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September 20, 2012, 16:18 
Wall shear stress in channel flowplus and minus

#1 
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Charlie Tan
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Hello,
Derivation of a wall shear stress for a channel flow results as a minus wall shear stress at the bottom of the wall and a plus at the top. Can someone please explain me how is this possible? Because intuitively thinking, they should be same, with a minus sign. Because at the top and the bottom, there is friction. Right?? Thank you very much! 

September 20, 2012, 17:28 

#2  
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Francesco Capuano
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Do they have opposite sign but are equal in magnitude? In this case, it may be due simply to how the solver calculates the derivative of the velocity at the wall, so just a matter of reference frame....
Quote:


September 21, 2012, 04:32 

#3 
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Charlie Tan
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Hello francesco_capuano,
Thank you very much for your reply. But actually this is a mathematical question, it's not a result that I get from any kind of software/simulation. The derivation of the wall shear stress is minus and plus at the channel bottom and top, respectively. I am searching for an anwser to that. How is that possible. Wall shear stress should have the same value at top and bottom, because the same shear force is being applied. Thank you very much. 

September 21, 2012, 04:59 

#4 
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Oliver Gloth
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Did you forget to multiply with the normal vector of the wall/control volume ...??


September 21, 2012, 05:03 

#5 
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Because the velocity gradient has opposite signs at the bottom and top walls.
Bottom: du/dy > 0 (velocity increases from 0 to u_bulk as we move along the vertical axis y) Top: du/dy < 0 (velocity decreases from u_bulk to 0 at the top wall, as we move along the y axis) Hence the difference in sign. Does that solve your question? Cheers, Michujo. 

September 21, 2012, 05:07 

#6 
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Oliver Gloth
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The normal vectors of your control volume also point into different directions  hence the difference in sign is cancelled and you get a negative momentum flux on both sides (compensated by the pressure in the channel).
Does that answer your question?? ;) EDIT @michujo: Sorry I didn't read who posted, so I assumed it was the original poster. The shear force on the wall does have the same sign, however! 

September 21, 2012, 08:58 

#7 
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Filippo Maria Denaro
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If you consider the simple case of fully developed laminar flow, the velocity is quadratic, the vorticity is linear and is zero at halfheigh of the channel (the well know butterfly profile). The wall stress is proportional to vorticity at wall, therefore is correct to have same magnitude and opposite sign


September 21, 2012, 09:03 

#8 
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Oliver Gloth
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cool, a channel with no pressure loss ... ;)


September 21, 2012, 09:09 

#9 
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Filippo Maria Denaro
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September 22, 2012, 09:56 

#10 
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Charlie Tan
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Hello,
Thank you very much ogloth and other members. This really answers my question. But I would like to ask more. Now I understand why the signs are turning out to be the same at the end. But could you please tell me, how this normal vector definition appears in a Finite Volume approach? Where in the discretization does it take place? Thank you very much guys and have a great Saturday. 

September 22, 2012, 14:26 

#11 
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Hi. In a finite volume approach this works like this:
Bottom wall: the momentum flux ENTERS the cell volume through the south face (bottom wall of the channel). Top wall: the momentum flux LEAVES the cell volume through its north face (top wall of the channel). Hence the sign difference. Cheers, Michujo. 

September 22, 2012, 14:42 

#12 
Senior Member

In a FV method you essentially work on the sum of the fluxes in and out of your control volume (i.e., the computational cell) following physical conservation principles.
When you write down this conservation law for a FV cell you end up with a sum over the faces of the cell, each term involving n scalar "something", where n is the face area vector (it has the direction of the outgoing unit normal and the magnitude of the face area) and "something" is usually the sum of the convective and diffusive fluxes. In the case of the channel flow (with positive x in the streamwise direction, positive y from the bottom to the top wall and positive z in the spanwise direction), at the bottom wall you have that n scalar the diffusive flux is negative (the velocity gradient being positive along the positive y direction and the face normal along the negative one). But, also on the top wall it is negative (the velocity gradient being positive along the negative y direction and the face normal along the positive one). So the whole contribution is indeed negative and balanced by the streamwise pressure gradient. 

October 9, 2012, 04:44 

#13 
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Charlie Tan
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Hello!
Sorry for my late answer. Thank you sooo much for helping me to figure this thing out. I almost got it! But I still need to do it myself to understand it. So do any of you know a website where I can see the FVM discretization, so that I can see the normal vector being involved in the control volumes? Thank you again and have a great day. 

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