Design and Simulation of Two Stroke Engines

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Chapter 2 - Gas Flow through Two-Stroke Engines
Appendix A2.1 The derivation of the particle velocity for unsteady gas flow
This section owes much to the text of Bannister [2.2], which I found to be a vital component
of my education during the period 1959-1962. His lecture notes have remained a model
of clarity and a model of the manner in which matters theoretical should be written by those
who wish to elucidate others.
The exact differential equations employed by Earnshaw [2.1] in his solution of the propagation
of a wave of finite amplitude are those established using the notation of Lagrange.
Fig. A2.1(a) shows a frictionless pipe of unit cross-section, containing gas at reference
conditions of po and po- The element AB is of length, dx, and at distance, x, from an origin of
time and distance. Fig. A2.1(b) shows the changes that have occurred in the same element AB
by time, t, due to the influence of a pressure wave of finite amplitude. The element face, A,
has now been displaced to a position, L, farther on from the initial position. Thus, at time t,
the distances of A and B from the origin are no longer separated by dx but by a dimension
which is a function of that very displacement; this is shown in Fig. A2.1(b). The length of the
f
element is now 1 + — dx. The density, p, in this element at this instant is related by the fact
dx)
that the mass in the element is unchanged from its initial existence at the reference conditions
and that the pipe area, A, is unity;
_c
D5
o
o \J
<
^
>
p0Adx = pA 1 + — dx (A2.1.0)
•w
W
< ^ >
^. ^
A B gas properties y R f\
reference po To / |
L+dx+ *P= dx
dx
pipe area=unity \J
(a) initial conditions
B
P+^dx
dX
(b) after time t
Fig. A2.1 Lagrangian notation for a pressure wave.
197