Fig. 2.7 has lines of constant AK expressed as
Chapter 2 Gas Flow Through Two-Stroke Engines
AREA,F1
Xil + Xr2—t»
•«• Xr1 -* Xi2
(A) SUDDEN EXPANSION IN AREA IN A PIPE
AREA.F1
AREA,F2
\ !
AREA,F2 Xi1 *- Xr2 N-
•Xr1 •* Xi2
(B) SUDDEN CONTRACTION IN AREA IN A PIPE
Fig. 2.9 Sudden contractions and expansions in area in a pipe.
As Xsl equals Xs2, this reduces to:
Fl*(Xil-Xrl)=F2*(Xr2-Xi2) (2.3.21)
Joining Eqs. 2.3.21 and 2.3.19, and eliminating each of the unknowns in turn, Xrl or Xr2, and letting FR=F2/F1:
Xrl=[(l-FR)*Xil+2*FR*Xi2]/(l+FR) Xr2=[2*Xil-Xi2*(l-FR)]/(l+FR)
(2.3.22) (2.3.23)
To get a basic understanding of the results of reflections of compression and expansion waves at sudden enlargements and contractions in pipe area, consider an example using the two pressure waves, PRe and PRi, previously used in Sect. 2.1.4.
PRe is a compression wave of pressure ratio 1.2 and PRi is an expansion wave
of pressure ratio 0.8. Such pressure ratios are shown to give pressure amplitude
ratios of 1.02639 and 0.9686, respectively. Each of these waves in turn will be used
as data for Xil arriving in pipe 1 at a junction with pipe 2, where the area ratio will be either halved for a contraction or doubled for an enlargement to the pipe area. In each case the incident pressure amplitude ratio in pipe 2, Xi2, will be taken as unity, which means that the incident pressure wave in pipe 1 is facing undisturbed conditions in pipe 2.
(a) An enlargement, FR=2, for an incident compression wave Xil=l.02639 From Eqs. 2.3.22 and 2.3.23, Xrl=0.9912 and Xr2= 1.01759.
Hence, the pressure ratios, PRrl and PRr2, of the reflected waves are:
PRrl=0.940 and PRr2=1.130
The sudden enlargement behaves like a slightly less effective "open end," as a completely open ended pipe from Eq. 2.3.6 would have given a reflected pressure ratio of 0.829 instead of 0.940. The onward transmitted pressure wave in pipe 2 is one of compression, of reduced pressure ratio 1.13.
(b) An enlargement, FR=2, for an incident expansion wave Xil =0.9686 From Eqs. 2.3.22 and 2.3.23, Xrl=1.0105 and Xr2=0.97908.
Hence, the pressure ratios, PRrl and PRr2, of the reflected waves are:
PRrl = 1.076 and PRr2=0.862
As above, the sudden enlargement behaves as a slightly less effective "open end"
because a completely bellmouth open end to a pipe in Sect. 2.3.2 was shown to produce a reflected pressure ratio of 1.178, instead of 1.076 found here. The onward transmitted pressure wave in pipe 2 is one of expansion, of diminished pressure ratio 0.862.
(c) A contraction, FR=0.5, for an incident compression wave Xi 1 = 1.02639 From Eqs. 2.3.22 and 2.3.23, Xrl=1.0088 and Xr2=1.0352.
Hence, the pressure ratios, PRrl and PRr2, of the reflected waves are:
PRrl=1.063 and PRr2= 1.274
The sudden contraction behaves like a partially closed end, sending back a partial "echo" of the incident pulse. The onward transmitted pressure wave is also one of compression, but of increased pressure ratio 1.274.
(d) A contraction, FR=0.5, for an incident expansion wave Xil~0.9686 From Eqs. 2.3.22 and 2.3.23, Xrl=0.9895 and Xr2=0.9582.
Hence, the pressure ratios, PRrl and PRr2, of the reflected waves are:
PRrl=0.929 and PRr2=0.741
As in (c), the sudden contraction behaves like a partially closed end, sending back a partial "echo" of the incident pulse. The onward transmitted pressure wave is also one of expansion, with an increased expansion pressure ratio of 0.741.
The theoretical presentation here, due to Benson(2.4), is clearly too simple to be completely accurate in all circumstances, and the subject was studied recently by Richardson(2.20), who produced some alternative theoretical treatments for these sudden area changes. While the new treatment by Richardson(2.20) is more accurate, it is also more complex.
2.3.5 Reflections of pressure waves at branches in a pipe
The simple theoretical treatment for this situation was also suggested by Benson(2.4) and in precisely the same form as for the sudden area changes found in the previous section. A sketch of a typical branch is shown in Fig. 2.10. The sign convention for the branch theory is that inwards propagation of a pressure wave towards the branch will be regarded as "positive." Benson(2.4) postulates that the superposition pressure at the junction, at any instant of wave incidence and reflection, can be regarded as a constant. Therefore, the theoretical solution involves expansion of Eqs. 2.3.19 and 2.3.21 to deal with the superposition state and mass flow rate of the extra pipe 3 at the junction, thus:
Xs 1 =Xi 1 +Xr 1 -1 =Xs2=Xi2+Xr2-1 =Xs3=Xi3+Xr3-1 (2.3.24) The net mass flow rate at the junction is zero:
Fl*(Xil-Xrl)+F2*(Xi2-Xr2)+ F3*(Xi3-Xr3)=0 (2.3.25) There are three equations to solve for the three unknowns, Xrl, Xr2 and Xr3. It
is presumed that in the course of any computer calculation one knows the values of all incident pressure waves from one calculation time step to another.
The solution of the above simultaneous equations gives:
FT=F1+F2+F3
Xrl=[(2*F2*Xi2)+(2*F3*Xi3)+((Fl-F2-F3)*Xil)]/FT (2.3.26) Xr2=[(2*Fl*Xil)+(2*F3*Xi3)+((F2-F3-Fl)*Xi2)]/FT (2.3.27) Xr3=[(2*Fl*Xil)+(2*F2*Xi2)+((F3-F2-Fl)*Xi3)]/FT (2.3.28) where FT is the sum of the pipe areas.
The Basic Design of Two-Stroke Engines AREA.F3
AREA.F1 to
-Xr1 Xi1
AREA,F2 Xr2-
-Xi2 Fig. 2.10 Pressure waves at a branch in a pipe.
Perhaps not surprisingly, the branched pipe can act as either a contraction of area to the flow or an enlargement of area to the gas flow. Consider these two cases where all of the pipes are of equal area:
(a) A compression wave is coming down to the branch in pipe 1 and all other conditions in the other branches are "undisturbed." The compression wave has a pressure ratio of PRil=1.2, or Xi 1=1.02639. The result of the calculation is:
Xrl=0.9911, Xr2=Xr3=l.01759, PRrl=0.940 and PRr2=PRr3=1.13
As far as pipe 1 is concerned the result is exactly the same as that for the 2:1 enlargement in area in the previous section. In the branch, the incident wave divides evenly between the other two pipes.
(b) Compression waves of pressure ratio 1.2 are arriving as incident pulses in pipes 1 and 2. Pipe 3 is undisturbed as PRi3=1.0. Now the branch behaves as a 2:1 contraction to this general flow, for the solution of Eqs. 2.3.26-28 shows:
Xrl=Xr2=1.0088, Xr3=1.0352 and PRrl=PRr2=1.0632 while PRr3=1.274 These numbers are already familiar as computed data for pressure waves of identical amplitude at the 2:1 contraction discussed in the previous section.
When one has dissimilar areas of pipes and a mixture of compression and expansion waves incident upon the branch, the situation becomes much more difficult to interpret by the human mind. At that point the programming of the mathematics into a computer will leave the designer's mind free to concentrate more upon the relevance of the information calculated and less on the arithmetic tedium of acquiring that data.
It is also physically obvious that the angle between the several branches must play some role in determining the transmitted and reflected wave amplitudes. This subject was studied most recently by Bingham(2.18) and Blair(2.19) at QUB. While the branch angles do have an influence on wave amplitudes, it is not as great as might be imagined and can be neglected in most calculations. The discussions of Bingham(2.19) and Blair(2.19) deserve further study for those who wish to achieve greater accuracy for all such calculations.
2.4 Computational methods for unsteady gas flow
The original calculation method for unsteady gas flow is known as the method of characteristics and is usually attributed to De Haller(2.8). As this was in an era before the advent of the digital computer, this was a graphical method. It is not well suited for computational methods, although Jones(2.9) did succeed in transferring the mathematical theory of the graphical solution to a digital computer. Since the 1960's the most popular computational method has been that based on Riemann variables. It was first suggested as applicable for unsteady gas flow in ic engine ducting in a paper by Benson(2.10), and is based on earlier work by Hartree(2.11) and Rudinger(2.3). Since that time there have been further mathematical methods pursued in the search for a more rapid or more accurate solution of the basic theory.
Such techniques are variations of finite-difference theory(2.12) and the Lax- Wendroff approach(2.13).
2.4.1 Riemann variable calculation of flow in a pipe (a) 3 and 6 characteristics
As already discussed in Sect. 2.1.3, a pressure wave is propagated along a pipe in a relatively "pure" manner with little deterioration of its signal due to either friction or other viscosity effects. The Riemann variable technique is based on this knowledge and establishes dimensionless values of 3 and B to denote the rightward and leftward characteristics, respectively. The values of 3 and B are called the Riemann variables and appear from the mathematical solution of the equations of continuity and momentum applied to the unsteady flow regime. The result of this analysis is that the collection of thermodynamic terms, represented by 3 or B, are constants for a point on a propagating pressure wave in a parallel pipe, as illustrated in Fig. 2.11. They are discussed more fully by Rudinger(2.3) and by Benson(2.4).
The values of 3 and B are as follows:
3=a + ((g-l)/2)*u -for air 3=a + u/5 (2.4.1) B=a-((g-l)/2)*u -for air B=a - u/5 (2.4.2) The values of a and u are the dimensionless values of the local acoustic and
particle velocities of a wave point, where:
a=A/Ao (2.4.3) u=CGs/Ao (2.4.4)
The Basic Design of Two-Stroke Engines
c
slope of the line is dS/dZ=u+a
position-time characteristic
a=a+(g-1)*u/2
d is constant along this line
DISTURBANCE PROPAGATING RIGHTWARDS IN PIPE
D
Fig. 2.11 Position-time characteristic.
As the pressure amplitude ratio on a wave point is evaluated as, Xs=A/Ao,
and the particle velocity as,
CGs=(2/(g-l))*Ao*(Xr-Xl), or for air CGs=5*Ao*(Xr-Xl).
Then Eqs. 2.4.1 and 2.4.2 become:
d=Xr+Xr-l=2*Xr-l.
B=X1+X1-1=2*X1-1
(2.4.5) (2.4.6) The values of Xr and XI are the rightward and leftward values of pressure amplitude ratio which are superposed at the same instant in time and space as the
3 and B characteristics. Consequently, the superposition pressure and particle velocity can be deduced from the superposition pressure amplitude ratio Xs:
Xs=Xr+Xl-l=(3+B)/2 (2.4.7) CGs= (2/(g-l))*Ao*(Xr-Xl)=Ao*(d-B)/(g-l) (2.4.8)
For air, where the value of g is 1.4, this reduces to:
CGs= 5*Ao*(Xr-Xl)=2.5*Ao*0-B)
The density during superposition, Ds, originally derived in Eq. 2.1.16, can be evaluated by the Riemann variables as follows:
Ds=Do*Xs
(2/s-
1>=Do*[0+B)/2]
l2/8-" (2.4.9)
For air, where the value of g is 1.4, this reduces to:
Ds=Do*[(3+B)/2]
5Consequently, the mass flow rate, MGs, at any point in a pipe of area, FPs, becomes:
MGs=Ds*CGs*Fs = {Do* [(3+B)/2]
12*'») * {Ao*0-B)/(g-1)) *FPs (2.4.10) For air, where the value of g is 1.4, this reduces to:
MGs= (Do*[(3+B)/2]
5) * (2.5* Ao*(3-B)} *FPs
The characteristics can be plotted in the time-distance plane, just as for a graphical solution to the method of characteristics, and this is illustrated in Fig. 2.11.
The distance and time values are also declared as dimensionless parameters, S and Z, respectively, where:
S=s/L (2.4.11) Z=Ao*t/L (2.4.12) The values of s and t are distance and time, while L is the length of a calculation
segment in the distance plane, and is called the mesh length. As can be seen in Fig.
2.11, the slope of the 3 characteristic is given by:
AZ/AS=l/(a+lul) (2.4.13) The use of the absolute value of u, i.e., lul, allows for the calculation of the slope
of the B characteristic in the leftwards direction.
(b) The mesh layout in a pipe
The pipe is that segment of a duct in an engine between two boundaries where a reflection process is taking place at both ends. For example, it could be straight, tapered, or partially straight and partially tapered, between an exhaust port and an open end to the atmosphere. At each end reflections take place, but in the pipe the 3 and 8 characteristics are tracked from mesh to mesh in individual time steps. The layout in a pipe for such a mesh calculation is shown in Fig. 2.12.
0'1 131
a-2 G"2 02 i32
Ei-3 8-3 03 (33
o'4 13-4 64 84
a-5
(3-5
as
(35
0'6 (3-6
&6 136
o"7 13-7 07 87
0 8 13-8 0 8 138
d
PIPE BETWEEN TWO BOUNDARIES0"9 8-9 09 89
9 MESH POINTS
ARRAY NOMENCLATURE
I)
Fig. 2.12 Mesh calculation of Riemann variables between boundaries.
For pictorial reasons, nine meshes are displayed. Obviously, the number of meshes is linked to the mesh length, L, and the total pipe length. To put a number on a typical mesh length found to give acceptable accuracy in two-stroke engines, a value of L between 20 and 40 mm would be sensible. For four-stroke engines or medium-speed two-stroke diesel engines this value could be between 30 and 60 mm. In fact, the mesh length is usually adjusted to give a time step of between 1 and 2 crankshaft degrees in an application for any engine.
In a computer calculation, the 3 and B values are stored in arrays. For the example cited in Fig. 2.12, this would give 3(j) and B(j), for j=l to 9. At the start of any time step, all of these values would be known. The objective is to calculate the new values of 3 and B at each mesh point by the end of the time step, AZ. The new values of 3 and 6 at each mesh point at the end of the tjme step are shown in Fig. 2.12 as 3'(j) andB'(j), forj=l to 9.
(c) Reflection of characteristics at the pipe ends
However, not all values are given simply from the mesh calculation. There are two which emanate from the reflection of the appropriate characteristic at each end of the pipe. For example, 3V9 is the new value of the 3 characteristic arriving at the extreme right-hand boundary. Therefore, the value of Bv9 will be dependent on the
Chapter 2 - Gas Flow Through Two-Stroke Engines type of boundary involved. For instance, assume that the right-hand end of the pipe is a plain open end to the atmosphere. If the value of the 3*9 characteristic is greater than unity it means that it is a wave of compression and the solution for B'9 is based on Eq. 2.3.6:
X s = l
From Eq. 2.4.7, Xs=(3+B)/2.
Hence, the solution for B"9 is given by:
Xs=l=(3~9+B-9)/2 6-9=2-3-9
At the left-hand end of the pipe, the value of R" 1 is known but the value of 3V1 will depend on the type of boundary encountered there. Suppose that it is an exhaust port open to a cylinder. In this case the boundary condition chart UFLO given in Fig.
2.7, and contained in the computer file of the same name, must be used for evaluation of 3V1. It will be observed in Fig. 2.7 that the axes are also labelled in terms of 3 and B values. The x-axis has values of Bl/Xc, where 61 is the value of B* 1 in question.
The methodology of solution is exactly as discussed in Sect. 2.3.3, in terms of the port-to-pipe AK value and the cylinder pressure amplitude ratio, Xc. The y-axis is also plotted in terms of (3V l+6~ l)/2*Xc so that the parameter can be evaluated by interpolation, and 3~ 1 determined.
For any particular type of boundary (all of the relevant ones have been discussed in Sect. 2.3), the equations have already been presented for reduction to 3 and 6 parameters instead of the X values previously employed.
(d) Interpolation of 3 and B values at each mesh point
The selection of the time step, AZ, is a vital factor at each stage of the calculation.
It is not possible to select a time step which will allow all 3 and 6 characteristics to perfectly intersect the mesh-time point. Therefore, as illustrated in Fig. 2.13 for a 3 characteristic between the j and thej+1 mesh points, it will be necessary to linearly interpolate a value of 3w between the 3(j) and 3(j+1) values at time Z l , which would have the precise slope to intersect the j+1 position at time Z2. In this case, for a
"perfect" transmission of 3 characteristics, 3"(j+1) would equal 3w.
First, the time step AZ must be determined. This is done by searching through all of the mesh points for the fastest wave propagation characteristic. In this manner the value of AZ/AS, or l/(a+lul), is determined at each mesh position and the minimum value found for AZ is used, as that is the fastest wave propagation characteristic. This is known as the "stability criterion" in the paper by Courant, et al(2.13). This allows all wave point values to be determined by interpolation, for it cannot be done by extrapolation.