+/dy+ without knowing u* ?

LuckyTran · November 27, 2022, 12:10

Either you do a crude CFD simulation first and get an estimate or the wall shear stress (and might as well get y+ from it while you're there) or you get your favorite empirical correlation for the wall shear stress from your favorite fluids handbook.

We often define friction velocity as the square root of the wall shear stress (divided by density of course) so there is no ambiguity and not using this du/dy stuff, they are the same of course if you hand-wave the notation.

Note that when you are doing an order-of-magnitude estimate for the friction velocity, a reference is hardly ever given because you should just report the actual mesh size or wall y+ after you've done the CFD.

kinematic_presser · November 29, 2022, 18:30

Quote:

Originally Posted by kinematic_presser

To answer my own question: A good estimate is known. Based on Fluid Mechanics by Frank M White,

$u_*(x)\approx\sqrt{ 0.027~\left(\frac{Ux}{\nu}\right)^{-1/7}\cdot \frac{1}{2}\nu U^2}$

Which is obtained by combining the approximate expression (ignoring turbulent fluctuations) for the wall shear stress " $\tau_{\text{wall}}$ " $=\mu~(\partial_y u)(x,0)$ (though this is a very bad choice of notation) with the definition of the friction velocity

Sorry, this expression is wrong (obviously, because the units are wrong.) The kinematic viscosity nu should not be in there on the right.. THe correct expression is
$u_*(x)\approx \sqrt{0.027 \left(\frac{Ux}{\nu}\right)^{-1/7}\frac{1}{2}U^2}=0.11619 \left(\frac{Ux}{\nu}\right)^{-1/14}U$

LuckyTran · November 30, 2022, 09:26

Btw there is a Y+ calculator provided on the wiki, the links briefly lists over a dozen correlations for the friction coefficient, and gives some more references. So to the answer the original question, yes there are tons of available sources.

November 27, 2022, 06:42	Good estimates for the friction velocity?	#1
kinematic_presser New Member K Join Date: Oct 2022 Location: Madison, WI Posts: 19 Rep Power: 3	Hello guys, I am making a mesh for an airofoil and I need to figure out how close to the wall I am going to put the first grid point for suitability for LES. Of course the usual recommendation is to use a y+ of about 1, but I don't know how to translate a y+ of 1 into physical (y) units. We model the scenario with a flat plate of length . y is the coordinate normal to the plate and x is the coordinate along the plate, with x=y=0 on the left edge of the plate. The plate is immersed in a velocity field having far-field velocity . The Reynolds number for this scenario is defined as $Re=\frac{UL}{\nu}$ and is this case is quite large (approx 600 000) so we consider the flow to be turbulent. We know, $y^+=\frac{yu_(x)}{\nu}$ Where $u_(x)=\sqrt{\nu~(\partial_y u)(x,0)}$ Is the friction velocity. So in order to estimate the value of y that corresponds to y+=1 , I need to have a good estimate for the friction velocity. I searched online but the only thing I found was this video (https://www.youtube.com/watch?v=SpkYBNdSOhQ) at 8:27, but he gives no reference. Anybody know a good source? mdtamim11 likes this.

November 27, 2022, 08:25		#3
kinematic_presser New Member K Join Date: Oct 2022 Location: Madison, WI Posts: 19 Rep Power: 3	But surely some decent estimate must be already known? My scenario is the standard flat-plate setup for boundary layer theory. How was the expression in the linked video obtained? EDIT: Yes, known estimates exist. Scroll down here: https://www.cadence.com/ko_KR/home/t...cs/y-plus.html mdtamim11 likes this.

November 27, 2022, 08:31		#4
kinematic_presser New Member K Join Date: Oct 2022 Location: Madison, WI Posts: 19 Rep Power: 3	To answer my own question: A good estimate is known. Based on Fluid Mechanics by Frank M White, $u_*(x)\approx\sqrt{ 0.027~\left(\frac{Ux}{\nu}\right)^{-1/7}\cdot \frac{1}{2}\nu U^2}$ Which is obtained by combining the approximate expression (ignoring turbulent fluctuations) for the wall shear stress " $\tau_{\text{wall}}$ " $=\mu~(\partial_y u)(x,0)$ (though this is a very bad choice of notation) with the definition of the friction velocity mdtamim11 likes this.

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November 27, 2022, 11:12		#6
kinematic_presser New Member K Join Date: Oct 2022 Location: Madison, WI Posts: 19 Rep Power: 3	How can we estimate du/dy from du+/dy+ without knowing u* ?

November 27, 2022, 12:10		#7
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,728 Rep Power: 66	Either you do a crude CFD simulation first and get an estimate or the wall shear stress (and might as well get y+ from it while you're there) or you get your favorite empirical correlation for the wall shear stress from your favorite fluids handbook. We often define friction velocity as the square root of the wall shear stress (divided by density of course) so there is no ambiguity and not using this du/dy stuff, they are the same of course if you hand-wave the notation. Note that when you are doing an order-of-magnitude estimate for the friction velocity, a reference is hardly ever given because you should just report the actual mesh size or wall y+ after you've done the CFD.

November 30, 2022, 09:26		#9
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,728 Rep Power: 66	Btw there is a Y+ calculator provided on the wiki, the links briefly lists over a dozen correlations for the friction coefficient, and gives some more references. So to the answer the original question, yes there are tons of available sources.