TITLE.PM5

892 ENGINEERING THERMODYNAMICS

dharm

\M-therm\Th16-2.pm5

16.11.Shock Waves

Whenever a supersonic flow (compressible) abruptly changes to subsonic flow, a shock
wave (analogous to hydraulic jump in an open channel) is produced, resulting in a sudden rise in
pressure, density, temperature and entropy. This occurs due to pressure differentials and when
the Mach number of the approaching flow M 1 > 1. A shock wave is a pressure wave of finite
thickness, of the order of 10–2 to 10–4 mm in the atmospheric pressure. A shock wave takes place in
the diverging section of a nozzle, in a diffuser, throat of a supersonic wind tunnel, in front of
sharp nosed bodies.
Shock waves are of two types :

1. Normal shocks which are almost perpendicular to the flow.


2. Oblique shocks which are inclined to the flow direction.

16.11.1.Normal shock wave

Consider a duct having a compressible sonic flow (see Fig. 16.11).
Let p 1 , ρ 1 , T 1 , and V 1 be the pressure, density, temperature and velocity of the flow (M 1 > 1)
and p 2 , ρ 2 , T 2 and V 2 the corresponding values of pressure, density, temperature and velocity after
a shock wave takes place (M 2 < 1).

V>C 11 V<C 22

p, ,T 111 r p, ,T 222 r

Normal shock wave

M>1 M<1

Fig. 16.11. Normal shock wave.
For analysing a normal shock wave, use will be made of the continuity, momentum and
energy equations.
Assume unit area cross-section, A 1 = A 2 = 1.
Continuity equation : m = ρ 1 V 1 = ρ 2 V 2 ...(i)
Momentum equation : ΣFx = p 1 A 1 – p 2 A 2 = m (V 2 – V 1 ) = ρ 2 A 2 V 22 – ρ 1 A 1 V 12
for A 1 = A 2 = 1, the pressure drop across the shock wave,
p 1 – p 2 = ρ 2 V 22 – ρ 1 V 12 ...(ii)
p 1 + ρ 1 V 12 = p 2 + ρ 2 V 22
Consider the flow across the shock wave as adiabatic.

Energy equation :

γ

γρ−

F

HG

I

KJ

+

12

1

1

pV (^12) = γ
γρ−
F
HG
I
KJ

• 12
  2
  2
  pV (^22) ...[Eqn. (16.7)]
  (z 1 = z 2 , duct being in horizontal position)
  or,
  γ
  γ− 1
  pp VV 2
  2
  1
  1
  12 22
  ρρ 2
  −
  F
  HG
  I
  KJ
  = − ...(iii)
  Combining continuity and momentum equations [refer eqns. (i) and (ii)], we get
  p 1 +
  ()ρ
  ρ
  11
  2
  1
  V
  = p 2 + ()ρ
  ρ
  222
  2
  V ...(16.42)