# calculate y+ and u+ -how to get wall shear stress?

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 March 27, 2006, 11:47 calculate y+ and u+ -how to get wall shear stress? #1 Ralf Schmidt Guest   Posts: n/a Hi! I guess, that's a beginner question, but how to calculate u+ and y+??? Ok, I know the formula, for both I need the wall shear stress. But how to get this one??? I found some empirical formulas for pipe flow, but how to figure out the wall shear stress for any other kind of flow like flow around a bluff body??? Any help is highly appreciated Ralf

 March 27, 2006, 15:16 Re: calculate y+ and u+ -how to get wall shear str #2 Biga Guest   Posts: n/a Wall shear stress = u_tau = sqrt{ tau_wall / rho }. For a 2-D flat plate flow, you can assume tau_wall = tau_xy = mu du/dy_wall. For a 3-D flow, you need to compute tau_wall from the stress-strain tensor (tau). You can do this by: tau_wall = sqrt{ Rx^2 + Ry^2 + Rz^2 }, where Rx = tau_xx * nx + tau_xy * ny + tau_xz * nz and so on... Here, {nx; ny; nz} is the face normal direction attached to the wall. Did it help?

 March 28, 2006, 02:12 Re: calculate y+ and u+ -how to get wall shear str #3 Ralf Schmidt Guest   Posts: n/a ok, thats theoretical equations. For all of them, I need the actuall velocity gradient close to the wall. I want to generate a grid, thats first element in the BL is at about y+ = 5. How to set this distance, when I don't know the velocity gradient a prior?? Ralf

 March 28, 2006, 05:58 Re: calculate y+ and u+ -how to get wall shear str #4 Christoph Guest   Posts: n/a I'm using the estimation of the Numeca manual: y = 6 (Re/L)^(-7/8) (L/2)^(1/8) y+ Re = Reynolds number L = Length scale y = 1st grid point distance y+ = what you want to reach Normally that gives useful results. Christoph

 March 28, 2006, 06:29 Re: calculate y+ and u+ -how to get wall shear str #5 Ralf Schmidt Guest   Posts: n/a thanks a lot for that! For a bluff body (cylinder in axial flow), the Re number is made from the free stream velocity and the cylinder diameter D? Is D=L? The wall shear stress (and therefore y+) depends on x, the distance from the beginning of the body. That's not in the equations... Can you send me the reference? Maybe as pdf? I can not access the Nemeca manual. Ralf

 March 28, 2006, 09:33 Re: calculate y+ and u+ -how to get wall shear str #6 Christoph Guest   Posts: n/a In this case L=D. I didn't manage to send you an email with attachement, so here's the text I'm refering to (it isn't really that much!): Regarding your name I suppose that you don't need a translation of it... ------------------------------- " Die Blasius-Gleichung bietet die Möglichkeit, den Zusammenhang zwischen dem Wandabstand der ersten Netzzelle und dem korrespondierenden y+ Wert vor der Rechnung abzuschätzen: y = 6 (Re/L)^(-7/8) (L/2)^(1/8) y+ mit: Re Reynoldszahl L einer typischen Referenzlänge y dem Wandabstand der ersten Netzzelle Größere Wandabstände können vom Löser ohne Probleme behandelt werden, führen jedoch zu einem Verlust an numerischer Genauigkeit. " Quelle: NUMECA

October 1, 2012, 22:30
#7
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Naimah
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Quote:
 Originally Posted by Biga ;41614 Wall shear stress = u_tau = sqrt{ tau_wall / rho }. For a 2-D flat plate flow, you can assume tau_wall = tau_xy = mu du/dy_wall. For a 3-D flow, you need to compute tau_wall from the stress-strain tensor (tau). You can do this by: tau_wall = sqrt{ Rx^2 + Ry^2 + Rz^2 }, where Rx = tau_xx * nx + tau_xy * ny + tau_xz * nz and so on... Here, {nx; ny; nz} is the face normal direction attached to the wall. Did it help?

Hi,
Im not really understand about the explanation for 3D case. Can you explain more details for 3D case?

Thank you.

Regards,
Naimah

 September 21, 2015, 09:20 #8 Member   Join Date: Sep 2011 Posts: 40 Rep Power: 6 Hi to everyone...I'm bringing this topic a bit out of the dust I assume, but I have a question about wall stress calculation. On site http://www.cfd-online.com/Wiki/Wall_shear_stress one can see a simple definition for wall shear stress, which as I presume needs only a simple derivative of x direction velocity in respect to y. 2 points needed, distance between them, job done. Or is it? Because in this topic and some others, I saw that for a 3d case, multiple components of the stress tensor should be used. So I'm a bit confused what is right thing to do. Use simple equation on the mentioned webpage, or use this square root value of shear stress tensor. And if the use of the latter is required, I'd be interested in reasons why? Because I presume that in the below written equation for shear stress on a wall in a flow between parallel plates, the non streamwise velocities will give on average 0 values (x is streamwise, y is wall normal direction). (dv/dx+du/dy)^2+(2dv/dy)^2+(dv/dz+dw/dy)^2 so, I assume only du/dy should be different than 0 on average. moreover, an additional question would be-does anyone know for some equations/laws which would give me an estimation for friction reynolds number at certain channel reynolds number? As I see from some DNS simulations for friction reynolds numbers of 180,395,590 and 1000 the connection might appear linear at first glance, but it's not like if Re_channel increases for 2x, the Re_friction would also increase 2x...I'm asking this as I have a case with a certain value of Re_channel and would like to know what Re_tau I should have... appreaciate any help and all the best!

September 21, 2015, 10:34
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Lucky Tran
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The relationship is fairly well-known to the channel flow community but is still pretty obscure. See Eq. 5.3 in:

George, W. K. "Is there a universal log law for turbulent wall-bounded flows?," Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Vol. 365, No. 1852, 2007, pp. 789-806.
doi: 10.1098/rsta.2006.1941

Quote:
 Originally Posted by Komon moreover, an additional question would be-does anyone know for some equations/laws which would give me an estimation for friction reynolds number at certain channel reynolds number? As I see from some DNS simulations for friction reynolds numbers of 180,395,590 and 1000 the connection might appear linear at first glance, but it's not like if Re_channel increases for 2x, the Re_friction would also increase 2x...I'm asking this as I have a case with a certain value of Re_channel and would like to know what Re_tau I should have... appreaciate any help and all the best!
It is a property of channel flows, where the wall shear stress at the wall can be directly related to the pressure drop (by a force balance).

Using this force balance, you can relate the friction Reynolds number to the channel Reynolds number. The friction Reynolds number contains a friction velocity (in terms of wall shear stress, which can also be written in terms of a pressure gradient), whereas the channel Reynolds number contains the bulk velocity. Some neat relations between the two Reynolds number can also be obtained by introducing a friction factor of some sort, such as through the Darcy-Weisbach equation. It turns out the ratio of friction Reynolds number to channel Reynolds number is given by: u_τ=√(τ_w/ρ)=U√(f/8

September 21, 2015, 10:47
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Filippo Maria Denaro
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Quote:
 Originally Posted by LuckyTran The relationship is fairly well-known to the channel flow community but is still pretty obscure. See Eq. 5.3 in: George, W. K. "Is there a universal log law for turbulent wall-bounded flows?," Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Vol. 365, No. 1852, 2007, pp. 789-806. doi: 10.1098/rsta.2006.1941 It is a property of channel flows, where the wall shear stress at the wall can be directly related to the pressure drop (by a force balance). Using this force balance, you can relate the friction Reynolds number to the channel Reynolds number. The friction Reynolds number contains a friction velocity (in terms of wall shear stress, which can also be written in terms of a pressure gradient), whereas the channel Reynolds number contains the bulk velocity. Some neat relations between the two Reynolds number can also be obtained by introducing a friction factor of some sort, such as through the Darcy-Weisbach equation. It turns out the ratio of friction Reynolds number to channel Reynolds number is given by: u_τ=√(τ_w/ρ)=U√(f/8

as very rude estimation, a factor 15 - 16 gives the magnitude of ReH once the Retau is given

 September 21, 2015, 16:05 #11 Member   Join Date: Sep 2011 Posts: 40 Rep Power: 6 Hi and thanks both for comments so far! I see the point in connection between the pressure drop/bulk velocity and friction velocity, but so far I fail in getting equations around to connect then channel reynolds with friction reynolds. For instance, if we'd be looking on equation 4.8, I assume the dP/dx would be the pressure drop (local, as this is derivative) at certain friction re? But how to then transform it to bulk velocity with the use of those equations? Namely, there is also term etc... but, if anyone would know how to do this, I'd also have this question-from my estimations, channel reynolds of 16700 would equal 850 friction reynolds number. Correct? regards

 September 23, 2015, 12:27 #12 Member   Join Date: Sep 2011 Posts: 40 Rep Power: 6 so, just to ask again about the first half of my question, the definition/calculation of wall stress. If anyone could specify a bit better if the derivative of streamwise velocity in regard to wall normal direction suffices, I'd appreciate it as I'm confused with definitons given before. moreover, I did a comparison of wall stress calculation with using only streamwise velocity derivative at the wall or using whole stress tensor at the wall multiplied with wall normal direction vector. I get pretty much different results (20x is the factor of difference)...of course I'd then go for the results which are closest to expected wall shear stress, but tbh, I might be doing something wrong and would like to get clarifications. best regards to all

September 23, 2015, 12:38
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Filippo Maria Denaro
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Quote:
 Originally Posted by Komon so, just to ask again about the first half of my question, the definition/calculation of wall stress. If anyone could specify a bit better if the derivative of streamwise velocity in regard to wall normal direction suffices, I'd appreciate it as I'm confused with definitons given before. moreover, I did a comparison of wall stress calculation with using only streamwise velocity derivative at the wall or using whole stress tensor at the wall multiplied with wall normal direction vector. I get pretty much different results (20x is the factor of difference)...of course I'd then go for the results which are closest to expected wall shear stress, but tbh, I might be doing something wrong and would like to get clarifications. best regards to all

What about your aim? you have a solution in dimensional form and you want to extract u_tau?

In a DNS you will find not only du/dy at wall different from zero but also other derivatives as they are point-wise (time and space) values. If you average in time the velocities, then only d<u>/dy is relevant as one gets <v>=<w>=0 everywhere

 September 23, 2015, 13:07 #14 Member   Join Date: Sep 2011 Posts: 40 Rep Power: 6 Hey! Well, I have a flow at certain Re number for a channel, let's say 7000, which corresponds to friction reynolds 400 following the theory. What I want is to check if simulations will give me this friction reynolds number. But first of all I need the proper definition of wall shear stress. As I understood at first, it should be just du/dy*viscosity. But then I found some topics here and otherwise arguing or showing that, as I wrote, whole stress tensor should be considered, not only streamwise velocity. Which made me confused However, I use time (and space) averaged values of velocities to define this stresses. And thank you for confirming my assumption that and should be 0, it's only logical that it should be like this. Then again-as I use averaged values, it should follow that only du/dy should be used for wall stress definition, but-as I say, I'm not convinced fully. Here an important reason is also this-that using only du/dy to define wall shear stress and then friction reynolds, I get 10% lower value than expected. Which bugs me. And just to add, I do LES simulations, first point in y is at y+=0,2-0,3 and 2 or 3 pts are below y+=1. Here I started to get another question-is definition of wall shear stress numerically, using such points, even ok? From discussion with different people, I got impression it is, but I assume it's always best to be as close to the wall as possible to make such calculations...

 September 23, 2015, 13:35 #15 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 2,411 Rep Power: 31 have a look here http://www2.mech.kth.se/courses/5C1218/bdrylayers.pdf to compute accurately the normal derivative at wall, you need a resolved boundary layer, therefore at least some computational nodes are necessary below y+=1

 September 23, 2015, 13:42 #16 Member   Join Date: Sep 2011 Posts: 40 Rep Power: 6 thanks for this!

 August 2, 2016, 22:32 #17 Senior Member   Vaze Join Date: Jun 2009 Posts: 140 Rep Power: 9 Hi FMDenaro Can you please share http://www2.mech.kth.se/courses/5C1218/bdrylayers.pdf thank you

 August 3, 2016, 06:54 #18 Administrator     Jonas Larsson Join Date: Jan 2009 Location: Gothenburg, Sweden Posts: 370 Rep Power: 10 Check out archive.org: https://web.archive.org/web/20070626...bdrylayers.pdf

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