Star-CCM+ Tutorial 5: Supersonic Flow Through a Nozzle

Copyright 2005, Sergio E. Perez. No part of this document may be reproduced in any

way without the author’s consent after downloading. This document is available free-of-

charge from www.computationalfluids.com .

Background:

In Tutorials 2 we introduced the concept of the shock. Basically, a shock will form when

supersonic flow is obstructed in any way, including pressure increases due to geometry

changes. In tutorials 2 and 3, shocks were seen to happen in external flow: flow over a

bullet, and flow over an airfoil.

Shocks can also form in converging-diverging nozzles. A converging- diverging nozzle is

used to accelerate flow from subsonic to supersonic speeds. Under the right conditions,

flow will accelerate in the converging portion of the nozzle, reach Mach 1 (the speed of

sound) at the throat, and then accelerate to supersonic speeds in the diverging portion of

the nozzle. This is completely contrary to our subsonic experience, where flow slows

down in a diverging duct.

A shock will form in the diverging portion of the nozzle if the expansion is made too

large, or if the pressure at the outlet of the nozzle (the back pressure) is too high.

Basically, the supersonic flow does not “find out” that the pressure is supposed to

increase until the flow is already there. Once a shock is encountered, the flow will no

longer be supersonic downstream of the shock.

As a gas flow accelerates, two things happen: the pressure drops, and the temperature

drops. You should be able to see both phenomena when you run your simulation.

Instructions:

Load the nozzle.sim file from www.computationalfluids.com, and run it at a speed of 150

m/s, from left-to-right. Supersonic speed is about 330 m/s, and the nozzle should

accelerate the flow to beyond this.

You will need to experiment with the Courant number to get this simulation to run (good

way to spend an afternoon!) – suffice it to say that the default setting of 5 is too high (see

Tutorial 2 for instructions on how to change this).

Use the k-e model with ideal gas and no-slip walls for this simulation. Since the speeds

are high, you should couple the energy and momentum equations – there will be

significant temperature changes which can affect viscosity and density.

As you run the simulation, observe the plots of Mach number and pressure. If you look at

only the residuals, you might be tempted to end the simulation too soon. By observing the

Mach number and pressure, and waiting until there are no longer any significant changes,

you may have a better idea of when convergence is reached.

1) For your report, you should include screenshots of the Mach number, pressure,

temperature and velocity vectors.

2) Study the plots carefully and make any observations you can about the location of the

shock (as evidenced by sudden increase in pressure and decrease in Mach number).

3) Study any trends in velocity, temperature, pressure, separation and Mach number

through the nozzle.

4) Use the first law of thermodynamics to explain why the temperature should drop.

5) Compare pressure values before and after the shock to shock table values.

6) Discuss any conclusions/ observations about changing the Courant number. Answer:

what is happening exactly when you change the Courant number?

7) Re-run the simulation at a lower pressure at the exit. What happens to the location of

the shock? Include screen shots comparing the two.