1. Introduction to Compressible Nozzle Flow

Definitions:

• Mach Number (M=u/c): the speed of fluid flow relative to the speed of sound of that fluid

• Nozzle: passage with variable cross sectional area designed to accelerate or decelerate flow

Assumptions:

• Flow is isentropic (has constant enthalpy - i.e. is both adiabatic and reversible)

• Flow is one dimensional

With compressible flow the relationship between the velocity, density, and flow areas are complex. To investigate this further we will consider 1-D isentropic nozzle flow

Mass balance for a steady-flow process:

In differential form:

Dividing through by uA:

Now we can use the conservation of momentum equation:

(this is a differential form of Bernoulli’s equation where there is negligible potential energy. For a more detailed derivation see here)

and the relation for the speed of sound:

To eliminate density from the differential contsant mass flow rate equation to give us:

Using the mach number definition M=u/c we get the final relation between area and velocity:

This equation tells us how the velocity of the flow changes as the area of the nozzle changes, and also that the velocity depends on the mach number:

• • M < 1 (subsonic): 1-M^2 > 0 therefore an increase in the area (dA>0) will produce a negative change in velocity (du<0) and vice versa

i.e. a diverging nozzle decreases velocity, a converging nozzle increases velocity

• • M > 1 (supersonic): 1-M^2 < 0 therefore an increase in the area (dA>0) will produce a positive change in velocity (du>0) and vice versa

i.e. a diverging nozzle increases velocity, a converging nozzle decreases velocity

So we get the opposite effect based on whether the flow is sub or super sonic

This is because when we have a compressible fluid both velocity and density can change

This equation describes the relationship between the the change in velocity and change in density and the mach number:

When flow is subsonic and M < 1, the change in velocity is much greater than the change in density.

i.e. when flow is subsonic, the flow is actually fairly incompressible so a reduction in area leads to an increase in velocity to maintain mass flow rate without much change in density

However when flow is supersonic and M > 1, the change in density is much greater than the change in velocity

i.e. when flow is supersonic, the flow becomes compressible so reducing area will lead to an increase in density and therefore a reduction in velocity to maintain the same mass flow rate

The same effect can be seen using a shallow water analogy:

 For more information see here