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Old   March 2, 2014, 19:32
Default Dimensionless wall distance
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Hi there!

I would like to ask about the dimensionless wall distance. In many papers are three-dimensional domains described using x^+,y^+,z^+. I am not sure how to compute it for other axis (x, z).

Wall y^+ can be computed as
y^+ \equiv \frac{u_* \, y}{\nu}.
Where u_* is the friction velocity at the nearest wall, y is the distance to the nearest wall and \nu is the local kinematic viscosity of the fluid.
** u_* \equiv \sqrt{\frac{\tau_w}{\rho}}
Where \tau_w is the wall shear stress and \rho is the fluid density at the wall.
*** \tau_w = \mu \left(\frac{\partial u}{\partial y} \right)_{y=0}
Where \mu is the dynamic viscosity, u is the flow velocity parallel to the wall and y is the distance to the wall.

Is it computed for let's say z^+ as follows?
z^+ \equiv \frac{u_* \, z}{\nu}
u_* \equiv \sqrt{\frac{\tau_w}{\rho}}
\tau_w = \mu \left(\frac{\partial u}{\partial y} \right)_{y=0}
are without change?

I am not sure about the wall shear stress component (gradient of u)
Thank you for any hints or recommendations.
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Old   March 3, 2014, 03:26
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As far as I know the dimensionless wall distance is measured orthogonal to the wall which is usually called yplus and sometimes zplus. Never heard of xplus but it might just be dependent on how you set you coordinate system.
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Old   March 3, 2014, 07:41
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Thank you. I thought as you do, but you can find x^+, z^+ widely used in domain descriptions. For example as shown here: in LES of three-dimensional wing ( It is possible to find it in other papers too.

I am not sure if x^+ and z^+ are "normalized" with respect to the wall y^+. I could not find any source with explanation. Probably, there is a trivial answer (that's why..).

Any idea guys? Thank you.
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Old   March 13, 2014, 04:33
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Hi Messik,

I guess there are two issues in your post requiring consideration:
1. The "u" should actually be interpreted as the velocity parallel to the wall, not the x-velocity component.
2. The "y" in wall shear stress needs be interpreted as the normal-to the wall coordinate rather than the y-coordinate.
Then the x^+ and the z^+ follow naturally by merely changing components.

I hope this helps,
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Old   March 15, 2014, 06:55
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Hi messik,

In 3D flows u may have the other commonents of shear stress, which IMHO makes sense to have the dimless in all three components. Although the boudary layer develops mainly normal to the wall, the resolution of the mesh in all the other two components is also important for the accuracy of your results. If you consider that u are resolving the laminar sub-layer (y+=1) it may be also important that your mesh in the other components fits within (x+/z~30). I think that u use the half of the mesh size and the velocity gradients in the right directions u can find those dimless.

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