Normal Shock Waves
In our quasi-1D flows, shocks can occur from a supersonic-to-subsonic state.
These shocks are discontinuous in our inviscid flow model (recall that shocks are
very thin and their thickness scales with 1 ):
Re
M
ML
shock
1
Upstream Mach: M L > 1
Downstream Mach: M R < 1
MR
Throat
The shock jump relationships come from the conservation equations we have
seen before:
ρ LU L AL = ρ RU R AR
AR
( ρ L u L2 + p L ) AL = ( ρ R u R2 + p R ) AR − ∫ pdA
AL
ρ L u L ho AL = ρ R u R ho AR
L
R
Since the jump is discontinuous (i.e. it has zero thickness)
AL = AR = A ⇒ dA = 0
Ranhine-Hugoniot
Shock Jump
ρ LuL = ρ RuR
Relationships
ρ L u L2 + p L = ρ R u R2 + p R
⇒
ρ L u L h0 L = ρ R u R hoR
Here are some important things to know about shock waves from these
relationships:
∗ Mathematically, “shocks” exist which jump from subsonic-to-supersonic flow.
However, these “shocks” can be shown to violate the 2nd Law ( ∆s < 0 ).
∗ Only shocks which jump from supersonic-to-subsonic states satisfy the 2nd
Law. The Mach number downstream of shocks is given by :
1
1 + (γ − 1) M L2
2
M R2 =
where M L > 1
1
2
γM L − (γ − 1)
2
And it can be shown that ∆s > 0 .
Normal Shock Waves
∗ The stagnation enthalpy (and therefore the total temperature) is constant
through a shock (shocks are adiabatic).
⇒
h0 L = h0 R
or, equivalently,
T0 L = T0 R
∗ Total pressure decreases through a shock (this is a direct result of the entropy
Increasing while T 0= const. ):
⇒
p0 R
p0 L
16.100 2002
= e−( sR − sL ) R
2