Page 112 - 48Fundamentals of Compressible Fluid Mechanics

74 CHAPTER 5. NORMAL SHOCK
2
be uniform . The conditions to the right of the shock wave are uniform, but different
from the left side. The transition in the shock is abrupt and in a very narrow width.
The chemical reactions (even condensation) are neglected, and the shock
occurs at a very narrow section. Clearly the isentropic transition assumption is not
appropriate in this case because the shock wave is a discontinued area. Therefore
the increase of the entropy is fundamental to the phenomenon and understanding.
It is further assumed that there is no friction or heat loss at the shock
(because the heat transfer is negligible due to the fact that it occurs at a relatively
small surface.). It is customary in this field to denote as the upstream condition
and as the downstream condition.
The mass flow rate is constant from the two sides of the shock and there-

fore the mass balance reduced to
(5.1)

In the shock wave, the momentum is the quantity that remained constant because

there are no external forces. Thus, it can be written that
(5.2)

The process is adiabatic, or nearly adiabatic, and therefore the energy equation

can be written

(5.3)

The equation of state for perfect gas reads

(5.4)

If the conditions upstream are known, then there are four unknown con-
ditions 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 (5.3). 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 flow from the pressure on both sides of the shock wave. The only tool
that brings us to the direction of 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
supersonic flow to 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:
(5.5)

2 Clearly the change in the shock is so significant compared to the changes in medium before and

after the shock that the changes in the mediums (flow) can be considered uniform.