Introduction to CFD: HW #$1$

$\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{t}}+\frac{\text{d}\text{e}\text{l}}{del{x}_j}(u_j)=0$

$\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{t}}(u_j)+\frac{\text{d}\text{e}\text{l}}{del{x}_i}(u_i{u}_j)=-{}_{ij}\frac{\text{d}\text{e}\text{l}\text{p}}{del{x}_i}+f_j+\frac{1}{R{e}_{}^{\text{'}}}\frac{\text{d}\text{e}\text{l}}{del{x}_i}[{}_{ij}\frac{del{u}_k}{del{x}_k}+(\frac{del{u}_i}{del{x}_j}+\frac{del{u}_j}{del{x}_i})]$

$\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{t}}(e_t)+\frac{\text{d}\text{e}\text{l}}{del{x}_i}(e_t{u}_i)=-{}_{ij}\frac{\text{d}\text{e}\text{l}(u_{j}p)}{del{x}_i}+u_j{f}_j+\frac{1}{R{e}_{}^{\text{'}}}\frac{\text{d}\text{e}\text{l}}{del{x}_i}[{}_{ij}{u}_j\frac{del{u}_k}{del{x}_k}+u_j(\frac{del{u}_i}{del{x}_j}+\frac{del{u}_j}{del{x}_i})]+\frac{1}{(-1)R{e}_{}^{\text{'}}}\frac{\text{d}\text{e}\text{l}}{del{x}_i}[\frac{}{Pr}\frac{\text{d}\text{e}\text{l}\text{T}}{del{x}_i}]$

Expand the equations using the Einstein summation convention.

Do the expansion for just $1$ of the momentum equations in addition to the mass and energy conservation equations.

For one component of the momentum equation, use the breakdown of velocity into the mean and fluctuating components and carry out Reynolds averaging. Assume density is not fluctuating, i$.$e$.,{}^{\text{'}}=0$