$(20$ pts$)$ Problem $5$

A converging$-$diverging nozzle is used in aerospace applications to accelerate a gas to generate

thrust. A sketch of the nozzle is given in Fig. $1.$ Here, $A^{*}$ is the area of the smallest section called

the throat. Depending on the local area $A,$ the Mach number $M,$ the non$-$dimensional measure of

how fast the flow is with respect to the speed of sound in the fluid at that location, is found by the

following relation

$\frac{A}{A^{*}}=\frac{1}{M}[\frac{2}{+1}(1+\frac{-1}{2}{M}^2){]}^{\frac{+1}{2(-1)}}$

Here, $$ is the specific heat ratio of the gas. Given an area ratio $\frac{A}{A^{*}}$ and a specific heat ratio $,$

there are two solutions for $M$; one is the supersonic solution $M>1,$ the other is the subsonic

solution

$($i$)(5$ pts$)$ Derive a fixed$-$point iteration method that finds the subsonic solution.

$($ii$)(5$ pts$)$ Derive a fixed$-$point iteration method that finds the supersonic solution.

$($iii$)(10$ pts$)$ Write a Matlab function that finds the Mach number $M$ given the values for the

area ratio $\frac{A}{A^{*}},$ specific heat ratio $,$ and the choice of the flow solution. Use the following

convention in your script:

Define the variable Arat as the area ratio $\frac{A}{A^{*}}.$

Define the variable gamma as the specific heat ration $.$

Define the variable M$0$ as the initial guess for the solution. In your code, return the

supersonic solution if MO is greater than unity and return the subsonic solution if M$0$ is

smaller than unity.

The function declaration must be function $M=$ CDNozzle$($Arat$,$ gamma, M$0).$ Do not

forget to name your script CDNozzle.m

For the fixed$-$point iteration calculations, use the following default values: imax$=100$ and

es $=0\mathrm{.}1.$

Figure $1$: Sketch of a converging$-$diverging $($de Laval$)$ nozzle.

Note: After deriving the methods for Parts $($i$)$ and $($ii$),$ you may need to test them to determine

what solution they find, but there is no need to prove their convergence. Assume that the user

passes the correct numbers to the function CDNozzle. Your function will be tested by using the

following commands

$M=$CDNozzle$(2,1\mathrm{.}4,0\mathrm{.}5)$

$M=$CDNozzle$(2,1\mathrm{.}4,3)$

Include all derivations and results in your report.

Grading rubric

$5$ pts for the correct derivation in Part $($i$)$

$5$ pts for the correct derivation in Part $($ii$)$

$10$ pts for the correct code and results in Part $($iii$)$