Design and Simulation of Two Stroke Engines

Chapter 2 - Gas Flow through Two-Stroke Engines
What is evident, however, is that during any simulation of unsteady gas flow or of the
thermodynamic processes within engines, it is imperative for its accuracy to use the correct
value of the gas properties at all locations within the engine.
2.2 Motion of oppositely moving pressure waves in a pipe
In the previous section, you were asked to conduct an imaginary experiment with Fred,
who produced compression and expansion waves by exhaling or inhaling sharply, producing
a "boo" or a "u...uh," respectively. Once again, you are asked to conduct another experiment
so as to draw on your experience of sound waves to illustrate a principle, in this case the
behavior of oppositely moving pressure waves. In this second experiment, you and your friend
Fred are going to say "boo" at each other from some distance apart, and af the same time.
Each person's ears, being rather accurate pressure transducers, will record his own "boo"
first, followed a fraction of time later by the "boo" from the other party. Obviously, the "boo"
from each passed through the "boo" from the other and arrived at both Fred's ear and your ear
with no distortion caused by their passage through each other. If distortion had taken place,
then the sensitive human ear would have detected it. At the point of meeting, when the waves
were passing through each other, the process is described as "superposition." The theoretical
treatment below is for air, as this simplifies the presentation and enhances your understanding
of the theory; the extension of the theory to the generality of gas properties is straightforward.
2.2.1 Superposition of oppositely moving waves
Fig. 2.3 illustrates two oppositely moving pressure waves in air in a pipe. They are shown
as compression waves, ABCD and EFGH, and are sketched as being square in profile, which
is physically impossible but it makes the task of mathematical explanation somewhat easier.
In Fig. 2.3(a) they are about to meet. In Fig. 2.3(b) the process of superposition is taking place
for the front EF on wave top BC, and for the front CD on wave top FG. The result is the
creation of a superposition pressure, ps, from the separate wave pressures, pi and p2. Assume
that the reference acoustic velocity is ao- Assuming also that the rightward direction is mathematically
positive, the particle and the propagation velocity of any point on the wave top,
BC, will be ci and ah From Eqs. 2.1.18-20:
ci=5ao(Xi-l) ai=ao(6Xi-5)
Similarly, the values for the wave top FG will be (with rightward regarded as the positive
direction):
c2 = -5a0(X2 - 1) a2 = -ao(6X2 - 5)
From Eq. 2.1.14, the local acoustic velocities in the gas columns BE and DG during
superposition will be:
ai = arjXi a2 = arjX2
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