$4$ Two$-$point velocity correlation

Let's consider an instantaneous velocity field $u(x)=(u_1,u_2,u_3)$ coming from a direct numerical simulation of a three dimensional homogeneous turbulent flow. The computational domain is a cube of physical size $0\mathrm{.}1280\mathrm{.}1280\mathrm{.}128m^3$ discretized on a ${128}^3$ grid with periodic boundary conditions. The kinematic viscosity of the fluid is

$u=2\mathrm{.}13*{10}^{-5}{m}^2*s^{-1}.$ We note $u^{\text{'}}(x)=(u_1^{\text{'}},u_2^{\text{'}},u_3^{\text{'}})$ the fluctuating velocity field. The turbulent kinetic energy defined as $e_t=($:$u_i^{\text{'}}{u}_i^{\text{'}}$:$)$ is equal to $0\mathrm{.}0127m^2*s^{-2}.$

Q$\mathrm{.}1)$ Considering the number of grid points, give an estimate of the maximum Reynolds number that can be reached by such a simulation. We will consider that the grid size in equal to the Kolomogorov length scale, $x=$ and the width of the domain is ten times larger than the largest length scale of the turbulent flow.

We define the three longitudinal correlation functions

where

$Q_{ij}(r{e}_k)=($:$u_i^{\text{'}}(x)u_j^{\text{'}}(x+r{e}_k)$:$).$

The values of those three correlation functions are available here within the file correlation. dat. Download the file to access the data. This text file $($ASCII$)$ contains $4$ columns corresponding the the following variables :

$r,f_{11}(r),f_{22}(r),f_{33}(r)$

Q$\mathrm{.}2)$ Give a physical interpretation of the correlation function, $f_{11}(r).$

Q$\mathrm{.}3)$ Plot $f_{11}(r),f_{22}(r)$ and $f_{33}(r)$ on the same graph for rin$[0.\mathrm{.}0\mathrm{.}063]$

Q$\mathrm{.}4)$ Do you think that the simulated turbulence can be consider as an isotropic turbulence $?$ Do you think that your conclusion holds $($justify your answer$).$

only for the large scales

only for the small scales

for all scales of the turbulent flow Argue your choice.

For the following, we defined, for rin$[0.\mathrm{.}0\mathrm{.}063],$ the longitudinale correlation functions $f(r)=f_{11}(r).$

Q$\mathrm{.}5)$ Compute the value of

the longitudinal integral scale, $L_f$

the longitudinal Taylor microscale, ${}_f$

How the integral scales compare with the size of the computational domain?

Q$\mathrm{.}6)$ Plot, on the same graph $f(r)$ and the osculating parabola of $f(r)$ at $r=0.$ Indicate $L_f$ and ${}_f$ on this graph.

$3$

Q$\mathrm{.}7)$ Give the value of the Reynolds number based on $u$ and $L$ and compare to estimate you made at Q$\mathrm{.}1.$

Q$\mathrm{.}8)$ Give the value of the dissipation rate of turbulent kinetic energy $$ computed from the r$.$m$.$s velocity, $u$ and the integral scale $L_f.$

Q$\mathrm{.}9)$ Give the value of the dissipation rate of turbulent kinetic energy $$ computed from the r$.$m$.$s velocity, $u$ and the Taylor microscale ${}_f.$

Q$\mathrm{.}10)$ Give an estimate of the Kolmogorov length scale, $.$ Do you think the simulation is able to resolve the Kolmogorov length scale?