Solving 1D Unsteady Homentropic Flow using Method of Characteristics

Question

I am looking to solve the 1D Unsteady Homentropic Flow equations using the Method of Characteristics (MOC). I am using Zucrow and Hoffman's Gas Dynamics Volume 1 as a reference and I am currently stuck.

In chapter 13, the authors begin describing the use of MOC to solve the 1D unsteady flow equations. I understand their methodology till I get to the step that states to substitute equation (13.16) back to equation (13.15).

For this example: $$ u = u(x,t) $$

and $$ p = p(x,t) $$

For homentropic flow, $$ \rho = \rho(p) $$

Equation (13.15) is the following:

$$ (\rho u \sigma_1 + \rho a^2 \sigma_2) ( u_x + \frac{\rho \sigma_1}{\rho u \sigma_1 + \rho a^2 \sigma_2}u_t) + (\sigma_1 + u\sigma_2 )(p_x + \frac{\sigma_2}{\sigma_1 + u\sigma_2} p_t) = 0 $$

Equation (13.16) is the following: $$ \frac{du}{dx} = u_x + \lambda u_t $$

$$ \frac{dp}{dx} = p_x + \lambda p_t $$

The resulting equation (13.18) when you substitute (13.16) into (13.15) is:

$$ (\rho u \sigma_1 + \rho a^2 \sigma_2) du + (\sigma_1 + u\sigma_2 ) dp = 0 $$

I understand that $$ \lambda = \frac{\sigma_1}{u\sigma_1 + a^2\sigma_2} = \frac{\sigma_2}{\sigma_1+u\sigma_2} $$

However, I have not been able to get (13.18) from back-substituting (13.16) into (13.15).

Does anyone have any suggestions for me to consider, look into, etc.?

Thank you!

Answer 1

In (13.15) use the first equation for $\lambda$ in the first term, and the second equation in the second term. That gives $$(\rho u \sigma_1 + \rho a^2 \sigma_2) \frac{du}{dx} + (\sigma_1 + u\sigma_2 )\frac{dp}{dx} = 0.$$

Then use the fact that $$\frac{du}{dx} \Bigm / \frac{dp}{dx} = \frac{du}{dp}.$$