### Problem 1

Turns Chapter 9 review question 1

### Problem 2

Turns 9.10

### Problem 3.

In class (and in Turns) we discussed Reynolds averaging of the Navier-Stokes equations. This works great for cold flows with constant density. For combustion however, density varies a lot, and it is common to do a so-called Favre averaging (which is a mass-weighted average).

Consider a term like $\rho uv$.

#### Part a

Apply a Reynolds decomposition to each variable and find the Reynolds average of the result. How many additive terms are there?

#### Part b

Apply a Favre decomposition to each variable and find the Favre average of the result. The Favre decomposition is given by $\phi=\tilde{\phi}+\phi^{\prime\prime}$, for some variable $\phi$, where $\widetilde{\phi^{\prime\prime}}=0$. The Favre average is defined as $\tilde{\phi} = \overline{\rho\phi}/\bar{\rho}$, or $\bar{\rho}\tilde{\phi}=\bar{\rho\phi}$. (Overbars denote Reynolds averages). So, to apply to $\rho uv$, first, draw an overbar on $\rho uv$, then apply the definition of the Favre average, then do a Favre decomposition, and simplify. How many additive terms are there now?

### Problem 4

Turns 12.2

### Problem 5

Turns 12.7