Computational Fluid Dynamics - Basic (e-2)

Compressible fluid through a converging nozzle

In the previous sub-section, an incompressible fluid evolution in nozzles is discussed. Now a compressible fluid through a converging nozzle is going to be discussed.

The flow is assumed to be 1-Dimensional, steady-state, and isentropic. Furthermore, we assume the fluid to be an ideal gas:
\begin{equation} P = \rho RT \ , \ c = \sqrt{\gamma RT} \end{equation} And the isentropic relations for temperature and pressure can be written as:
\begin{equation} \frac{T_0}{T} = 1 + \left(\frac{\gamma-1}{2}\right)M^2 \ , \ \frac{P_0}{P} = \left[1 + \left(\frac{\gamma-1}{2}\right)M^2\right]^{\gamma/(\gamma-1)} \end{equation} where $ c $ is velocity, $ M $ is Mach number, $ \gamma $ is isentropic expansion factor, $ T $ and $ P $ are static, and $ T_0 $ and $ P_0 $ are stagnation temperature and pressure respectively.

When the Mach number is unity, the fluid properties are called critical properties (representing by the superscript asterisk) and can be calculated setting $ M = 1 $ in Eq. (2):
\begin{equation} T^* = T_0 \frac{2}{\gamma+1} \ , \ P^* = P_0 \left(\frac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)} \end{equation} If the back pressure is lower than the critical pressure ($ P_b < P^* $), the mass flow is in its maximum value and the flow is choked (see Fig. e.2.1). In this case, all the flow properties will be equal to the critical properties respectively.

Using the Matlab code provided in this sub-section, it is possible to set the nozzle dimensions, characteristic gas constant ($ R $), value of $ \gamma $, inlet velocity, temperature and pressure, and finally the back pressure to find the flow properties through the nozzle.

Fig. e.2.1. Choking condition in a converging nozzle.

The more advanced topics are listed in the main section 1-D flow model in nozzles.

The relevant Matlab codes can be downloaded using the following link:
Matlab Codes Bank

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