Viscous Flow Assignment 8 — Turbulence Modeling  
Question #1
Starting from the Reynolds-averaged constant-density momentum equation $$ \rho \left( \frac{\partial \bar{v}_i}{\partial t} +\sum_j \frac{\partial \bar{v}_i\bar{v}_j}{\partial x_j} \right) = -\frac{\partial \bar{P}}{\partial x_i} +\sum_j \frac{\partial \overline{\tau_{ji}}}{\partial x_j} -\sum_j \rho\frac{\partial \overline{v_i^\prime v_j^\prime}}{\partial x_j} $$ Derive the modeled form of the momentum equation (through the Boussinesq approximation): $$ \rho \left( \frac{\partial \bar{u}}{\partial t} +\sum_j \frac{\partial \bar{v}_i\bar{v}_j}{\partial x_j} \right) = -\frac{\partial \bar{P}}{\partial x_i} +\sum_j \frac{\partial}{\partial x_j} (\mu+\mu_t)\left(\frac{\partial \bar{v}_i}{\partial x_j}+\frac{\partial \bar{v}_j}{\partial x_i} \right) $$ with the turbulence eddy viscosity defined as: $$ \mu_t \equiv \frac{1}{2} \rho V_t L_t $$
Question #2
Starting from the definition of the turbulence eddy viscosity: $$ \mu_t \equiv \frac{1}{2} \rho V_t L_t $$ Explain as well as possible the Prandtl mixing length model for a flow over a flat plate: $$ \mu_t \approx \rho L_m^2 \left| \frac{\partial u}{\partial y}\right| $$ with $$ L_m = 0.435 y \left(1- \exp(-y^+/26) \right) $$ and with $$ y+\equiv \frac{y \sqrt{\rho \tau_w}}{\mu} $$ with $\tau_w$ the shear stress at the wall.
Question #3
Consider the following two-dimensional channel with smooth walls:
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Modify the function “find_coefficients_and_rhs” within the attached C code that solves the Prandtl algebraic turbulence model using a TDMA algorithm:
/* finds the discretization coefficients a,b,c in kg/m3s, mut in kg/ms, and the RHS rhs in Pa/m for all nodes given
   N -> number of nodes
   H -> half height of the channel in m
   rho -> density in kg/m3
   mu -> viscosity in kg/ms
   mut -> eddy viscosity in kg/ms
   dPdx -> pressure gradient in Pa/m
   u[] -> velocity on all nodes in m/s */
void find_coefficients_and_rhs(long N, double H, double rho, double mu, double dPdx, double *u, double *mut, double *a, double *b, double *c, double *rhs){

Then, do the following:
(a)  Compare the bulk velocity obtained numerically to the one obtained through the Jones correlation. If you implemented the algorithm well, both should be within 20% of each other.
(b)  Find $N$ that yields a “grid-converged” solution. To do so, start with $N=100$, and then keep on doubling $N$ until $u_{\rm b}$ obtained numerically does not vary by more than 1% compared to its value on the coarser grid. What is the minimum value of $N$ that achieves this?
(c)  Plot the velocity along $y$ for $N$ found in part (b).
Use $H=0.1$ m, $\rho=1000$ kg/m$^3$, $\mu=10^{-3}$ kg/ms, and $dP/dx=-2$ Pa/m.
The Jones correlation applicable within the range $2000 \le {\rm Re}_{D_{\rm H}} \le 2 \times 10^5 $ corresponds to: $$ \frac{1}{\sqrt{f_{\rm D}}}=2.0 ~{\rm log}_{10} \left[ \frac{2}{3} {\rm Re}_{D_{\rm H}} \sqrt{f_{\rm D}}\right]-0.8 $$ where $H$ is the half height of the channel, $D_{\rm H}$ the hydraulic diameter and $f_{\rm D}$ is the Darcy friction factor: $$ f_{\rm D} = \left.-\frac{d P}{d x} D_{\rm H}\right/ \left( \frac{1}{2} \rho u_{\rm b}^2\right) $$ where $u_{\rm b}$ can be written as a function of the Reynolds number as follows: $$ {\rm Re}_{D_H}=\frac{\rho u_{\rm b} D_H}{\mu} $$
3.  The correct bulk velocity obtained with the Prandtl model lies between 0.25 and 0.28 m/s.
Due on Monday December 10th at 9:00. Do all parts.
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