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A generalized thermal/dynamic wall function: Part 3

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A generalized thermal/dynamic wall function: Part 3

Posted October 17, 2016 at 11:25 by sbaffini
Updated November 18, 2018 at 05:57 by sbaffini

In this post i summarize the initial problem and the procedure to determine the wall function value (i.e., the solution) for given y^+,F_T^+,Pr and Pr_t.

We looked for a solution T^+\left(y^+,F_T^+,Pr,Pr_t\right) to the problem:

\frac{dT^+}{dy^+}=\frac{Pr\left(1+F_T^+y^+\right)}{\left[1+\left(\frac{Pr}{Pr_t}\right)\left(\frac{\mu_t}{\mu}\right)\right]}

with:

\frac{\mu_t}{\mu}=\frac{\left(ky^+\right)^3}{\left(ky^+\right)^2+\left(ka_0\right)^3-\left(ka_0\right)^2}

for given a_0. This represents, in nondimensional form, a certain set of viscous boundary conditions for velocity/temperature. In practice, T^+ is a wall-function. The solution is given by:

T^+=Pr \left[f^+\left(1 +F_T^+y^+\right)-g^+\right]

Where, f^+ is given by:

f^+=\int_0^{y+}{\frac{1}{\left[1+\left(\frac{Pr}{Pr_t}\right)\left(\frac{\mu_t}{\mu}\right)\right]}dz^+}=\frac{Pr_t}{kPr}\ln\left(\frac{y^++a}{a}\right)+

\frac{\alpha}{a+4\alpha}\left\{\left(a-4\alpha\right)\ln\left[\frac{a\left[\left(y^+-\alpha\right)^2+\beta^2\right]}{2\alpha\left(y^++a\right)^2}\right]+\gamma\left[\arctan\left(\frac{y^+-\alpha}{\beta}\right)+\arctan\left(\frac{\alpha}{\beta}\right)\right]\right\}

\gamma = \frac{2\alpha\left(5a-4\alpha\right)}{\beta}

\beta^2=2a\alpha-\alpha^2

2\alpha=a-\frac{Pr_t}{kPr}

a=\frac{\left(1+\eta+\eta^2\right)a_0}{\theta \eta}

\eta = \sqrt[3]{1+\theta\left(\psi+\sqrt{\psi^2+\phi}\right)}

\psi=\frac{\theta \phi}{2}

\theta = 3 k a_0 \frac{Pr}{Pr_t}

\phi = 3\left(k a_0-1\right)

and g^+ is given by a (typically) straightforward integration:

g^+=\int_0^{y+}{f^+\left(F_T^++z^+\frac{dF_T^+}{dz^+}\right)dz^+}

In practice, given f^+ above, the integral in g^+ is easily computable for F_T^+ in the form of a polynomial of arbitrary degree. For the sake of conciseness we only considered F_T^+ as constant, the resulting integral being F_T^+ times the following function:

https://www.wolframalpha.com/input/?...Batan(b%2Fc)))

The whole procedure is implemented in the attached MATLAB/Octave script, where the analytical solution above (ta) is compared with the numerical one (tn), obtained by numerically integrating dT^+/dy^+.

The usage should be straightforward:

1) Pick up values for k,Pr,Pr_t and F_T^+ (lines 8-11).

2) Choose plotting options (lines 5-6)

3) Choose how many points you want to use to integrate numerically dT^+/dy^+ (line 7).

4) Run the script.

Note that the constant a_0=11.489 has been calibrated using as reference the mixing length model with a constant A^+=19. The relative turbulent mixing length viscosity has been left commented in the script (line 60), so that you can use it to calibrate the model for different values of the constant (line 13). Note that the calibration is always done once using Pr=Pr_t=1 and F_T^+=0. After that, any value of those parameters will be taken into account automatically by the formula.

In general, the value a_0 is such that \mu_t/\mu=1 at y^+=a_0. Hence, it could be also used as a mean to introduce roughness effects in the formulation.

In conclusion, it is worth mentioning what are the limitations of the present wall function with respect to the full solution of the turbulent equations with, say, the SA model:

1) The equations are not in their full form, but in the incompressible TBLE form, with constant RHS.

2) The underlying turbulent viscosity model is not formally sensitized to the pressure gradient.

However, note that the same limitations also apply to, say:

K. Suga et al. / Int. J. Heat and Fluid Flow 27 (2006) 852866

which provides a relation similar to the present one but with a higher approximation on the turbulent viscosity.

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