Homework 7

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 $\varphi =\tilde{\varphi }+{\varphi }^{\prime \prime }$, for some variable $\varphi$, where $\widetilde{{\varphi }^{\prime \prime }}=0$. The Favre average is defined as $\tilde{\varphi }=\overline{\rho \varphi }/\overline{\rho }$, or $\overline{\rho }\tilde{\varphi }=\overline{\rho \varphi }$. (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