Computational Fluid Dynamics - Basic (e-1)

Incompressible fluid through a converging nozzle

In this sub-section, 1-Dimensional equations of motion for an incompressible fluid through a converging (or a diverging) nozzle is explained. This simple problem is used as an introduction to more complicated problems such as compressible and multiphase flows which will be discussed in the following sub-sections.

A conical nozzle can be introduced by its inlet and outlet diameters ($ D_{in} $ and $ D_{out} $). The input data for the problem is the inlet pressure, $ P_{in} $, the inlet flow motion which can be given as mass flow rate, $ \dot{m}_{in} $, or velocity, $ V_{in} $. Since the flow is incompressible, the density is constant; therefore, the temperature does not appear in the equations. Using the above-mentioned input data, it is possible to analyze the fluid flow. Fig. e.1.1 shows a sketch of the problem.

Fig. e.1.1. Flow in a converging nozzle.

The outlet velocity can be calculated from the continuity equation. When the density is constant, the continuity equation is simplified to a volume continuity equation:
\begin{equation} \nabla \cdot V = 0 \end{equation} and solving for outlet velocity, we have:
\begin{equation} V_{out} = \frac{V_{in} A_{in}}{A_{out}} \end{equation} Now the only unknown remain in the problem is the pressure at the outlet, which can be derived using the Bernoulli equation:
\begin{equation} P_{in} + \frac{1}{2}\rho V_{in}^2 + \rho g h_{in} = P_{out} + \frac{1}{2}\rho V_{out}^2 + \rho g h_{out} \end{equation} From the equations above, it is concluded that decreasing the area leads to increase in velocity and consequently the pressure is decreased:
\begin{equation} A_{out} < A_{in} \Longrightarrow V_{out} > V_{in} \Longrightarrow P_{out} < P_{in} \end{equation} 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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