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4 Compressible Fluids
As long as the outlet pressure of a convergent nozzle exceeds the critical pressure
( p 3 = p * ), the velocity in the outlet section increases as the outlet pressure decreases.
With an outlet pressure equal to the critical pressure p 3 = p * , a sonic state is attained in the outlet section. An additional decrease of the outlet pressure ( p 6 ) cannot
change the state within the nozzle compared to the state attained at p 3 , as the Mach
number cannot exceed M = 1. This means that an expansion from p 3 to p 6 must occur
within the space downstream of the nozzle. As the outlet pressure decreases, staying
above p 3 however, the mass flow through the nozzle increases. It is blocked at the
value corresponding to p 3 if pressure is lower than p 3 . Such a state is termed choked.
The choking mass flow rate equals ρ * v * A * .
We consider a convergent-divergent channel, as shown in Fig. 4.4 (right).
From the relations demonstrated in Fig. 4.3 we learn that, with a subsonic flow
within a convergent-divergent channel, the flow accelerates within the convergent part and decelerates within the divergent part. Velocity and Mach number
become maximal in the throat section. The channel is said to function as a subsonic Venturi tube. Note that, if the flow enters the channel supersonically, a
similar flow is possible where flow is supersonic everywhere and velocity and
Mach number attain a minimum in the throat section. This is termed a supersonic
Venturi tube. From the relations in Fig. 4.3 follows that subsonic flow in the convergent part may turn into supersonic flow in the diverging part. The critical state
is then attained in the throat section. The backpressure p 3 in Fig. 4.4 (right) corresponds to a subsonic evolution with M = 1 being attained in the throat section.
Pressure p 5 corresponds to a subsonic-supersonic evolution. For a backpressure
between p 3 and p 5 , the corresponding pressure evolution is no longer continuous
everywhere. The supersonic branch of the pressure evolution in the divergent
part is followed partially or completely, followed by a discontinuous compression phenomenon (shock wave). Dependent on the pressure value p 4 , the shock
wave may occur within the divergent part or downstream of the outlet section
(see fluid mechanics). With backpressure p 6 lower than p 5 , there is an expansion
downstream of the nozzle. In that case, the pressure evolution corresponding
to p 5 is followed within the nozzle (see fluid mechanics). With a convergentdivergent nozzle, the mass flow is blocked at the choking value ρ * v * A * , when
the backpressure drops under p 3 , as the critical state is then attained in the throat.
The density ρ 3 is linked to the outlet pressure by an isentropic relation. The outlet
velocity v 3 is linked to the outlet pressure according to the Saint Venant formula.
Expressing constant mass flow rate, this implies a non-linear equation in outlet
pressure, with two solutions, namely p 3 and p 5 .
4.5 Nozzle with Initial Velocity
In the foregoing we assumed that the velocity at the nozzle inlet equals zero, with an
inlet state being equal to the total state (state with subscript 0). From now on, we assume that a velocity v 0 ≠ 0 may be present and we indicate the inlet section with the
subscript 0 (state ≠ total state). The above equations so must be adapted somewhat.
4 Compressible Fluids
As long as the outlet pressure of a convergent nozzle exceeds the critical pressure
( p 3 = p * ), the velocity in the outlet section increases as the outlet pressure decreases.
With an outlet pressure equal to the critical pressure p 3 = p * , a sonic state is attained in the outlet section. An additional decrease of the outlet pressure ( p 6 ) cannot
change the state within the nozzle compared to the state attained at p 3 , as the Mach
number cannot exceed M = 1. This means that an expansion from p 3 to p 6 must occur
within the space downstream of the nozzle. As the outlet pressure decreases, staying
above p 3 however, the mass flow through the nozzle increases. It is blocked at the
value corresponding to p 3 if pressure is lower than p 3 . Such a state is termed choked.
The choking mass flow rate equals ρ * v * A * .
We consider a convergent-divergent channel, as shown in Fig. 4.4 (right).
From the relations demonstrated in Fig. 4.3 we learn that, with a subsonic flow
within a convergent-divergent channel, the flow accelerates within the convergent part and decelerates within the divergent part. Velocity and Mach number
become maximal in the throat section. The channel is said to function as a subsonic Venturi tube. Note that, if the flow enters the channel supersonically, a
similar flow is possible where flow is supersonic everywhere and velocity and
Mach number attain a minimum in the throat section. This is termed a supersonic
Venturi tube. From the relations in Fig. 4.3 follows that subsonic flow in the convergent part may turn into supersonic flow in the diverging part. The critical state
is then attained in the throat section. The backpressure p 3 in Fig. 4.4 (right) corresponds to a subsonic evolution with M = 1 being attained in the throat section.
Pressure p 5 corresponds to a subsonic-supersonic evolution. For a backpressure
between p 3 and p 5 , the corresponding pressure evolution is no longer continuous
everywhere. The supersonic branch of the pressure evolution in the divergent
part is followed partially or completely, followed by a discontinuous compression phenomenon (shock wave). Dependent on the pressure value p 4 , the shock
wave may occur within the divergent part or downstream of the outlet section
(see fluid mechanics). With backpressure p 6 lower than p 5 , there is an expansion
downstream of the nozzle. In that case, the pressure evolution corresponding
to p 5 is followed within the nozzle (see fluid mechanics). With a convergentdivergent nozzle, the mass flow is blocked at the choking value ρ * v * A * , when
the backpressure drops under p 3 , as the critical state is then attained in the throat.
The density ρ 3 is linked to the outlet pressure by an isentropic relation. The outlet
velocity v 3 is linked to the outlet pressure according to the Saint Venant formula.
Expressing constant mass flow rate, this implies a non-linear equation in outlet
pressure, with two solutions, namely p 3 and p 5 .
4.5 Nozzle with Initial Velocity
In the foregoing we assumed that the velocity at the nozzle inlet equals zero, with an
inlet state being equal to the total state (state with subscript 0). From now on, we assume that a velocity v 0 ≠ 0 may be present and we indicate the inlet section with the
subscript 0 (state ≠ total state). The above equations so must be adapted somewhat.