Simulation of flow through a Converging-diverging nozzle

Definition

A converging-diverging nozzle, also known as a de Laval nozzle, is a tube that pinches in to a narrow "throat" and then flares out again. It is designed to accelerate a pressurized gas to supersonic speeds by converting the thermal and pressure energy of the fluid into kinetic energy.

Main Content

1. Gas Dynamics Principles

The flow behavior depends on the Mach number ($M$), which is the ratio of local fluid velocity to the speed of sound.
For subsonic flow ($M < 1$), the velocity increases as the cross-sectional area decreases (Bernoulli’s principle).

2. The Choking Phenomenon

As the gas accelerates in the converging section, it reaches the speed of sound ($M = 1$) at the minimum area section, known as the throat.
Once the flow reaches $M = 1$ at the throat, it is said to be "choked," meaning the mass flow rate cannot be increased further by lowering the downstream pressure.

3. Expansion in the Diverging Section

In the diverging section, if the flow is supersonic ($M > 1$), the velocity continues to increase as the area increases.
The expansion allows the high-pressure gas to convert its internal energy into directed exhaust velocity.

    Converging      Throat      Diverging
      Section                    Section
     \        \    ____    /        /
      \________\  /    \  /________/
                \/      \/
                /\______/\
      ________/  \____/  \________
     /        /          \        \
     /        /            \        \
     Flow ->

Working / Process

1. Geometry Modeling and Meshing

Create a 2D or 3D CAD model representing the nozzle profile and define the inlet, outlet, and wall boundaries.
Generate a high-quality computational mesh, ensuring finer grid density near the throat where pressure and velocity gradients are steepest.

2. Physics Setup and Boundary Conditions

Select a compressible flow solver suitable for high-speed aerodynamics (e.g., Density-Based Solver).
Set the inlet as a "Pressure Inlet" (stagnation pressure) and the outlet as a "Pressure Outlet" (back pressure) to control the pressure ratio.

3. Iteration and Data Analysis

Run the simulation until the residuals reach convergence (constant values).
Extract data to plot pressure, temperature, and Mach number profiles along the nozzle axis to verify if the flow correctly reaches supersonic speeds.

Advantages / Applications

Used extensively in rocket engine nozzles to produce high-velocity thrust by expanding combustion gases.
Utilized in supersonic wind tunnels to study aerodynamic behavior of objects at high speeds.
Employed in steam turbines to accelerate steam for driving high-efficiency blades.

Summary

The simulation of a converging-diverging nozzle involves analyzing how pressure and area changes affect gas velocity. By achieving sonic conditions at the throat, the nozzle enables the efficient conversion of pressure into kinetic energy for industrial and aerospace applications. Important terms include Mach number, choked flow, stagnation pressure, and nozzle expansion ratio.