Design and Simulation of Two Stroke Engines

Chapter 2 - Gas Flow through Two-Stroke Engines
r2 , 2
M s + 111
M - 2Y 2 r 2 = —7 M s 7 i3_—— ^ij r4 = xsr2 f
y-l
1
s
i + r4 + r3r4 . v _i + r4-r3r4
then Xlnew= * and X2new = * {2225)
The new values of particle velocity, Mach number, wave pressure or other such parameters
can be found by substitution into Eqs. 2.2.20 to 2.2.24.
Consider a simple numeric example of oppositely moving waves. The individual pressure
waves are pi and p2 with strong pressure ratios of 2.3 and 0.5, and the gas properties are
air where the specific heats ratio, y, is 1.4 and the gas constant, R, is 287 J/kgK. The reference
temperature and pressure are denoted by po and To and are 101,325 Pa and 293 K, respectively.
The conventional superposition computation as carried out previously in this section
would show that the superposition pressure ratio, Ps, is 1.2474, the superposition temperature,
Ts, is 39.1 °C, and the particle velocity is 378.51 m/s. This translates into a Mach number,
Ms, during superposition of 1.0689, clearly just sonic. The application of the above theory
reveals that the Mach number, Ms neW) after the weak shock is 0.937 and the ongoing pressure
waves, pi new an d P2 new, have modified pressure ratios of 2.2998 and 0.5956, respectively.
From this example it is obvious that it takes waves of uncommonly large amplitude to
produce even a weak shock and that the resulting modifications to the amplitude of the waves
are quite small. Nevertheless, it must be included in any computational modeling of unsteady
gas flow that has pretensions of accuracy.
In this section we have implicitly introduced the concept that the amplitude of pressure
waves can be modified by encountering some "opposition" to their perfect, i.e., isentropic,
progress along a duct. This also implicitly introduces the concept of reflections of pressure
waves, i.e., the taking of some of the energy away from a pressure wave and sending it in the
opposite direction. This theme is one which will appear in almost every facet of the discussions
below.
2.3 Friction loss and friction heating during pressure wave propagation
Particle flow in a pipe induces forces acting against the flow due to the viscous shear
forces generated in the boundary layer close to the pipe wall. Virtually any text on fluid
mechanics or gas dynamics will discuss the fundamental nature of this behavior in a comprehensive
fashion [2.4]. The frictional effect produces a dual outcome: (a) the frictional force
results in a pressure loss to the wave opposite to the direction of particle motion and, (b) the
viscous shearing forces acting over the distance traveled by the particles with time means that
the work expended appears as internal heating of the local gas particles. The typical situation
is illustrated in Fig. 2.4, where two pressure waves, pi and p2, meet in a superposition process.
This make the subsequent analysis more generally applicable. However, the following
analysis applies equally well to a pressure wave, pi, traveling into undisturbed conditions, as
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