Basics of Fluid Mechanics, 2014a

406 CHAPTER 11. COMPRESSIBLE FLOW ONE DIMENSIONAL
The net force is obtained by utilizing equation (11.69)
F net = P 2 A 2
P 0 A ∗
P 2 A 2
(1 + k)
( k +1
2
) k (
k−1
F 2
F ∗ − F )
1
F ∗
= 500000 × 1
2.1 × 2.4 × 1.23.5 × (2.1949 − 0.96666) ∼ 614[kN]
End Solution
11.5 Normal Shock
In this section the relationships between the two
sides of normal shock are presented. In this discussion,
the flow is assumed to be in a steady state,
and the thickness of the shock is assumed to be
very small. A shock can occur in at least two different
mechanisms. The first is when a large difference
(above a small minimum value) between the
two sides of a membrane, and when the membrane
bursts (see the discussion about the shock tube).
Of course, the shock travels from the high pressure
flow ρ x
direction
P x
T x
c.v.
P y ρ y
T y
Fig. -11.11. A shock wave inside a
tube, but it can also be viewed as a
one–dimensional shock wave.
to the low pressure side. The second is when many sound waves “run into” each other
and accumulate (some refer to it as “coalescing”) into a large difference, which is the
shock wave. In fact, the sound wave can be viewed as an extremely weak shock. In
the speed of sound analysis, it was assumed the medium is continuous, without any
abrupt changes. This assumption is no longer valid in the case of a shock. Here, the
relationship for a perfect gas is constructed.
In Figure 11.11 a control volume for this analysis is shown, and the gas flows from
left to right. The conditions, to the left and to the right of the shock, are assumed to be
uniform 6 . 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. Therefore,
the increase of the entropy is fundamental to the phenomenon and the understanding
of it.
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 on a relatively small surface). It
is customary in this field to denote x as the upstream condition and y as the downstream
condition.
The mass flow rate is constant from the two sides of the shock and therefore the
mass balance is reduced to
ρ x U x = ρ y U y (11.70)
6 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.