Viscous Flow Design Project — Viscous Flow Code Optimization  
Consider the same fully developed turbulent flow in a channel that was solved in Assignment #8:
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Now, improve the Prandtl algebraic code that you wrote in Assignment #8 and submit a new final code that makes the following ratio as high as possible: $$ \Phi=\frac{1}{N \times n} $$ In the latter, $N$ is the number of nodes and $n$ is the number of iterations needed to obtain convergence.
To maximize $\Phi$, make the following changes:
(a)  Take advantage of the symmetry of the problem: reduce the domain from $0\le y \le 2 H$ to $0 \le y \le H$ by imposing a symmetrical boundary condition at $y=H$.
(b)  Implement variable grid spacing and cluster more nodes near the wall than near the axis of symmetry.
(c)  Adjust the relaxation factor alpha for optimal convergence rates
(d)  Use higher-order polynomials to extrapolate the properties at the cells interfaces (optional: extra bonus given).
You need to edit the functions init_y_u() and find_coefficients_and_rhs() only:
/* initialize y[], u[], k[], omega[] on all nodes given
   N -> number of nodes
   H -> height of half channel in m
void init_y_u(long N, double H, double dy1, double *y, double *u){
  long j;


/* finds the discretization coefficients a,b,c in kg/m3s 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
   dPdx -> pressure gradient in Pa/m
   u[] -> velocity on all nodes in m/s */
void find_coefficients_and_rhs(long N, double *y, double rho, double mu, double dPdx, double *u, double *a, double *b, double *c, double *rhs){

Everything else should be left untouched and will work fine (assuming that node 1 is a wall boundary and node N is a symmetry boundary).
1.  The number $n$ must correspond to the number of iterations needed for all properties to reach a relative error of $10^{-7}$. I.e., in the code, relerrmax must be set to 1E-7.
2.  The parameter $\Phi$ must be obtained for a “grid converged” solution. That is, you must demonstrate that should $N$ be divided by 2, the bulk velocity $u_{\rm b}$ does not vary by more than 1%.
Due on Monday December 10th at 9:00. Do all parts.
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