Basics of Fluid Mechanics, 2014a

11.5. NORMAL SHOCK 407
In a shock wave, the momentum is the quantity that remains constant because
there are no external forces. Thus, it can be written that
P x − P y = ( ρ x U y 2 − ρ y U x
2 ) (11.71)
The process is adiabatic, or nearly adiabatic, and therefore the energy equation can be
written as
C p T x + U x 2
2 = C p T y + U y 2
2
The equation of state for perfect gas reads
(11.72)
P = ρRT (11.73)
If the conditions upstream are known, then there are four unknown conditions
downstream. A system of four unknowns and four equations is solvable. Nevertheless,
one can note that there are two solutions because of the quadratic of equation (11.72).
These two possible solutions refer to the direction of the flow. Physics dictates that
there is only one possible solution. One cannot deduce the direction of the flow from the
pressure on both sides of the shock wave. The only tool that brings us to the direction
of the flow is the second law of thermodynamics. This law dictates the direction of the
flow, and as it will be shown, the gas flows from a supersonic flow to a subsonic flow.
Mathematically, the second law is expressed by the entropy. For the adiabatic process,
the entropy must increase. In mathematical terms, it can be written as follows:
s y − s x > 0 (11.74)
Note that the greater–equal signs were not used. The reason is that the process is
irreversible, and therefore no equality can exist. Mathematically, the parameters are
P, T, U, and ρ, which are needed to be solved. For ideal gas, equation (11.74) is
( )
Ty
ln − (k − 1) P y
> 0 (11.75)
T x P x
It can also be noticed that entropy, s, can be expressed as a function of the
other parameters. These equations can be viewed as two different subsets of equations.
The first set is the energy, continuity, and state equations, and the second set is the
momentum, continuity, and state equations. The solution of every set of these equations
produces one additional degree of freedom, which will produce a range of possible
solutions. Thus, one can have a whole range of solutions. In the first case, the energy
equation is used, producing various resistance to the flow. This case is called Fanno flow,
and Section 11.7 deals extensively with this topic. Instead of solving all the equations
that were presented, one can solve only four (4) equations (including the second law),
which will require additional parameters. If the energy, continuity, and state equations
are solved for the arbitrary value of the T y , a parabola in the T − s diagram will be
obtained. On the other hand, when the momentum equation is solved instead of the