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Advanced Transport Phenomena
Module 4 Lecture 13
Momentum Transport: Shock Waves
Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras
Momentum Transport: Shock Waves
STEADY 1D COMPRESSIBLE FLUID FLOW
 Steady, frictionless flow of a nonreacting gas mixture in a
constant-area duct with heat addition:
 Conservation equations may be written as:
G  const
(mass),
du
dp
u

d
d
dh0
u
 q '''
d
(momentum),
(energy ),
Gdu  dp
Gc p dT0 
(q ''' Ad  )
 Gdq
A
STEADY 1D COMPRESSIBLE FLUID FLOW
Mass & momentum conservation equations yield:
p  Gu  const  p 0
Since u  G /   Gv,
p  G   p0
2
This locus on the p-v (or corresponding T-s) plane 
Rayleigh line (locus)
STEADY 1D COMPRESSIBLE FLUID FLOW
Local stagnation (or total) temperature
u2
T0  T 
2c p
For a constant cp-gas mixture between any two duct
sections 1 & 2, change in T0 is governed by heat addition
per unit mass:
q 12  c p (T0,2  T0,1 )
Since
Tds  c p dT   dp
(Gibbs ),
Entropy change & Ma at each point along Rayleigh line
may be calculated.
STEADY 1D COMPRESSIBLE FLUID FLOW
Steady one-dimensional flow of a perfect gas (with g1 .3) in a constant area duct,
frictionless flow with heat addition
STEADY 1D COMPRESSIBLE FLUID FLOW
 Steady compressible flow of a nonreacting gas mixture in
a constant-area duct with friction but without heat
addition:
 Conservation equations may be written as:
G=constant
(mass),
du dp
P
G

 w
d d
A
ho =constant
(energy)
2
(Fanno Locus)
1G 1
2
C p T0  T )      Gv )
2   2
STEADY 1D COMPRESSIBLE FLUID FLOW
Steady one-dimensional flow of a perfect gas (with g1 .3) in a constant area
duct, adiabatic flow with friction
SHOCK WAVES
 Discontinuity separating two adjacent continua
 e.g., mixture of perfect gases, same EOS valid on both
sides of discontinuity
 How do we apply conservation laws on field variables?
 Assume locally planar discontinuity, fixed in space, fed
by a gas stream with known velocity normal to it,
known thermodynamic state properties
SHOCK WAVES
Control volume and station nomenclature for applying conservation principles
across a gas dynamic discontinuity separating two regions of flow in which
diffusion processes can be neglected
SHOCK WAVES
 Consider a macroscopic control volume, shrunk down to
a “pillbox” of unit area straddling the figure
 Field variables “jump” across discontinuity
 “jump operator”
   ) 2  ) 1
SHOCK WAVES
 Conservation equations across a discontinuity without
chemical reaction:
(total mass ),
 u   0
  ui   0 ( species mass),
  uu     p  (momentum),
  uh0   0 (energy )
(entropy )
  us   0
SHOCK WAVES
 Mass flux
G  u ,
G   0 (total mass),
i   0 ( special mass),
G u     p  (normal momentum),
(energy ),
 h0   0
(entropy ),
s  0
SHOCK WAVES
 In general, G, wi, h0 are continuous (no jump) across
discontinuity; u, p, T, v (≡ 1/), s jump.
 Compatible with conservation principles, relevant EOS
 Combining total mass & normal momentum relations:
p

u 1 u 2
 
SHOCK WAVES
 When
discontinuity
becomes
a
sufficiently
weak
compression wave, positive entropy jump is negligible;
hence
1/2
 p 
u 1 u2 a 
   sconst
For a perfect gas, then:
1/2
 p 
a 
   s
 g RT 


M


1/2
SHOCK WAVES
 For a discontinuity of arbitrary strength, final state must
lie on intersection of Fanno and Rayleigh loci passing
through initial state on T-s diagram, corresponding to
common mass flux G
 Rayleigh line links all states with same p + Gu
(irrespective of heat addition)
 Fanno locus links all states with same stagnation
enthalpy irrespective of viscous dissipation
SHOCK WAVES
Fanno and Rayleigh loci for the same mass flux G, displayed on the T-s plane. The normal
shock transition goes from the supersonic intersection to the subsonic intersection
SHOCK WAVES
 Rankine – Hugoniot interrelation:
1
 h   p .2  1 ) .
2
 Defines a locus (“shock adiabat”) on p-v plane along
which final state must lie
 For a perfect gas, this relation is given by
 g 1  2



g 1   1
p2


,

p 1  g 1 2
 g 1 .  1

 1
exp
 s
R/M

p0, 1
p0,
2
1/ g 1)
 2g
g 1 
2

.Ma 1 

g

1
g

1



2
g 1
.


2
 g  1) Ma 1 g  1 
g /(g 1)
SHOCK WAVES
 For
strong compression waves, upstream flow is
supersonic, downstream subsonic
 Rules out possibility of rarefaction Ma 1  1 shocks
SHOCK WAVES
Rankine-Hugoniot “shock adiabat” on the p-v plane
SHOCK WAVES
 In terms of upstream (normal) Mach number:
p2
2g
 1
Ma 21  1
p1
g 1

T 2
T1
and
)
2
2
g
Ma
  g  1)
 g 1
2
1
 2 1 
Ma 1 
,
2
2
2

 g  1 Ma

)
1
 2  g  1)

 1  g  1) Ma 21  2
Ma 21
As Ma1  ∞, p2/p1 and T2/T1 also  ∞; however,  2 / 1
approaches the finite limit (g1)/g1)
SHOCK WAVES
Normal shock property ratio as a function of upstream
(normal) Mach number Ma ( for g =1.3 )
DETONATION / DEFLAGRATION WAVES
 Abrupt transitions accompanied by chemical reactions

i   0
 h must include chemical contributions
 Reaction may be seen as adding heat q per unit mass
to a perfect gas mixture of constant specific heat g

e.g., many fuel-lean/ air mixtures
DETONATION / DEFLAGRATION WAVES
 Generalized R-H conditions then become:
G   0,
G u     p  ,
c p T0   q,
q
s 
T2
DETONATION / DEFLAGRATION WAVES
 Detonation adiabat is above shock adiabat by an
amount depending on heat release q
 Detonations propagate with an end-state at or near
Chapman-Jouguet point CJ (figure on next slide)
 Singular point at which combustion products have
minimum possible entropy, and normal velocity of
products is exactly sonic (Ma 2 = 1)
DETONATION / DEFLAGRATION WAVES
Rankine -Hugoniot “detonation adiabats” on the p-v plane
DETONATION / DEFLAGRATION WAVES
 Imposing Ma2 = 1 yields:
Ma 1 1  H )  H 1/2
1/2
{+ sign  upstream Mach number for a CJ-detonation
(compression wave)
- sign  upstream Mach number for a CJ-deflagration
(subsonic combustion wave)}
where
g 2  1 Mq
H
.
2g RT 1
MULTIDIMENSIONAL INVISCID STEADY FLOW
 Even neglecting diffusion & non-equilibrium chemical
reaction, equations governing local conservation of mass,
momentum & energy for steady flow of a perfect gas
remain PDE’s
 In field variables
v ,  ,T and p
 Must be solved subject to conditions “at infinity” & vn
along body surface
DETONATION / DEFLAGRATION WAVES
 Simpler procedure:
 Reduce PDEs to one (higher order) PDE involving
only one unknown– velocity potential, v  x ) ,where:
v =  gradv
 All inviscid compressible flows admit such a potential,
with constant gradient far from the body

v / n  0 everywhere along body surface
MULTIDIMENSIONAL INVISCID STEADY FLOW
Scalar function v  x ) must satisfy non-linear 2nd order
PDE:
2
1
2
a div(gradv )  gradv .grad   gradv ) 
2

where
a 
2
g RT0
M

g 1
2
 grad v )
2
when a2  ∞(e.g., incompressible liquid):
div  grad v )  0
(Laplace’s Equation)
MULTIDIMENSIONAL INVISCID STEADY FLOW
 Hence,
many
simple
inviscid
flows
can
be
constructed using “potential theory” & analytical
methods (rather than numerical)
 Nature of solution depends on whether local flows are
supersonic or subsonic
 In case of upstream supersonic conditions, shock
waves can appear within flow field– piecewise
continuous

Energy can then be dissipated even in inviscid fluids

Interplay of diffusion & convection determines structure of
discontinuities