AA210A Fundamentals of Compressible Flow

Chapter 10 - Gasdynamics of nozzle flow

10/27/20 1 10.1 The area-Mach number function

Quasi-1D equations of motion

Assume

10/27/20 2 Equations of motion in fractional differential form

10/27/20 3 Equation of state

Effect of area change on velocity

10/27/20 4 Effect of area change on other flow variables

10/27/20 5 Effect of area change on Mach number

10/27/20 6 10/27/20 7 Mass Conservation

10/27/20 8 10/27/20 9 Between any two points in a channel with zero mass addition

If the flow is adiabatic

Pt1A1 f (M1 ) = Pt 2 A2 f (M 2 )

If the flow is adiabatic and isentropic

10/27/20 10 10.2 A Simple convergent nozzle

If the flow is subsonic

10/27/20 11 For subsonic flow the exit Mach number can be determined from

Thus

The exit Mach number reaches one when

10/27/20 12 If the pressure ratio is very high the flow from the nozzle will spread rapidly.

10/27/20 13 10.3 Converging-diverging nozzle

Determine two critical exit Mach numbers from

10/27/20 14 The corresponding critical exit pressures are determined from

10/27/20 15 10.3.1 Case 1 - Isentropic, subsonic flow in the nozzle

10/27/20 16 10.3.2 Case 2 - Non-isentropic flow - shock in the nozzle

10/27/20 17 The exit flow is subsonic and so the exit pressure matches the ambient pressure.

Solve for the exit mach number

Now determine the stagnation pressure ratio across the nozzle.

10/27/20 18 The shock Mach number is now determined from

10/27/20 19 As the nozzle pressure ratio is increased the shock moves downstream until it sits at the nozzle exit.

The Mach number behind the shock is

10/27/20 20 This condition is reached when

In summary, the shock-in-nozzle case occurs over the range

10/27/20 21 10.3.3 Case 3 - Isentropic supersonic flow in the nozzle

i) Over expanded flow

10/27/20 22 ii) Fully expanded flow

10/27/20 23 iii) Under expanded flow

10/27/20 24 Figure 5.4 from Liepmann and Roshko

10/27/20 25 10.4.3 Gasdynamics of a double throat - Supersonic Wind Tunnel Start and Unstart

10/27/20 26 10/27/20 27 10/27/20 28 The shock moves downstream until a point is reached where it sits just upstream of station 2.

10/27/20 29 M 2 = M 3 = 3.368 M e = 2.99

All supersonic flow is established in the wind tunnel - the tunnel is said to have started.

10/27/20 30 10.5 Problems

10/27/20 31 10/27/20 32 10/27/20 33 10/27/20 34