Shock waves
The devil is in the detail
Jiˇr´ı Krtiˇcka
Masaryk University
Motivation
Compression of a gas: advancing piston
Let us study a tube filled with a gas bounded by a piston advancing
subsonically.
The piston starts at O and moves
towards righ causing an compression
of the gas. The gas adjanced to a
piston moves with the same velocity
as the piston. Only one set of
characteristics drawn from piston
propagates into the flow. The flow
in a zone I is not influenced by a
moving piston. Intersecting
characteristic form an envelope. The
solution is not unique at the
intersection. This leads to a
formation of a shock wave.
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Shock conditions
Laws of conservation
Laws of conservation of mass, momentum, and energy do not necessary
assume continuity of flow variables. A discontinuity can be regarded as
the limiting case of very large gradients of the flow variables across a very
thin layer. Viscosity an heat conduction may become important within
such a thin layer. Nonviscid flow equations are not valid in this region.
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Laws of conservation
Let us consider a column of gas bounded by a0(t) < x < a1(t) containing
a discontinuity. The conservation of mass requires
d
dt
a1(t)
a0(t)
ρ dx = 0.
The change of momentum is given by forces acting on the column
d
dt
a1(t)
a0(t)
ρv dx = p(a0, t) − p(a1, t).
The change of energy is given by work of acting forces
d
dt
a1(t)
a0(t)
ρ
1
2
u2
+ e dx = p(a0, t)u(a0, t) − p(a1, t)u(a1, t).
The entropy shall not decrease:
d
dt
a1(t)
a0(t)
ρs ≥ 0.
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Limit of thin layer
Assuming that the discontinuity apprears at coordinates x = ξ(t) and
moves with speed ˙ξ(t) = U(t), the integrals of type
J =
a1(t)
a0(t)
ψ(x, t) dx
change with time as
dJ
dt
=
d
dt
ξ(t)
a0(t)
ψ(x, t) dx +
d
dt
a1(t)
ξ(t)
ψ(x, t) dx
=
a1(t)
a0(t)
ψt (x, t) dx +ψ0
˙ξ(t)−ψ(a0, t)v(a0, t)+ψ(a1, t)v(a1, t)−ψ1
˙ξ(t).
Therefore, in the limit of a0(t) → a1(t), we have
dJ
dt
= ψ1V1 − ψ0V0,
where Vi = vi − U are relative velocities with respect to shock.
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Rankine-Hugoniot jump conditions
The application on the laws of conservation of mass, momentum, and
energy gives the Rankine-Hugoniot jump conditions
ρ1V1 − ρ0V0 = 0,
ρ1v1V1 − ρ0v0V0 = p0 − p1,
ρ1
1
2
v2
1 + e1 V1 − ρ0
1
2
v2
0 + e0 V0 = p0v0 − p1v1,
ρ1s1V1 − ρ0s0V0 ≥ 0.
These equation can cast in the conservative form
ρ1V1 = ρ0V0,
ρ1V 2
1 + p1 = ρ0V 2
0 + p0,
1
2
V 2
1 + h1 =
1
2
V 2
0 + h0,
s1 ≥ s0,
where we used equation of continuity, selected the coordinate frame,
where the shock is stationary, U = 0, and where h = e + p/ρ. 5
Rankine-Hugoniot jump conditions for perfect gas
For a perfect gas with constant γ the specific enthalpy is
h = γ/(γ − 1)p/ρ and Rankine-Hugoniot jump conditions give
ρ1
ρ0
=
(γ + 1)M2
0
γ + 1 + (γ − 1)(M2
0 − 1)
=
V0
V1
,
p1
p0
=
γ + 1 + 2γ(M2
0 − 1)
γ + 1
,
T1
T0
=
γ + 1 + 2γ(M2
0 − 1) γ + 1 + (γ − 1)(M2
0 − 1)
(γ + 1)2M2
0
,
where the Mach number is M0 = V0/a0 and the speed of sound is
a2
0 = γp0/ρ0. These relations express the post-shock quantities ρ1, V1,
p1, and T1 in terms of pre-schock quantities and M0.
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Compression shocks
Compression shocks
For compression shocks M0 > 1 from the Rankine-Hugoniot jump
conditions follows that p1 > p0, T1 > T0 (examine T1/T0 as a function
of p1/p0, where we have substituted M2
0 = (γ + 1)/ (2γ) (p1/p0 − 1) + 1,
this shows that T1/T0 is monotonically increasing function with
T1/T0 = 1 for p1/p0 = 1), ρ1 > ρ0 (ρ1 has minimum as a function of M0
for M0 > 1 at M0 = 1) and therefore V1 < V0. Therefore also
M2
1
M2
0
=
V 2
1
V 2
0
a2
0
a2
1
=
V 2
1
V 2
0
p0
p1
ρ1
ρ0
=
V 2
1
V 2
0
T0
T1
< 1,
and M1 < 1. Gas flows into the discontinuity with a supersonic velocity
M0 > 1 and flows out with a subsonic velocity M1 < 1. Equivalently, the
shock propagates at a supersonic velocity with respect to the undisturbed
gas and at a subsonic velocity with respect to compressed gas behind the
shock.
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Infinitely strong shock M0 → ∞
In the limit of infinitely strong shocks (M0 → ∞) the Rankine-Hugoniot
jump conditions give p1 → ∞ and T1 → ∞ (extremely high pressure and
temperature), but the density remains finite
lim
M0→∞
ρ1
ρ0
=
γ + 1
γ − 1
,
which gives ρ1/ρ0 = 4 pro monoatomic perfect gas with γ = 5/3.
For very strong shocks M0 ≫ 1, the post-shock temperature is
T1 =
2γ(γ − 1)M2
0
(γ + 1)2
T0.
Since M0 = V0/a0 and a2
0 = γkT/µ, the post-shock temperature does
not depend on the pre-shock temperature. Evaluating the post-shock
temperature for a monoatomic gas with γ = 5/3,
T1 =
3
16
µV 2
0
k
= 14 MK
V0
1000 kms−1
2
.
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Rarefaction shocks
Rarefaction shocks
Another type of shocks allowed by Rankine-Hugoniot jump conditions are
rarefaction shocks, in which the existence of discontinuities leads to the
expansion of the gas rather than to the compression. For M0 < 1 we
have p1 < p0, T1 < T0, ρ1 < ρ0 and therefore V1 > V0 and using the
same reasoning M1 > 1. Gas flows into the discontinuity with a subsonic
velocity M0 < 1 and flows out with a supersonic velocity M1 > 1.
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The change of entropy
The entropy of the gas with constant specific heats is S = cV ln(pρ−γ
),
therefore the entropy jump is
S1 − S0 = cv ln
p1ργ
0
p2ργ
1
From the Rankine-Hugoniot jump conditions we have
M2
0 =
γ + 1
2γ
p1
p0
− 1 + 1
and inserting into the expression for ρ1/ρ2 follows that
S1 − S0 = cv ln
p1
p0
(γ − 1)p1/p0 + γ + 1
(γ + 1)p1/p0 + γ − 1
γ
.
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The change of entropy
The analysis of behaviour of function S1 − S0 shows that the function is
monotonic and that
S1 − S0



< 0, for p1 < p0,
= 0, for p1 = p0,
> 0, for p1 > p0.
The entropy increases across the compression shock, whereas the entropy
decreases across the rarefaction shock. Therefore, the rarefaction shocks
are not termodynamically allowed. This statement holds for majority of
real fluids.
Rarefaction shocks are not mechanically stable: any distaurbance in the
pre-shocked gas travels with speed of sound and outruns the shock wave.
After a certain time the rarefraction region will include the gas in front of
the discontinuity and the discontinuity will dissapear.
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Shocks are everywhere...
Mach cone around bullet
Schlieren photography of bullet taken in Prague by Mach (1887).
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Mach cone around supersonic planes
Schlieren photography of Mach cone around supersonic plane.
13
Initial accretion phase
(Hartmann 2016)
14
Shocks in hot star winds
0.0
0.5
1.0
1.5
2.0
2.5
1 2 3 4 5 6 7
v(r)/1000kms-1
r / R∗
(Feldmeier et al. 1997)
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Interaction of stellar wind with interstellar environent
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Interaction of supernovae with circumstellar environment
The plots show interaction of
supernova blast wave with
circumstellar environment in the
form of disk (Kurf¨urst et al. 2020).
The plots show density, radial and
tangential velocity, and temperature
in three time steps.
17
Suggested reading
R. Courant & K. O. Friedrichs: Supersonic Flow and Shock Waves
A. Feldmeier: Theoretical Fluid Dynamics
D. Mihalas & B. W. Mihalas: Foundations of Radiation Hydrodynamics
F. H. Shu: The physics of astrophysics: II. Hydrodynamics
Y. B. Zeldovich, Y. P. Raizer: Physics of Shock Waves and
High-Temperature Hydrodynamic Phenomena
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