1D flow and nozzle

Question 1

Describe the underlying assumptions of a one-dimensional isentropic flow in a nozzle or a duct.

Question 2

Write down basic relation for a one-dimensional isentropic flow in a nozzle or a duct.

Question 3

Applying the integral form of mass conservation laws for a compressible gas to a one-dimensional steady-
state flow in a pipe, show that the mass flow rate is the same in every cross-section of the pipe.

Question 4

Applying the integral form of energy conservation laws for a compressible gas to a one-dimensional
adiabatic steady-state flow in a pipe, show that the stagnation enthalpy is constant in the pipe

Question 5

Explain why the momentum equation for a compressible flow in a constant cross-section pipe 𝑝+𝜌𝑣^2=𝑐𝑜𝑛𝑠𝑡 and the Bernoulli equation 𝑝+(1/2)𝜌𝑣^2=𝑐𝑜𝑛𝑠𝑡 do not contradict each other.

Question 6

Derive formula for the sound speed in an ideal gas

Question 7

Derive area/velocity variation in a 1D isentropic flow of compressible gas

Question 8

Derive area/Mach number variation in a 1D isentropic flow of compressible gas

Question 9

Explain the term stagnation properties of the gas flow

Question 10

Derive relation between static and stagnation temperatures for an adiabatic flow of a perfect gas

Question 11

Derive relation between static and stagnation pressure for an isentropic flow of a perfect gas

Question 12

For which model of gas and which type of flow (adiabatic, isentropic) the following relation between static
and stagnation temperature is valid
𝑇0=𝑇(1+(𝛾−1/2)𝑀^2)

Question 13

For which model of gas and which type of flow (adiabatic, isentropic) the following relation between static
and stagnation pressure is valid 𝑝0=𝑝(1+((𝛾−1)/2)𝑀^2)^(𝛾/(𝛾−1))

Question 14

What does it mean for the flow if the static pressure is below the critical pressure?

Question 15

Describe the regimes of adiabatic flow in the converging nozzle for various values of background pressure.
Sketch the pressure and Mach number variation along the nozzle.

Question 16

Explain why the exit velocity cannot exceed the sound speed in a converging nozzle

Question 17

A supersonic flow is allowed expanding indefinitely in a diverging channel. Does the flow velocity approach a
finite limit, or does it continue to increase indefinitely? Justify your answer. Assume a perfect gas with
constant specific heats

Question 18

Describe the regimes of adiabatic flow in the converging-diverging nozzle for various values of background
pressure. Sketch the pressure and Mach number variation along the nozzle. In which regime is the
background pressure optimal?