## A de Laval Nozzle

A de Laval nozzle is a tube that is pinched in the middle as in the above figure. It is often used to accelerate low speed high pressure gases to supersonic speed and is applied to variety of aerospace devices. Using Flowsquare, the flow through the nozzle can be simulated straightforwardly. The shape of the nozzle is just taken from wikipedia using Microsoft Paint. Easy Peasy!

Computational time:
Boundary Condition: bc.bmp
Initial Condition:
Parameters: grid.txt

During the simulation, the Mach number field transients as in the below figures.

Mach number field at 2000 step.

Mach number field at 4000 step.

Mach number field at 6000 step.

Mach number field at 8000 step.

Mach number field at 12000 step.

Mach number field at 40000 step.

At the location where nozzle area is the smallest, the Mach number reaches to 1 and the fluid moves at a sonic speed. After that, the velocity continues to increase due to gas expansion to reach supersonic speed near the exit. The exit fluid speed can be obtained theoretically using the following equation.

Exit velocity calculation.

Here, ve is exit velocity (m/s), T is inlet temperature, R is Universal gas constant (8314 J/kmol K), M is the gas molar weight (g/mol), gamma is isentropic expansion factor (=1.4 in flowsquare), pe is absolute exhaust pressure (Pa), and p is the absolute inlet pressure. As in the figures in below, if we neglect the inlet boundary effect, we obtain T=410 (K), R=8314.5 (J/kmol K), M=29 (air, g/mol), gamma=1.4, pe=13,800 (Pa), and p=231,000 (Pa), yielding ve=681.1 (m/s) from the above equation. If you compare the exit velocity obtained from Flowsquare simulation, the difference is only 1.7% of the theoretical value!

Pressure variation along the central axis.

Fluid speed variation along the central axis.

Temperature variation along the central axis.

Mach number variation along the central axis.