Prof. dr. A. Achterberg, Astronomical Dept Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit Gas Dynamics, Lecture 7 (shocks: theory) see: www.astro.ru.nl/~achterb/

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

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

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!

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

V + Cs Supersonic flow past a sphere V - Cs

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

The marble-tube analogy for shocks

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

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

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

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

Steepening of Sound Waves:

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

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

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

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

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

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

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)!

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

Weak shock:

From jump conditions:

Weak shock ~ strong sound wave! Sound waves:

Very strong normal shock

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

Conclusion for a strong normal shock:

Very strong normal shock

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

Conclusion for a strong shock:

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

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

Oblique shocks: tangential velocity unchanged!

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

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