neuma

Structure of the repository

The present repository 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 present README.md

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 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 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 $y=0$, $\partial_y \phi = 1$. At $y=2$, $\partial_y \phi = -1$.

Wall-units

Statistics in the xls file and in the csv folder 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:

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