Design and Simulation of Two Stroke Engines

Chapter 2 • Gas Flow through Two-Stroke Engines
Appendix A2.2 Moving shock waves in unsteady gas flow
The text in this section owes much to the often-quoted lecture notes by Bannister [2.2].
The steepening of finite amplitude waves is discussed in Sec. 2.1.5 resulting in a moving
shock wave. Consider the case of the moving shock wave, AB, illustrated in Fig. A2.2. The
propagation velocity is a and it is moving into stationary gas at reference conditions, po and
po. The pressure and density behind the shock front are p and p, while the associated gas
particle velocity is c. Imagine imposing a mean gas particle velocity, a, on the entire system
illustrated in Fig. A2.2(a) so that the regime in Fig. A2.2(b) becomes "reality." This would
give a stationary shock, AB, i.e., the moving front would be brought to rest and the problem is
now reduced to one of steady flow. Consider that the duct area is A and is unity.
The continuity equation shows across the now stationary shock front:
or
(a - c)p A = apoA (A2.2.1)
The momentum equation gives, where force is equal to the rate of change of momentum,
This can be rearranged as:
UJ
cr.
= >
OT
CO
LU
DC
0.
(a - (a - c))ap0A = (p - p0)A
capo = P ~ PO
P _ Po capp
P P P
B
•flL>
v p oT0po
DISTANCE
gas properties y R
reference po TQ
pipe area=unity
(a) shock wave AB moving rightwards
(b) shock wave imagined to be stationary
Fig. A2.2 The moving shock wave.
201
a
(A2.2.2)
(A2.2.3)