Calculation of Wall Shear Stress
I am trying to understand how wall shear stress components are calculated in CFX. I have carried out a steady-state analysis of incompressible laminar fluid flow over a flat plate and validated the solution against the Blasius solution. As far as I am aware (I could not find a definition in the theory manual) the components and magnitude of the wall shear stress vector are defined as follows:
Tx = [Nx*Mu*(2*Du/Dx)]+[Ny*Mu*(Du/Dy+Dv/Dx)]+[Nz*Mu*(Du/Dz+Dw/Dx)]
Ty = [Nx*Mu*(Dv/Dx+Du/Dy)]+[Ny*Mu*(2*Dv/Dy)]+[Nz*Mu*(Dv/Dz+Dw/Dy)]
Tz = [Nx*Mu*(Dw/Dx+Du/Dz)]+[Ny*Mu*(Dw/Dy+Dv/Dz)]+[Nz*Mu*(2*Dw/Dz)]
T = sqrt(Tx^2+Ty^2+Tz^2)
In CFX-post I have exported the following variables for a sample node located roughly halfway along the length of the plate:
Dynamic Viscosity = 8.90E-04
Normal X = 0.00E+00
Normal Y = -1.00E+00
Normal Z = 0.00E+00
Velocity u.Gradient X = -5.27E-02
Velocity u.Gradient Y = 4.13E+02
Velocity u.Gradient Z = 0.00E+00
Velocity v.Gradient X = -9.79E-06
Velocity v.Gradient Y = 5.93E-02
Velocity v.Gradient Z = 0.00E+00
Velocity w.Gradient X = 0.00E+00
Velocity w.Gradient Y = 0.00E+00
Velocity w.Gradient Z = 0.00E+00
Using these variables and the equations above I calculate the following values for the components and magnitude of the wall shear stress vector:
Tx = -3.68E-01
Ty = -1.06E-04
Tz = 0.00E+00
T = 3.67E-01
Unfortunately, when I compare the calculated values to those exported from CFX I notice some differences:
Tx = 3.67E-01
Ty = -1.21E-19
Tz = -4.84E-25
T = 3.67E-01
Although the magnitude comes out the same I am clearly getting something wrong. If anyone has a better idea of how the wall shear stress components are calculated I would really appreciate some advice.
-4.84E-25 = -1.21E-19 = zero with a bit of numerical noise. So your calculation of Ty and Tz are correct within numerical accuracy. You have a sign error in Tx, but the magnitude is correct.
Thanks a lot for replying. Your explanation makes sense for Tz = 0 but from my hand calculations should Ty not be slightly greater than zero?
The sign error for Tx occurs because, according to the results exported from CFX, the y-component of the normal vector Ny = -1. As the plate is flat I expected this component to be positive and if it was I guess the calculations would match up. Any ideas why it comes out as negative?
Also, do you think I have the overall methodology correct?
I do not have time to check your methodology. For you sign error check the definition of axes. Are you sure you have the directions correct?
Don't forget that you are returning the value of a volume centred slightly off the wall, and that it is a representative average of the conditions of that volume. So you will always get small errors associated with that.
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.
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