$(40$pt$)$

$($a$)(10$pt$)$ Describe the convection$-$diffusion form of the conservation law for any conserved quantity, say u$.$

Define the Peclet number and its physical relevance in a flow.

$($b$)(10$pt$)$ Consider the following $1$D Burger's equation in a linear form:

$($delu$)/($delt$)+$a$($delu$)/($delx$)=\backslash$alpha $($del$^(2)$u$)/($delx$^(2))$

Discretize this equation using appropriate finite difference schemes for the individual terms. Use your

numerical schemes to derive a Crack$-$Nicholson scheme for the same equation.

$($c$)(10$pt$)$ What are the consistency, stability, and convergence of a numerical scheme? How can we predict

the convergence of a numerical solution without performing simulations?

$($d$)(10$pt$)$ Obtain Reynolds Averaged Navier$-$Stokes equations by applying the Reynolds decomposition to

the momentum equation $($of the Navier$-$Stokes equations$).$ How will you use the Reynolds stress tensor to

compute the turbulent kinetic energy?