HW 8

Problem 1

Derive the expression for the maximum mixture fraction variance in terms of the mean mixture fraction.

Problem 2

Part a

Write a code to generate an equilibrium table for some variable $\eta$, such as temperature, in terms of mixture fraction. This is a one-dimensional table. Pick some number of mixture fractions and then populate an array $\eta_i$ at $Z_i$ for some index $i$.

Part b

Feed these into interp1D to generate a function $\eta(Z)$ that can be evaluated for any $Z$.

Problem 3

Part a

Write a function to evaluate the $\beta$ PDF, $P_\beta(Z;\widetilde{Z},\widetilde{Z^{“2}})$. That is, the function takes arguments Z, Zm, Zv (Zm for mean, and Zv for variance) and returns $P_\beta$.

Part b

Write a function to evaluate

$$\tilde\eta = \int_Z\eta(Z)P_\beta(Z;\widetilde{Z},\widetilde{Z^{“2}})dZ.$$

Use the functions developed above for $\eta(Z)$ an $P_\beta$. You can do the integration with quad where you pass it your $P_\beta$ function. You might need to use the args parameter of quad to let you pass in Zm and Zv.

Problem 4

Create a two dimensional table for $\widetilde{\eta}(\widetilde{Z},i_{Zv})$, where $f_{Zv}$ is the fraction of $\widetilde{Z^{“2}{max}}$, which depends on $\widetilde{Z}$. That is, $f{Zv}=0$ when $Zv=0$, and $f_{Zv}=1$ when $Zv=\widetilde{Z^{“2}_{max}}$.

You can convert this table to a function (like you did in Problem 2b), by using from scipy.interpolate import RegularGriddInterpolator as RGI. This will let you call $\widetilde{\eta}(\widetilde{Z},i_{Zv})$ as a function for any $\widetilde{Z}$ and $i_{Zv}$, where $i_{Zv}$ is no longer an integer.

Problem 5

Compare your table to the experimental results of Flame D for Favre average temperature profiles.