m ;;00 VlO m() C ;;03: :> ..m " Z - j , ""0
>
O::j ;;om ~ m ZDESIGN OF THE RECOIL SYSTEM FOR THE 4" x 7" HYPERSONIC SHOCK TUBE
TECH
ISCHE HOGESCHOOL
DElFT
VLlEGTUJ';Bouwn; .m: by Michiel de Ruylerweg 10 - DHFT,. j.
1961
A. K. RobertsHYPERSONIC SHOCK TUBE
by
A. K. Roberts
"
ACKNOWLEDGEMENTS
The author wishes to express his appreciation to Dr. G. N. Patterson, Director, for his interest and the opportunity to pursue the present work.
This project was supervised by Dr. 1.1. Glass whose advice and assistance are gratefully acknowledged.
The discussions with Mr. R. James of Dowty Equipment of Canada Limited are greatly appreciated.
This work was supported by the Defence Research Board of Canada.
The 4" X 7" hypersonic shock tube incorporates two recoil sections which allow the driver and dump tank to move independently of the channel and test section. In conjunction with these recoil sections a damping system, consisting of liquid springs, has been designed to absorb the recoil energy and to limit the recoil distance.
The design problem consisted of: (1) the calculation of the pressure-time histories acting on the driver end-plate and the rear wall of the dump tank, {2) the design of the damping system for each end based on the above histories, (3) the design of the footings and connecting struts to anchor the channel and test section to the floor.
The calculations of the pressure-time histories have been com-pleted in som e detail in this work using a perfect, inviscid gas analysis. The design of the liquid springs was carried out by Dowty Equipment of Canada Ltd. , but a description of the springs along with their performance characteristics is presented for completeness. For the same reason, passing mention is made of the design of the footings and struts, but no details are inc1uded.
( i )
TABLE OF CONTENTS
Page No.
NOTATION ii
1. INTRODUC TION 1
1. 1 The Recoil System 1
1. 2 Dynamic Loads in a Shock Tube 3
1. 3 Design Procedure 4
2. CALCULATION OF THE PRESSURE-TIME DATA 5
2. 1 Characteristic Relations for a Perfect Gas 5
2. 2 Wave Interactions 7
2. 3 Constfuction of the Characteristic Diagrams 13 2.4 Description of the Diagrams for Ms ::: 10" 8, 6 and 20 15 3. THE USE OF PRESSURE-TIME DATA IN THE LIQUI;JD
SPRING DESIGN 16
3. 1 Construction of a Liquid Spring 16
3. 2 Some CharacterJistics of a Liquid Spring 17 3.3 Outline of the Design and Performance Calculations 18
3.4 Maximum Operating Range 19
3. 5 Final Form of Pressure-Time Histories 19
3. 6 Results of the Calculations 29
4. DISCUSSION AND CONCLUSIONS 21
REFERENCES 24
TABLES 1 to 8 FIGURES 1 to 18
a
-
a .I al - I a C F L M m secs. M Ms p p .. 1J P Q R s S S t u u V L NOTATION Speed of sound.2:... nondimensional speed of sound a",
Speed of sound behind a shock wave
~'
nondimensional speed of sound behinda ...
Damping coefficient (constant) Applied load
Length of shock tube Mass of driver
Milliseconds
ui
a, Flow Mach numberw/a1' shock Mach number
Pressure
Pi/Pj , nondimensional pressure ratio (i, j
=
1, 2, 3 .. .. . . ) Nondimensional Riemann variable as defined by Eq. (3) Nondimensional Riemann variable as defined by Eq. (4) Reaction force Stroke Entropy S ~ R' nondimensional entropy Time Velocity of gasui
a4' nondimensional velocity Velocity of driverw
w
x Subscripts D.T. e L R 1, 2, 3, 4 ( iii )Velocity relative to shock tube YL, nondimensional shock velocity
Q'I-Length
Ratio of specific heats (Cp/Cv) x/L, nondilnensional length a4t/L, nondimensional tim e
Dump tank Channel exit
Left side of contact surface Right side of contact surface
Regions defined in Fig. 5
1. INTRODUCTION 1. 1 The Recoil System
In combustion-driven shock tubes of large cross-sectional area very high axialloads can occur. Two shock tube configurations can arise:
(1) The complete tube (driver, channel, test section and dump tank) can be constructed as asolid assembly capable of with-standing the peak axialloads (see Fig. 1).
(2) The driver and the dump tank are not rigidly attached to the channel and test section. Under these circumstances the driver and dump tank will recoil, effectively isolating the channel and test section from any axial loads (see Fig. 2).
The UTIA 4" x 7" hypersonic, combustion-driven shock tube has a nominal shock Mach number range of 6
4
Ms ~ 20. Over the upper half of this range, peak pressures of 10, 000 psi are encountered in the combustion chamber, and such pressures correspond to axial loads of about 280,000 Ibs. In addition, pressures up to 15,000 psi (425,000 lb. loads) could be experienced if detonation of the combustion gases should occur. (Since a blow-out diaphram in the driver will break at 15, 000 psi, this represents the highest possible axial load).A shock tube constructed as asolid assembly, capable of withstanding axial loads of this magnitude, would be a massive, un
-wieldy_ and expensive piece of equipment. As aresult the principle of the second alternative was adopted and two telescopic, recoil sections were designed to allow the driver and dump tank to recoil up to 3"
before breaking a vacuum seal. (For design details of the tube, see Ref. 1). The principle of test-section isolation was also desirabie because the major instrument for determining the flow-density distri-bution in the tube, a Mach-Zehnder interferometer (see Ref. 2), is most sensitive to even the slightest movement of the tube.
Once the shock tube configuration has been chosen it must next be decided how the driver is to recoil. Several methods are available:
(1) The driver may be allowed to freely recoil with only the force of friction to stop it (see Fig. 2).
(2) The driver may be held by a buttress arrangem ent which would be capable of withstanding the dynamic load (see Fig. 3). (3) Energy absorbing, damping jacks may be installed which would allow the driver to recoil some prescribed distance while
-2-applying a reaction to oppose the load (see Fig. 4).
Before a choice can be made between these three systems, the
limitations, if any, imposed by the shock tube design itself must be con
-sidered. In this instanee there is only one such restriction, and that is the 3" maximum travel of the driver before a vacuum seal is broken. How-ever, if the blowout rupture disc in the driver end-plate should break
under detonation conditions, then motion towards the channel would arise if, for some reason, the diaphragm did not break. This was taken into account by centering the recoil section over itE! 3" length and, therefore,
allowing the driver to recoil 1. 5" either forwards or backwards when
necessary.
Returning to the possible recoil system s as previously listed, the first alternative was unacceptable because the recoil distance would
be larger than 1. 5". In addition there was astrong possibility that align
-ment would be lost if the recoil were too long, causing the sliding joints
to bind. The second alternative was acceptable but would have involved a somewhat awkward attachment problem because different driver lengths
are used over the shock Mach number range. (For low shöck:.Mach
numbers a long driver must be used in order to ensure a reasonable test-ing time. At higher Mach numbers this problem no longer exists so shorter drivers are used to conserve gas). Also a buttress big enough to handle the loads involved would require a very large foundation, which could prove awkward in the somewhat limited floor space available. As a result of these considerations the final alternative was adopted for the 4" x 7" shock tube.
The entire recoil system may now be divided into three separate sections:
(l) The jacks (with their nttings), which must restriet the
movement of the driver to less than 1. 5".
(2) The floor foundations, which must be capable of
with-standing the maximum jack reaction and the corresponding mom'ent that
will occur at the floor.
(3) The struts, which will connect the channel and test section unit to the floor foundation.
The design of the jacks and fittings was undertaken by Dowty Equipment of Canada Limited using the axialload (pressure) - time
histories at the driver end-plate as calculated in this report. The design of the floor was carried out by the Superintendent's Office at the
Univer-sity of Toronto and the design of the struts was don:e at UTIA, by G. F.
Bremner (see Ref. 1). Finally, it was necessary to decide on the type of
and low operating and maintenance demands. Upon consultation with
Dowty, it was decided that liquid springs would be em ployed.
The channel consists of 5 sections, each 8 feet long and the
test section is 4 feet long (see Ref. 1). These 6 sections are bolted
to-gether through flanges that are designed to withstand a 20, 000 lb. axial
load. In order to eliminate a complicated and bulky floor attachment
through which the spring reaction would act, it was decided to attach the
liquid springs directly to the channel and test section unit. With such an
arrangement, of course, the channel and test section must be rigidly
attached to the floor as shown in Fig. 4. Hence, it becomes immediately
apparent that the reaction of the dam ping system cannot exceed the
maximum axial load of 20, 000 Ibs. that the channel and test section unit
can take.
It is now c1ear that the channel and test section unit is no
longer isolated from the remainder of the tube as was originally desired. This leads to the possibility of the test section moving slightly during the
operation of the shock tube because it wil! be subject to some axial
loading. However, with properly designed floor supports, along with
the fact that the channel and test section unit is very rigid, no deflection
of the test section is anticipated when the tube is put into operation.
1. 2 Dynamic Loa\is in a Shock Tube
When the diaphragm in Fig. 2 is broken there is an unbalanced
force acting on the driver end-plate which tends to move the driver away
from the channe 1. In order to predict how far and how fast the driver
will recoil, it is necessary to know the load-time history acting on the
driver end-plate. This will be a rapidly changing, dynamic load because
all important shock tube phenomena are over in less than a second. With
-out energy dissipation the total load acting on the dump tank would be the
same as that experienced by the driver, although its time history would
be more complex. Since energy is dissipated the loading wil! be less
severe in the dump tank ~ut, for convenience, it was decided to use the
same recoil system at the dump tank end as at the driver. Therefore,
only the conditions at the driver end wil! be considered in this report. However, a detailed analysis of the conditions in the dump tank can be found in Ref. 3.
When the diaphragm is broken, a shock wave propagates into
the low pressure region (P1) and a rarefaction wave into the high pressure
region (P4) as shown in Fig. 5 for a long channel. Wh en the head of the rarefaction wave reaches the driver end-plate, the combustion pressure
(P4) is immediately relieved and the pressure acting on the end-plate drops off rapidly and monotonical!y as the remainder of the rarefaction
wave is reflected off the end-plate and moves back down the channel.
-4-discontinuity at the enjLrance to the dump tank. Then, upon entering the
dump tank, it diffracts and tends to becom e spherical with a resulting
decrease in strength as its radius increases. It was assumed that the
strength of this shock when it re-enters the channel, af ter reflection
off the rear wall of the dump tank, is so weak that it is negligible. However, there is a further mechanism capable of producing
ä_shock wave that will propagate upstream towards the driver. As mass
flows into the dump tank the pressure there win build up until, at some-time, it will be such that supersonic outflow into the dump tank can no longer exist. At th is time the outflow becomes subsonic and character-istic lines win start to move upstream from the area discontinuity into
the flow to form a compression wave. This wave will coalesce into a
shock which eventually reaches the driver end-plate and is reflected. The resulting double compression of the gas then yields a step increase in the pressure that will cause the driver to recoil. The pressure after the double compression depends on the pressure before the shock arrives and the strength of the shock. The strength of the rarefaction
wave relieving the combustion pressure (P4) and the length of time it has
to act solely determine the pre-shock pressure, while the dump tank
conditions will determine at what time the shock will reach the end-plate.
It should also be pointed out that various other interactions
occur within the shock tube besides the shock wave hitting the driver
end-plate. Inflow from the dump tank will comrnence at some time, causing a contact surface to move upstream towards the driver. This contact surface win interact with the shock wave reflected from the
driver end-plate, with the result tha t another shock is propagated towards
the driver. In this report these are the only interactions that were
con-sidered to be of importance, and af ter they have occurred a linear pressure decrease was assumed at the driver end-plate for convenienci9
and simplicity in further calculations.
1. 3 Design Procedure
The recoih.: system must be capable of handling the most
severe initial conditions th at the shock tube will experienc e in operation.
These conditions were found by calculating the maximum P1 and P4
values (see Ref. 4) th at can be used to produce shock Mach numbers
over the operating range of6 to 20 under the design limitations of the
tube (see Ref. 1). Knowing these conditions it was then necessary to
carry out the following three steps before construction of the system couid be undertaken.
(1) Calculation of the pressure-time histories acting on
the driver end-plate for representative shock Mach numbers over the
:.
(2) Determination of which pressure-time history im-posed the most severe loading upon the recoil system.
(3) Design of the liquid springs based on the critical
pressure-time history as found in (2) above. (The struts and footings
were designed to handle the maximum allowable reaction of 20, 000 Ibs. ).
2. CALCULATION OF THE PRESSURE-TIME DATA
2. 1 Characteristic Relations for a Perfect Gas
In order to obtain the pressure-time histories acting on the
driver·· end-plate, the m ethod of characteristics was employed to draw
the wave diagram for each shock Mach number. Once such a diagrapl .. l
was constructed, the pressure-time history was immediately obtainable.
The following assumptions were made in the construction of the wave
diagrams:
(1) The shock tube has a constant cross-sectional area. (2) The gas in region (4) achieves constant volume
com-bustion properties and is perfect and inviscid. AIso, isentropic frozen
flow exists throughout the rarefaction fan.
(3) The air in region (2) is in chem ical equilibrium. (4) The entropy of the gas in region (4) and in the dump tank has been referred to a state of the same temperature and pressure as in region (1).
(5) The shock wave originating from the area discontinuity
(see, for example, Fig. 7) is so weak that no appreciable entropy
change exists across it. This essentially means, if it is noted that the
air in region (2) is quickly swept out of the channel, that all the gas to the left of the contact surface originating at point 5 on Fig. 7 has the same entropy as the gas in region (4).
(6) Isentropic flow exists behind the contact surface
created when inflow comme nces. The value of the entropy to the right
of this front is that of the dump tank and the value of )( across the
contact surface is unchanged.
In an assessment of these assumptions it will be noted that
assumption (1) neglects viscous effects and these could become
apprec-iable at high shock Mach numbers. The importance of assumption (2)
is not known precisely but from other work (see Ref. 4) it does not appear to be too serious in this instance. Next, the effects on the cal-culations of region (2) are not great and as no characteristic lines are drawn in th is region it need not be dealt with at length. Assumption (4),
-6-~"lOwever, is of a more serious nature because it essentially means that the state of the combustion gases is being referr.ed to the air in region (1).
This assumption, along with the final two, are not too valid and have been
used to simplify the design calculations . Consequently, the pressures in
the combustion chamber and the dump tank pressures will be measured experimentally. This data will be published in Ref. 3, along with the experimentally determined recoil distances.
Using a4 to non-dimensionalize the equations that apply to an isentropic flow in a duct of constant, cros s -s ectional area, the following relations arise (see Refs. 4 and 5):
u
:p
=
Q
:P-Q
2
--L
Q.'t't - \
2
0.~'t
- \ ( 1 ) ( 2 )+
U. ( 3 )u...
( 4 )The nondimensional time and length, are:
o...t
L
L
( 5 )
( 6 )
Here L is the distance from the driver end-plate to the area discontinuity
at the entrance of the dump tank. The driver length varies with the shock
Mach number, hence
(5
)diaphragm also varies wi th this parameter.This variation is given in Tab!e 1.
the
Finally, the relation between the pressure at any position in
isentropic region for an ideal gas
is
.given by)~ _ h _ \ ~'\-\
"
=
la. )
..p'l-
( 7 )J
2. 2 Wave Interactions
In Sec. 1. 2 the assumption was made that only interactions anel flow phenomena occurring at the entrance to the dump tank could cause waves to propagate back upstream towards the driver end-plate (i. e. reflections from the dump tank walls were ignored). These interactions and flow phenomena are:
(1) Interaction Between the Incident Shock Wave and the Area Discontinuity
When a shock wave passes through an area discontinuity a rare-faction wave is reflected upstream if the flow behind the shock is subsonic. This rarefaction wave is required to match the pressures across it. How-ever, in all cases under consideration in this report the flow in region (2) is supersonic, hence no reflected wave arises from this interaction.
(2) Interaction Between the Contact Surface and the Area Discontinuity
If supersonic flow exists in both regions (2) and (3) then no reflected wave will re.sult from this interaction (see Ref. 5). If, how -ever, the flow in region (2) is supersonic while the flow in region (3) is subsonic, then a rarefaction wave will move upstream from the area discontinuity. Tables 2 and 3 indicate that this reflected rarefaction wave will occur only at shock Mach numbers of about 6 and less. In all other cases the flow in region (3) is supersonic.
(3) Supersonic Outflow into the Dump Tank
As previously indicated in Sec. 1. 2, the outflow is supersonic and at a maximum in region (3). This flow is gradually slowed down thereafter by the rarefaction wave pulses moving towards the dump tank. If the pressure in the tank (PD. T. ) is less than the pressure at the entrance to the dum p tank (Pe) then this supersonic outflow cannot be affected by conditions in the dum p tank. However, if PD. T . is grea ter than Pe' a shock wave will form and travel into the channel towards the driver end-plate when (see Ref. 5),
( 8 ) The question now arises as to what the values and time varia -tions of the dump tank pressures are. Rather than conduct a detailed calculation of these quantities which would require a long and complex iteration procedure (see Ref. 3 for the details of suc ha calculation), it was decided to assume a quasi-static dump tank pressure . This m eans that the pressure in the dump tank was taken as constant after the plane
-8-shock wave enters the dump tank. This pressure should be greater than,
but as close as possibleto)the maximum p D. T . will reach in reality because
the greater the pressure in the dump tank )the stronger the shock wave that returns to the dump tank will beo
From Table 3 the value of P2 is 500 psi for 6
6:-
Ms ~ 11. 5. Inthis range, therefore, the quasi-statie value of p D. T. was chosen as 500
psi because the pressure behind the diffracted shock will not be higher thai':l
the pressure behind the plane shock (P2) in the channel. For Ms "'>' 11. 5
Table 2 shows that P4 = 10, 000 psi for 11. 5 ~ Ms :::;. 20, which means
that the pressure -time histories in this range will differ only af ter the
time when the shock wave from the area discontinuity (see Sec. 1. 2)
arrives at the driver end-plate. This time can be determined if a
quasi-static pressure is known, as for the cases when Ms
L
11. 5. Over thehigher Mac_h number range (Ms> 11. 5), however, P2 de_cr_ease.B as Ms
increases (see Table 2), making it difficult to be sure that an assumed
quasi-static pressure is large enough. Fortunately a detailed analysis
of the Ms
=
6, 8 and 10 cases showed a definite trend with regard to thetime s of shock arrival at the end-plate (see. for example, Fig. 7 wh~re
the shock arrives at t
=
28.4 msecs. when Ms = 10, and Fig. 8 where itarrives at t
=
23. 0 msecs. when Ms=
8). Hence, by comparison withthese cases, it was possible to predict the pressure-time histories for Ms
>
11. 5 without actually using a dump tank pressure at all.This sim plification arose because the trends showed that the
arrival of the shock wave at the end-plate occurred at increasingly later
times as the shock Mach number was increased. Calculation of tbe Ms
=
20 pressure-time history for a long channel also showed that after t
=
28.4msecs. the pressure at the end-plate became very low by comparison to
the Ms
=
6, 8 and 10 cases. Hence it was concluded that the arrival of ashock wave of strength comparable to those arising in the lower shock Mach number cases would not be capable of raising the pressure to a level that would be significant. Therefore, no further calculations were carried out.
The method of choosing a limiting, quasi-static dump tank pressure leads to a sharp, artificial jump from supersonic to subsonic
outflow on the Ms = 10 and Ms = 8 wave diagrams (see Figs. 7 and 8 and
Sec. 2.4). Although this discrepancy would not occur in a real flow,
detailed calculations of the pressure-time variations in the dump tank
(see Ref. 3) show that the condition of Eq. (8) is met, hence a shock wave
will move upstream from the entrance of the dump tank. In tQis reference
the treatment of the dump tank conditions is very detailed, but the
pressure-time histories, if worked completely out, would be similar to
those of this report, particularly over the lower shock Mach number range. Since the assumption of a quasi-static dump tank pressure permits the construction of a more straight-forward and simpier wave
diagram than the method of Ref. 3, it would appear that its use is
(4) Sonic Outlow into the Dump Tank
The assumption of a quasi-statie dump tank pressure allows thè sonic point to be caIcuIated independently of the wave diagram. However) when this point is Iocated on the wave diagram, in some instanees it lies in the supersonic outflow region. In this manner the discrepancy pre-viously mentioned arises. Since the flow is continually slowing down, sonic outflow will exist only momentarily at the time when
( 9 )
(5) Subsonic Outflow into the Dum p Tank and Formationof a Shock Wave from the Area Discontinuity
Again, because of the assumed quasi-statie dump tank pressure, a subsonic outflow region is imposed on the wave diagram. Now, under these conditions, a compression wave will form at the area discontinuity. This waye will move upstream towards the dr~ver end-plate, and while so doing it steepena into a shock wave.
Once Iocated, this shock wave divides the wave diagram into two parts which must be treated separateIy, and be properly matched at the discontinuity. Since the changes of the flow variables across a shock wave can be considered as taking place instantaneously, the flow conditions can be matched if the flow is steady. The ~etho~of obtaining a direct relation between the wave diagram variabIe~ Pand Q and th~,
strength of the shock is detaile!! in Ref. 5._ Here
All/a:.
'
Ms and%
are tabulated as functions ofA:t>jä:..
andAo/o:.
Such a table allows all shock w.,ê-ve problems to be solv~<J quite rapidIy. At any point on a shock waveÀ~êl..
for a P-shock and À)Vci. for a Q -shock may be found directly from the wave diagram constructed to that point, therefore the strength of the shock at that point is now directly obtainable from this reference. In order to avoid an iterative procedure for the Iocation of each point on the shock wave, a mean shock velocity has not been used. Little error is introduced under this simplification and considerable tim e is saved.The velocity of a P -shock wave relative to the shock tube is
w
=
u..
-+
0.Ms
( 10 ) and for a Q -shock wave is-1û
-During subson~c ~utflow the instantaneous pressure at the channel exit must be equal to the instantaneous pressure in the dump tank., or
( 12 ) This condition is achieved by the characteristics which are now capable of travelling upstream to adjust the oncoming subsonic flow. Also, if the flow up to the channel exit is isentropic, Eq. (7) gives,
( 13 )
(6) Inflow into the Channel and Formation of a. Contact Surface at the Area Discontinuity .
The transition between subsonic outflow and inflow is also momentary and the boundary between the two will occur when
u
,.Q..::
0
( 14 )Since the mixed gas in the dump tank wil! be at a different entropy level than the gas in the channel, a contact surface will form at this time. This contact surface wil! then proceed upstream towards th~ driver, so it is important to locate this time accurately. This may be easily done when a quasi-statie dump tank. pressure is assumed.
Before any characteristics behind the contact surface can be drawn, the variation of the conditions at the entrance with respect to time must be determined. Since Pe wil! be known from the diagram )only one of
a
e, u e or Qe need be independently calculated. A combination of Eq. (3) ~pd the energy equation yielkls a value of a e (see Ref. 5) of-~P.
+
- 2.
~~
2. Jl.( 15 ) However, before this equation can be used to calculate a e , the value of aD. T. must be determined. This was done on the basis of assumption
o..1),T.
CL,
( 16 )
whereS1
=
0 has been chosen as a reference. Now Sn. T. is still un-known and it must be estimated before the calculations can proceed. Itshould be noted that under the assumption of isentropic flow both to the Ieft and right of the contact surface as it moves upstream, the entropy to the Ieft will be S4 and to the right SD. T.' For design purposes, the value of SD. T. should be chosen to give a conservative pressure dis
-tribution. As I;lhown in Fig. 7, the contact surface will interact with the shock wave that has reflected off the driver end-plate. Such an inter-action can yield either a reflected shock wave or a reflected rarefinter-action wave (see Fig. 6) depending on the initial conditions. Therefore, a con~
servative condition will arise when the reflected wave is a shock wave. Assuming a constant value of ~ across the contact surface, it was found that this condition is ensured when
( 17 )
This must meet one further restriction because the planar shock, u~on
entering the dump tank, will weaken as it diffracts and tend to become spherical. Hence,
( 18 )
The value of S4 is found by using Eg. (16) with different end states:
Jl
( 19 )
The contact surface created by the inflow was only caIcuIated for the Ms
=
10 case (the reasons for which are detailed Sec.' 2.4). For this case Eg. (19) and Fig. 2. 2. 17 of Ref. 4 for a perfect gas yields~
::
it
·
02
and ( 20 )Considering the limitations of Egs. (17) and(18) with the values of Eg. (20) the following value of SD. T. was arbitrarily taken as
12-(
s~",)
=
. ~~ :\0
\,00
( 21 ) It should be emphasizedthat this value merely ensures the formation of a shock wave towards the driver af ter the interaction between the contact
surface and the reflected shock wave from the end-plate (see Fig. 7). lf,
however, the dump ta~k entropy should be such that a rarefaction wave
results from the interaction (see. Fig. 6), then the pressures at the
end-plate would be relieved rather than increased. This is an example of choosing a conservative case for the design analysis. Also, the effect of S D. T. on the calculations is not marked, hence this choice proved to be quite adequate.
Dnce
the conditions at the entrance to the channel are deter-mined at any point, the characteristic line emanating from that point maybe constructed on the wave diagram, Also, the conditions on either side
of the contact su:dace are found usJ:ng the relations of Ref. 5,
--0.L.
-:PL
+Q'R
( 22 )( 23 )
where
~":....
=~S\
= 1.4~has
been assumed for simplicity. In Eqs. (22)and (23) aL; aR and QR are the unknowns, hence a trial and error
process must be used if a solution is to be obtained. The value of S4 is
previously obtained from Eq. (19).
(7) Reflection of the Shock Wave from the Driver End-Plate
. This phenomenon will cause a double compression of the gas
adjacent to the end plate due to the incident and reflected shock waves. (8) Interaction Between Shock Wave and Contact Surface The reflected shock wave of (7) wil! iIlteract with the oncoming
alternatives arise and these are depicted in Fig. (6). The method for solving such an interaction that best suits this analysis is given in Ref.
5
.
2. 3 Construction of the Characteristic Diagrams
The results of these calculations show that the pressure-time histories over the entire shock Mach num ber range can be accurately estim ated from the detailed calculations of shock Mach num bers 6, 8, 10 and 20. Hence only these cases are considered. Also, the rare -faction fans for the high shock Mach number cases (11. 5 to 20) are so strong that by the time any shock wave reaches the driver end-plate, the pressure there, is very low. Thus, even after this low pressure has undergone a double compression, its ab_solute value is still smal! and does not play an important part in the design of the recoil system. This proves to be most convenient because at these high shock Mach numbers the tail of the rarefaction wave is swept right out of the chan,nel, and when th is happens, arealistic assumption of the dum p tank
pressure is very difficult to make. Hence only shock Mach numbers of 6, 8 and 10 have been analysed considering the conditions in ,the dump tank. The pressure-time history for the Ms
=
20 case was ~olved by assuming a long channel for the initial part of the curve and by a com -parison with the three detailed cases for the late pressure increases caused by a shock wave hitting the driver end-plate (see Sec. 2.2, Part3).
Before the solution for the pressure-time histories can be started it is first necessary to find the maximum value of P1 and P4 that can be used to produce a given shock Mach number. Only one value of P 41 will produce a given shock Mach number, but this value may be obtained in an infinite number of ways si.rnply by changing Pl. However, P1 cannot be increased indefinitely because there will always be certain structural lim itations imposed by the apparatus being used. In the 4" x 7" UTIA tube the design is such that the driver pressure (P41. must not exceed 10, 000 psi and the channel pressure (P2) must not exceed 500 psi (see Ref. 1).
The calculation of (P1)max\and the corresponding (P4)max~
makes use of assumptions {2) and (3) of Sec. 2.1 to give,
( 22 )
-14'~
Since the value of P is constant through a Q -rarefaction wave (see Ref. 4), Eq. (3) becomes,
( 24 ) AIso, across the contact surface (see Ref. 4),
( 25 )
( 26 ) For a given shock Mach number the values of P2 and u2 can be obtained from the equilibrium air plots of pressure ratio (P21) and the velocity ratio (U21) as given in Ref. 4 (i. e. imperfect gas values were used across
the shock wave). Applying the initia I conditions (see Table 1) and the
limit-ing structural pressure conditions to Eqs. (22) to (26) the values of (P1)max and (P4)max can be calculated for each shock Mach number. The results are listed in Tables 2 and 3 and they show that, as P1 is increased at low shock Mach numbers, the value of p2 reaches 500 psi before P4 reaches 10, 000 psi. Under these conditions the maximum channel pressure limits the upper value of Pl. At higher Mach numbers, however, P4 will reach
10, 000 psi before the corresponding value of P1 allows P2 to reach 500 psi. For these Mach numbers the maximum driver pressure limits the upper value of Pl' The dividing line between the driver and channel pressure
limiting cases occurs at a shock Mach number of 11.5.
The next step is the calculation of the head and tail character-istics for the Q-rarefaction wave that forms when the diaphragm is broken. From Tables 1, 2 and 3 the values of u4' a4' u3 and a3 are known. Hence using Eqs. (3) and (4) the required calculations were made with the
results listed in Table 4.
The choice of characteristic lines to be used between the head and tail is purely arbitrary although enough should be chosen so tha t the diagram will be reasonably wen defined throughout. Those characteristics
along which sonic outflow into the dump tank and the ~eginning of inflow
into the shock tube occur should be included. Since Q characteristics
reflecting from ..!..he closed end of the driver (leit side of the wave diagrams) simply become P characteristics of the same value, these two dividing
characteristics (see Eqs. (9) and (14» are easily found when a
quasi-static dump tank pressure is assumed (see=8ec. 2. 2, Part 3). From Eqs.
(3), (9) and Wr.ana Table 3 the value of (Pe') oni was_calculated while
Eqs.
(~),
(13) and (14) and Table 3 were use9 to lind (Pe)inflo . Theresults of these calculations are listed in Table 5 for Ms
=
6,r
and 10.for Ms
=
20 need not be considered in detail). The characteristics used on the wave diagrams, including those for sonic outflow and the commence -m ent of inflow, are listed in Table 6.2.4 Description of the Diagrams for Ms
=
10, 8, 6 and 20The Ms
=
10 wave diagram was the only one completely drawn, consequently it will be dealt with first and in some detail. All procedureB used in the construction of other shock Mach number wave diagrams we re included in this one. Also, the results of the completed diagram were such that they could be used to estimate certain features of the Ms=
6 and 8 cases, thereby making it unnecessary to fully draw these latter two cases.The Ms
=
10 calculations followed the normal procedure for representing a rarefaction wave reflecting from a closed duct until point 1 of Fig. 7 was reached. However, under the assumption of a quasi-static dump tank pressure, the flow at this point became sonic even though the wave diagram indicates that the flow should be supersonic(see Sec. 2. 2, Part 3). Once this point was located alL flow at succeeding times became subsonic so that there is a sharp jump from supersonic outflow to subsonic outflow commencing at Point 1 on Fig. 7. With the subsonic region established the construction again became standard with
Q characteristics, eme.nating at points 2 and 3, steepening inio a Q
shock wave at point 4. The boundary ~onditions for points 2 and 3 were those for the subsonic outflow where P was known from the diagram and ~ calculated from Eq. (13). The value of Q behind the shock had to be determined by interpolating between the knoèfn characteristics already constructed on the wave diagram. Once L1
Yä:
was known, then the shock wave was drawn using the tables of Ref. 5 and Eq. (11) as noted in Sec. 2.2, Part 5.At point 5 inflow commences with the assumed constant entropy
~onditions existing on both sides of the interface. Using the value of S 'l>.T. given in Eq. (21) in conjunction with the equations and assumptions of Sec. 2.2, Part 6 and the standard trial and error procedureet.of Ref. 5, the contact surface and its interactions were plotted on the wave diagram. The completed wave diagram for Ms
=
10 is shown in Fig. 7, and the corresponding pressure-time history is given in Table 7.On the Ms
=
8 and 6 diagram s (see Figs. 8 and 9) the con-struction of the contact surface was omitted to save time. Hence only the shock wave going back towards the driver end-plate is shown, but by a direct comparison with the Ms=
10 pressure results the effect of the contact surface interactions on the pressure-time histories was estimated. This is shown in Table '7.Finally, the pressure-time history for the Ms
=
20 case was calculated for a long channel only (i. e., no dump tank) as previously-'16-explained in Sec. 2.2:- 'Part 3 and Sec. 2.3. The wave diagram is shown
\ .
in Fig. 10 and the corresp'onding pressure-time history in Table 7. Actually a wave diagram is not necessary in this si.mple situation pro-viding an ilnalytical solution exists, as illustrated in Ref. 6, where the
~
=
7/5 and 5/3 cases have been solved in closed form.3. THE USE OF PRESSURE -TIME DATA IN THE LIQUID SPRING DESIGN
3. 1 Construction of a Liquid Spring
A schematic representation of a liquid spring is shown in Fig. 11. The cylinder is com pletely filled with oil)therefore when the piston moves with respect to the cylinder the oil wiU be com pressed and
forced through the orifices to give both a springing and damping action respectively (see Sec. 3. 2). Since the oil is loaded to 7500 psi initially and will reach pressures of 40, 000 psi during operation, the question of
seals, both statie and dynamic, imrnediately arises. This problem ios ,
solved by the "gland assembly" which provides astatic seal against the inside wall of the cylinder and adynamie seal around the m oving piston rod. The gland is constructed on the unsupported area principle which ensuresthat the gland pressure is greater than the fluid pressure. This principle is depicted in Fig. 12 and described in detail in Ref. 7. Briefly, the fluid pressure acts over the entire annular area of the gland assembly whereas the reaction acts over the annular area less the area of the pegs. Therefore, since the axial forces in both directions are equal, the gland
pressure must be greater than the fluid pressure .
Although the absolute pressures experienced by the fluid are very high, the dam ping action is conventional. This occurs because the action depends only on the pressure drop across the orifices, and this
is of the usual magnitude. Therefore, the design of the piston is standard
other than ensuring that the material is strong enough.
In Sec. 1. 1 it was pointed out that the liquid springs at the driver end (see Fig. 4) must be capable of handling the normal recoil
away'from the channel as well as recoil towards the channel if the safety
diaphragm should break when the diaphragm did not. In order to use the same piston design at both the driver and dump tank ends the m ethod of attaching the liquid springs to the shock tube was set up to allow: (1) the piston to .move within a stationary cylinder under normal circu.mstances
or (2) the cylinder to move around a stationary piston when the safety
diaphragm breaks. Therefore, as far as the piston orifices are
3.2 Some Characteristics of a Liquid Spring
In general the resultant (inertial) force acting on a one degree of freedom system with damping (such as the one under discussion) is given by,
mass x acceleration
=
Applied Force+
Restoring Force+
Damping Force ( 29 ) As long as the system is not critically or over dam ped, periodic motion will arise. In the context of this report the mass involved is that of the driver and it is constant. The load-time history acting on the driver end-p1ate is the applied force on the mass,and the restoring force is produced by the compressibility of the fluid which, being elastie, will at all tim es oppose the direction of the applied force. Since this force is elastic no net work is done on the fluid, hence no energy is absorbed by the fluid. Finally, the dam ping force arises when the oil flowa;through the orif.ices in the piston, but a net amount of work is done in this instance, and energy is dissipated throughout the oi! in the form of heat causing the temperature of the oH to rise.From the above considerations the two characteristiceoèfficients of a liquid spring must be the same as those in an ordinary one degree of freedom spring-dashpot system, that is, the spring and dam ping coefficients. Thus the four parameters that must be dealt with in the design and
per-formance of a liquid spring are:
(1) The spring coefficient (2) The dam ping coeffieient (3) The maximum reaction (4) The maximum stroke
When a liquid spring is statically loaded and then unloaded, a hysteresis loop is plotted for load against stroke or deflection. The upper curve of this loop is known as the "closing spring curve" (see Fig. 13) and is the spring curve used in the ca1culations. The area under this curve is the energy absorbed by the fluid in compression. When the load is
relieved, this energy is resfdlred because the fluid will expand back to its original volume, hence no net work is done. However, when the spring is rapidly loaded, as it is when the shock tube is fired, the liquid is com-pressed more quickly than in the statie case, and the shape of the "per-formance curve" (reaction against stroke) changes to that shown in Fig. 14. When the piston reaches its maximum stroke (point Z) it will return slow1y to its starting position {point X) and the aet amount of energy absorbed is the area
KYZ
·
.
The efficiency of the spring is defined as the-18
-area AXYZD divided by the rectangular -area ABCD. The eff~ciency of a
spring is normally determined by its internal components, that is, whether
it has a mechanical spring or not, the type of ports, etc. A liquid spring
with a steep spring curve is said to be "hard", hence the more energy that must be absorbed, for a given maxim urn reaction, the softer the spring must be to give the necessary longer stroke.
3. 3 Outline of the Design and Performance Calculations
Since the action encountered is essentially that which an aircraft undercarriage experiences, first attempts at design by Dowty Equipment were modelled af ter standard undercarriage procedures. However, it was soon found that a more fundamental approach had to be taken because
of a basic difference between the two cases. This difference involved the
calculation of the input energy to the recoil system. In the case of an aircraft, the vertical landing speed and the weight of the aircraft are
known in advance, hence the input kinetic energy is immediately known.
However, the velocity of the driver is not known beforehand and the energy
input must, therefore, be computed from a work consideration. But a
difficulty now arises because the applied force is a function of time while the stroke and reaction are not known. As a result of this complication,
the calculations became longer and more tedious than at first anticipa ted.
For a given shock Mach number the calculations consisted of the following three stages:
(1) Calculation of the Closing. Spring Curve
The start of the curve occurs at zero stroke and depends simply on the loading pressure of the liquid spring. This pressure may vary from
2000 psi to 10, 000 psi under conditions wher:e no temperature cyç;:ling occurs.
Next the percent compression of the fluid is determined from an assumed p~ston rod diameter and total stroke. Then, from experim ental curves of the physical behaviour of oil under pressure, the closing spring curve may
be calculated point by point. These experim ental curves correspond to the
spring coefficient mentioned in Sec. 3.2.
(2) Calculation of the Performance Curve
In a liquid spring, the relationship between the velocity of the
piston and the reaction is given by ~
2
"Rl*-)
=
C
V(+.)
( 30 )Where C is an unknown constant (the damping coefficient) which determines
the size of the ports. If at a given time the reaction is assumed, then the
acceleration may be found by using the known applied load-time history.
From this the incremental velocity at that time may be immediately
should agree with the assumed reaction. lf not, the procedure must be
repeated until agreement is reached. The stroke may now be calculated
from the following work-energy relation applied to the driver mass,
( 31 )
Where F (t) is the applied load. This entire procedure is now repeated
point by point until the performance curve (Fig. 14) is completed. (3) Check Maximum Reaction and Stroke
Once the performance curve has been drawn it must meet the
maximum reaction and stroke specification as first laid down (i. e. Rmax =
20,000 Ibs. and s'max
=
1. 5" as given in Sec 1. 1). lf either specificationis exceeded then step (2) must be repeated with a new value of the dam ping
coefficient. lf no value of C can be found to do the job then a new spring
curve must be chosen and all three steps in the procedure repeated.
3.4 Maximum Operating Range
The pressure-time (load-time) histories of Table 7 represent
the theoretically applied loads that the driver will experience under norm al
ope rating conditions . However, if the recoil system had been designed
simply to handle norm al operating conditions it would be damaged if these
conditions were exceeded, and this would happen if the combustion mixture
detonated rather than burning as it should. Therefore, the design figures for the recoil system must be those for the most severe detonation
con-ditions (i. e. detonation occurring but not breaking the safety diaphragm).
Although detonation waves can give rise to very high local pressure ratios
(see Ref. 8), it was assumed that detonation would uniformly increase the
driver pressure ~) by a factor of 1. 5. However, the peak pressure that
can be effectively achieved by detonation is 15, 000 psi because the safety
diaphragm is designed to break at this time (see Sec. 1. 1). Therefore,
the maximum conditions the recoil system should be capable of handling are found by multiplying all values in Table 7 by 1. 5. Subsequently, these
load-time histories will be referred to as the "detonation cases" and are
shown in Fig. 15.
3.5 Final Form of Pressure-Time Histories
Af ter the detonation load-time histories of Fig. 15
(correspond-ing to the histories of Table 7) were given to Dowty, the preliminary calculations showed tha t strokes far exceeding an inch or two would be
obtained for Ms = 6 to 9. Two alternatives were suggested by Dowty:
(1) Allow a greater maximum reaction
-20-(preferably not greater than la, 000 Ibs. ) so that the maximum reaction
woold always be greater than the applied load at times af ter 20 millisecs.
As previously indicated-in Sec. 1. 1 the maximum reaction could not be
increased because of the flange strength. Hence it was necessary to adopt the second alternative and the necessary pressure-time histories were revised.
For a given shock Mach mach nurn ber P 41 is essentially con-stant, hence if P1 is lowered, P4 must also decrease. This means that when P1 has been lowered sufficiently the applied load will not exceed
18,000 Ibs. af ter 20 millisecs. That value of P1 which just allowed a late
load increase to reach 18,000 Ibs. was taken as the maximum operating p . A plot of Ms against p is shown in Fig. 16. The values for Ms =
10
to 20 are taken from Ta6le 2 and 3 while those for Ms=
6 and 8 fromthe restrictions given above. This figure also shows the region of lost
performance arising from the necessity of reducing P1 below the channel pressure limiting maximum. However, it should be noted, in view of the assumptions made in computing the pressure-time histories, that the
calculated values of operating channel pressures (see Fig. 16) are subject
to revision after the shock tube is calibrated experimentally. Any such revisions will be publishe d in Ref. 3.
Finally, Dowty requested the details of the applied load-time histories for some shock Mach num bers between 10 and 20 as well as the
details of the Ms
=
20 case at time s greater than 20 msec. It may be seenfrom Fig. 15 that the higher the shock Mach number, the lower the late
pressure increases. Since no detailed calculations for Ms = 20 were
carried out at the dump tank, it was assumed that the same pressure-time
histories existed as for Ms
=
10 at times greater than 25 msecs. Thiswas, of course, severe because the actual values will be lower, but for the" purposes of design this assumption acted simply as a safety factor.
Conceming shock M-ach numbers between 10 and 20, Table 2
shows that P4
=
10,000 psi for Ms greater thanUS.
Since the driverlength is the same over this range, the initial drop off in pressure will
be exactly the same as the Ms = 20 case. Th get the final part of the
load-time history, the last part of the Ms = 10 history was used for the same
reasons given above. In other words the pressure-time histories that
Dowty received for shock Mach numbers of 12 to 20 were identical. The applied load-time histories to which Dowty designed the liquid spring are
now listed in Table 8 and shown in Fig. 17.
3. 6 Results of the Calculations
For each spring curve and dam ping coefficient combination the performance curves for shock Mach numbers 6, 8,c~.J.Q and 12 to 20 had to be calculated. This was necessary because the load-time histories vary
so radically over the operating range (see Fig. 15) tha t it was impossible
severe action on the liquid spring. All the calculation s we re carried out by the engineering staff of Dowty and the procedure they followed has already been outlined in Section 4.2. No details of their calculations have been inc1uded in this report.
The final results, however, are shown in Fig. 18 and indicate that
the Ms
=
10 case gives a maximum stroke of 1. 15 inches. Since twoliquid springs will operate together, the maximum reaction of one will be only 10,000 Ibs. as shown in Fig. 18. On the basis of these results Dowty designed and manufactured four liquid springs(two for the driver and two for the dump tank end) for the UTIA 4" x 7" hypersonic shock tube.
As far as the action of the recoil system is concerned, several
interesting characteristics may be seen from Fig. 18. Consider first
the Ms
=
10 case. The initial part of the curve, up to point A, is caused by the high combustion pressure in the driver and its subsequent relief by the rarefaction wave, that is, the first 28 millisecs. of the load-tim e history. Since point A does not lie on the c10sing spring curve, the driver is still moving at this time. At A the shock wave from the dumptank hits the end-plate causing the driver to accelerate again because the
applied force is greater than the liquid spring reaction. At point B,
however, the reaction becomes greater than the applied force and decel
-eration of the driver commences. This is indicated by the inflection
point at B. Therefore, fOE shock Mach numbers fr om 13 to 20 most of
the stroke is the result of the high initial pressure and only the last 0.2" is caused by the late step increases in pressure. However, for the low
shock Mach number cases it may be sefm that exactly the opposite
situa-tion arises. For the Ms
=
8 case, about 0.1" of the stroke is caused by the initial combustion pressure, and for Ms=
6 less than 0.05" is due tothe initial pressure. In these cases almost all the stroke is caused by
the shock wave which hits the driver end-plate af ter some 20-25 msec. and
the subsequent interactions .
4. DISCUS SION AND CONCLUSIONS
When shoc k tubes of large croo s -sectional area are constructed
(particularly those which are com bustion drive~) the question of recoil
must be seriously considered. The manner in which a recoil system is
designed may vary widely, depending on such factors as the available floor space, size of the recoil system, cost and the degree of
sophisti-cation desired. For example, the AVCO-RAD tube employs several air
jacks (see Ref. 9) in a mannar analogous to the liquid springs under
dis-cussion, while the G. E. hypersonic shock tube has an anchored driver and
a dump tank which is capable of recoiling but is restricted from doing so (see Ref. 10).
When firing into atmosphere under conditions where no shock wave wi11 return to the driver end-plate, the pressures causing recoil
-22-may be eithe.r determined graphically using the method of characteristics, or analytically using a solution of closed form (see Ref. 6), assuming the channel to be infinllelly, long. The former approach was used to cal
-culate the loading in the R. A. E . 6" high-pressure shock tube (see Ref. 11), and in some driver recoil studies carried out by English Electric (see Ref. 12). Both these tubes yielded good experimental agreement to the theory but neither invol ved pressures close to the 10, 000
-15, 000 psi peaks discussed herein. In fact, the R. A. E. tube is mainly used with a cold driver and the peak pressures do not exceed 2300 psi.
In this report the method of characteristics has been used to analyze a high-pressure, combustion-driven shock tube. Although the major assumptions postulate isentropic flow and perfect gas behaviour throughout the rarefaction wave, the results are probably acceptable in this instance in comparison wi th the other assumptions made. In
reality, recombination effects will be present throughout the rarefaction wave with the result that the local temperatures and sound speeds will be higher than those predicted for a frozen flow (see Ref. 4).
The assumption neglecting the original shock wave af ter it enters the dump tank merits some justification. There is the possibility of this shock reflecting off the rear dum p tank wall and re- entering the channel. It would then overtake the shock wave that formed from the area discontinuity, and the resultant shock which hits the driver end-plate would be somewhat stronger than the shock wave considered in this work. However, the increased pressure from this source would not be very significant.
The shock Mach number range over which the qUjisi-static dump tank pressure is valid has yet to be determined experimentally, and the results wi11 be given in Ref. 3. The analysis of Ref. 3, although not yet com pleted, appears to substantiate the simplified quasi-static approach for shock Mach numbers near 10, but it also shows that around Ms
=
20 this approach becomes over-sim plified. Fortunately, this lattér development should not adversely affect the ca1culations of this report because of the way in which the Ms=
20 pressure-time history was finally presented (see Sec. 3.5).The recoil system design limits the maximum channel pressures as shown in Fig. 16, but these values are subject to experimental veri-fication once the tube is in operation and any revisions will also be reported in Ref. 3.
In conclusion, a recoil system containing liquid springs has been designed for the UTIA hypersonic shock tube over the Mach number range 6 ~ Ms ~ 20. The calculations were based on the assumptions of a perfect, inviscid gas in an isentropic flow where applicable. In
(detonation case) and the liquid springs were designed to handle this severe case. The use of liquid springs as the major components of a
recoil system yields a very compact, but not inexpensive unit. However,
as it is completely self contained it will require no day to day maintenance
and it is reputed to be very dependable, as evidenced by the widespread
1. Bremner, G. F. 2. Hal!, J. G. 3. Lau, J. 4. Glass 1. 1. 5. Rudinger, G. 6. Stekettee, . J. A. 7. Bingham, A. E. 8. Martin, F . J . White, D . R. 9. Offenhartz, E. 10. Nagamatsu, H. T. Geiger, R, E . Sheer, Jr., R. E 11. Woods, B .. A. 12. Freeman, A. J. M. -24-REFERENCES
UTIA Master' s Thesis, 1960
Design of a 4 x 7 Inch Hypersonic Shock Tube.
The Design and Performance of a 9-inch Plate Mach-Zehnder Interferometer,
UTIA Report No. 27 (1954)
UTIA Master' s Thesis (to be published)
Theory and Performance of Simple
Shock Tubes, UTIA Review No. 12, Part I (1958 )
Wave Diagrams for Nonsteady Flow in Ducts, D. van Nostrand Co., New York
(1955)
On the Interaction of Rarefaction Waves in a Shock Tube, UTIA Review No. 4 (1952)
Liquid Springs: Progress in Design and Application. A paper presented to a General Meeting of the Institution of Mechanical
Engineers in London, 13th May 1955 and published by the Institutiop.
The Formation and Structure of Gaseous Detonation Waves. A paper presented to the Seventh Symposium on Combustion at London and Oxford, 28 August - 3
September, 1958.
Private Communication, Avco RAD,
Wilmington, Mass.
Hypersonic Shock Tunnel, General Electric Electric Report No. 59-RL-2164 (1959)
Calculation of the Recoil of a Shock Tube,
R. A. E . Tech. Note, k r o. 2627 (1959).
Dynamic Forces Acting on a Shock Tube Structure, English Electric Co., Guided Weapons Division, Report LM. u. 007 (1956).
(1) Initial Operating Conditions
Combustion Mixture is 80% He, 20% H2 - 02 (stoichiometrie)
0
~4
Tl=
300 K=
1. 49~1
=
1.4 (air) a 4=
7450 fps al=
1141 fps u 4 = 0 uI = 0(2) Some Design Details of the Hypersonic Shock Tube
2
Channel Cross-Section
=
4"
x7"
or 28 in.Channel Length
=
44' Dump Tank Diam eter = 3'Test Seetion Length
=
4' Dump Tank Volume=
33.4 ft. 3Driver Cross-seetion
=
6 11 diameter or 28. 3 in. 2Dump Tank Mass
=
47 slugsMaximum Operating Driver Pressure
=
10, 000 psi (driver pressureis relieved at 15,000 psi at which time a rupture diaphragm wil! break).
Maximum Channel Pressure = 500 psL
(3) Possible Driver Configurations M s 20 10 8 6
Length (ft. ) Mass (Slugs)
4.5 148 7.5 196 10.5 252 13. 5 310 REQUIRED DATA TABLE I
(~
)DiaPhragm 0. 086 O. 135 O. 180 0. 220TABLE 2
DRIVER PRESSURE LIMITS
M
U
21
uJ2
=
u3
a3
~4
4>3
=
1?2 P 21
.f1
s
(fps) (fps) (psi) (psi) fusi)
20
18.4
21,000
2,300
10, 000
7.8
520
0.015
18
16
.
9
19, 300
2, 720
10,000
22
425
0
.
052
16
14
.
6
16,700
3,360
10,000
78
330
O. 236
14
12
.
7
14,500
3, 900
10,000
200
250
0.800
12
10
.
7
12, 200
4,460
10,000
440
185
2.38
11. 5
10.3
11, 800
4,560
10, 000
500
165
3.03
TABLE 3CHANNEL PRESSURE LlMITS
Ms
U
21
u2
=
u3
a3
4>2 =.t>3
1>4
P 21
1>1
(fps) (fps) (psi) (psi) (psi)
11. 5
10.3
11,800
4,560
500
10,000
165
3.03
11
9. 9
11, 300
4,680
500
8,300
150
3.33
10
8. 9
10,200
4, 950
500
6, 150
130
3
.
85
8
6.9
7,870
5, 520
500
3,090
80
6.25
6
5. 1
5,820
6, 020
500
1,850
43
11. 6
VALUES OF QAT HEAD AND TAIL OF RAREFACTION WAVES .t1.r.i1UJ TAIL
--Ms a4 uT4 a 4 QHead u3 a3 QTail (fps) 20 7450 0 1. 00 4.08 2.82 .309 -1. 56 10 7450 0 1. 00 4.081 1. 37 . 664 1. 34 8 7450 0 1. 00 _:4. • .08 1. 06 .741 1. 97 6 7450 0 1. 00 4. 08 .781 .808 2.52 TABLE 5THE DIVIDING
P
CHARACTERISTICS1
-
-Ms a e
=
u e (P e)sonic (Pe)inflow10 . 622 3. 16 2.54
8 .741 3.76 3. 02
TABLE 6
RAREFACTION FAN Q - CHARACTERISTICS
M s
=
20 M s=
10 Ms=
8 M=
6s
4. 08 (Head) 4.08 (Head) 4.08 (Head) 4.08 (Head &
Sonic) 3. 48 3.62 3. 76 (Sonie) 3.76 2.86 3. 16 (Sonie) 3.46 3.45 I
