Prof dr A Achterberg Astronomical Dept IMAPP Radboud
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 1. Shocks are sudden transitions in flow properties such as density, velocity and pressure; 2. In shocks the kinetic energy of the flow is converted into heat, (pressure); 3. Shocks are inevitable if sound waves propagate over long distances; 4. Shocks always occur when a flow hits an obstacle supersonically 5. 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 V - Cs Flow velocity V Forward propagating sound wave V + Cs
V + Cs V - Cs Supersonic flow past a sphere
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:
1 2 Flux = density x velocity Incoming flux: Outgoing flux:
Conclusions: 1. The density increases across the shock 2. 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 (1 D case) Generic conservation law:
Change of the amount of Q in layer of width 2 e: 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 1 D flow Starting point: 1 D 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 1 D case: Shocks can only exist if Ms>1 ! Weak shocks: s=1+�with � << 1; Strong shocks: s>> 1.
Weak shock:
From jump conditions:
Weak shock ~ strong sound wave! Sound waves:
Very strong normal shock
2 Strong shock: P 1<< � V 1 1 Approximate jump conditions: put P 1 = 0!
Conclusion for a strong normal shock:
Very strong normal shock
2 Strong shock: P 1<< � V 1 1 Approximate jump conditions: put P 1 = 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
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