gxay

Structure of the repository

The present repository contains our results for the conjugate cases. The folder g1a1 present here is a symbolic link to the folder g1a1 found in the master tree.

The folders are named gXaY. $X$ is $G$, the fluid-to-solid thermal diffusivity ratio. $Y$ is $\frac{1}{G_2}$, the fluid-to-solid thermal conductivity ratio.

The folder g1a1 contains:

• The source code used to run the simulation (Makefile, .f90 and .prm files)
• A raw folder with raw statistics (1D text files). Our code outputs statistics averaged in the streamwise direction in a binary format. The statistics in the raw folder are averaged spanwise and preprocessed. As a result, the file vphim1d.dat contains $\overline{v \phi} - \overline{v} \overline{\phi}$.
• A scilab script (.sce) that reads the raw statistics and output quantities in wall-units.
• A csv folder with statistics in wall-units.
• A xls file with statistics in wall-units.

The other folders contain:

• A raw folder with raw statistics (1D text files). Our code outputs statistics averaged in the streamwise direction in a binary format. The statistics in the raw folder are averaged spanwise and preprocessed. As a result, the file vphim1d.dat contains $\overline{v \phi} - \overline{v} \overline{\phi}$.
• A csv folder with statistics in wall-units.
• A xls file with statistics in wall-units.

The simulations were performed with the source code present in the folder g1a1, except for the file user_module.f90 that is case dependent and can be found in each corresponding directory. Please note that the scilab scripts are not included as they are almost identical to the one in folder g1a1.

Configuration of the turbulent channel flow, dynamic part.

Here, $[x,y,z]$ and $[1,2,3]$ will be used for the streamwise, wall-normal and spanwise directions, respectively.

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The fluid domain is a parallelepiped: $[0,0,0] \leq [x,y,z] \leq [25.6, 2, 8.52]$. The mesh is streched in the wall-normal direction (istret = 2 and beta = 0.225). Periodic boundary conditions are used in the directions $x$ and $z$. At $y=0$ and $y=2$, the velocity is null and the pressure satisfies an homogeneous Neumann boundary condition. The number of nodes in the $[x,y,z]$ directions is $[256, 193, 256]$.

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The solid domains are parallelepipeds located on top and on bottom of the fluid domain : $[0,2,0] \leq [x,y,z] \leq [25.6, 3, 8.52]$ and $[0,-1,0] \leq [x,y,z] \leq [25.6, 0, 8.52]$. In the streamwise and spanwise directions, the grid in the solid domain is identical to the fluid one. In the wall-normal direction, a Chebyshev grid with $129$ interior nodes is used.

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The momentum equation solved reads:

$$\partial_t u_i = - \frac{\partial_j \left( u_i u_j \right) + u_j \partial_j u_i}{2} - \partial_i p + \nu \partial_{jj} u_i + f_i$$

The kinematic viscosity $\nu$ is the inverse of the bulk Reynolds number, which is equal to $2280$ here. The source term is present only in the streamwise direction. Its amplitude is exactly $0.0042661405$, which leads to a unit bulk velocity.

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The time step is $0.002$. After the flow reached a statistically steady state, statistics were gathered for $1,500,000$ time steps.

Configuration of the turbulent channel flow, thermal part.

The scalar conservation equation in the fluid domain reads:

$$\partial_t \phi = - \partial_j \left( \phi u_j \right) + \frac{\nu}{Pr} \partial_{jj} \phi + \frac{\nu}{Pr} u_x$$

The value of the Prandtl number is $0.71$. At the fluid-solid interfaces, the scalar satisfies $\phi = \phi_s$ and $\partial_y \phi = G_2 \partial_y \phi_s$. $G_2$ is the ratio of solid-to-fluid thermal conductivities and $\phi_s$ is the scalar in the solid domain. There, the scalar conservation equation reads:

$$\partial_t \phi_s = \frac{\nu}{G Pr} \partial_{jj} \phi_s$$

Where $G$ is the ratio of fluid-to-solid thermal diffusivity. At $y=-1$, $\partial_y \phi_s = G_2$. At $y=3$, $\partial_y \phi_s = -G_2$.

Wall-units

Statistics in the xls files and in the csv folders are in wall-units. The conversion from computational units to wall-units is performed in the scilab script (.sce). This conversion is briefly described here. For further details, please consult a good book on Turbulence and (Computational) Fluid Mechanics. For instance Turbulent flows by S. B. Pope, The theory of homogeneous turbulence by G. K. Batchelor or A first course in turbulence by H. Tennekes and J. L. Lumley.

At the wall $y=0$, the friction velocity $u_\tau$ verifies:

$$u_\tau = \sqrt{ \nu \partial_y \overline{U_x} \left( y=0 \right) }$$

And the friction temperature $T_\tau$ verifies:

$$T_\tau = \frac{\overline{q_w}}{\rho \; C_p \; u_\tau} = \nu \frac{\partial_y \overline{\phi} \left( y=0 \right)}{Pr \; u_\tau}$$

The velocity is converted to wall-units when divided by $u_\tau$. The temperature is converted to wall-units when divided by $T_\tau$. Distances are converted to wall-units when multiplied by $\frac{u_\tau}{\nu}$. Application of dimensional analysis should easily allow one to convert time or pressure to wall-units.

For the budgets of the Reynolds stresses, please see equation (1) in Mansour, Kim and Moin

For the budgets of the turbulent heat fluxes, please see equation (12) in Kozuka, Seki and Kawamura

For the budget of the temperature variance, some look at the budget of $\overline{{\phi'}^2}$ and some look at the budget of $\frac{\overline{{\phi'}^2}}{2}$, by analogy with $k$, the turbulent kinetic energy, which also contains a factor 2. Below is the budget equation of the latter:

$$\partial_t \frac{\overline{{\phi'}^2}}{2} + \partial_k \left( \overline{u_k} \frac{\overline{{\phi'}^2}}{2} \right) = - \overline{u'_k \phi'} \partial_k \overline{\phi} -\partial_k \left( \overline{u'_k \frac{{\phi'}^2}{2}}\right) + \frac{1}{Pr} \partial_{kk} \left( \frac{\overline{{\phi'}^2}}{2} \right) - \frac{1}{Pr} \overline{\partial_k \phi' \partial_k \phi'}$$

In the solid domain, the budget equation is similar but much simpler. To obtain it, set the velovity to null and replace $Pr$ by $G \; Pr$.