Design and Simulation of Two Stroke Engines

Chapter 2 - Gas Flow through Two-Stroke Engines
2.18.3 The wave transmission during the time increment, dt
This is illustrated in Fig. 2.21. Consider the mesh J of length L. At the left end of the mesh
the rightward-moving pressure wave, PR, is not propagating fast enough at (XSR to reach the
right end of the mesh in time dt. Consequently, a value of superposition propagation velocity,
Op, from a wave point of pressure amplitude ratio, Xp, linearly related to its physical position
p and the PR and PRI values, will just index the right end of the time-distance point at time dt,
while it is being mutually superposed upon a leftward pressure wave, aq, emanating from
physical position q. This leftward-propagating pressure wave point of amplitude Xq will just
indent the left-hand intersection of the time-distance mesh during the time increment dt. The
values of Xq and ocq are also linearly related to their physical position and in terms of the
leftward wave pressures, XL and XLI, at either end of mesh J.
In short, the calculation presumption is that between any two meshes there is a linear
variation of wave pressure, wave superposition pressure and superposition propagation velocity,
both leftward and rightward, and that the values of Xp and Xq will, should no other
effect befall them, become the new values of rightward and leftward pressure wave at either
end of mesh J at the conclusion of time increment dt.
2.18.4 The interpolation procedure for wave transmission through a mesh
Having determined the time increment for a calculation step, and knowing the gas properties
within any mesh volume for that transmission, the simulation must now determine the
values of Xp and Xq, within the terms outlined above. The situation is as sketched in Fig.
2.21.
The propagation of rightward wave Xp through leftward wave Xq is conducted at superposition
velocities, ocp and aq, respectively. Retaining the sign convention that rightward
motion is positive, then from Eqs. 2.2.9 and 2.2.10, these values of propagation velocity are
determined as,
«p = a o( G 6 X P - G4Xq - l)
ar = -a0(G X„ - GAX
6^q 4^p -i)
(2.18.8)
(2.18.9)
The time taken from their respective dimensional starting points, p and q, is the same, dt,
where dt is equal to the minimum time step inferred from the application of the stability
criterion in Sec. 2.18.2.
Therefore, and determining the arithmetic values of the lengths xp and xq,
xp = apdt (2.18.10)
xq = ar
dt (2.18.11)
The dimensional values xp and xq also relate to the numeric values of Xp and Xq as linear
variations of the change of wave pressure between the two ends of the mesh J boundaries.
147