# CFD: Can one calculate the dimensionless wall distance $y^+$ a priori?

perhaps the most idiotic question I ever asked.

people throw around the parameter $y^+$, as if it is the be all and end all of CFD (Computational Fluid Dynamics) parameters and use it as if everyone and their dog know how to calculate it. (Specially in CFD literature)

Now the question,

$u_\tau = (\tau/\rho)^{1/2}$

where $u_\tau$ = frictional velocity and $\tau$ is wall stress.

also $\tau = \mu*\left(\dfrac{du}{dy}\right)$

$\mu$ = co-efficient of viscosity

$\dfrac{du}{dy}$ = velocity gradient normal to wall (this formula holds near wall)

and

$y^+ = u_\tau \dfrac{y}{\nu}$

where

$\nu = \mu/\rho$

I am totally down with all this (please correct any of the equations if I am wrong).

is there a way to calculate values of $y^+$ before running the simulation? I mean can we get good guess for $u_\tau$? If not what is the point of all this?

OK, maybe we can't reliably, it has to be done after/while simulation runs, again after how many time-steps, should I consider that I am getting overall correct values of $y^+$ (both (min,max) and avg)

openFoam provides such function called yPlusRAS / yPlusLES.

Now if we need to calculate all the values to get $y^+$, what is the point of log law? (What are we modelling exactly? because we calculated $u$ and $du/dy$ anyway)

$$u^+ = (1/\kappa) \ln(y^+) + C^+$$

• Maybe, if you have a desired value of yplus, this can help to estimate wall distance (cfd-online.com/Tools/yplus.php). I'll give a longer answer below, the thing is that cfd solver mostly work with ystar calculated using u* as more convenient to avoid singularity u u_tau in some regions. Dec 27, 2013 at 9:30
• @JohntraVolta it would give wall distance for RAS? what about LES and what if I have rotatory machinery in a close container, freeStream is 0 but I have max velocity at the tips of 22 m/s. Yes,please give longer answer :) Dec 27, 2013 at 9:49
• Scroll down to the bottom of that page, and you will find theoretical explanation how did they estimate wall distance. Change step five of that procedure, and instead of finding wall distance knowing y+, try to find y+ knowing the wall distance. That will give you an estimate of y+ for a given mesh. Dec 27, 2013 at 9:56
• @JohntraVolta please write answer including Cf and most preferable for different Re (perhaps can say something about LES), I will accept it.Thank you. Dec 27, 2013 at 10:16

The White-Christoph formula can be used to estimate the skin-friction coefficient as:

$$C_f\approx\frac{0.455}{\ln^2(0.06Re_x)}$$

From this correlation, one can relate $C_f$ to $y^+/y_1$. Assuming that $x=1$, the following estimate is given for the initial spacing $y_1$ off the wall as:

$$y_1=\frac{y^+}{Re}\left(\frac{0.2275}{\ln^2(0.06Re_x)}\right)^{-\frac{1}{2}}$$

I have studied and validated this on a flat plat for Reynolds numbers up to $10^9$, for a nominal submarine hull at full scale Reynolds numbers, and also for a ducted propulsor using both $q-\omega$ and Spalart-Allmaras turbulence models. In the case of the propulsor it's important to use the tip velocities when computing the $y^+$ estimate.

Ref: W.H. Brewer, On simulating tip-leakage vortex flow to study the nature of cavitation inception, 2002, pp. 40-43, pp. 70-73.

• This was a long time ago, but I really appreciate your answer. The comments made me realize how I totally missed the modelling aspect. Jan 18, 2017 at 18:34