Question

For ideal gases undergoing isentropic flows, obtain expressions for \(P / P^{*}, T / T^{*},\) and \(\rho / \rho^{*}\) as functions of \(k\) and \(\mathrm{Ma}\)

Short answer

Answer: The expressions for the ratios of pressure, temperature, and density as functions of specific heat ratio (k) and Mach number (Ma) are: Pressure ratio, \(P/P^{*}\): \(P / P^{*} = \left(\frac{1 + \frac{k - 1}{2} \text{Ma}^2}{1 + \frac{k - 1}{2} \text{Ma}^{*2}}\right)^\frac{k}{k-1}\) Temperature ratio, \(T/T^{*}\): \(T / T^{*} = \frac{1 + \frac{k - 1}{2} \text{Ma}^2}{1 + \frac{k - 1}{2} \text{Ma}^{*2}}\) Density ratio, \(\rho/\rho^{*}\): \(\rho / \rho^{*} = \left(\frac{1 + \frac{k - 1}{2} \text{Ma}^2}{1 + \frac{k - 1}{2} \text{Ma}^{*2}}\right)^\frac{1}{k-1}\)

Step-by-step solution

Recall the isentropic flow equations and ideal gas equation

For isentropic flows, we have the following equations: 1. \(p v^k = \text{constant}\) 2. \(p^{(1 - k)/k} T^{k/(\gamma-1)} = \text{constant}\) where \(p\) is pressure, \(v\) is specific volume (\(1/\rho\), where \(\rho\) is density), and \(T\) is temperature of the ideal gas. Also, we have the ideal gas equation: 3. \(p v = R T\) where \(R\) is the specific gas constant.

Express the isentropic flow equations in terms of Mach number

To find expressions for \(P/P^*\), \(T/T^*\), and \(\rho/\rho^*\), we will express the isentropic flow equations in terms of Mach number (Ma). We know that: Mach number, \(\text{Ma} = \frac{u}{c}\) where \(u\) is the flow velocity, and \(c\) is the speed of sound. For an ideal gas, the speed of sound is given by: \(c = \sqrt{k R T}\) Using Equation 3, we can express the speed of sound in terms of pressure and density as: \(c = \sqrt{\frac{k p}{\rho}}\) Now, let's express the isentropic flow equations in terms of Mach number.

Find expressions for P/P*, T/T*, and ρ/ρ* as functions of k and Ma

We will now use the isentropic flow equations (in terms of Mach number) to derive expressions for \(P/P^*\), \(T/T^*\), and \(\rho/\rho^*\) as functions of \(k\) and \(\mathrm{Ma}\). From the isentropic flow equations, we have: For Pressure ratio (\(P/P^*\)): \(p = p^* \left(\frac{1 + \frac{k - 1}{2} \text{Ma}^2}{1 + \frac{k - 1}{2} \text{Ma}^{*2}}\right)^\frac{k}{k-1}\) For Temperature ratio (\(T/T^*\)): \(T = T^* \frac{1 + \frac{k - 1}{2} \text{Ma}^2}{1 + \frac{k - 1}{2} \text{Ma}^{*2}}\) For Density ratio (\(\rho/\rho^*\)): \(\rho = \rho^* \left(\frac{1 + \frac{k - 1}{2} \text{Ma}^2}{1 + \frac{k - 1}{2} \text{Ma}^{*2}}\right)^\frac{1}{k-1}\) These are the expressions for \(P/P^*\), \(T/T^*\), and \(\rho/\rho^*\) as functions of \(k\) and \(\mathrm{Ma}\) for ideal gases undergoing isentropic flows.