7
ME I 1984
AN ANALYTJCAL AND NUMERI CAL STUDY OF THE INTERACTION
OF RAREFACTION WAVES WITH AREA CHANGES IN DUCTS
PART 1:
AREA REDUCTIONS
.
.
November
1983TECHNiSCHE HOGESCHO
OL DELFT
LUCHTVAART -E RUlI,lTEVAARTTECHNIEK b Y C~,-,:.J01"~Er.:~{Kluyverweg 1 -
DELFT
J. J.
Gottlieb and T. Saito
UTIAS Report No. 272
CN ISSN 0082-5255
AN ANALYTICAL AND NUMERI CAL STUDY OF THE INTERACTION OF RAREFACTION WAVES WITH AREA CHANGES IN DUCTS
PART 1: AREA REDUCTIONS
by
J. J. Gott1ieb and T. Saito
Submitted March 1983
November 1983
UTIAS Report No. 272 CN ISSN 0082-5255
Acknowledgements
The authors would like to express their gratitude to both Dr. O. Igra and Mr. P. M. Ostaff for their capable assistance with the quasi-steady flow analysis and computations.
The assistance of Mr. R. L. Deschambault with the work proces-sing and use of his microcomputer and line printer to produce the man-uscript for this report are deeply appreciated.
The encouragement received from Professor I. I. Glass was very helpful.
Thanks are also extended to Miss H. S. Murty for proof reading the manuscript for this report.
The financial support received mainly from the Defence Research Establishment Suffield, Ralston, Alberta, Canada, and partly from the Natural Sciences and Engineering Council of Canada is also gratefully acknowledged.
J
Abstract
The interaction of a rarefaction wave with a gradual monotonic area reduction of finite length in a duct, which pro-duces transmitted and reflected rarefaction waves and other pos-si bIe rarefaction and shock waves, was studied both analytically and numerically. A quasi-steady flow analysis which is analyti-cal for an inviscid flow of a perfect gas was utilized first to determine the domains of and boundaries between four different wave patterns that occur at late times, af ter all local transient
disturbances from the interaction process have subsided. These
boundaries and the final constant strengths of the transmitted, reflected and other waves are shown as a function of both the incident rarefaction-wave strength and area-reduction ratio, for
the cases of perfect diatomic and monatomic gases. The
random-choice method was then used to solve numerically the nonstation-ary one-dimensional equations of gas motion for many different combinations of rarefaction-wave strengths and area-reduction ratios, for the case of perfect diatomic gases and air with a speGific-heat ratio of 7/5. These numerical results show clearly how the transmitted, reflected and other waves develop and evolve with time, until they eventually attain constant strengths, in agreement with quasi-steady flow predictions for the asymptotic
wave patterns. Note that in all of this work the gas in the area
reduction is initially at rest.
'
"
-
"
Table of Contents Title Page. Acknowledgements Abstract. Table of Contents. Notation. • 1. INTRODUCTION2. ANALYTICAL AND NUMERICAL ANALYSES 2.1 Quasi-Steady Flow Analysis • •
2.2 Nonstationary Flow Analysis.
3. RESULTS AND DISCUS SION • 3.1 Quasi-Steady Flow.
3.2 Nonstationary Flow.
4. CONCLUDING REMARKS
5. REFERENCES •
Figures • •
Appendix A: COMPUTER PROGRAM LISTING OF THE QUASI-STEADY FLOW ANALYSIS Appendix B: COMPUTER PROGRAM LISTING OF THE
NONSTATIONARY FLOW ANALYSIS
iv Page i i! • ii! iv v 1 1 1 4 5 5 8 13 14 17 Al
BI
Ir a C P Cv e h R-H N P R-s S Sd Su t tc l1t Notation speed of sound
specific heat at constant pressure specific heat at constant volume
total energy per unit volume [p/(Y-!)
+
pu2/21
specific enthalpy (CpT)
length of the area change or transition in the duct flow Mach number (u/a)
total number of grid zones for the flow field static pressure
gas constant specific entropy
local cross-sectional area of the duct duct area downstream of the area reduction duct area upstream of the area reduction time
time for the nonstationary flow to become quasi-steady time interval between successive spatial
distributions of pressure, flow velocity, density, and entropy
T temperature u flow velocity x distance
Notation (continued)
y specific-heat ratio (Cp/Cv)
p density
T nondimensional time (alt/~)
TC nondimensional time for the nonstationary flow to
becorne quasi-steady
ÓT nondimensional time interval (aiÓt/~) between
successive spatial distributions of pressure, flow velocity, density, and entropy
vi
. ,
1. INTRODUCTION
A number of basic analytical and numerical studies were done in the past for the interaction of a shock wave with an area change of finite length in a duct, in which the wave motion and flow were treated as one-dimensional. This interaction process that eventually establishes a well-defined quasi-steady flow pattern is now quite weIl understood (e.g., see Refs. 1 to 8 and especially Ref. 9). By contrast, no work of a similar nature is available for the case of a rarefaction wave interacting with an area change, although Refs. 1, 3, 10 and 11 contain relevant basic information. Yet, the passage of a rarefaction wave through an area change is a common feature of nonstationary gas flows encountered in both engineering practice and research. For example, these flows occur in the piping system of reciprocating engines and pumps, in gas transportation pipelines, and in shock tubes and blast-wave simulators that have an area change in the driver or at the diaphragm station (Refs. 10 to 14).
The case of a rarefaction wave (and other waves) moving in a duct with or without area changes can, of course,be hand led routinely by well-developed mathematical methods, as is illustrated by the follówing examples for solving general one-dimensional nonstationary flow problems. In early work the method of characteristics was employed to predict nonstationary flows in ducts; for example, see the work of Kudinger (Kef. 3) and Bannister & Hucklow (Ref. 11). In this work the flow through constant-area duct segments was treated as non-stationary and, for simplicity needed in hand calculations, the flow across each area change between constant-area segments was treated simply as steady. ~li th the advent of high-speed computers, however, modern gasdynamic computer codes based on the random-choice method, finite-difference methods, and the method of characteristics now handle the pipe flows much more realistically as
nonstationary everywhere, including each gradual area change (e.g., see Ref. 9 and Refs. 13 to 15). These general-purpose computer codes treat any rare-faction wave interaction with an area change as an integral part of the whole flow problem. They have not been applied, however, with the specific intent of making a thorough study of such an interaction process.
The purpose of the present extensive study is to present basic detailed results that apply in general to the specific case of a rarefaction wave inter-acting with an area reduction. Special attention is devoted to understanding the nature of the transient flow phenomena which will eventually establish a quasi-steady flow at late times, af ter all transient disturbances from the in-teraction process have subsided or decayed. A clear picture is then provided of the entire nonstationary interaction process. Note that the complementary investigation of the interaction of a rarefaction wave with an area enlargement has been completed (see Ref. 16).
2. ANALYTICAL AND NUHERICAL ANALYSES 2.1 Quasi-Steady Flow Analysis
A rarefaction wave that moves through a quiescent gas toward an area reduction in a duct is illustrated in Fig. 1. This wave produces a flow that moves in the opposite direction, as shown. Depending on the magnitudes of the area-reduction ratio (Su/Sd) and the incident rarefaction-wave strength (or pressure ratio P2/ Pl across this wave), the rarefaction wave interaction with the area reduction will result in only one of four different postulated wave
patterns shown schematically in Fig. 2. Transmitted and reflected rarefaction waves are depicted in wave patterns A, H, C and D.Additionally, an upstream-facing shock wave appears also in the area change in pattern B or downstream of it in pattern C, whereas an upstream-facing rarefaction wave appears
down-stream of the area change in pattern D. Owing to the existence of a shock wave
in patterns Band C, a contact surface or contact region also occurs in these patterns, as indicated by the double dashed lines.
The boundaries between patterns A, B, C and D are defined simply in the
following manner. For the boundary between patterns A and B, the tail of the
transmitted rarefaction wave of pattern A becomes vertical and the gas flow in
region 7 which enters the area change becomes just sonic (M7 = -1), whereas the
stationary shock wave of pattern B is only a Mach wave at the flowentrance to the area change (Su). For the boundary between patterns Band C, the station-ary shock wave of pattern Band the downstream-swept shock wave of pattern C become stationary at the flow exit of the area change (Sd), where they have the
same strength. Finally, for the boundary between wave patterns C and D, the
downstream-swept shock and rarefaction waves of patterns C and D, respectively,
become Mach waves. Hence, at each boundary the adjacent wave patterns have the
same limiting pattern.
The rarefaction wave interaction with the area reduction is initially a nonstationary flow process, and the flow solution obtained from the one-dimensional equations of motion will be numerical (see section 2.2). However, as local transient disturbances subside through wave reflection and coalescing processes, the flow will eventually become quasi-steady or steady. That is, the rarefaction and shock waves will eventually become distinct and develop constant strengths, and these waves and the contact region will eventually
separate developed regions of steady flow. The solution for the quasi-steady
flow for each asymptotic wave pattern (A, B, C and D) can be obtained in an analytical manner, and quite readily, as outlined in the rest of this section.
The concept and application of a quasi-steady flow analysis are both quite weIl known in gasdynamies (e.g., see Ref. 3). However, they are not presented in a form suitable for convenient utilization in obtaining all of the flow properties of the quasi-steady waves and steady-flow regions for patterns
A to D of the present paper. Consequently, the method of solution for pattern
A will be presented briefly, for illustration purposes.
For an inviscid flow of a perfect gas, the flow properties in regions 1 and 7, on either side of the transmitted rarefaction wave (see pattern A in Fig. 2), are connected by an equation for a negatively sloped characteristic line crossing a simple expansion wave (Ref. 3 or 17),
=
(With u1
=
0), and the isentropic relations2 - - a y-l 1 =
(::)2
Y/(Y-U (1) (2)The symbols p, T, a, P, u and Y denote the statie pressure, temperature, sound
speed, density, flow velocity and specific-heat ratio, respectively. If the
strength of the transmitted rarefaction wave P7/ Pl is specified initially (for
convenience instead of P2/Pl)' all flow properties in region 7 can be obtained directly from Eqs. 1 and 2, because the flow properties in region 1 are known initial conditions. Note that the flow velocity in region 1 is taken to be zero in the present work.
For a steady, one-dimensional, isentropic flow through the area change, from region 7 to region 3, the continuity and energy equations (Ref. 3 or 17)
=
along with the sound-speed relation a2 enthalpy h=
CpT YRT/(Y-l), yield12
= =P7
=
(3)
+
(4)yRT Yp/p, and the definition of the
(
2
+
(y-l)M~)y/(y-l).2
+
(y-l)M~(5)
The symbols h, Rand H denote the specific enthalpy, gas constant and flow Hach
number (u/a), respectively. Hecause H7 is dictated by previously determined information (i.e., u7 and a7 ) and the duct cross-sectional areas upstream (Su) and downstream (Sd) of the area reduction are specified, M3 can be obtained from the latter part of Eq.
S.
Values for P3' T3 , a3, and P3 then follow fromthe former part of Eq. S, and u3 is obtained from the product a3H 3•
Region 2 lies behind the incident rarefaction wave and ahead of the reflected rarefaction wave (see pattern A in Fig. 2). The flow properties in this region are connected to region 1 by an equation for a negatively sloped characteristic line,
2
y-l
(6)
and to region 3 by a similar equation for a positively sloped characteristic line,
+
+
(7)These two equations yield a2 and u2• Isentropic expressions like those given by Eqs. 2 then yield the values of P2' T2 and P2 • This completes the method of solution for obtaining all flow properties and wave strengths for pattern A.
The procedure for obtaining a complete set of flow properties for wave patterns A, B, C and D, for some specific area-reduction ratio, is outlined here. The solution for pattern A covers a limited range of strengths of the incident and transmitted rarefaction waves. For the transmitted wave the ratio P7/ Pl takes on the values from unity, for which there is no flow and all waves
are Mach waves, to a minimum value of [P7/Pl]min
=
[2/(Y+l)]2Y/(Y-l),
when the transmitted wave is strongest. At [P7/ P l]min the tail of the transmitted rare-faction wave has become vertical, as shown in pattern B of Fig. 2, and the flow entering the area change from region7
is just sonic with M7=
-1.
For pattern A, the pressure ratio P7/Pl is first decreased in small increments from the maximum value of unity to its minimum value of [P7/Pl]min=
[2/(y+l)]2Y/(Y-l),
and the flow properties in regions 2 and 3 are then calculated at each step by using the previously described method and equations. Note that ifp /Pl is specified the flow properties in regions 2, 3 and 7 (including P2/ P l) can be obtained directly, whereas if P2/Pl was specified initially the flow properties (including P7/Pl) could be determined only by iteration.
For pattern B, a stationary upstream-facing shock wave with a strength ps/ps greater than unity is stationary in the area change. In the calcula-tions its location can be changed in small increments from the smallest area Su for which ps/ps
=
1 to the largest area Sd for which Ps/ps=
[Ps/Ps]max. This procedure is again convenient to obtain the flow properties (including P2/ Pl) without using an iterative methode The sudden change in the flow properties across this stationary shock wave, from supersonic to subsonic flow conditions, has to be included in the calculations by using the well-known Rankine-Hugoniot relations (Ref. 3 or 17). The flow properties in regions 2, 3 and 4 of pattern Bare then calculated in a manner similar to that for pattern A. The contact region separating regions 3 and 4 is handled easily because both the pressure and flow velocity are unchanged across it.For pattern C, an upstream-facing shock wave that is swept downstream by the oncoming supersonic flow occurs with a contact region also being swept downstream of the area change. In the calculations the strength Ps/Ps of this downstream-swept shock wave is reduced in small increments from its maximum value of (Ps/Ps)max to its minimum value of unity when this shock wave becomes a Hach wave. The flow-property changes across this shock wave and the speed or velocity at which it is swept downstream follow directly from the well-known Rankine-Hugoniot relations.
Finally, for pattern D, an upstream-facing rarefaction wave that is swept downstream by the oncoming supersonic flow replaces the downstream-swept shock wave, and a contact region does not appear. In the calculations, the strength P3/ PS of this downstream-swept rarefaction wave is reduced increment-ally from its upper value of unity, for the case of a Mach wave, to its minimum value of zero, for which this rarefaction wave is strongest. The flow-property changes across this downstream-swept rarefaction wave are determined by using expressions like those given by Eqs. 1 and 2. Note that if P3/PS is reduced to zero, P2/ Pl also goes to zero, and the incident rarefaction wave now becomes a complete expansion wave with its fan of characteristics spread out to its maxi-mum extent. It would be much wider spread than originally sketched in wave pattern D of Fig. 2.
2.2
Nonstationary Flow AnalysisThe continuity, momentum and energy equations for one-dimensional, non-stationary, compressible gas flows, in conservation form, are (Refs. 3 or 17)
d
a-t(p)
+
d dX (p u)4
_ 1:.
dS (p u) ,t
a
at
(p u)a
- (e) at+
+
a
ax (u e+
up)=
(9) 1 dS- S
dx (u e + up) , (10)where the new symbols x, t, S and e denote distance, time, duct cross-sectional area (local), and total energy per unit volume, respectively. The total energy is the sum of the internal energy PCvT and the kinetic energy pu2/2, which is of ten expressed as p/(Y-1)
+
pu2/2. This set of equations is completed by thethermal equation of state for a perfect gas, that is, p
=
pRT.For the solution of the problem of a rarefaction wave interaction with an area reduction in a duct, the specific area variation Sex) must be known. In the present work the area change between two constant-area ducts of upstream area Su and downstream area Sd is specified by
sex)
(ll )where x
=
0 at the large end (Sd) and x=
~ at the small end (Su). This par-ticular area transition is monotonic, smooth and applies equally weIl for both an area enlargement and reduction. It was chosen because1 dS
S
dx (12)is a symmetric, sinusoidal distribution, which is advantageous over asymcet-rical variations in reducing numeasymcet-rical noise in computed flow properties.
Equations 8 to 12 are solved numerically in the present study by using the random-choice method (RCM) invented by Glimm (Ref. 18) and first applied by Chorin (Ref. 19), ·which is weIl sui ted for solving such problems. Shock waves and contact surfaces with sharp fronts are weIl defined by this method, unlike in finite-difference methods where they are smeared out over many mesh zones, owing to the effects of explicit artificial and implicit numerical viscosities. The operator-splitting technique introduced to the RCM by Sod (Ref. 20), in order that one-dimensional flow problems with area changes could be solved, is also used in the present study. Note that the RCM is a first-order, explicit numerical scheme that repeatedly solves a Riemann or shock-tube problem between adjacent grid points, to get the solution at the next time level, and details of this method can be found in Refs. 18 to 21.
3. RESULTS AND DISCUS SION 3.1 Quasi-Steady Flow
For a rarefaction wave interacting with an area reduction, four dif-ferent wave patterns shown schematically in Fig. 2 were postulated, and the quasi-steady flow analysis for these patterns was outlined in section 2.1. By using the quasi-steady flow analysis, the domains and boundaries for patterns
A, B, C and D are now calculated as a function of the pressur~ ratio P2/ P l of
the incident rarefaction wave and the area-reduction ratio SuiSde The results
for perfect diatomic gases and air with Y
=
7/5 are presented first in Fig. 3.Additional results are presented in Fig. 4 for the cases of perfect monotomic
gases with
Y
=
5/3
(solid lines) and a perfect polyatomic gas withY
=
11/10
(dashed lines), in order to show the effects of different specific-heat ratios.
The results for diatomic gases with Y
=
7/5 lie about midway between the solidand dashed lines. The effect of the specific-heat ratio is important but not
unduly pronounced. Note that, if the specific heat ratio takes on the value of
unity, the resulting boundaries are displaced only slightly to the right of the dashed lines for
Y
=11/10.
The boundaries for each different gas have a confluence point at the
top of the graph where Su/Sd = 1 and P2/Pl =
[2/(Y+l)]2Y/(Y-l).
From thispoint they separate and run downward, and each one ends at a different point on the bot tom of the graph where Su/Sd
=
O. The values of P2/ Pl at these three end points can be obtained by using the appropriate quasi-steady flow equations and taking the limit as Su/Sd goes to zero. The end-point values are given by[
1/212Y/(Y-O
t
+
;[Y!I)
,
(13)[
~
.!.[r..:l:.)
2y+l
3/21 2Y /(Y-O
(14)[
~
.!.[r..:l:.)
2y+l
1/212Y/(Y-O
(15)for the boundaries between wave patterns A
&
B, B&
C and C&
D, respectively. Consequently, for any given area ratio in the range 0<
Su/Sd<
1, thequasi-steady flow analysis predicts that all four wave patterns can occur, depending
on the incident rarefaction-wave strength P2/ Pl. For the limiting and trivial
case for which Su/Sd = 1 (no area change), only patterns A and D are possible. In the case of pattern A, th.e incident rarefaction wave merely becomes the
transmitted wave, and no reflected wave occurs. In the case of pattern D, the
first part of the incident rarefaction wave, from its head (zero flow) to a characteristic line which is vertical (sonic flow), becomes the transmitted
wave. The latter part, from the vertical characteristic to the tail (where
the flow is supersonic) becomes the upstream-facing but downstream-swept
rare-faction wave, and no reflected wave exists.
The strength P7/Pl of the transmitted rarefaction wave is now presented
in Fig. 5, in terms of the incident rarefaction-wave strength P2/Pl and
area-reduction ratio Su/Sd, for perfect diatomic gases and air with Y
=
7/5. Withina triangular-shaped region, corresponding to the case of wave pattern A, P7/Pl
is a unique function of P2/P 1 and SuiSde Along the line on the left-hand side
where Su/Sd = 1, for the case of no area change, the strength of the transmit-ted rarefaction wave is, of course, equal to that for the incident rarefaction
wave. For all other area-reduction ratios within the triangular region, P7/Pl
is less than P2/ Pl' showing that the transmitted rarefaction wave is stronger
6
than the incident rarefaction wave. The transmitted wave is strongest at the
right-hand side where Su/Sd
=
0, that is, the duct of area Sd is infinitelylarger than the duct of area Su. As P2/Pl is reduced from unity, for a given
value of the area ratio, P7/Pl decreases from unity within th~ tr!angular
region and eventually reaches its lower limit of [2/(Y+l)]2Y/~Y-l) or 0.279
for the bottom border of the triangular region. Further reductions in P2/ Pl'
which will produce wave patterns B, C and then D, do not alter P7/ Pl from its
lower limit, since the transmitted rarefaction wave cannot accelerate the flow
beyond its sonic speed at the flowentrance to the area change. This is
sim-ilar to a steady flow from a constant-pressure reservoir which is choked at the nozzle throat if the downstream pressure is sufficiently low.
The strength P3/ P2 of the reflected rarefaction wave is now shown in
Fig. 6, as a function of both P2/Pl and Su/Sd. AIso, the boundaries
corres-ponding to wave patterns A to D are indicated by the dashed lines. For each
given area ratio in the range 0
<
Su/Sd<
1, a continuous variation in P3/P2from unity to zero occurs through regions A to D as P2/Pl decreases from unity
to zero. Over most of the range of Su/Sd not too close to zero, P3/P2 is
lar-ger than P2/Pl and the reflected wave is weaker than the incident wave. The
reverse situation occurs for very small values of SuiSde For the simple case
of no area change, if Su/Sd
=
1, the reflected rarefaction wave is simply notpresent, because P3/P2
=
1, and only patterns A and D are possible as mentionedin section 2.1. On the other hand, for the case of Su/Sd
=
0, one can showeasily that P
3/P2 = [2 - (Pl/P2)(Y-l)/2Y]2Y/(Y-l), and only patterns A and B
can occur. This corresponds to the case for which the incident rarefaction
wave is reflected from the closed end of a duct, because the flow from the duct of area Su into the infinitely larger duct of area Sd has a negligible effect on the wave reflection process. However, this relatively minor flow from the small to the large duct can be entirely subsonic and shock wave free, as for pattern A, or partly supersonic and then subsonic af ter the stationary shock wave, as for pattern B.
It is interesting to observe that changes in P2/ Pl produce relatively
small changes in P3/ P2 in region C of Fig. 6, whereas they produce much larger
changes in P3/ P2 in region D. This behavior is a direct result of the
occur-ence of the downstream-swept shock wave in pattern C and the downstream-swept
rarefaction wave in pattern D. The shock wave causes an increase in pressure
in region 3 of the wave pattern, whereas the rarefaction wave ca~ses a decrease
in pressure.
The strength PS/P6 of the stationary shock wave in pattern Band the
strengths PS/P6 and P3/ P6 of the downstream-swept shock and rarefaction waves
of patterns C and D, respectively, are shown in Fig. 7, for the case of
per-fect diatomic gases and air with Y
=
7/5. For a given area ratio Su/Sd anddecreasing values of P2/Pl' the stationary shock first appears with a strength
PS/P6 = 1 at the lower boundary of region B, and its strength then increases monotonically to its maximum value at the boundary between regions Band C.
For a further reduction in P2/ Pl' the strength PS/P6 of the downstream-swept
shock wave decreases from this maximum value to unity at the boundary
separ-ating regions C and D. Thereafter, the strength P3/ P6 of the downstream-swept
rarefaction wave decreases rapidly in region D to zero as P2/Pl goes to zero.
For the case of monatomic gases with Y
=
5/3, graphical results for thestrengths of the transmitted rarefaction wave, reflected rarefaction wave, and upstream-facing shock and rarefaction waves are now presented in Figs. 8, 9 and 10, respectively. These results are, in general, very similar to those given
in Figs. 5 to 7 for diatomic gases with Y
=
7/5, and any further discussion ofthese results is, therefore, unnecessary.
The listing of the computer program for the quasi-steady flow analysis is given for interest and possible future use in appendix A. This program was employed to generate all of the data used to plot the graphical results given in Figs. 3 to 10.
3.2 Nonstationary Flow
Numerical results obtained by the RCM for the interaction of a rare-faction wave with an area reduction are now presented graphically and discus-sed, in order to illustrate how the transmitted, reflected and other waves form, evolve with time, and eventually attain constant strengths as they be-come quasi-steady, in agreement with the quasi-steady flow predictions for the asymptotic wave patterns. Computations were made for many different combina-tions of the incident rarefaction-wave strength and area ratio, and the numer-ical results for sixteen different cases are presented and discussed.
A
very convenient graphical summary of the locations of the sixteen cases in theSu/Sd-versus-P2/PI plane is given in Fig. 11. This figure will be very helpful later, in quickly locating the position of a particular case; that is, in de-termining quickly whether it is in the center of a particular domain or near some particular boundary.
Numerical results for the sixteen cases are given in Figs. 12 to 27, in the form of separate sets of spatial distributions at successive time levels for nondimensional pressure P/PI' flow velocity u/al' density p/p l and entropy (s-sl)/R. Each successive distribution"is displaced upward slightly from the previous one, both for clarity and to produce the effect of a time-distance diagram. The nondimensional time interval between adjacent distributions is given by ~T
=
al~t/~, and the nondimensional value of ~T for each case is given in the caption of the corresponding figure. The location of the area reduction of length ~ is shown by the two vertical dashed lines in each set of spatial distributions. Furthermore, the flow field is computed with 720 grid zones, of which 60 grid zones are allocated to the area reduction.The first set of numerical results for pressure, flow velocity, density and entropy is presented in Fig. 12 (a to d), for the case of P2/Pl
=
0.65 and Su/Sd = 0.75, corresponding to a point located in the upper part of the domain of pattern A in Fig. 3. The incident rarefaction wave is shown in the bottom distribution, to the left of the area reduction or vertical dashed lines, just prior to its impingement on the area reduction. lts subsequent interaction can be observed in the following time sequential distributions for pressure (a), flow velocity (b), density (c) and entropy (d), where the formation and the evolution of the transmitted and reflected rarefaction waves, as weIl as the eventual development of steady subsonic flow in and on both sides of the area change, can he clearly seen. For the present case of a small area reduction, the reflected rarefaction wave is quite weak, relative to the incident wave, and barely noticeable. It would have been stronger and more visible if Su/Sd had been lower than 0.75. Note that the steady flow in the area change at la-ter times decelerates (becomes less negative) and the pressure increases as the gas moves from right to left through an area enlargement. This is the ex-pected flow behavior for such a subsonic diffuser. Furthermore, note that the flow in the duct is entirely isentropic, since shock waves are not present, as can be seen from the constant entropy distributions.Other numerical results for different values of P2/Pl and Su/Sd, cor-responding to the upper part of the domain of pattern A where Su/Sd
>
0.45, are similar to those given in Fig. 12. However, numerical results for the lower part of this domain showed one interesting anomaly. A compression wave that might be initially smooth or have one or more coalescing shocks was found to form in the area reduction and then follow the tail of the transmitted rare-faction wave. This can be seen in the second set of results given in Fig. 13, for which P2/ Pl=
0.80 and Su/Sd=
0.25. In this case the overpressure change(P-Pl)/Pl across this compression wave with two coalescing shocks is approxi-mately 0.08, which is not negligible when compared to the pressure changes of 0.20 and 0.35 across the incident and transmitted waves, respectively.
When the results shown in Fig. 13 were extended to later times (not shown), it was found that the compression wave with the two shocks steepened into one shock wave with one steep front. This shock wave also overtook and interacted with the transmitted rarefaction wave. It was obvious that this interaction process would proceed slowly with time, continuously eroding the tail of the rarefaction wave and simultaneously reducing the strength of the overtaking shock wave, until only a weaker transmitted rarefaction wave is eventually left (Refs. 22 and 23). The final strength of this transmitted wave should then agree with the quasi-steady flow prediction.
It is worth pointing out th at the formation of a compression wave which results in a steep-fronted shock wave at the tail of a rarefaction wave that is moving into an area convergence is not some new phenomenon, but rat her a simple variation of a known feature of spherical explosions. For example, the sudden release of a high-pressure sphere of gas into another surrounding gas not only produces an outward moving shock wave and inward moving rarefaction wave, but it also generates an imploding shock wave at the tail of the rarefaction wave
(Ref. 21). This imploding shock wave, resulting from a steepening compression wave owing to the spherical geometry, reflects at the origin to form the second outward moving shock wave. In the present problem, however, the final area convergence to a focus is absent. Thus, for sufficiently large area reductions for which a compression or shock wave forms behind the rarefaction wave, a transmitted rarefaction wave followed by a compression or shock wave should be expected.
Numerical results for four additional cases, for which P2/Pl and Su/Sd correspond to different points in the domain of wave pattern A (Fig. 11), are now given in Figs. 14 to 17. Besides substantiating a number of the previous comments, these results provide much additional insight into the nature of the transient flow behavior. The transmitted rarefaction wave shown in Fig. 14 is not widely spread out, and a quickly growing quasi-steady region between its tail and the area reduction occurs, because P2/Pl = 0.80 and Su/Sd = 0.75 cor-respond to a point quite far away from the boundary between patterns A and B. The opposite happens in the results given in Fig. 15, because P2/Pl
=
0.47 andSu/Sd
=
0.80 now correspond to a point very close to the boundary between patterns A and B. Similar results to those in Fig. 15 can be seen in Fig. 17, for which P2/Pl=
0.66 and Su/Sd=
0.30 correspond to a point that is also very close to the boundary between patterns A and B. In this case, however, the quasi-steady region that develops between the area reduction and the tail of the transmitted rarefaction wave contains fairly large fluctuations. These fluctuations are mostly due to the numerical method, and the fluctuations can be reduced by using a finer grid containing more than 720 grid zones. How-ever, these fluctuations mayalso suggest that the flow field is on the verge of containing a stationary upstream-facing shock wave in the area reduction,because the initial conditions chosen correspond to a point that is very close
to the domain of pattern B. Finally, we come to the numerical results given in
Fig. 17, for which P2/Pl = 0.80 and Su/Sd = 0.50 for a point that is far away
from the boundary between patterns A and B. These results are interesting in
that they show a weak but noticeable compression wave following the tail of the
transmitted rarefaction wave.
It is apparent from the results shown in Figs. 12 to 17 that the wave pattern that emerges at late times is pattern A, even for those points taken
very close to the boundary to pattern B. Furthermore, quasi-steady wave
pat-tern A emerges fairly quickly in the numerical results, and the wave strengths
c·onverge fairly rapidly to those predicted by the quasi-steady flow analysis.
This occurs shortly af ter the tail of the incident rarefaction wave enters the area change and shortly af ter the tails of the transmitted and reflected rare-faction waves leave the vicinity of the area change, af ter which quasi-steady regions of increasing extent begin to develop on each side of the area change. The flow properties in these growing regions, as computed by the RCM, are gen-erally within 5% of the quasi-steady flow predictions by the time that the tail of the last wave leaving the area change has moved about two area-transit ion
lengths (2~) away from the area change. Also, at this time, the strengths of
both the reflected and transmitted waves are finally within 5% of the quasi-steady flow predictions, except for the case when a compression or shock wave
overtakes the transmitted wave. In this case the strength of the transmitted
wave will take a much langer time to decay to within 5% of its final quasi-steady flow prediction.
The seventh set of numerical results for pressure, flow velocity,
den-sity and entropy are presented in Fig. 18, for the case of P2/P 1 = 0.45 and
Su/Sd =.0.40, which now corresponds to a point near the center of the domain
of pattern B. The first part of the incident rarefaction wave moves all of the
way through the area reduction and establishes the transmitted rarefaction
wave. The tail of this transmitted rarefaction wave becomes stationary at the
flowentrance to the area change where the flow becomes sonic, and na steady-flow region develops upstream of the area change. The lat ter part of the in-cident rarefaction wave cannot move through the area reduction, against the high-speed oncoming flow that is at least sonic at the flowentrance to the area change. However, this latter part of the incident wave produces a low pressure in the area change, which causes the flow to accelerate from sonic speed to increasing supersonic speeds as it passes through the area change, like the flow in a supersonic nozzle. Any compression wave that farms at the tail of the incident rarefaction wave as it moves into the area reduction would
also be stalled by the high-speed oncoming flow. Consequently, a
sufficient-ly strong upstream-facing shock wave develops in the area change and eventual-ly becomes stationary near the center (for the present case). This shock wave then terminates the oncoming supersonic flow and a subsonic diffuser flow then follows in the downstream part of the area change. The reflected rarefaction wave in the present example is relatively weak compared to the incident wave, but it is visible in the numerical results.
During the formation of the upstream-facing shock wave a contact region of changing density and entropy is produced, which is swept downstream by the flow at the local flow velocity (see Fig. 18c and 18d). When the upstream-fac-ing shock wave is stationary in the area change, it then produces a new, grow-ing quasi-steady flow region with a constant density and entropy between the contact region and the area reduction. Note that the large fluctuations in
the entropy distributions in Fig. 18d stem mainly from the RCM. Small random
variations in placing the upstream-facing shock wave in the area change lead to large entropy fluctuations. These random variations and therefore the entropy fluctuations can be reduced by using a finer grid for the computations, or a more appropriate random-number algorithm (see Ref. 16), but this was not deemed necessary for the present study.
Three complementary sets of numerical results are presented in Figs. 19 to 21, for values of P2/ Pl and Su/Sd that also lie in the domain of pattern B. The results shown in Fig. 19 for the conditions P2/ P l
=
0.45 and Su/Sd=
0.65 are quite similar to the previous case, because only the area-reduction ratio has been changed, and not significantly, from 0.40 to 0.65. Because P2/Pl and Su/Sd correspond to a point that lies midway between boundaries A and B, the upstream-facing shock wave again forms near the center of the area reduction. However, the shock strength is weaker for this latter case, because the area reduction is less severe. For the next case of P2/Pl=
0.65 and Su/Sd=
0.20, for a point nearthe boundary between patterns A and B, the results in Fig. 20 show that the upstream-facing shock wave forms near the flowentrance to the area change. On the other hand, for the last or final case of P2/Pl=
0.38 and Su/Sd=
0.45, for a point near the other boundary to pattern C, the results in Fig. 21 now show that the stationary shock wave is located near the flow exit of the area change.In all of the numerical results shown graphically in Figs. 18 to 21, the wave pattern that emerges at late times is always pattern B, even if points are taken very close to the boundary of pattern either A or C. Furthermore, quasi-steady wave pattern B emerges fairly quickly in the numerical results, but not as quickly as that for pattern A. Shortly af ter the tail of the in-cident wave enters the area reduction, the tail of the reflected wave leaves the area change and the upstream-facing shock wave becomes stationary. Shortly thereafter the flow becomes quasi-steady, with the wave amplitudes and the flow properties in the steady regions being within 5% of the predicted quasi-steady values.
The eleventh set of numerical results for the pressure, flow velocity, density and entropy are presented in Fig. 22, for the case of P2/Pl
=
0.15 and Su/Sd=
0.20, which now corresponds to a point in the lower part of the domain of pattern C. As for the previous case of pattern B, the first part of the incident rarefaction wave again propagates right through the area reduction and establishes the transmitted rarefaction wave with a stationary tail at the flowentrance to the area change. The center part of the incident wave also reduces the pressure in the area change, such that an upstream-facing shock wave with a contact region is formed. However, in this case, the additional, latter part of the strong incident rarefaction wave slowly overtakes the down-stream-swept shock wave. This wave interaction process results in a gradual weakening of the upstream-facing shock wave and the gradual disappearance of the overtaking rarefaction wave (Refs. 23 and 24). As the shock wave weakens it can no longer maintain its stationary position, and it is swept downstream by the oncoming supersonic flow, as shown in Fig. 22. For this particular example, the wave interaction process will end when the tail of the incident rarefaction wave will eventually overtake the shock wave (not shown in the figure). A weaker upstream-facing but downstream-swept shock wave will then emerge with a constant strength. This was sh9wn to be true for the present case, in spite of the lengthy time and high cost of the numerical calculations.It is worth mentioning that, af ter the upstream-facing shock wave is swept downstream of the area reduction, the flow through the area change is
entirely supersonic. Eecause this supersonic flow moves through an area that
is increasing in the flow direction, it is a supersonic nozzle flow.
Three complementary sets of numerical results are presented in Figs. 23
to 25, for values of P2/Pl and Su/Sd that also correspond to points within the
domain of pattern C. These results, especially those of Fig. 23, are similar
to those of Fig. 22, but they do differ in detail. In the case of the results
given in Fig. 24, for P2/Pl
=
0.35 and Su/Sd= 0.4
5, for a point that is quiteclose to the boundary between patterns Band C, the upstream-facing shock wave
that forms in the area change is just about to be slowly swept out of the area
reduction, in the last spatial distributions for pressure, flow velocity,
den-sity and entropy. On the other hand, the graphical results given in Fig. 25, for P2/Pl
=
0.10 and Su/Sd=
0.45 at the boundary between patterns C and D, do depict that the upstream-facing shock wave is quickly swept downstream of thearea reduction. Also obvious from the graphical results is that its strength
is diminishing in successive distributions and with time.
It should be clear from the numerical results presented in Figs. 22 to 25 and their discussion that quasi-steady wave pattern C will eventually be
established for each case. The time required for pattern C to be established,
however, is long relative to those to establish patterns A and B, because the
wave interaction process for the overtaking of the upstream-facing shock wave
by the lat ter part of the incident rarefaction wave proceeds relatively slow-ly with time.
The two final sets of numerical results are now presented in Figs. 26
and 27, for the two cases of P2/Pl = 0.020
&
Su/Sd = 0.25 and P2/Pl = 0.050&
Su/Sd
=
0.80, corresponding to points in the bottom and top parts of the domainof pattern D, respectively. The transient flow development with time is
simi-lar to the previous case for pattern C, but with one very important exception.
The incident rarefaction wave is now sufficiently strong that the wave interac-tion process for the overtaking of the upstream-facing shock wave by the lat-ter part of the incident rarefaction wave proceeds very slowly to a different conclusion. The shock wave will be gradually reduced to a Mach wave (that is,
eliminated) in the interaction process and a weaker upstream-facing rarefaction
wave will. eventually emerge with a constant strength (Refs. 23 and 24). This
is not shown in the results of Figs. 26 and 27, owing to the excessive time and cost required to continue the numerical calculations. However, it should be clear from the results given in these figures that the shock-wave strength is diminishing and this shock wave is swept downstream more quickly with time.
It is fairly obvious from these results and their discus sion that quasi-steady wave pattern D will eventually be established if the incident rarefaction-wave strength and area-reduction ratio correspond to any point in
the domain of wave pattern D. The time required for pattern D to be
estab-lished, however, is long relative to that to establish pattern C, and even
longer relative to those to establish patterns A and B. Note that the reason
for the increase in the times to establish patterns A, B, C and D is that a sequence of events always occurs before the final quasi-steady wave pattern is eventually established. Pattem A is always formed first. If the rarefaction wave is sufficiently strong, pattern A is changed into pattern B, pattern B is then altered to pattern C, and pattern D finally evolves from pattern C, as the incident rarefaction wave becomes stronger.
The characteristic time for the nonstationary flow to decay and become quasi-steady, thereby establishing pattern A, B, C or D, has been discussed
only qualitatively. In order to obtain quantitative results, a definition for
this characteristic time is needed. Let the characteristic time te be defined
as the time interval measured from when the incident rarefaction wave first en-counters the area reduction until the nonstationary flow in 'quasi-steady' flow regions between distinct waves are within five percent of the quasi-steady flow
predictions. Based on this definition, the nondimensional characteristic times
Tc = altC/~ obtained from the numerical results are shown versus the incident
rarefaction-wave strength in Fig. 28. The characteristic times that are shown
as a banded region increase for stronger incident rarefaction waves or
decreas-ing values of P2/ Pl. This should be expected because a stronger rarefaction
wave with a wider fan of characteristics would take longer to complete its in-teraction with the area reduction.
The characteristic times are presented in the form of a banded region
rather than a single curve or curves for the following two reasons. Firstly,
the choiee of a characteristic time from numerically predicted results for the nonstationary flow properties to come within 5% of the quasi-steady flow pre-dictions is somewhat arbitrary, because the numerical results contain numerical noise or random fluctuations, which are typical of the random-ehoice method. For this reason alone, precise values could not be obtained in this
investiga-tion. Finally, the characteristic times were found to be weakly dependent on
the area-reduction ratio, which could not be determined with precision from the numerieal results. However, the discovered trend was that characteristie times were always slightly longer for more severe area reductions, for a given value of P2/Pl .
For the numeri cal results presented in Figs. 12 to 27, the spatial ex-tent of the incident rarefaction wave was always taken to be five-sixths of the length of the area reduction. When the spatial extent of the incident wave was decreased, it was found that the transient-flow behavior did not really change appreeiably, and the resulting time required to establish a particular pattern did not become significantly shorter. When the spatial extent was increased, however, it was found that the transient-flow behavior was similar but extended proportionally in time.
The listing of the computer program for the nonstationary analysis is
given for interest and possible future reference in appendix B. This program
was employed to generate and plot all of the numeri cal results presented in Figs. 12 to 27.
4. CONCLUDING REMARKS
The interaction of rarefaction waves with area reductions in ducts has
been studied successfully with two complementary analyses. The quasi-steady
flow analysis that describes the flow behavior at late times was instrumental in establishing the asymptotic wave patterns, including the asymptotic values of the quasi-steady flow properties and the asymptotic strengths of the trans-mitted, reflected and other waves, as a function of the incident rarefaction-wave strength and area-reduction ratio. The nonstationary flow analysis was necessary for the determination of the nonstationary or transient flow behavior from early to late times and showing how the quasi-steady flow was eventually established. This provided a lot of fundamental insight into the interaction process of a rarefaction wave with an area reduction. The random-choice method was found to be excellent for such nonstationary flow computations.
The nonstationary flow analysis showed that the asymptotic wave pat-terns were established very rapidly for pattern A, quite rapidly for pattern B, and very slowly for patterns C and D (see Fig. 22). Consequently, the quasi-steady flow analysis would give a good estimate of the flow properties and the strengths of the transmitted, reflected and other waves at fairly early times for patterns A and B, but not for patterns C and D. For the cases of patte~s
C and D, or when a detailed study of the transient wave behavior is needed, the nonstationary flow is required to obtain accurate flow-field calculations.
In the present work the gas flow in the duct and area change has been assumed inviscid and one-dimensional for simplicity. The resulting gas flow calculations are, therefore, not always good approximations for actual flows in ducts. For example, actual gas flows through a large or rapid increase in area are normally two-dimensional with a thickening boundary layer, flow separation from the duct walls, and oblique upstream-facing shock waves if the gas flow is initially supersonic. Also, since the gas has been assumed perfect, real-gas effects like liquifaction at low temperatures behind strong rarefaction waves have been neglected. The reader should be reminded of these limitations of the present study, even though these limitations are not set out here in detail, mainly for the sake of brevity.
5. REFERENCES
1. A. Kahane, W. R. Warren, W. C. Griffith and A. A. Marino, "A Theoretical and Experimental Study of Finite Wave Interactions with Channels of Varying Area", Journalof Aeronautical Sciences, Vol. 21, No. 8, pp. 505-525, Aug-ust 1954.
2. O. Laporte, "On the Interaction of a Shock with a Constriction", LASL Re-port No. LA-1740, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, U.S.A., August 1954.
3. G. Rudinger, "Wave Diagrams for Nonsteady Flow in Ducts", D. van Nostrand Company, New York, New York, U.S.A., 1955. Also, "Nonsteady Duct Flow: Wave-Diagram Analysis", Dover Publications, New York, New York, U.S.A.,
1969.
4. G. A. Bird, "The Effect of Wall Shape on the Degree of Reinforcement of a Shock Wave Moving into a Converging Channel", Journalof Fluid Mechanics, Vol. 5, Part 1, pp. 60-66, January 1959.
5. A. K. Oppenheim, P. A. Urtiew and R. A. Stern, "Peculiarity of Shock Im-pingement on Area Convergence" , Physics of Fluids, Vol. 2, No. 4, pp. 427-431, July-August, 1959.
6. G. Rudinger, "Passage of Shock Waves through Ducts of Variable Cross Sec-tion", Physics of Fluids, Vol. 3, No. 3, pp. 449-455, May-June 1960. 7. W. Chester, "The Propagation of Shock Waves along Ducts of Varying Cross
Section", Advances in Applied Mechanics, Vol. 6, pp. 119-152, 1960. 8. D. A. Russel, "Shock-Wave Strengthening by Area Convergence", Journalof
Fluid Mechanics, Vol. 27, Part 2, pp. 306-314, 1967.
9. D. R. Greatrix and J. J. Gottlieb, "An Analytical and Numerical Study of a Shock Wave Interaction with an Area Change", UTIAS Report No. 268, Univer-sity of Toronto Institute for Aerospace Studies, Downsview, Ontario, Cana-da, November 1982.
10. F. Shultz-Grunow, "Nichtstationaire, Kugelsymmetrische Gasbewegung und Nichtstationaire Gasstromung in Dusen und Diffusoren", Ingenieur Archiv, Vol. 14, pp. 21-29,1943.
11. F. K. Bannister and G. F. Mucklow, "Wave Action Following Sudden Release of Compressed Gas from a Cylinder", Proceedings of Industrial and Mechanical Engineering, Vol. 159, pp. 269-300, 1948.
12. J. J. Gottlieb and J. W. Funk., "Effects of Different Shock-Tube Driver Ge-ometries on a Simulated Blast-Wave Signature", Proceedings of the Seventh International Symposium on Military Applications of Blast Simulation, Vol. 1, pp. 1.6-1 to 2.6-19, sponsored by the Defence Research Establishment Suffield, Ralston, Alberta, Canada, symposium held on the 13-18 July 1981 in Medicine Hat, Alberta, Canada.
13. A. D. Jones and G. L. Brown, "Determination of Two-Stroke Engine Exhaust Noise by the Method of Characteristics", Journalof Sound and Vibration, Vol. 82, No. 3, pp. 305-327, 1982.
14. J. J. Gottlieb, O. Igra and T. Saito, "Simulation of a Blast Wave with a Constant-Area Shock Tube containing Perforated Plates in the Driver", Pro-ceedings of the Eighth International Symposium on Military Applications of Blast Simulation, Vol. 2, pp. 7-1 to 7-21, sponsored by the Gruppe for Rus-tangsdienste, AC-Laboratorium Spiez, Spiez, Switzerland, sypmosium held on the 20-24 June 1983 in Spiez, Switzerland.
15. R. F. Warming and R. M. Beam, "On the Construction and Application of Im-plicit Factored Schemes for Conservation Laws", Proceedings of the SIAM-Al1S Symposium on Computational Fluid Dynamics, held in New York, New York, U.S.A., in April 1977.
16.
o.
Igra, J. J. Gottlieb, and T. Saito, "An Analytical and Numerical Study of the Interaction of Rarefaction Waves with Area Changes in Ducts - Part 2: Area Enlargements", UTIAS Report No. 273, University of Toronto Insti-tute for Aerospace Studies, Downsview, Ontario, Canada, to be printed in 1984.17. H. W. Leipmann and A. Roshko, "Elements of Gasdynamics", JohnWiley and Sons, New York, New York, U.S.A., 1957.
18. J. Glimm, "Solution in the Large for Nonlinear Hyperbolic Systems of Equa-tions", Communications of Pure and Applied Mathematics, Vol. 18, pp. 697-715, 1965.
19. A. J. Chorin, "Random Choice Solution of Hyperbolic System", Journalof Computational Physics, Vol. 22, Part 4, pp. 517-533, 1976.
20. G. A. Sod, "A Numerical Study of a Converging Cylindrical Shock", Journal of Fluid Mechanics, Vol. 83, Part 4, pp. 785-794, 1977.
21. T. Saito and I. I. Glass, "Application of Random-Choice Method to Problems in Shock and Detonation-Wave Dynamics", UTlAS Report No. 240, University of Toronto Institute for Aerospace Studies, Downsview, Ontario, Canada, Octo-ber 1979.
22. G. F. Bremner, J. K. Dukowicz and I. I. Glass, "One-Dimensional Overtaking of a Rarefaction Wave by a Shock Wave", American Rocket Society Journal, pp. 1455-1456, October 1961. Also, UTIA Technical Note No. 33, University of Toronto Institute for Aerospace Studies, Downsview, Ontario, Canada, 1960.
23.1. I. Glass and J. G. Hall, "Shock Tubes", Section 18 of the Handbook of Supersonic Aerodynamics, NAVORD Report 1488, Vol. 6, u.S. Government Printing Office, Washington, D.C., U.S.A., December 1959.
24. I. I. Glass, L. E. Heuckroth and S. Molder, "One-Dimensional Overtaking of a Shock Wave by a Rarefaction Wave", American Rocket Society Journal, pp. 1453-1454, October 1961. AIso, UTIA Technical Note No. 30, University of Toronto Institute for Aerospace Studies, Downsview, Ontario, Canada, 1959.
rAreo
Sd
t-
(Area Su
_4-FIO_W
~IIIII_.
:::_--..
R'
I +Fig. 1. I11ustration of a rarefaetion wave (Ri) moving toward an area reduetion in a duet.
t',
s
I
~
1
17
17
'5~
1
/
1
/
~
1
..
I
I
...
!~
:~
~
~
2
.~
:~
~
:~
1
R'/'"
I
~
II
I
I
1I
I •PATTERN
C
PATTERN
0
Fig. 2. Four different sehematie quasi-steady wave patterns for the interaetion of a rarefaetion wave with an area reduetion in a duet.
·
,
1.0---000y0----...
- - - r - - - ,o
0.6
c
8
0.4
0.2
0.4
0.6
Sd Su21111
...
==_-R·
IA
0.8
1.0
Fig. 3. Domains and boundaries for wave patterns A, B, C and D for the interaction of a rarefaction wave of incident pressure ratio P2/Pl with an area reduction of ratio
1.0
,
\\ I \\ SdI
\\
2111
1 ::: SuI
\\
,
,
\
D
I,
,
,
\\
lt·
I I,
\0.8
I I,
,
\ \ I I \ I I \Su
I I \ \-
Sd
,
I,
,
\ \ I,
\I
,
\,
,
\I
,
\,
C
I
\,
I,
A
,
I\
I
I \,
,
\I
,
\,
,
\,
I \0.4
I,
I I \ \I
I
\ I IB
\,
I
\I
I \,
I \,
I \,
,
\,
,
\0.2
,
,
I
\ I \,
I \,
I \,
I \,
I \,
/,
/,
~ \ ~,
/..,"
,
0
0
0.2
0.4
0.6
0.8
1.0
P2
/P
l JFig. 4. Domains and boundaries for wave patterns A, B, C and D for the interaction of a rarefaction wave of incident pressure ratio P2/Pl with an area reduction of ratio Su/Sd, for perfect monatomic gases wi th Y = 5/3 ( )
,
0.6
0.2
ScI
Su21111:_-
...
R'
I y = 7/50.2
0.4
0.6
0.8
P2 /Pt
1.0
Fig. 5. Strength of the transmitted rarefaction wave P7/Pl as a function of the incident rarefaction-wave strength P2/Pl and area-reduction ratio Su/Sd, for perfect air and dia tomic gases wi th y
=
7/5.1.0
1.0-.-/..,~~... -
...
---r---r---...-...
~0.8
/
/
I1---
0.75---'1'
y=
7/5...
R·
I0.6
0.8
1.0
P2 /Pl
Fig. 6. Strength of the reflected rarefaction wave P3/P2 as a
function of the incident rarefaction-wave strength P2/Pl and area-reduction ratio Su/Sd' for perfect air and
diatomic gases with Y
=
7/5.,
0.1
o
o
1
1
I
0.75
I
//\
0.95/ 0.95
0.2
0.4
0.75
y=
7/5 Sd Su21111:_-
..
R'
I1.0
Fig. 7. Strengths of the stationary shock wave of pattern B
(PS/P6)'
downstream-swept shock wave of pattern C
(PS/P6)'
and down-stream-swept rarefaction wave of pattern D(P3/P6)'
for perfect air and diatomic gases with y=
7/5.P
7
-Pi
1.0
Sd
Su
2111
<::
-""Ri
0.8
y=
S/30.6
Su/Sd
0.4
0.2
o~____
~______
~______
~____
~____
~o
0.2
0.8
1.0
Fig. 8. Strength of the transmitted rarefaction wave P7/Pl as a
function of the incident rarefaction-wave strength P2/Pl
and area-reduction ratio Su/Sd' for perfect monatomic gases with Y = S/3.
P3
P
2
1.0
1.0 ".
~....
/"
I \"
I \"
0
I \ I0.8
I I I I I I I,
0.8
I0.6'
I I I I I I I I I I0.6
I I I IC
I I I I I I ISu/Sd
0.4
I I I I I I I I I I I Y=
5/3 1 1 1Sd
ISu
I2111
t ::
1 I 1...
IRi
I I I0
I0
0.2
0.4
0.6
0.8
1.0
P
2
/(1
Fig. 9. Strength of the reflected rarefaction wave P3/P2 as a
function of the incident rarefaction-wave strength P2/Pl
and area-reduction ratio Su/Sd' for perfect monatomic gases with Y = 5/3.
10---~~~~~~--~---~
5
Su/Sd
0.5
y = 5/3Qlti-____
~____
~____
~______
~__
~o
0.2
0.4
0.6
0.8
1.0
Fig. 10. Strengths of the stationary shock wave of pattern B (PS/P6)'
downstream-swept shock wave of pattern C (PS/P6)' and
down-stream-swept rarefaction wave of pattern D (P3/P6)' for
