Problem B.2: Shock Wave This year's qualification round featured a spaceship escaping from a shock wave (Problem B). The crew survived and wants to study the shock wave in more detail. It can be assumed that the shock wave travels through a stationary flow of an ideal polytropic gas which is adiabatic on both sides of the shock. Properties in front and behind a shock are related through the three Rankine-Hugoniot jump conditions (mass, momentum, energy conservation): Pio? + P1 = P2v3 + P2 +hị + h2 Pivi = P2V2 where p, v, p, and h are the density, shock velocity, pressure, and specific enthalpy in front (1) and behind (2) the shock respectively. Shock front v2, P2; P2; h2 v1, P1, P1, h1 (a) Explain briefly the following terms used (i) stationary flow (ii) polytropic gas (iii) specific enthalpy the text above: (b) Show with the Rankine-Hugoniot conditions that the change in specific enthalpy is given by: 1 - + P2 - P1 1 Ah = P1 P2 The general form of Bernoulli's law is fulfilled on both sides of the shock separately: + + + h = b 2 where d is the gravitational potential and ba constant. (c) Assuming that the gravitational potential is the same on both sides, determine how the con- stant b changes at the shock front. (d) Explain whether Bernoulli's law can be applied across shock fronts.
Problem B.2: Shock Wave This year's qualification round featured a spaceship escaping from a shock wave (Problem B). The crew survived and wants to study the shock wave in more detail. It can be assumed that the shock wave travels through a stationary flow of an ideal polytropic gas which is adiabatic on both sides of the shock. Properties in front and behind a shock are related through the three Rankine-Hugoniot jump conditions (mass, momentum, energy conservation): Pio? + P1 = P2v3 + P2 +hị + h2 Pivi = P2V2 where p, v, p, and h are the density, shock velocity, pressure, and specific enthalpy in front (1) and behind (2) the shock respectively. Shock front v2, P2; P2; h2 v1, P1, P1, h1 (a) Explain briefly the following terms used (i) stationary flow (ii) polytropic gas (iii) specific enthalpy the text above: (b) Show with the Rankine-Hugoniot conditions that the change in specific enthalpy is given by: 1 - + P2 - P1 1 Ah = P1 P2 The general form of Bernoulli's law is fulfilled on both sides of the shock separately: + + + h = b 2 where d is the gravitational potential and ba constant. (c) Assuming that the gravitational potential is the same on both sides, determine how the con- stant b changes at the shock front. (d) Explain whether Bernoulli's law can be applied across shock fronts.
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Transcribed Image Text:Problem B.2: Shock Wave
This year's qualification round featured a spaceship escaping from a shock wave (Problem B).
The crew survived and wants to study the shock wave in more detail. It can be assumed that
the shock wave travels through a stationary flow of an ideal polytropic gas which is adiabatic on
both sides of the shock. Properties in front and behind a shock are related through the three
Rankine-Hugoniot jump conditions (mass, momentum, energy conservation):
Pivž +P1 = P2v% +P2
+ hị
+ h2
Pivi = P2V2
2
2
where p, v, p, and h are the density, shock velocity, pressure, and specific enthalpy in front (1)
and behind (2) the shock respectively.
Shock front
v2, P2; P2; h2
V1, P1; P1, h1
(a) Explain briefly the following terms used in the text above:
(i) stationary flow
(ii) polytropic gas
(iii) specific enthalpy
(b) Show with the Rankine-Hugoniot conditions that the change in specific enthalpy is given by:
P2 – P1
Ah =
1
1
+ –
P1
P2
The general form of Bernoulli's law is fulfilled on both sides of the shock separately:
v2
+ D + h = b
2
where d is the gravitational potential and b a constant.
(c) Assuming that the gravitational potential is the same on both sides, determine how the con-
stant b changes at the shock front.
(d) Explain whether Bernoulli's law can be applied across shock fronts.
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