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Closing on wall functions - part 3: a standard wall function

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Closing on wall functions - part 3: a standard wall function

Posted April 23, 2022 at 05:41 by sbaffini
Updated May 15, 2022 at 03:20 by sbaffini

We make the following assumption for the turbulent viscosity ratio:

\frac{\mu_t}{\mu} = \left\{
            0 & \text{for } y^+ < y_v^+ \\
            \kappa y^+ & \text{for } y^+ \geq y_v^+

where \kappa is the von Karman constant and y_v^+ is, for the moment, an unspecified positive parameter. One can then show that the following results:

{s_T^i}^+ =\frac{{y_{min}^+}^{i+2}}{i+2} +\frac{\left(-1\right)^{i+1}}{\left(\kappa \frac{Pr}{Pr_t}\right)^{i+2}}\left[\ln{\left(\frac{1+\kappa\frac{Pr}{Pr_t}y_{max}^+}{1+\kappa\frac{Pr}{Pr_t}y_v^+}\right)+\sum_{j=1}^{i+1} a_j \left(\frac{Pr}{Pr_t}\right)^j}\right]

{p^i}^+ =\frac{\left(-1\right)^{i+1}}{\kappa^{i+2}}\left[\left(i+2\right)\ln{\left(\frac{1+\kappa y_{max}^+}{1+\kappa y_v^+}\right)+\frac{y_v^+-y_{max}^+}{\left(1+y_v^+\right)\left(1+y_{max}^+\right)}+\sum_{j=1}^{i+1} a_j \left(i+j+2\right)}\right]

where y_{min}^+=MIN\left(y_v^+,y^+\right), y_{max}^+=MAX\left(y_v^+,y^+\right), and:

a_j = \frac{\left(-\kappa \right)^j}{j}\left({y_{max}^+}^j-{y_v^+}^j\right)

The general problem with the standard formulation is that, while it is reasonable to pick up an y_v^+ value for the velocity case, even a second one for the TKE production (rigorous doesn't mean stupid, so if different y_v^+ values work better for {s_U^i}^+ and {p^i}^+, why not?), it is not reasonable to manually pick a value in {s_T^i}^+ for each value of the ratio Pr/Pr_t.

As it turns out, however, the above formulation requires modifying y_v^+ only for Pr>Pr_t, but just works in all the other cases. This statement just accounts of the fact that for Pr<Pr_t the turbulent viscosity ratio becomes less and less important, while it becomes more and more important in every detail when multiplied by the ratio Pr/Pr_t>1.

The Pr<Pr_t case is correctly accounted by the present formulation because it has not neglected the 1 at the denominator of the integrand function in {s_T^i}^+ and because the solution is not arbitrarily expressed in terms of the velocity one with all the logarithmic constants lumped in a single one (typically E). The fact that the formulation still doesn't work for Pr>Pr_t is just a statement of the fact that, in this case, the turbulent viscosity ratio becomes important for the temperature distribution before it does for the velocity one, so y_v^+ in {s_T^i}^+ must be reduced. Let's call this new value y_T^+ (yet, this is only needed for Pr>Pr_t).

As the Jayatilleke P term exactly embeds this exact same concept, it is easy to show that it can be used by computing y_T^+ from the following non linear, implicit equation (a similar one must be solved also for the classical standard wall function):

y_T^+ = \frac{y_v^+ + P + \frac{1}{\kappa}\log{\left(\frac{1+\kappa y_T^+}{1+\kappa y_v^+}\right)}}{\frac{Pr}{Pr_t}}

where y_v^+ is the original value used in the velocity profile and P is the mentioned Jayatilleke term (but any similar correlation can substitute it in the equation above, say the Spalding one). I have found that, initializing y_T^+ as y_v^+/\left(Pr/Pr_t\right)^{1/4}, 3 iterations of the Halley's method are sufficient to compute y_T^+ close to machine precision for any practical value of Pr/Pr_t.
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