# Calculation of Wall Shear Stress

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# Calculation of Wall Shear Stress

Posted February 1, 2014 at 21:53 by Nucleophobe
Updated February 2, 2014 at 01:10 by Nucleophobe

Quote:
Hi Glen,

Sorry, I wasn't asking you to check my calculations or anything. I was just wondering, as I cant find the definition in the theory manual, if the wall shear stress vector is definitely derived in CFX from the dot product of the viscous stress tensor and the unit normal vector as follows:

Tau = T . n

T = viscous stress tensor
n = surface normal vector
Tau = wall shear stress vector

As far as my axes are concerned, I generated my model using the global Cartesian axes. I have attached a picture to illustrate (the red vectors indicate the direction of the surface normal vector).

Thanks again for your input, its much appreciated.
Dave
It is interesting how some codes use :

Code:
`Tau_w = T.n`
where 'T' is the deviatoric or viscous stress tensor (for an incompressible, Newtonian fluid):

Code:
`T = 2 * mu * e_ij`
with 'e_ij' the strain rate tensor

Code:
`e_ij = 1/2 * (partial(u_i,x_j) + partial(u_j,x_i))`
Meanwhile, other codes use:

Code:
`Tau_w = T.n - ((T.n).n)n`
In most cases, the normal component of 'T.n' is small, so it doesn't matter. However, in highly curved geometries, the difference can be somewhat large (up to a 5-10% difference).

See this discussion for more details:
http://www.cfd-online.com/Forums/ope...bend-pipe.html

Unfortunately, in code documentation, wall shear stress is usually just defined as
Code:
`Tau = mu * du / dy`
Which is of course correct, but lacks a description of how this is actually implemented in the code.

-Nuc

Edit:

For clarity:
Code:
```t = T.n  % This is the traction vector
(T.n).n  % This is the magnitude (scalar) of the normal component of the traction vector, which is usually close to zero

((T.n).n)n % This a vector pointing in the direction of the normal vector 'n' with magnitude (T.n).n```
Therefore,
Code:
`Tau_wall = T.n - ((T.n).n)n`
Is the traction vector without it's normal component; this is also called the surface traction vector, and is a good estimate of the wall shear stress 'mu * du/dy'.

T.n is also a fairly good estimate of wall shear stress in most cases.

**************
I'm looking for more sources on this topic, so please let me know if you have any! I'll add them here as I collect them:

Meister et al. "Modelling and simulation of bottle rinsing". Pg. 1417, eqn. 11. International Journal of Food Science and Technology. 2012.
<- This article uses the definition 'T.n', which is the same one used by the OpenFOAM utility 'wallShearStress'! This is in reality wall traction without the normal contribution of pressure.

Matyka et al. "Wall orientation and shear stress in the lattice Boltzmann model", Appendix A, eqn. A.6. Computers and Fluids. 2013.

Stahl et al. "Measurements of wall shear stress with the lattice Boltzmann method and staircase approximation of boundaries", Appendix A, eqn. 36. Computers and Fluids. 2010.
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