Design and Simulation of Two Stroke Engines

Chapter 2 - Gas Flow through Two-Stroke Engines
value of XRI for mesh 1. To proceed with the next time step of the calculation, the value of
XLI is required. What then is the value of XLI ? The answer, as before, is that it will depend on
what the right-hand end of the pipe is connected to, i.e., a branch, a restricted pipe, the atmosphere,
etc.
The GPB modeling method stores information regarding the "hand" of the two end meshes
in every pipe, and must be able to index the name of that "hand" at its connection to its own
discontinuity, so that the numerical result of the computation of boundary conditions at the
ends of every pipe is placed in the appropriate storage location within the computer.
Consider each boundary condition in turn, a restricted pipe, a cylinder, plenum or atmosphere,
and a branch.
(a) a restricted pipe, as sketched in Fig. 2.22(c)
Imagine, just as it is sketched, that mesh 1 is at the right-hand end of pipe 1 and that mesh
2 is at the left-hand end of pipe 2. Thence:
!XR1 = Xpl and 2XL = Xq2 (2.18.40)
The unknown quantities are the values of the reflected pressure waves, iprj and 2PR,
represented here by their pressure amplitude ratios, namely IXLI and 2XR. The analytical
solution for these, and also for the entropy gain on the downstream side of whichever direction
the particle flow takes, is to be found in Sec. 2.12. In terms of the notation in Fig. 2.8,
which accompanies the text of Sec. 2.12, the value referred to as Xpi is clearly the incident
wave XJI, and the value of Xq2 is obviously the incident pressure wave Xj2.
(b) a cylinder, plenum, or the atmosphere as sketched in Figs. 2.16 and 2.18
A cylinder, plenum, or the atmosphere is considered to be a large "box," sufficiently large
to consider the particle velocity within it to be effectively zero. For the rest of this section, a
cylinder, plenum, or the atmosphere will be referred to as a "box." In the theory given in Sees.
2.16 and 2.17, the entire analysis is based on knowing the physical geometry at any instant
and, depending on whether the flow is inflow or outflow, either the thermodynamic state
conditions of the pipe or the "box" are known values.
As previously in this section, the "hand" of the incident wave at the mesh in the pipe
section adjacent to the cylinder is an important element of the modeling process. Imagine that
mesh 1, as shown in Fig. 2.22(a), is that mesh at the left-hand end of the pipe and attached to
the cylinder exactly as it is sketched in either Fig. 2.16 or 2.18. In which case at the end of the
time step in computation, to implement the theory given in Sees. 2.16 and 2.17, the following
is the nomenclature interconnection for that to occur:
lXL = Xqj (2.18.41)
and pi2 = p01xg 7 (2.18.42)
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