. CONFIDENTIAL
crpy No. 154
ACR No. L5D20
^
NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
ADVANCE CONFIDENTIAL REPORT
PRELIMINARY INVESTIGATION OF SUPSRSONIC DIPFUSERS
By Arthur Kantrowitz and Coleman duP. Donaldson
Langley Memorial Aeronautical Laboratory
Langley Field, Va.
Washington
May 19if5
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alty and discretion who of necessity must be informed thereof.
CONFIDENTIAL
OOCUMENtS DEPARTMENT
■ )
NACA ACR No. L5D20 CONFIDENTIAL
NATIONAL ADVISORY C0MITTS3 FOR AERONAUTICS
1. , 13 n
1
ADVANCE CONFIDENTIAL REPORT
PRELIMINARY INVESTIGATION OF SUPERSONIC DIPPU3ERS
By Arthur Kantrowitz and Coleman duP. Donaldson
SUMMARY
The deceleration of air from supersonic velocities
in channels has been studied. It has become apparent
that a normal shock in the diverging part of the diffuser
is probably necessary for stable flow, and ways of mini-
mizing the intensity of this shock have been developed.
The effect of various geometrical parameters, especially
contraction ratio in the entrance region, on the perform-
ance of supersonic diffusers has been investigated.
By the use of these results, diffusers v/ere designed
which, starting without initial boundary layer, recovered
90 percent of the kinetic energy in supersonic air streams
up to a Mach number of I.85.
INTRODUCTION
The deceleration of air from supersonic to subsonic
velocities is a problem, that is encountered in the design
of high-speed rotary compressors ai'^d supersonic air intakes.
The efficiency of the supersonic diffusers used to accom-
plish this deceleration has an im.portant effect on the
performance of these machines. The present study is
intended to provide inform.ation upon which to design
efficient supersonic diffusers for use in cases in which
the flow starts without initial boundary layer.
The available data on supersonic diffusers are very
meager and are reviewed by Crocco in reference 1. This
review indicates that, in the deceleration of air from
supersonic velocities, the total-head losses are so large
as to impair seriously the efficiency of machines employing
this process. The experiments reported in reference 1 were
primarily designed to serve the needs of supersonic wind
tunnels, and therefore only diffusers starting with initial
boundary layer were considered.
CONFIDENTIAL
2 CONFIDENTIAL NACA ACR No. L5D20
PLOW IN A SUPERS ONIG-DIPPUSER
Stability . - In a Laval-nozzle the gases start at a
low velocity, are accelerated to the velocity of sound in
the converging part of the nozzle, and are accelerated to
supersonic velocities in the diverging part. of the nozzle.
The supersonic velocities reached can be calculated approx-
imately from the isentroplc-mass^f low curve of figure 1 and
the geometry of the nozzle. It is Viiell known that, for
shock-free flow, experiment is in good agreement with this
one -dimensional isentrooic theory although, since the
boundary layer thickens in the diverging part of the nozzle,
the Mach numbers reached may be a little lower than the
values calculated. (For example, see reference 2.) Two-
dimensional nozzles can be designed by the Prandtl-Busemann
method (reference 3) to give essentially shock-free expan-
sions, which can be obtained experimentally provided nc
moisture-condensation effects are present (reference 2).
It might be supposed that the flow in a nozzle designed
by the Prandtl-Busemann method could be reversed and, if
proper allowance were made for boundary-layer displacement
thickness, a smooth deceleration through the speed of
sound obtained. A flow of this type is, however, unstable
in the sense that it is unattainable in practice. Consider
that a flow of this type has been established. (See
fig. 2(a),) In this flow pattern the mass flow per unit
area through the throat is the maximum possible for the
given state and velocity of the gas entering the diffuser.
As long as the flow entering the diffuser is supersonic,
the entering mass flow would be unaffected by events down-
stream, A transient disturbance prooagated upstream from
'the subsonic region would, however, reduce the mass flow
at least temporarily in the velocity-of -sound region.
Thus, a disturbance would result in an accumulation of
air ahead of the throat. The 'perturbation of the original
isentropic flow produced by this accumulation of air would
prevent the mass flow from returning to its initial max-
imum value; thus, air would continue to accumulate ahead
of the throat until the mass flow entering the diffuser
was reduced. In the case of a supersonic diffuser immersed
in a supersonic stream as in the experimental arrangement
described later, this would necessitate the formation of
a normal shock ahead of .the diffuser and, in other arrange-
ments, would likewise necessitate drastic changes in the
flow pattern. From the discussion of the starting of
supersonic flows in diffusors given later, it will be seen
COxMFIDENTIAL
NACA ACR No. L5D20 CONFIDENTIAL 3
that these changes are irreversible (certainly in the
experimental arrangement described later and probably in
most other arrangements). It therefore appears that
isentropic deceleration through the speed of sound in
channels is unstable and unattainable in practice.
In a series of preliminary attempts to produce an
approximation to isentropic deceleration through the speed
of soiond, it was found that supersonic flow could not be
started into diffusers designed to produce this flow. In
diffusers with a larger throat area, the normal shock
jumped from a position ahead of the diffuser to a position
in the diverging part of the diffuser. Flows of the type
shown in figure 2(b), which involve a normal shock in the
diverging part of the diffuser, were found to be stable.
Contraction ratio and losses ." An important part of
the los ses in a supersonic Jif fuser are associated with
the- dissipation accompanying the no:irmal shock in the
diverging part of the diffuser. It is therefore Important
to consider the factors that determine its intensity. As
in a Laval nozzle, the position of the shock v;ave is
controlled by the back pressure on the diffuser and moves
upstream as the back pressure is increased. When the back
pressure forces the shock to a point close to the minimum
area of the diffuser, the shock Mach number approaches
its lowest value and the associated losses are minimized.
The magnitude of these minimum losses depends upon how
much the air entering the diffuser is slowed up by the
time it reaches the minimum cross section. The more the
entrance area of the diffuser can be contracted, the lower
the Mach number of the normal shock and the greater the
efficiency of the diffuser. It is therefore valuable to
consider what determines the maximum contraction ratio
that can be used. (Contraction ratio is defined as the
ratio of the area at the entrance of a diffuser to the area
at its mlnimurii section. See fig. 2(b).)
In most applications, the establishment of supersonic
flow is ;Dr3ceded by a normal shock traveling dov/nstream..
If this normal shock is to move into the diffuser at a
given entrance Mach number and thus establish supersonic
flow, the throat of the diffuser must be large enough to
permit the oass age of the mass flow in a stream, tube
having an area that corresponds to the entrance area of
the diffuser and a total head that corresoonds to the
value behind a normal shock at the entrance Mach number.
Thus, if the throat area -has a minimum value for a given
CONFIDENTIAL
I|. CONFIDSNTIAL NAG A ACR No. L5D20
entrance Mech .number, the Mach number at the throat will
be close to 1 when there is a normal shock ahead of the
diffuser. An approximation to the contraction ratio' that
produces this condition can be found from conventiohal
one-dimenslonal-f lov/ theory. The conditions after the
normal shock are known from the usual normal-shock equations
and It is necessary merely to find, the- stream tube contrac-
tion, which Increases the Mach number at the .throat to 1.
Since th© mass flow per unit area at the Mach. .number of 1
for 3 given stagnation temperature is proportional to, the
total head, .the maximum permissible contraction ratio' is
equal to the contraction ratio that would be required for
an isentropic compression to the Mach number of 1 (from,
the initial supersonic conditions) mult.iplied by .the
total-head ratio across the normal shock. The maxijrium
theoretical contraction ratio that permdts starting of
supersonic flow is computed in this way in appendix A and
is shown In figure 3« If the throat area were reduced-
after supersonic flow had been established or if the flow
through • the diffuser were started by temporarily increasing
the entrance Mach number to a value greater than the [
design value, a less intense shock and lower losses could
probably be obtained. In these cases, the lowest' limit
of the shock intensity would be provided by stability
considerations.
For diff users in which the geometry (oarticularly
the throat area) cannot be varied .'and in which the ;sdpsr-
sonic flow cannot be started by temporarily incre.asing the
entrance Mach number, the minimum-^loss diffusion, occurs
with the shock' just downstream from the minimum section.
The Mach number preceding such a- shock (with isentropic
flow assumed) can be found from -the computed contraction
ratio (fig. j) and equation (2): of appendix A. The total-
he ^^d loss across a normal shock at this M?ch. number
(equation (1+), appendix A) is then an approximation to
the minimum losses (with boundary-layer " losses neglected)
in a supersonic diffuser subject to the foregoing' starting
restrictions. The performances of diffusers obtained in
this way are given in figure i|. ■
It should be pointed out that these theoretical
considerations are derived with the tacit assumption that
conditions in. a plane perpendi'cular to the axis of the
channel are constant; that is, one-dimensional flow is
assumed. For example, the occurrence of oblique shocks
at the entrance o'f a ^diffuser would slightly alter these
conditions; in particular, the normal shock in the diverging
C01-IPID3NTIAL.
NACA ACR NO. L5D20 CONFIDENTIAL 5
part of the dlffuser would have a somewhat reduced
intensity and the theoretical efficiency v;ould be some-
what higher. It is considered* however, that the general
features would not be much altered by the departures from
one-dimensional flow that would occur in diffusers such
as those discussed in the experimental part of this report.
EXPERILIENTAL TECENIQUE
In order to investigate experimentally the properties
of constant-geometry suoersonic diffusers, the apparatus
shown schematically in figure 5 '''^^^ designed and con-
structed. The settling chamber was connected to a supply
of dry compressed air controlled by a valve in such a way
that the cham.ber pressure could be held constant -at any
desired valu3. The air left the chamber through inter-
changeable two-dimensional nozzles that were designed to
give p.arallel flow at various desired Mach numbers. The
feather-edge tip of the diffuser (fig. 6) was held in the
center of the supersonic jet at the exit of the nozzle.
The experimental arrangement was designed to study the
operation of supersonic diffusers that started without
initial boundary layer. This condition was studied for
two reasons: (1) It is the simplest defined boundary-
layer condition to obtain experimentally, and (2) it is
considered to approximate more closely than any other the
boundary -layer conditions that occur at the entrance to
su;oersonic diffusers used in compressors. A long sub-^
sonic dlffuser cone behind the supersonic diffuser tip
was provided to complete the diffusion process. The valve
behind ' the cone was used to control the back pressure in
the subsonic oortion of the diffuser and an orifice was
used to measure the mass flow through the diffuser. The
surface in the suoersonic diffuser tios vifas machined
steel, whereas the cone in the subsonic oortion was rolled
and finished heavy sheet steel.
In order to compare the efficiencies of the various
diffuser combinations tested, two quantities were required;
(1) the percentage of the total head that the diffuser
recovered and (2) the entrance M^oh number at v/hich the
diffuser attained this recovery.
Because the losses in vi/ell-designed supersonic
nozzles are small, the absolute pressure in the settling
chamber was assumed to be the total heed before diffusion.
CONFIDENTIAL
CONFIDENTIAL NACA ACR No, L5D20
This pressure was measured v/ith a large mercury manometer.
The total head after diffusion can be assumed equal to the
static pressure at the end of the subsonic diffuser cone
without appreciable error, inasmuch as the kinetic energy
at the end of the cones was of the order of O.I6 percent
of the entering kinetic energy. A mercury manometer was
used to measure the difference between the total heads
before and after diffusion. These two measurements were
sufficient to determine the percentage of total head
recovered.
The mass flow per unit ar6a and the stagnation con-
ditions are sufficient to determine the Mach number at
any point. (See equation (2), appendix A.) The Mach
number at which a diffuser was operating was determined
by measuring the mass flow through the diffuser, which
had a known entrance area, and by measuring the settling
chamber uressure and temperature that correspond to
stagnation conditions.
Two other observations were made. The pressure just
inside the supersonic tip of the diffuser was measured to
make sure that the shock had passed down the diffuser and
that supersonic flow existed in the contracting portion.
The flow in the nozzle and into the diffuser was observed
with a schlieren system to check visually whether the
shock had entered the diffuser.
In order to make a test, the nozzle was brought up
to design speed by increasing the pressure in the settling
chamber p^ to some value that was held constant through-
out the test. The throttling valve behind the diffuser
cone was open and the "shock passed down the diffuser, if
the contraction ratio permitted, and stopped at some place
in the diffuser cone. The throttling valve was then
slov/ly closed, thus increasing the pressure at the end of
the cone p^ and pushing the shock upstream to lower and
lower Mach numbers. 'JVhen the shock had been moved upstream
as far as possible, that is, just downstream from the
minimum section of the diffuser, p^ reached its maximum
value. Although p^ was increased during this process,
the mass flow through the diffuser was not affected because
the flow was supersonic into the diffuser tip, Yi/hen the
valve was closed farther, the shock wave passed the mini-
mum section and suddenly moved out in front of the diffuser.
C0NP^ID3NTIAL
NACA ACR NO. L5D20 CONFIDENTIAL 7
The mass flow immediately dropped (and continued to drop
as the valve was closed farther) and the pressure inside
the diffuser tip immediately jumped to a subsonic value.
The results of a tyoical test are presented in fig-
ure 7* The breaks in the mass-flow and tip-pressure
curves give an excellent indication of when the diffuser
was operating at maximum efficiency and when it failed
to act as a supersonic diffuser. The slight change in
mass flow while the diffuser was operating was due to the
fact that the pressiire in the settling chainber varied
slightly from the beginning to the end of the test run.
The curves indicate that a given diffuser may have any
value of total-head recovery, up to a certain maximum,
depending uoon the position of the shock. Therefore, the
obvious method of comparing the performance of a number
of diffusers is to compare their maximiom recoveries.
RESULTS AND DISCUSSION
The primary design parameter of a suoersonic diffuser
is its contraction ratio, which determines the minimum
Mach. number at; which the supersonic diffuser -operates and
the amount of compression that the entering air undergoes
before it must negotiate the normal shock. If the con-
traction ratio of a diffuser is Increased, the ralnimiim
Mach number at which it operates theoretically increases
as shovfli in figure 5. The minimum Mach numbers at which
a number of diffusers were observed to ooerate and the
Mach numbers at which they first failed to operate are
shown in figure 5. The points so plotted give excellent
agreement with the theoretical contraction-ratio curve.
As was pointed out previously, the effect of contrac-
tion ratio uoon the performance of a supersonic diffuser
should be approximately as sho'/m in figure i^. The observed
performances of three diffusers v;lth different contraction
ratios are plotted in figure 8. The effect of contraction
ratio is very similar to the approximate theoretlca:l
results shovm In figure l+. The indicated discrepancy
between experimental and theoretical results is probably
chiefly due to losses in the subsonic portion of the
diffuser..
After the contraction ratio of a supersonic diffuser
has been fixed according to the minimum Mach number at
CONFIDENTIAL
8 CONFIDETJTIAL NACA ACR No. L5D20
which it must operate, two other parameters - the entrance-
cone angle and the exit-cone angle - may be considered.
Owing to' the dlfi'iculty of measuring the exact
entrance angles .:on the srjall diffusers tested, the data
evaluating the effact of the entrance-cone angle are not
considered quantitative and are not presented herein.
The trend observed,, however, was that the 'larger the
entrance-cone angle, the better the performance of the
diffuser. Further experiment is needed to determine the
optimum entrance-cone angles although, for the three
diffusers of figure 8, the entrance-cone arigles are
probably so close to the optimum that no large gain in
recovery could be expected from a change in this parameter.
In the diffusers tested, the internal shape was faired in
a smooth curve between the entrance cone and the exit cone.
The curve was close to a circular arc and started very near
the leading edge of the entrance cone.
Two diffusers of equal contraction ratio and entrance-
cone angle but different exit-cone angle were tested. The
performances of the two diffusers with exit-cone angles
of 5° a^d 5° are plotted in figure 9. The diffuser with
an exit-cone angle of 3° ^as found to give consistently
higher recoveries. As is pointed out in reference I|., the
boundary layer. is thick after a normal shock and therefore
the pressure recovery in the subsonic cone must be slow
to prevent separation. The slightly different shape of
the performance curve of these diffusers when compared
with the other diffusers reported (fig. 3) may be due to
the fact that, although the two diffusers correspond
closely to each other except for exit-cone angles, they
do not correspond to the other three diffusers.
The total-head recoveries measured in the experiments
were transformed into energy efficiencies. The energy
efficiency n is defined as the percentage of available
kinetic energy recovered in the diffusion process or the
kinetic energy of an expansion from the pressure at rest
after diffusion p« to the pressure at the entrance of
the diffuser pg divided by the kinetic energy of an
expansion from the initial chamber pressure p^ to pg.
Because no external work is done, the whole process of
expansion and diffusion is a throttling proces's and the
stagnation temoerature Tq is the sajne after diffusion
CONFIDENTIAL
NACA ACR No. L5D20 CONFIDENTIAL
as in the settling chamber. The equation for the energy
efficiency may be written
r\
2c.
To - Tol —
/PeY^'P
\pf/
Tn - Tr' — >
2c
P
sPo/
'P
The symbols are defined in appendix B. When ■ -r- = 5«5>
K
r, = 1 -
M'
- 1
(1)
where M
diffuser.
is the Mach number of the flow entering the
The efficiencies obtained by equation (1) are compared
in figure 10 with the typical efficiencies (converted to
efficiency as defined in equation (l))of the work previously
done with supersonic diffusers presented by Crocco in
reference 1, the efficiency of a normal shock (comibined
with compression to rest without further loss), and the
approximate maximum theoretical efficiency for constant-
geometry diffusers previously derived. Figure 10 shows
that the normal-shock efficiency may be exceeded and that
energy recoveries of over 90 percent can be obtained up
to a Mach number of 1.85; thus, the results presented for
supersonic diffusers in reference 1 are far too conservative
for diffusers that have no initial boundary layer.
CONCLUDING REMARKS
An investigation of the deceleration of air in channels
from supersonic to subsonic velocities was conducted. A
channel flow involving the shock-free deceleration of a
gas stream through the local s^eed of sound was found to
be unstable. A stable flow probably involves a normal
shock in the diverging part of the diffuser. The losses
CONFIDENTIAL
10 CONFIDENTIAL NAG A ACR No. L5D20
involved in this normsl shock can be minimized by making
the throat ares as small as possible for a given entrance
Mach number. The maximum contraction ratio that permits
starting of supersonic flow at a given entrance Mach
number has been calculated and checked very closely by
experiment.
With the use of these results, diffusers were designed
v/hich, starting without Initial boundary layer, recovered
over 90 percent of the kinetic energy in supersonic air
streams up to a Mach number of 1.85»
Langley Memorial Aeronautical Laboratory
National Advisory Committee for Aeronautics
Langley Field, Ve.
CONFIDENTIAL
NAG A ACR NO. L5i:)20 CONFIDENTIAL 11
APPENDIX A
CALCULATION OF MAXIMUM PERMISSIBLE CONTRACTION RATIO
It can be shown that the mass flow per unit 'area at
Mach number M is
i/V+i
= M(l + -4— M V / (2)
Po^o
where the symbols are defined in appendix B.
The isentropic area-contraction ratio from a Mach
number M to the local velocity of sound is then
(pv)m=i (5)
(PV)J,T
where pV is computed from equation (2).
When air crosses e shock wave, its stagnation
temperature is unchanged; hence, the reduction in possible
mass flow per unit ares, from equation (2) and the perfect
gas law, is proportional to the total-head loss across
the shock. The total-head ratio p^/p across a normal
shock wave can be shov/n to be
Y+1 2y
/ y + 1 \y-1 t;Y-1
£3
Pc Y 1
ik)
MultiTolying equation (I4-) by expression (5) gives the
maximum contraction ratio that permits supersonic flow to
start in a diffuser. This, quantity is plotted in figure 2.
CONFIDENTIAL
12 • CONFIDENTIAL NACA ACR No. L5L20
APPENDIX B
SYMBOLS
Y ratio of specific heat at constant pressure to
specific heat at constant volume
p density
a velocity of sound
V velocity
M Mach number
c-o specific heat at constant pressure
R gas constant
T) efficiency
Pg pressure at entrance of diffuser
p^ pressure at rest after diffusion
Pq initial chamber oressure
p-z total head after normal shock wave ' ':
p^ pressure at internal leading edge of supersonic
diffuser (see fig. 7) • •
U^ design Kach number of supersonic diffuser; that
is, minimum starting I.iach number of diffuser
with given contraction ratio
''" entrance angle of diffuser (see fig. 6)
9 exit angle of diffuser
b, c dimensions used in fig. 2
Cp^ contraction ratio (see fig. 2(b))
CONFIDENTIAL
NACA ACR Nc. L5D20 CONFIDENTIAL 13
S passage area
T temperature
The subscript o refers to Initial stagnation conditions.
REFERENCES
1. Crocco, Lui£;i : Gallerie aerodynamichu per alte velocita.
L' Aerotecnica, vol. XV, fasc. 3, March 1955> PP* 237-
275 Piid vol. XV, fasc. 7 &nd 8, July and Aug. 1955,
pp. 755-778.
2. Ksntroviritz, Arthur, Street, Robert S., and Erwin,
John R. : Study of the Two-Dimensional Flow through
a Converging-Diverging Nozzlo. NACA C3 No. 3'^^h,
1914-5.
5. Busemann, A.: Gasdynemik:. Hsndb, d. Experlmentalphys . ,
Bd. IV, 1. Toil, Akad. Verl^tisgesollschaf t m. b. H.
(Leipzig), l';31, pp. i|21-I^51 end kk-7-hh9'
l\., Doneldson, Coleman duP.: Effects of Interaction
between a Normr 1 Shock and Boundary Layer. NACA CB
No. i4.A27, 19i|l+.
CONFTDSNTIAL
NACA ACR NO. L5D20
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CONFIDENTIAL
Supersonic flow
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NATIONAL ADVISORY
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j:
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CONFIDENTIAL
(b) Stable supersonic diffuser flow. (For circular
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ratio . )
Figure 2.- Flow in a converging-diverging diffuser.
NACA ACR NO. L5D20
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CONFIDENTIAL
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UNIVERSITY OF FLORIDA
3 1262 08105 007
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UNIVERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
120 MARSTON SCIENCE LIBRARY
P.O. BOX 117011
GAINESVILLE. FL 32611-7011 USA