1. Prof. dr. A. Achterberg, Astronomical Dept
Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit
Gas Dynamics, Lecture 7 (shocks: theory) see:

2. Shocks: non-linear fluid structures
Shocks occur whenever a flow hits an obstacle
at a speed larger than the sound speed

4. Shock properties such as density, velocity and pressure;
Shocks are sudden transitions in flow properties
such as density, velocity and pressure;
In shocks the kinetic energy of the flow is converted
into heat, (pressure);
Shocks are inevitable if sound waves propagate over
long distances;
Shocks always occur when a flow hits an obstacle
supersonically
In shocks, the flow speed along the shock normal
changes from supersonic to subsonic

5. Why must the flow be supersonic?
Pressure and density fluctuations travel at the
sound speed Cs ;
In a supersonic flow the signal of pressure changes
can not travel upstream;
Only way a supersonic flow can adjust velocity to
“miss”an obstacle is through a shock!

6. Subsonic flow around sphere
backward propgating
sound wave
Forward propagating
sound wave
V - Cs
V + Cs
Flow velocity V

7. V + Cs
Supersonic flow
past a sphere
V - Cs

9. Bow shock
Earth’s magnetic axis
Earth’s magnetosphere
Solar wind
V ~ 350 km/s , CS ~ 70 km/s, Machnumber S = V/Cs ~ 5

11. The marble-tube analogy for shocks

12. Time between two `collisions’
`Shock speed’ = growth velocity
of the stack.

13. 1 2
Go to frame where the `shock’ is stationary:
Incoming marbles:
Marbles in stack:

14. 2 1
Flux = density x velocity
Incoming flux:
Outgoing flux:

15. Conclusions:
The density increases across the shock
The flux of incoming marbles equals
the flux of outgoing marbles in the
shock rest frame:

16. Steepening of Sound Waves:

17. Effect of a sudden transition on a general conservation law (1D case)
Generic conservation law:

18. Change of the
amount of Q in
layer of width 2e:
flux in flux out

19. Infinitely thin layer:
What goes in must
come out : Fin = Fout

20. Infinitely thin layer:
What goes in must
come out : Fin = Fout
Formal proof: use a
limiting process for   0

21. Summary of shock physics
Shocks occur in supersonic flows;
Shocks are sudden jumps in velocity, density and pressure;
Shocks satisfy flux in = flux out principle for
- mass flux
- momentum flux
- energy flux

23. Simplest case: normal shock in 1D flow
Starting point: 1D ideal fluid equations in conservative form;
x is the coordinate along shock normal, velocity V along x-axis!
Mass
conservation
Momentum
conservation
Energy
conservation

24. Flux in = flux out: three jump conditions
Three conservation laws means three fluxes for flux in = flux out!
Mass flux
Momentum flux
Energy flux
Three equations for three unknowns: post-shock
state (2) is uniquely determined by pre-shock state (1)!

25. Shock strength and Mach Number
1D case:
Shocks can only exist if Ms>1 !
Weak shocks: Ms=1+ with << 1;
Strong shocks: Ms>> 1.

26. Weak shock:

27. From jump conditions:

28. Weak shock ~ strong sound wave!
Sound waves:

29. Very strong normal shock

30. Strong shock: P1<< 1V12
Approximate jump conditions: put P1 = 0!

32. Conclusion for a strong normal shock:

33. Very strong normal shock

34. Strong shock: P1<< 1V12
Approximate jump conditions:
put P1 = 0!

36. Conclusion for a strong shock:

37. Jump conditions in terms of Mach Number: the Rankine-Hugoniot relations
Shocks all have
S > 1
Compression ratio:
density contrast
Pressure jump

39. Oblique shocks: four jump conditions!
(1)
(2)
(3)
(4)

40. Oblique shocks: tangential velocity unchanged!

42. From normal shock to oblique shocks:
All relations remain the same if one makes the
replacement:
θ is the angle between upstream velocity and normal
on shock surface

43. From normal shock to oblique shocks:
All relations remain the same if one makes the
replacement:
θ is the angle between upstream velocity and normal
on shock surface
Tangential velocity along shock surface is unchanged