Design and Simulation of Two Stroke Engines

Chapter 2 - Gas Flow through Two-Stroke Engines
reflected pressure, Xr, from a known value of the incident pressure, Xj, at the plain open end.
The solution is expressed as a function of the unknown quantity, Xr, as follows:
W4 x i + x r - J— x : z~rz f: ^GT
G5 i + G6(Xi + xr - i) v " X i _ 0 (2.8.16)
This is solved by the Newton Raphson method, i.e., by differentiation of the above equation
with respect to the unknown quantity, Xr, and iterating for the answer in the classic
mathematical manner. Thence the value of particle velocity, cr, can be derived "approximately"
from substitution into the equation below once the unknown quantity, Xr, is determined.
cs = ci + cr = G5ao(Xi-Xr) (2.8.17)
This is a far from satisfactory solution to an apparently simple problem. In actual fact, the
inflow process at a plain open-ended pipe is a singular manifestation of what is generally
called "cylinder to pipe outflow from an engine." In this case the atmosphere is the very large
"cylinder" flowing gas into a pipe! Thus, as this complex problem is treated in great detail
later in Sec. 2.17, and this present boundary condition of inflow at a plain-ended pipe can be
calculated by the same theoretical approach as for pipe to cylinder flow, further wasteful
simplistic explanation is curtailed until that point in the text has been reached.
2.9 An introduction to reflection of pressure waves at a sudden area change
It is quite common to find a sudden area change within a pipe or duct attached to an
engine. In Fig. 2.6 at position 10 there are sudden enlargements and contractions in pipe area.
The basic difference, gas dynamically speaking, between a sudden enlargement and contraction
in pipe area, such as at position 10, and a plenum or volume, such as at position 2, is that
the flow in the duct is considered to be one-dimensional whereas in the plenum or volume it
is considered to be three-dimensional. A subsidiary definition is one where the particle velocity
in a plenum or volume is so low as to be always considered as zero in any thermodynamic
analysis. This will produce a change in amplitude of the transmitted pulse beyond the area
change and also cause a wave reflection from it. Such sudden area changes are sketched in
Fig. 2.8, and it can be seen that the pipe area can contract or expand at the junction. In each
case, the incident wave at the sudden area change is depicted as propagating rightward, with
the pipe nomenclature as 1 for the wave arrival pipe, with "i" signifying the incident pulse,
"r" the reflected pulse and "s" the superposition condition. The X value shown is the conventional
symbolism for pressure amplitude ratio and p is that for absolute pressure. For example,
at any instant, the incident pressure pulses at the junction are pji and pi2 which, depending
on the areas Ai and A2, will give rise to reflected pulses, pri and pr2-
In either expansion or contraction of the pipe area the particle flow is considered to be
proceeding from the upstream superposition station 1 to the downstream superposition station
2. Therefore the properties and composition of the gas particles which are considered to
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