7 7- '. -
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL MEMORANDUM 1357
AIR ADMIXTURE TO EXHAUST JETS*
By E. Sanger
1. Introduction.- The customary jet engines rockets,.turbojet
engines, pulse jet engines show at actual flight speeds certain defects,
like low efficiencies, failure in case of increased flight velocities, etc.
Furthermore, all these power plants have, with the customary propeller-
piston power plants, the characteristic in common that they discharge
considerable heat quantities with the exhaust gases without utilizing them.
The ram-jet engine has no static thrust, and, in the subsonic region,
its thrust increases approximately with the square of the flight velocity;
thus, one is forced, in many cases, to combinations with other jet engines
for take-off and slow flight.
This situation gives rise to the theoretical investigation of how
far such disadvantages can be fundamentally reduced by air admixture to
the exhaust jet in special shrouds.
The problem of thrust increase for jet engines by air admixture to
the exhaust jet was introduced into aviation techniques by the suggestions
of Melot (ref. i). Due to a too general interpretation of several
theoretical investigations of A. Busemann (ref. 2), so far no practical
use has been made of these suggestions.
The following considerations show that, in the case of low-pressure
mixing according to Ml4ot's suggestions, probably no thrust increase of
technical significance will occur for the flight speeds of interest
(however, the low-pressure mixture is highly promising for ground test
setups and for special power plants of relatively slow sea and land
In contrast, application of the high-pressure mixing in ram-jet type
shrouds, where surrounding air is admixed to the exhaust jet, appears
advantageous throughout for some aeronautical power plants in the range
of high subsonic and supersonic speeds.
The relative increases become larger the higher the flight speed and
the less satisfactory the thermic efficiency of the jet engine used.
Luftzumischung zu Abgasstrahlen," Ingenieur-Archiv, vol. 18, 1950,
NACA TM 1357
Fundamentally, the four energy components of two gas jets mix in
such a manner that, for constant total energy, the sum of the enthalpies
increases at the expense of the sum of the kinetic energies, with the
total entropy increasing as well.
If the mixing of the jet takes place at a pressure exceeding the
undisturbed external pressure, part of the enthalpy developed may be
reconverted into kinetic energy (by the following expansion of the gas
mixture) and thus be made useful.
These conditions lead to combination of the ram-jet engine with other
power plants in the form of ram-jet type shrouds and mixing chambers,
with the purpose of utilizing part of the otherwise completely lost
exhaust gas energies, especially the heat energies of the core jets for
reheating of an additional ram-jet engine.
Since, in this kind of heating, kinetic energy is unadvoidably
supplied beside the thermic energy, we call this principle, in short,
impulse heating of the ram-jet engine.
For simplification of the following theoretical investigations, we
assume in the calculation that the mixing of the jet always takes place
at constant pressure; in practice, however, mixing at variable pressure
is applicable as well, particularly in low-pressure chambers for pressure
increase, in high-pressure chambers for pressure drop.
I. Constant-Pressure Mixing of Gas Jets
2. Theory of constant-pressure mixing.- A gas jet leaves an opening
of cross section F0 with the velocity v0 and the remaining state
points p, pO, TO. It mixes with the free surrounding air which flows
in the same direction with the velocity v2 and possesses the remaining
state points p, p2, T2 (fig. 1).
The following relations are valid for the mixing of a certain air
mass m2 = P2F2V2 flowing per second through the cross section F2 with
the corresponding exhaust gas mass m0 = pOFOvo:
Continuity theorem: The sum of the masses flowing per second through
the cross sections F2 and FO equals the mass flowing per second
through the cross section Fx
POFOVO + P2 F2v = PxFxVx
NACA TM 1357
Momentum theorem: Due to the pressure being constant throughout,
the total momentum remains the same in every cross section
POFV20 + P2F2 = PxFxv2 (2)
Energy theorem: The sum of the energies flowing per second through
the cross sections F2 and FO equals the energy flowing per second
through the cross section Fx with presupposition of a conversion of
kinetic energy into enthalpy free from relaxation as in the calculation
of the gas throttling
0Fo + gcpTo)+ 2F2v2( + gcpT2= xFxx + gcpTx) (3)
Equation of state: The values of the quantities px, Tx are related
to the known constant mixing pressure
p = gpxRTx (4)
In the jet core, the customary rectangular distribution of the state
quantities over the cross section with the same mass flow as the actual
distribution was assumed. The gas constants R and cp are assumed
approximately equal for the two gases to be mixed. The turbulence dissi-
pation into enthalpy is assumed to be free from inertia as was said before.
With these four equations, the four unknowns Fx, Px' Vx, Tx
may be expressed by the known quantities F2, p02 v2, T2, FO' PO'
For simplification of the notation, new symbols are introduced for
the known quantities
Mass sum p0Fov0 + P2F22 = M kgsec/m] (5)
Momentum sum pOFO2 + P2F222 = J kg] (6)
Energy sum pOFOvO v20 2 + gcpTo) + 2F2v2 2 +
gcpT2) = E kgm/sec] (7)
Temperature density p/gR = 0TO = P2T2 = PxTx = D kgsec2 o/m (8)
NACA TM 1357
Therewith the four determining equations become
pxFxvx = M
PxF,2 = J
PXFxVx + gCpTx = E
PxTx = D
From them, there follow the unknowns
ME 1 J2
Fx- = 2
ME 1 J2
ME 1 J2
Tx = 2
and the mixture Mach number
V;=\ 2 1
ax 1 2ME/J2 1
used later on.
3. Propulsive mixing efficiency.- The ratio of the kinetic energy
flowing off per second through Fx and the total kinetic energy which
has flowed in is
Tm = --- = ---
p0F + P2F2 2E 2gcp pOFOOTO + PFZ2vT2
S m2v2o \2
1 + m2o22
\ "Of A 2
therefore, it is only a function of the two ratios m2 /m and
NACA TM 1357
The propulsive mixing efficiency depends neither on the pressure p
at which the mixing takes place, nor on the densities, temperatures,
enthalpies, or Mach numbers of the mixing partners concerned or on the
relaxation of the vortex conversion into heat. The kinetic energies
disappearing in the mixing are converted via vortex motion into additional
enthalpy as in the process of gas throttling.
The mixing efficiency for m2lm0 -- 0 and m2 mO- c~ becomes of
course equal, nm = 1, since, when one of the mixing partners disappears,
a mixing, and therefore mixing losses, are no longer possible.
The mixing efficiency becomes for v2 vO = 0, that is, mixing with
surrounding air at rest, equal m = 1/(1 + m2 mo) = "mOrx thus for mixing
ratios in practical use very small; the jet energy is converted almost
entirely into vortices and heat.
The mixing efficiency becomes, furthermore, with v2/v0 = 1 equal
to Tl = 1, that is, when both gas jets have equal velocity of the same
direction, the mixing occurs by diffusion without losses in kinetic energy,
even when the temperatures of the two jets are different.
In the range of arbitrary values of m21m0 and v2/vo, the mixing
efficiencies show minima; these minima lie at values of m2/mO the higher,
the smaller v2/vo; their course may be calculated to be m2/mO = v0 v2,
that is, the losses become largest when both jet impulses are at first
equal. The course can be seen in figure 2.
If the gases to be admixed have a priori noticeable velocities in the
direction of the gas jet, the mixing efficiencies are throughout consider-
able and the higher, the more gases are admixed inasmuch as m2/mo > v0 v2.
Only for very small v2/v0 or m2/"O this rule is inverted, that is,
the efficiency then deteriorates with increasing admixture until
mOVO = m2v2 and rises again afterwards.
Since, in case of mixing efficiences below unity, warming of the gas
mixture occurs also when the temperature of both jets is equal, the supply
of momentum always has a heating effect as well.
NACA T 1357
For the mixture Mach number
i+ 2 a + o i+ 2 __
X_ -1_ xvl2 + X-1
2 a12 (1 + m-2V2 m
2 2 +- i
1+ 2 a1 ( + 2m 0 1 + 2 av
X-1v2 X J1 1v+
is valid. The mixture Mach number is therefore a function of the mass-,.
velocity-, and Mach number ratios of the components. When the Mach
number ratio of the components becomes unity, the ratio defined above
of the air Mach number expression and of the mixture Mach number expres-
sion is equal to the mixing efficiency.
4. Discussion of special cases.- The discussion of the limiting cases
of the mixing efficiency may be extended to the remaining properties of
the mixed jet and leads to remarkable characteristics.
The special case (frequently occurring in practice) v2/vO = 0, that
is, admixture gas at rest gives because of the disappearing velocity
component parallel to the axis of the admixed gas directly F2 = o, since
continuity and momentum theorem can be satisfied simultaneously only
when mixing occurs, thus PxFxvx 0 PoFovo. One may now consider Fx as
an arbitrarily selectable independent variable and assume it to be pre-
scribed since to every arbitrary Fx there pertains a F2 = m.
From the equations (1) to (4) there follows the quantity (undetermined
F2v2 1 X( 1 0 i + 1 Fx -
FOvO 2 a 2 TO TOFO /
S+ T+ 1 (14)
\ 2 o o
and therewith directly, from the equations (9) and (11) and (12), the
unknown quantities vx, px, Tx.
NACA TM 1357
A still more restricted special case not infrequent in practice
is v2/VO = 0 and T2/TO = 1. Due to the throughout equal pressures p,
one then has also p2/PO = 1, and the undetermined quantity F2v2 becomes
x 1- +L -1) + 1) (14a)
and hence furthermore the unknowns from equations (9), (11), and (12)
x 1 T + X 1 0 FOVO ox T
VO 1 + F2 v/v0 0 a 1 + F1V 2 V Tx
In the mixed jet temperature and density are, therefore, completely
different from the values of the gases (all equal) before the mixing; the
jet mixing has a heating effect as would be shown by a schlieren-optical
A third special case results with v2 = v0
M = Vo(PoF0 + P2F2)
J = v(PoFo + PF)
E = v0 FO( + gcpTO) + P2F2 + gpT2
D = pOTO = p2T2
and with the aid of equations (9) to (12)
vx = V0, the flow velocity remains the same after the mixing
Fx = FO + F2, the flow cross sections add
x = PF + 2F2 the densities mix in proportion to the masses
FO + F2
Tx = P0 + 2F2 the temperatures mix in proportion to the
PF0 + P2F2
NACA TM 1357
Similarly, all other special cases of the gas jet mixture may be
derived from section 2.
5. The constant-pressure mixing chamber.- The free jet mixing
treated in sections 2 and 3 takes place in the same manner in a closed
mixing chamber, if the walls of the chamber have the shape of the
streamlines of the admixed gas indicated in figure 1. In this case,
the mixing ratio m2/m0 also may be arbitrarily limited. Due to the
pressure in the admixed gas remaining constant, the flow velocity,
temperature, and density of that gas remain constant in the mixing
chamber before the mixture is achieved.
The individual, usually decreasing cross-sectional areas F of
the mixing chamber may therefore easily be calculated from the
decreasing quantity of the admixed gas. At the point on the mixing-
chamber axis where the mixing cross section is determined by
SM()E 1 2
the mixing-chamber cross section is, according to the continuity
F = F() + F2(x) F2() (15)
one uses therein the designations of figure 3 and the symbols have the
same significance as in section 2.
The independent parameter F2(,) may be selected arbitrarily
between FO and F2(x) and results in each case in a mixing-chamber
cross section F.
Whereas thus the mixing-chamber cross sections may be calculated
simply and unequivocally, the actual meridian form, that is, the
coordination of these cross sections to given points along the axis of
the device, is determinable only on the basis of empirical experiences
regarding the actual opening angle of the mixed jet, the meridian form
of the mixed jet, the velocity distribution in it, the rate of the
turbulence dissipation, etc.
NACA TM 1357
Without knowledge of the results of such tests, one may consider
that the velocity distribution in a conical jet with a total opening
angle of about 100 to 140 will be homogeneous.
Another important research problem concerning the mixing chamber
arises in the use of very hot core jets as are given off for instance
from rockets and where the very strong thermal dissociation, which may
contain in latent form more than half of the energy supplied to the
core jet, is reduced in the mixing with surrounding air and'thereby
will, in addition, very greatly heat the mixed jet.
The question how far the mixing can be accelerated by special guide
vanes in the mixed jet would have to be clarified separately.
As mentioned before, it was further presupposed in the present
consideration that the kinetic energy first converted into vortex
energy in the jet mixing is further converted, still within the mixing
chamber, practically completely into heat. This process, essentially
caused by internal friction, is greatly accelerated by the large dif-
ferences in velocity existing, the high viscosity of the hot combustion
gases, and the chemical reactions taking place simultaneously. Thus,
it appears justified to calculate as in the customary gas throttling;
experimental confirmation, however, is still lacking.
As the examples completely calculated later on show, this assump-
tion is unfavorable for low-pressure mixing chambers, of slight influence
for high-pressure mixing chambers which are heated by power plants of
small inner efficiency, and favorable for high-pressure mixing chambers
heated by power plants of high internal efficiency.
II. Thrust Increase of Jet Engines by Admixture of
Air to the Exhaust Jet
6. Theory of thrust increase.- One has
Core thrust: PO = p0Fov0 P2F202
Total thrust: P = P4F4 v pFv2
p4F 4v pF2
Factor of thrust distribution: -
PO PFov p2F20v2
0F0 0 2F20
10 NACA TM 1357
Of the characteristic parameters listed in figure 4, 14 are unknown
F, v2' P2' T2
F3, p3, v3, P3' T3
F4', p4, v4' P4 T4
The parameter p2 is assumed to be known since it is, by design assump-
tions, a priori arbitrarily selectable within certain limits.
Of flow equations, there are available:
Zone F F2: Continuity and energy theorem, adiabatic equation and
Zone F2 F3: See section 2; for the F2 there one has to put here
F2 F20; use is made of equations (9), (10), (11),
(12), and of the relation for constant-pressure
mixing p3 = P2
Zone F3 F: Continuity and energy theorem, adiabatic equation and
gas equation, and the pressure reduction condition
P4 = p
Thus one has, for the 14 unknowns, an equal number of determining
equations. The unknown quantities p4, v4, F4, v2 appearing in the
thrust factor P/PO are to be calculated from the prescribed quanti-
ties p2, p, v, p, T, FO, PO, Vo, pO, TO, F20, F2'
From the flow equations of the first zone there follow the relations
P2 (2) (16)
T2 = P2--- (17)
NACA TM 1357
V2= VZ+2gcpTrl-(j) (18)
F = F2 + gpT (19)
where the symbols defined as parameters for equations (5) to (8) become
where the symbols, defined as parameters for equations (5) to (8) become
M = P0FOF0 + p(F2 F20) ()
Energy sum: E
+ p(F2 20) ) 2 + 2gCpT
+ gcTo) +
(F F20) v2 + 2gcpT +
Temperature density: D =2 kgsec2 o/m4
12 NACA TM 1357
and finally the directly used unknowns themselves
P4 = gcp M2D15 X gsec2/m (20)
M 1 2 X-1
T4 = 2 p oK] (21)
1 2 X-1
4 = + 2 1 X m/sec (22)
M2 M 22 \P
F4 2 (P) [2 (3
gcpD F 2(M- 1 -P
Therewith, one may at last write the desired factor of thrust distribution
p F v2 pFv2
P 2 2
O Po0FO P2F20v2
X-( 1 X-1 2gcX- P2 1
2 oFvov 4S L2
S P2X 2 22 + P 1 2
2 2 + 2gcpT 22\ IP\X
pFO 2 p 20v 1 + -- 1 -)
*The NACA reviewer has pointed out that the quantity (p2/p)l/X in the
denominator of this equation was erroneously inverted in the German text.
NACA TM 1357
7. Discussion of the thrust increase.- The thrust factor P/PO
does not immediately signify the thrust increase of a jet engine by
addition of the shroud under otherwise equal circumstances. Rather it
describes, in the first place, the distribution of the thrust between the
core jet and the shroud for the respective state of flight p, v.
However, for this state of flight, the core jet without shroud,
might have a thrust essentially different from the thrust it would
have with a shroud. In practice, it will happen not infrequently that
the core-jet thrust without shroud is considerably smaller or even zero
so that the actual thrust increase for a certain state of flight by
addition of the shroud can be much larger than P/PO, even infinite.
This case occurs for instance for pulse jet tubes and high flight
velocities. On the other hand, the thrust of core jets might be
reduced by addition of a shroud, as in rockets, although this effect
will often remain negligibly small. In the last special case P/PO
then actually signifies the thrust increase of only the core jet by
addition of the shroud.
The physical technical significance of equation (24) will become
even clearer by discussion of a few special cases.
(a) Special case v = 0, that is, static thrust. Equation (24)
is specialized to the form
X 1 P0FO,
1 OFov -2o
PF20T /2)X P2
P(F2 F20) gcpT px
P _-; 2 2
+ 1 + 1
2p(F2 F20o) T(P
p = F0v i
*NACA reviewer s correction: The erroneous term in the
denominator of the German text was changed to the correct form PoFoo
+ p -
14 NACA TM 1357
With p2/P > 1 the solution becomes imaginary; this case, as is
immediately clear, is physically not realizable. With p2/p = 1 one
obtains P/PO = 1; this case is the same as in the jet mixing with
free surrounding air and an increase in thrust does not occur.
With p2/p < 1, real solutions with P/PO > 1 are possible as
long as the sum of the three terms under the large square root sign
remains positive. Since the expression
L (p/p) X
is always negative, the second term will always be negative and the
first, too, becomes negative when the enthalpy of the core jet is
large compared to its kinetic energy, thus, the Mach number vo/a0
is small. One understands immediately, by means of the following
consideration, that, for the static case, even for P2 < p, cases may
exist where the flow is physically not realizable.
From the elementary gas dynamic relation
p 1 X-l 2 -1
P2 2gcpT2) 2
there follows that the pressure ratio for the flow either in the static
or dynamic case depends only on the Mach number. If there becomes for
instance v/a <2 v2/a2, that is, if, in the mixing process, the enthalpy
increases more than the kinetic energy, thus decreasing the Mach number,
the higher external pressure can no longer be attained at all after the
constant-pressure mixing. This case will occur in practice quite fre-
quently in the mixing of hot slow combustion gas jets with cold air,
particularly when, in addition, reverse dissociation occurs.
Generally, the static thrust increase will be zero, thus P = PO,
2 -1 /2 2
P 2ME J P
P -- or v2 gcpT2 2
P2 \ 2ME 2
NACA TM 1357 15
that is, one will have to work sometimes with very high mixing chamber
inflow velocities v2 in order to attain high mixing efficiencies if
the static thrust value is important. The v2 optimum for static
thrust may be immediately determined for any given p2/p from the
equation for P/PO0
With the aid of equation (12a), one finds generally that, for an
efficiency BD of the end-diffuser different from unity the undisturbed
external pressure p is again attainable when
VO I X-12
vo X 1 v2
v2r X I 1
- + 1
(b) Special case v0 = 0, that is, momentum less heat supply, with
the static thrust becoming immediately zero, as known from the standard
The effect of the fuel injection as a core jet will be mostly
negligible in this case; that is, the core thrust PO becomes zero.
For the pure ram-jet thrust, there then follows with
M = MO + pF2(2 2 + 2gCpT 1
J = pF2 )f2 + 2gcpT
m2V2 2 2
1 j + --2-- 1
2 X 1 V2
NACA TM 1357
equation (7a):E = MOgH + pF2 P21X
when M0 is the mass of the added fuel and H its thermal value in
kgm/kg, from equation (24)
+ J21\ X
With the known assumptions F2/F--.o, M--O0, this formula may be
transformed into the simple approximation formula for the thrust of
the ram jet given before
P = pFv2 1)
It differs, in addition, by the constant values of the specific heat
from the exact calculation of the ram-jet engine according to ZWB-UM 3509
(NACA TM 1106).
(c) Special case T = TO without further peculiarities.
(d) *Special case (F2 F20) = 0. With
M = p0FOvO,
J = PO'A ,
E = POFOVO + gcpTO)
-P= 1 P2)X
*ACA reviewer's footnote: The symbol F20 was misprinted in the
German as F2.
NACA TM 1357
IFO 2g 2 T ] 2gc T X
POF + L.- F + 2
POO P 2 X-1 ( 1
pOFOv2 PF2o V2 1 + 2 1 ~ X -(PX
Thus after eliminating the admixed air, in general, a thrust
increase remains, which follows from the formulation, since the
thrust PO of the core jet had been referred to the moderate flow
velocity within the shroud and additional useful pressure drops
originate due to the higher flight velocity. Only with p = pO one
finally has P/PO = 1, for instance, in case of rockets as core jets,
when with F20 = 0 also p/p2 = 1.
(e) Dependence of the P/PO on F2. In equation (24), in the
V2ME 1 (1 + J2P (
the terms quadratic in F2 become equal to the second numerator term pFv2,
and thus disappear for very large F2 so that P/PO then becomes, for
very small v/vO, approximately proportional to F). It therefore
becomes proportional to the square root of the admixed mass whereas
otherwise P/PO, independently of F2 and the admixed mass, tends
toward a fixed limiting value.
This behavior is known from the elementary theory of the Melot
device as well as of the ram-jet device.
Generally, one may, by augmenting F2, increase the thrust not
without limit but only up to a fixed limiting value. Only at the flight
speed zero it increases theoretically with /f without limit.
NACA TM 1357
(f) Dependence of the P/PO on p/p2. For p/p2 = 1 thrust
increase does not occur in any case; equation (24) yields P/PO = 1.
For P/P2 < 1 there originate throughout real values for P/PO which
are larger than unity, whatever may be the flight speed v, ratios of
masses, velocities, or temperatures. Under these circumstances, there
exists therefore no range of flight speed or of the other operational
conditions where the shroud would lose its thrust-improving effect.
For P/P2 > 1 the situation changes completely. Here, cases with
P/Po 1 as well as cases with P/P0 = 0 are possible. The term
chiefly responsible for the thrust
22 ME p X
P4F4v4 =M + M2 1
contains under the square root sign two energy terms: the total kinetic
energy J2/MP present at the end of the mixing chamber F3 and the
addition kinetic energy 2 originating during the process
of expansion in the discharge nozzle between F3 and F4 from the
total enthalpy present in F3 when p/P2 < 1, or kinetic energy recon-
verted into enthalpy when p/p2 > 1; thus, compression flow exists as
is here to be considered.
When p/p2 is larger than unity but still
compression flow so little kinetic energy
so small that in the
is consumed that there still remains
(PF1v 2- pFv2)- (P0Fv2 p2Fv2) > 1
we obtain again a positive thrust increase P/PO > 1.
NACA TM 1357
When p/p2 has exactly the magnitude that
p +J gcpT
1 X-1 X- x-1
S(P X] + 2(P) X 2l/ 2gcPT )X =
2E + j2 pF2v2 + V-r2-1 (
SoFov -PFo2 2gcT v2 ( X 2 \
0 OY0 PF20 + 2 P P
the thrust increase disappears, that is, P/PO = 1.
If p/P2 further increases, the left side of the above equation
becomes smaller than the right side; due to the shroud, there appears
immediately a loss in thrust, that is, P/PO < 1.
If P/P2 is augmented still further, for instance, until
V X -1 X 1
P2 P2p/ v2p/
is valid, the thrust of the core jet is exactly used up by the processes
in the shroud; there remains P/PO = 0 and the total thrust of the
arrangement becomes zero.
All these cases are definitely realizable design, and have probably
been actually realized as proved by the numerous failures of related
This holds true also for the case of still larger p/p2, for instance
for the case that exactly all kinetic energy J2/M2 present in F3 is
used up during the recompression to the external pressure which is
P 1 + M2
F2 2(ME 1 J2)
NACA TM 1357
In spite of all the energy absorption, the entire power plant behaves
the same as a pitot tube, has therefore on the whole, only drag.
From these considerations, there results that the jet mixing must
take place, if possible, at pressures exceeding the static pressure of
the outer air so that expansion to external pressure occurs after the
mixing, whereby a compression to external pressure is not necessary.
That is, fundamentally, melot type low-pressure mixing nozzles are
less favorable or quite useless, and ram-jet type high-pressure mixing
nozzles more favorable.
Since low-pressure shrouds therefore promise advantages only for
the static case and at very low speeds of motion, we shall deal with
them here only by comparison or for ground setups.
(g) Dependence of the P/PO on the inner efficiency of the core
jet. The ratio between the kinetic jet energy and the total jet energy
of the different jet engines, characterized by the jet Mach number
varies to an extreme extent.
For rockets, this ratio approaches sometimes 50 percent, for turbo-
jets it lies around 10 percent, and for pulse jet tubes it drops to a
few percent. For the basic application of momentum heating to the
exhaust gases of these power plants, it is therefore worth knowing how
far the attainable thrust increase P/PO depends on these inner effi-
ciencies. One may write equation (24) for this purpose in the form
X-X-1 X-1 1
J 1- (P-+ pF2v + 1 -
P22 F2 )2 X v )
1 2gcT p9) X (P2 X
J pF2v2 + 2gc~T -
In many cases which are important in practice, the ratio P/Po
is determined predominantly by those terms which contain M, E, and J,
that is, it will increase with
NACA TM 1557
Large values of ME/J2, that is, large ratios between total and kinetic
energy or small jet Mach numbers according to equation (12a), however,
signify nothing else but low thermal efficiencies.
For otherwise equal conditions, one may therefore expect higher
thrust improvements with core jets of low inner efficiency, that is,
small jet Mach number (for instance, pulse jet tubes) than for instance
with turbojets or rockets.
Since the thrust of the shroud is determined by the thermal energy
loss, and the core thrust by the "useful" kinetic energies, the above
statement is quite plausible.
The influence of the thermal efficiency frequently is greater than
the influence, large in itself, of the mixing-chamber pressure. In
practice, the values of ME/J2 lie approximately between 10 and 40
whereas the values of
(at the high subsonic speeds treated here first) vary between 0.05
and 0.1, and the second term under the square root sign always remains
close to unity.
Thus, the differences in the thrust factors may become larger due
to the selection of different core jets than due to different mixing
8. Air admixing shroud heated by rocket exhaust gases.- Let a
rocket be prescribed with the jet characteristics PO = 0.02 kgsec2/m4,
V0 = 2000 m/sec, FO = 0.04 m2, TO = 17500 K; hence P0FOv0 = 1.6 kgsec/m,
pOFv2 = 3200 kg, p0FO0v/2 + gepTo) = 6,720,000 kgm/sec (= 90000 hp).
Let the dissociation heat still bound at the mouth of the rocket be
3,620,000 kgm/sec, corresponding to a lower mixture-thermal value of
the rocket propellants of 1540 kcal/kg. The jet Mach number is
VO/aO = 2.63, the expansion ratio of the rocket Pi/PO = 23. Favorable
arrangements of thrust increasing air-admixing shrouds at flight
speeds 0 and 800 km/h are to be indicated.
For v = 0 km/h naturally only a low-pressure mixing chamber comes
into question. Since drag of the shroud itself here matters little and
NACA TM 1557
the static thrust is known to increase without limit with the admixed
masses, an optimum magnitude for this case cannot be stated as is also
shown by the differentiation of equation (24) with respect to F2.
If one chooses arbitrarily a diameter of 2m for the F2 cross
section, and the state of the outer air to correspond to the standard
atmosphere, and does not at first assume a value for the mixing chamber
pressure P2 which is likewise arbitrarily selectable, there follow
for F20 = 0 and v = 0, the values for M, J, and E as functions
of p/p2 from the equations (5a) to (7a), and the corresponding thrust
coefficient from equation (24).
It is known that P/PO = 1 for p2/p = 1, likewise for small
values of p2/p. Hence, there must necessarily exist an optimum value
for p2/p with respect to thrust increase which may be determined by
differentiation of equation (24) with respect to P2/P to be about 0.7
as also shown by the representation of this equation in figure 5 (with-
out reverse dissociation). P/PO there becomes approximately 2.5, that
is, the total thrust is increased by the shrouding from 3.2 to 7.5 tons.
If the rocket nozzle is adjusted to the new pressure ratios, P/Po
becomes 33, the exhaust velocity increases to 2080 m/s, and there
follows a small additional core jet thrust of 0.15 tons with which the
entire calculation would have to be repeated. If the end diffuser of
the shroud has an efficiency different from unity, one attains only a
slightly higher mixing chamber pressure P2 which, according to fig-
ure 5, has only a small effect on P/PO because of the flat thrust
Corresponding to the pressure drop of 0.7 p there is an inflow
velocity of 238 m/s and an inflow Mach number of 0.72. The mixing
efficiency becomes about 55 percent and the mixture Mach number,
according to equation (12a), remains sufficiently far above the inflow
Mach number to insure the reincrease to the external pressure.
With the aid of equations (16) to (23), the remaining desired
quantities may be calculated. The dimensions of this shroud are
represented in figure 6 and are reminiscent of wind-tunnel proportions.
Possibilities of its application for very slow aircraft, launching
devices, or water and land vehicles seem therefore dubious. On the
other hand, these low-pressure shrouds are interesting for ground test
setups, for instance, for investigation of rocket engines operating at
low nozzle opening pressures or of entire rocket devices with high
approach flow velocities and for subsonic and supersonic wind tunnels
of very large test cross section with rocket propulsion, particularly
by high-pressure low-temperature (for instance, water vaper) rockets.
NACA TM 1557
In the calculation, so far, the possibility was disregarded that
the considerable dissociation of the rocket jet does not stop during
the relatively lengthy mixing procedure, in spite of the very consider-
able cooling of the combustion gases, but does reassociate so that the
dissociation energy mentioned at the beginning additionally heats the
M and J remain, of course, unchanged, whereas E increases
correspondingly; hence, the thrust coefficient also varies according
to equation (24) in the manner represented in figure 5 (with reverse
The optimum of the mixing chamber pressure now lies at p2/P = 0.9
and reduces the thrust quotient optimum there to P/PO = 1.6; thus, the
total thrust then increases from 3.2 tons to only 5.1 tons. The further
heat supply by reverse dissociation and possibly afterburning of the
combustion gases has therefore, in the low-pressure mixing chambers, a
very deteriorating effect on the thrust improvement. In this case,
minimizing the vortex conversion into heat would therefore be advantageous.
At a flight speed above zero, one finds, for the shroud, a region
with two optimum mixing chamber pressures, one below and one above the
external pressure of the air at rest. The first, with increasing v,
soon becomes insignificant so that one is concerned only with the high-
pressure mixing chambers at all flight speeds of practical interest.
The optimum thrust increases become largest in the static case or
for very low flight speeds (region of good Melot effect), pass through
a region of very moderate values at medium subsonic speeds, and finally,
approaching sonic velocity (region of good Lorin effect), increase
again to higher values which, however, remain far behind the high
initial values. Only in the supersonic region does the high-pressure
shroud become equivalent to the low-pressure shroud in the static case.
In flight, at v = 800 km/h, the air drag W of the shroud must
be taken into consideration in determining the optimum conditions. One
can again express the thrust increase (P W)/PO as a function of the
quantities to be determined, F2, and p2/p, and find their optimum
values. For example, for the conditions represented in figure 7 it is
found to be P2/P = 1.2, and a thrust increase of about 20 percent is
found which corresponds to a thrust coefficient of the shroud of approxi-
mately c, = 0.2. Thus with the assumptions made here (full reconver-
sion of the dissociation and turbulence into heat) one finds this thrust
increase is 30 percent of the maximum thrust possible from a prescribed
NACA TM 1357
Whereas, in the low-pressure mixing, one could observe expansions
of the entrance and exit cross section, the high-pressure mixing chamber
showed the known narrowing of these two cross sections.
On the whole, one will conclude from the moderate thrust increases
of this example that the momentum heating of the ram-jet engine by rocket
exhaust gases probably will have technical significance only for special
9. Ram-jet tube shroud heated by exhaust gases of turbojet or
pulse-jet tubes.- Let the necessary characteristic parameters of the
entering and leaving gas jet of a jet engine of moderate jet Mach num-
ber at the flight speed of 200 km/h be prescribed. The arrangement of
a ram-jet shroud with maximum thrust is desired; the flow velocity at
inlet and outlet of the power plant is to remain unchanged, and the
flight speed is to be v = 800 km/h.
One will choose an arrangement according to figure 4 where the
core jet operates within the shroud in the adiabatically decelerated
air in a medium of the same surrounding velocity as in the initial
state (200 km/h) but with increased values of pressure, density, and
temperature. Due to these changes alone, the thrust increases by
15 percent, the fuel consumption by 17 percent.
With consideration of the drag of the shroud, one may again
represent, with the aid of equations (Sa) to (7a) and (24), the thrust
coefficient as a function of the cross section area F2 of the shroud,
and one obtains a flat maximum for instance in the neighborhood of
F2/FO = 2.3 of (P W)/PO = 1.2. The total thrust increases due to
the shrouding from the standard value at 200 km/h to a value by 20 per-
cent higher at 800 km/h without noticeable change in the fuel consump-
tion, thus, with a multiple of the total efficiency. The practical
importance of this arrangement lies not so much in the moderate thrust
increase in itself as in the fact that the increased thrust may still
be expected for a flight speed at which the operation of a simple pulse
jet tube is altogether questionable. The shroud simulates at 800 km/h
flight speed the conditions of 200 km/h which are more favorable for
the core jet, particularly the lower diffuser entrance velocity. The
internal-pressure level of the core jet increased by almost the whole
free-stream impact pressure and therewith provided the compensation of
the high additional opening pressures on the air control valve flaps,
and improved the air supply from the rear for the frontal ram effect
of the air.
Whereas the efficiency of the jet tube mentioned as an example
amounted to about 1.5 percent for v = 200 km/h, it can be increased
by the shrouding, for v = 800 km/h, to 8 percent so that it comes
quite close to the known efficiencies of simple turbojets and the range
of flying missiles thus equipped may become very considerable.
NACA TM 1357
Figure 8 shows the approximate thrust variation against the flight
velocity of a pulse jet tube without shroud, with the previously men-
tioned free shroud, and, finally, with a shroud attached to an existing
airplane fuselage or wing in such a manner that no additional drags
are caused by it.
The shrouding causes the thrust variation of the pulse jet tube
to become similar to that of a turbojet. This circumstance which is
very favorable to the pulse jet tube shows simultaneously that the
effect of the shrouding on a turbojet remains by far smaller since the
standard equipment of the latter anticipates a large part of the
effects utilized by the shrouding.
10. Air admixing shroud heated by exhaust gases of piston power
plants.- A 000-hp piston power plant yields for 800 km/h flight speed
about 950 kg propeller thrust and approximately 3.8 kg/sec exhaust
gases of 6000 C temperature. The free-stream impact pressure near the
ground is q = 3080 kg/m2. If one wanted to mix this quantity of
exhaust, gas with an air quantity as large as possible, for instance
100 times as large, the required Fl would be Fl = 1.39m2, and the
temperature after the mixing would be
T3 = (873 + 100(288 + v2/2000))/101 = 3180 K
in order to make the shroud thrust P become
P =2qF(iT3/T2 1) = 81.3 kg
that is, 8.5 percent of the propeller thrust.
The natural drag of the high-pressure shroud was disregarded as
well as that of the entire piston power plant; a possible direct thrust
of the exhaust gas jet remains practically unaffected by the additional
equipment. For lower flight speeds, this thrust ratio deteriorates,
for higher ones, it improves.
The admixing of surrounding air to exhaust gas jets of power plants
may have a thrust-increasing effect, if it takes place at a pressure
other than the surrounding pressure.
NACA TM 1557
Admixing in low-pressure mixing chambers, it is true, is limited
to such moderate speeds of motion that the flight impact pressure does
not yet have too much of a disturbance effect on the low pressure.
In these cases, there result very considerable thrust increases
which make the low-pressure mixing interesting for fixed installations
in subsonic and supersonic wind tunnels, altitude test chambers, and
water or land vehicles.
Admixing in high-pressure chambers, in contrast, becomes the more
effective the higher the speed of motion so that this type is suitable
especially for aircraft in the high subsonic or in the supersonic range.
The relative thrust increase, under otherwise equal circumstances,
is the greater, the smaller the Mach number of the original exhaust
gas jet. This increase rises therefore sharply in the following order:
rocket, turbojet power plant, pulse jet power plant.
For the last type of power plant, the operation of which is sensi-
tive against high approach flow velocities and low air densities, one
may in high-pressure mixing chambers attain an improvement in climate
by which the flight speed range and flight altitude range for pulse
jet power plants are extended.
One may expect from this type of power plant, under these circum-
stances and at high flight speeds, total efficiencies which approach
those of the turbojet power plants.
The relative thrust increase for low-pressure mixing chambers is
greatest in the static case; with increasing speed of motion it drops
very rapidly and disappears at fractions of the Mach number 1.
The relative thrust increase for high-pressure mixing chambers is
zero in the static case, rises very rapidly with increasing speed of
motion, and attains maximum values in the supersonic range.
Translated by Mary L. Mahler
Committee for Aeronautics
NACA TM 1557 27
1. Melot, H. F.: French patents 522163, 1919; 523427, 1920; 571863, 1922.
2. Busemann, A.: Schriften der Deutschen Akademie fur Luftfahrtforschung,
Heft 1071/43. Berlin 1943.
28 NACA TM 1557
Figure 1.- Free mixed jet surrounding air in motion.
NACA TM 1557
U 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Figure 2.- Propulsive mixing efficiency of constant-pressure mixing.
NACA TM 1557
Figure 3.- Constant-pressure mixing chamber.
NACA TM 1557
Core jet ch
(kgsec 2/m4) P0
Figure 4.- Ram-jet power plant with momentum heating.
NACA TM 1557
Figure 5.- Dependence of the thrust quotient P/PO of a low-pressure mixing
chamber in the static case on the mixing chamber pressure with and with-
out reverse dissociation of the rocket jet at 17500 K.
NACA TM 1557 33
I b f cI)
-i M f
o I &.,I-
2 o '
NACA TM 1557
F11 4 -
NACA TM 1557
With attached shroud ..
--- "-' With free shroud
Without shroud -
0 200 400
Figure 8.- Thrust variation of a
600 800 1000
pulse jet tube with and without shroud.
NACA-Langley 7-13-53 -1000
m w o-
G L: a o
~ M i.!,
mUL f a c4 -
Qa< P W w B-wZ
3 fn o
E w -i
i- -^ u 0
Q tQ -4 "
z z ",-
r( k c S
co S q- Es"
- Q1 3R e .9
% 3 c ^ MC
N -0 2 eo ro" '
*s 1 0 mb a
t ma U
I d w 0 )U a ff V
E 1- E-
4: ro -.1 in
I 0. ca
8 | I V3
^i e in .>-
4' i- g -i -^ t a ^
01 01 a 0 a 1
0C8< i 4in U3i<
fi in C
I ^ j S ^
o o' b 0
O < ~s 6 s E
*a^ 5~" Q30
OI-v -: -
C- r~- Q|- ~ O ,m "
Qa '50 i0 a, 3 L S c
.em 3 3 2 ow .n Z t
Q) 4, ; 4, W$4
^ ~~ *c2 ~C -
CO~ (L & & Qj OI lM
0000 0 C
Q .CW C al Wra .
3PE 0 c C C c < '
Q~( 0, < dU WmV.
"0 0M r
4,i i u4 y4 di H
0 0 0.
W F. M m
(Wr s .
t- o- 2Uc
" 2 *5.*E ;
Raq ;; 0 .6" 50
V a -
S- .0 E
-.0 s II 1
t a r -4 co u EL u
43"2- o,-- &.0a<
0 C 7B 10 D .[
W 0 44Of Q/ Obb.....
= 11 z c C = a < ', <
aL- < w0- w- E
cm m w Le D
0 c m to to
(04,4 4, 4,$4 43
~~0J 04 e 112
I m 5.0
W W.- 0
430 -X o-'a co
1 .-* 0 $4 ii
So a.- t
g'S g k
M a C~ t-
^'QJ .^1 Ci 0
e,6 s Q l
it 0430 g
S S T hg
S 4S.g~ "*
.*ss^s.ss > .8o
04 :g 44 too
-S o 0?15 4.
S 5 a a -
- I "2 -a i
Jr r m E a a S
w (0 42 ii
43; 43 43 r~d~
0, C o
lE Mo ILO
z z C
r- o g '
M; I a"5
r 72 Q Oi
0 0 2
L S 1 P6 .
I0 .0 5 -
-a c- d q, ai
dS" S i.
" o ~ ii
!!. tj Er
0 w w1 V
p> E a0
00 0 w *
0 -a oE Er
,C6 n a1 M
cc w M
3 co in
t : >
2 0 A I-' n
0,1 s bO-e6,
3 ~ a -E A
LokS 1 5
W.s o~js 1 0
(* -a m
-43 A.I -- Vq--
.0 B 430 a
0I- ci B. a 0
I -. -- .c.
x t- a 1
wi-I- C. i '
S0 E 0
E WO r i- z
C4 i i a o vii;
?' w ~ u Qi
to G cic i ci ci cii
taQ~ ce e c a<< .
Q tyL]DdUcd 17Z
-.nc WI 4* if ri 5
0 0 <
,. z -< 6
K Z Li a s-
0 0 0
- I i
E E- i C
gf^a S 5
l '*.B i 0 01
C- U J*F I aB
0 LW -i
c i V
Sca*i- U 1
0.0 3 C 8
Zrm a 'ii- o,d
al,- a to O-
-*I -- Ns
i i s ]Issll '
~Ad d ~ MSi."l
C cn ~
t3 ~ ~ : 4,M m a f<3 z,
U~g| g S E^-r. \/
~aW 136 = r- l
t^ OD U i~d"
tQ 05 < Q i
0-p f ^ n (_rS
R' a q a ;; p
'I -- 0-1--
4 m .4 .,
*C 0 4
!C I W ill-
o ti ai 6j o
c 0 < < ca *
-an CTi a i D z H
L n n ;
Mu ILO. -
Li Ir- 0
-A 0 V3
(O Q3 i
u l9 r,6
Sm E T
. Ea a"1
- t- GA m
0j 0j v L .s
-U C.ai -U 0
s S E
.a**5 t *0t 'S
5' "~ Od S L.*
0 i- 601
-L. cM -~ S04
0.3 c L,
E- o1S S Q* r
iaq R a p se
Sa 4 4 a 4 s-
g a ,- -.a 2 /
QS :1 9 a-:5 <
'0 w w 4 Z
aii v i 4 c ci
oH W _oV
I~CiN N N NrZ.S
- E- la 0
|Z Z i46
5< o..5 Sl ci2
T I43 5 i
10 s s
10 1 C ci
.5l -a E
UNIVERSITY OF FLORIDA
I 1111 III 1 1111 111
3 1262 08105 818 1