2. Convergent-Divergent Nozzles

Using the fact that convergent nozzles increase the velocity of subsonic compressible fluid flow and divergent nozzles increase the velocity of supersonic compressible fluid flow, we can combine the them to form a convergent-divergent nozzle (sometimes known as a de Laval nozzle):

The convergent section accelerates the fluid to Mach 1 at the throat at which the flow is said to be choked - the flow velocity will no longer increase with a decrease in cross sectional areaand this means that the throat dictates the mass flow rate through the nozzle. Adding the divergent section of the nozzle allows for further increases in speed above Mach 1.

(You don't need to know this section for exams, but it provides some context to how CD nozzles can be useful)

This is particularly useful in propulsion technology such as rockets, ramjets, and scramjets. This is because the amount of thrust produced by an engine is dependent on the mass flow rate of the exhaust, the exhaust velocity, as well as the pressure at the exit of the engine:

Ve = Exit Velocity

pe = Exit Pressure

p0 = Free Stream Pressure

Ae = Area of Nozzle Exit

For a given mass flow rate we can therefore increase the thrust produced by the engine by increasing the exhaust velocity as much as possible (the efficiency will also increase, as it is a function of exhaust velocity). CD nozzles are therefore very important in achieving high engine performance.

However the energy needed to increase the exhaust velocity has to come from somewhere, so it results in a decrease in pressure and temperature:

It’s often useful to express properties of the flow at an arbitrary section of the nozzle relative to the value of that property at the nozzle throat where M = 1 (indicated by a superscript *).

Since the throat dictates mass flow rate:

ρ*/ρ and v*/v can be expressed in terms of mach number using the isentropic relations defined below:

Multiplying these two together gives the relationship for A/A*:

Similarly for pressure and temperature we have the following relations:

These plots should hopefully now make some intuitive sense:

• • Area decreases in size (converges) until the throat (A/A*=1) then increases in size again (diverges)

  • Velocity continually increases  throughout both the converging and diverging sections

  • The energy needed to increase the velocity (by increasing the kinetic energy of the flow) needs to come from somewhere. This energy is sourced from the thermal and pressure energy of the flow

  • Increases in velocity results in a decrease in density (-M^2 dv/v = dρ/ρ) hence there is a continual decrease in density