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Chapter 2
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Design and Dynamics of Jet and Swirl Injectors

Vladimir Bazarov*
Moscow Aviation Institute, Moscow, Russia
and
Vigor Yang1" and Puneesh Puri*
Pennsylvania State University, University Park, Pennsylvania

Nomenclature
A = geometric characteristic parameter of swirl injector, A = Rm/A-m\ area
a = nondimensional parameter of swirl injector, defined in Eq. (75)
b = nondimensional parameter of swirl injector, defined in Eq. (71)
C = coefficient of nozzle opening, C = R[n
C* = characteristic velocity
c = specific heat
D = diameter of injector element
d = diameter of substance (drop, spray)
/= functional symbol; frequency
h — liquid film thickness
Im = imaginary part of complex variable
K = momentum-loss coefficient; O/F ratio
k = ratio of specific heats, CP/CV
I — length of injector element
M = Mach number
m = mass
m = mass flow rate

Copyright © 2004 by the Authors. Published by the American Institute of Aeronautics and
Astronautics, Inc., with permission.
*Professor and Head, Dynamic Processes Division of the Rocket Engines Chair. Member AIAA.
'Distinguished Professor, Department of Mechanical Engineering. Fellow AIAA.
"Graduate Student, Department of Mechanical Engineering.

19
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20 V. BAZAROV, V. YANG, P. PURI

n = number of passages
p = pressure
A/7 = pressure drop
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Q = volumetric flow rate

q (A) = gasdynamic function; tangent of nozzle surface inclination to
injector axis
R = gas constant; radius of injector element
Re = Reynolds number
r — radius of jet element; radial location of liquid film
5 = surface
Sh = Strouhal number
T = temperature, K
t = time; spacing between injectors
U = velocity
V = volume
x, y, z = coordinates
a = tilt angle of inlet passage; spreading angle of liquid spray
/3 = tilt angle of wall
A = increment
8 = wall thickness, clearance
e = coefficient of jet contraction
rj = pressure-loss coefficient; efficiency; acoustic admittance function
N = polytropic exponent
A = velocity coefficient; drag coefficient; wavelength
^i = mass flow coefficient
l^d — dynamic viscosity
v = kinematic viscosity
II — response or transfer function
£ — hydraulic-loss coefficient; fluctuation of liquid film thickness
TT = nozzle expansion ratio
p = density
cr = surface tension
r = time interval
<& = phase angle of individual process
(p — coefficient of passage fullness, i.e., fractional area occupied by liquid
in nozzle
^ — phase angle of element in the assembly
fl = amplitude of liquid surface wave
(o — radian frequency

Subscripts
a — axial
c = combustion chamber
eq = equivalent
e = nozzle exit
exp = experimental
ext = external
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 21

/= propellant feed system

fl = flow
fr — friction
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g-gas
gg = gas generator
gl — gas-liquid
id = ideal
/ = injector
in = inlet
7 = Jet
k = head end of vortex chamber
/ — liquid
m = liquid vortex
mix = mixing
N, n = nozzle
out = outlet
r = radial
s = vortex chamber
sp = spray
sw = swirl
T = inlet passage
t = tangential spacing between injectors
th = nozzle throat
u = circumferential
w = wall, wave
v = saturated vapor
vc = vortex chamber
X = total
0 = initial conditions
1 = exit conditions
oo == infinite value

Superscripts
- = dimensionless parameter
' — pulsation component

I. Introduction

M IXTURE formation is one of the most important processes in liquid

rocket combustion devices because it determines combustion efficiency,
stability, and heat transfer characteristics. This process is implemented through
the use of propellant injectors, which not only accomplish their main missions
of propellant atomization and combustible mixture formation, but also represent
elements of an engine as a complex dynamic system operating under various
conditions. Any change in engine operating conditions (such as startup, thrust
variation, and shutdown) or flow paths in the feed line and combustion
chamber (such as turbulence and pulsations) may lead to a drastically different
injector behavior.
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22 V. BAZAROV, V. YANG, P. PUR!

The operating conditions of propellant injectors are complicated, and as such

the types of injectors and injector assemblies as a whole are numerous. The
selection and design of a specific injector depends on the situations of concern
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and must fulfill the following basic requirements:

1) To provide high combustion efficiency, injectors should ensure high-quality
propellant mixing with uniform distributions of mixture ratio and flow intensity
over the combustion chamber as much as possible. Minimum consumption of
energy is required for propellant atomization and mixing.
2) To protect combustor walls against excessive thermal loading, injectors
should provide nonuniform distributions of mixture ratio and flow intensity in
regions of concern.
3) To suppress combustion instability in the chamber and to achieve staged
combustion, injectors should achieve prescribed distributions of atomization
dispersivity, mixture ratio, and flow intensity in the mixture-formation zone.
For gas injectors, provisions should be made to remove acoustic energy from
the chamber.
4) To suppress flow instabilities, the acoustic conductivity of an injector
should be minimized, with a smallest possible response to disturbances arising
from variations of the flow rate and other parameters.
5) For more complicated situations such as pulse-triggered instability and
thrust transients, injectors must feature prescribed dynamic characteristics
within preset limits with fixed steady-state characteristics.
6) For a liquid rocket engine (LRE) operating in a pulse mode, additional
requirements of minimizing the volume of injector cavity must be fulfilled. For
LRE with a wide range of thrust variation, prescribed mixture-formation
parameters should cover the entire operating envelope.
To meet these requirements, injectors should provide pre-specified liquid-
sheet thickness, spray-cone angle in the range of 36-120deg, and dynamic
characteristics. In addition, the fabrication procedure should be simplified to
achieve reliable designs. Because many of these requirements vary for LRE of
different types, a great number of injector types have been developed and
implemented. The selection for a specific application is a result of a compromise
between the preceding requirements and to a great extent depends on technologi-
cal expertise, design tradition, and development experience.

A. Classification of Injectors and Methods of Mixture Formation

Liquid propellant injectors and methods of mixture formation can be classified
on the following basis:
1) applications: low-thrust engines, gas generators, medium-thrust engines,
and boost engines of launchers;
2) propellants: earth-storable, hypergolic, and cryogenic propellants;
3) pressure drop: high-pressure and low-pressure drops across injectors;
4) design features: dimensions and configurations of flow passages; and
5) propellant mixing: external and internal mixing.
To disperse liquid into droplets and distribute them over the mixture-
formation zone, various kinds of energies can be used. The choice, however, is
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 23

limited for injectors. The classification of propellant injectors from the standpoint
of the type of energy used for propellant atomization and mixture formation is
discussed here.
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1) The most often used concept for an LRE injector is the conversion of poten-
tial energy of the liquid in the form of pressure drop across the injector into
kinetic energy of a liquid jet, which subsequently produces a spray of droplets.
This atomization principle is used for jet, film, and swirl injectors. The energy
efficiency of such injectors, however, is rather low. A major part of the energy
is consumed in increasing the droplet velocity, rather than overcoming the
surface tension. The pressure drop for a commercial LRE typically ranges
between 0.29 and 3.75 MPa. The upper limit is determined by chamber stability
considerations instead of the required atomization quality.
2) In the case of gas-liquid injectors, the kinetic energy of the gas flow is used
to generate wavy motions at the liquid-gas interface, separate wave crests from
the liquid core, and to accelerate drops. The pressure drop across the gas
passage for achieving required atomization quality is significantly lower than
that of liquid injectors. It is mainly determined by the requirements of operation
stability and uniformity of propellant distribution between injectors. In the case
of increased requirements of atomization quality, for example, in low-thrust
engines, gas-liquid injectors are used for liquid propellant atomization with the
help of additional high-pressure gas.
3) Thermal energy is often used to heat the liquid being atomized, in order to
change its surface tension and viscosity, and in the limiting case (e.g., hydrazine)
to evaporate and decompose the liquid to provide gaseous-phase mixing. In most
LREs, the heating and evaporation of liquid propellants take place in cooling
jackets or in heat exchangers. Only in hydrazine thermal-decomposition reac-
tions, atomizers with developed surface of thermal contact with the liquid are
used for initiating reactions.
4) Acoustic energy is used in acoustic and ultrasonic injectors. The resultant
flow oscillations promote the formation of surface waves in the liquid stream,
which then disintegrates into droplets with a low energy consumption rate.
5) Mechanical energy is used in injectors with reciprocating atomizers or rotors.
The former has been employed only in experimental low-thrust engines (up to 1 N)
using piezoelectric or magnetostriction vibrators. The latter has found application in
low-thrust jet engines and in some designs of liquid propellant rocket engines.1
6) Electric energy has been used to activate piezoelectric and magnetostrictive
vibrators for injectors of low-thrust engines operating in the pulse mode. In the
latter case, the electric energy is directly converted into the potential energy of
liquid. In addition, there have been proposals to use electric discharges in non-
conducting liquids to obtain pressure pulses, so-called electrohydraulic effect.
These methods have not found practical applications in commercial LREs.
7) Combined atomization methods using several different kinds of energy.
Different conversion techniques can be used simultaneously to intensify
atomization, such as combined utilization of the potential energy of liquid and
kinetic energy of atomizing gas. Application of vibrational energy during
pneumatic and hydraulic atomization produces the most significant effect.
This combination allows considerable savings of the energy spent for
atomization.
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24 V. BAZAROV, V. YANG, P. PURI

B. Liquid Injectors
1. Jet and Slit Injectors
The jet injector is the simplest device for converting the potential energy of
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liquid in the form of pressures drop to the kinetic energy of a jet. The injector
also represents a local contraction connecting the propellant manifold to the
atomization zone. The ideal-liquid exhaust velocity t/id is determined by
Bernoulli's equation:

(1)

where AP/ is the pressure drop across the injector, and p is the liquid density. The
effect of viscous loss is usually taken into account by using either empirical coef-
ficients or expressions derived from the boundary-layer theory. The liquid flow
rate is determined by the following equation:

(2)

where AN is the injector cross-sectional area. The discharge coefficient JULN mainly
depends on the injector shape.
The slit injector has a flow passage formed by either flat or concentric
surfaces. The hydraulics of slit injectors are well understood and reliable
design methods are available.2 The atomization mechanism of a liquid stream
involves disintegration into drops due to the loss of flow stability under the
effects of aerodynamic forces on the liquid surface.3
The jet and slit injectors are exclusively used for bipropellant applications,
with their sprays formed by the intersection of liquid streams. Figure 1 shows
an injector with five intersecting jets (four oxidizer jets and one fuel jet). It

A-A

Fig. 1 Bipropellant spray injector with intersecting fuel and oxidizer jets; 1-casing,
2-fuel passage, 3-oxidizer passage.
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 25

contains a casing (1) brazed into the assembly bottom, an axial hole (2), and four
axially symmetric holes (3) for the other propellant tilted towards its axis. The
number of holes (3) may vary from two to four. In case two holes are employed,
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no axial hole is needed, and one of the holes (3) should be connected to the mani-
fold of the other propellant. This situation is referred to as a doublet injector.4 As
the propellants flow out, a thin flat bipropellant lobe providing high-quality
mixing and sufficiently fine dispersion of propellants is formed, as illustrated
in Fig. 2. Injectors of the "triplet" type (with two lateral holes for one of the
propellant) have the same scheme of liquid-propellant atomization and mixing.
In other cases, a sharp bipropellant spray cone with rather coarse atomization
is formed at the place of jet intersection.
The advantages of injectors with intersecting jets are:
1) simplicity of design and reliability of hydraulic analysis;
2) satisfactory atomization quality and uniform propellant mixing for doublet
and triplet injectors; and
3) short flow residence time in the injector and short ignition delay.
The disadvantages of jet injectors are:
1) stingy requirements imposed on the fabrication technology. A small devi-
ation in dimensions may considerably change the atomization and mixing quality;
2) significant non-reproducibility of the resultant spray property;
3) considerable differences of propellant-mixing pattern between flow tests for
model liquids and real propellants in case of hypergolic propellants. This is attrib-
uted to the liquid sheet repulsion and deterioration of the mixing conditions,
caused by gaseous products produced in liquid-phase reactions between the inter-
secting propellant flows;
4) the spray fan formed in the propellant-flow intersection region is extremely
sensitive to local flow fluctuations, especially in the transverse direction. Injector
assemblies consisting of doublet injectors are very susceptible to high-frequency
transverse instabilities.

Fig. 2 Formation of fine dispersion of propellants in a doublet swirl injector;

1-casing, 2-fuel channel, 3-oxidizer channel.
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26 V. BAZAROV, V. YANG, P. PURI

To reduce the effect of fabrication errors, slit injectors with flat holes producing
intersecting sheets are made by spark erosion machining. For example, the platelet
manufacturing technology was used for the impinge jet multi-injector assembly for
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the Space Shuttle Orbit Maneuver System (OMS),5 as shown in Fig. 3. Although
the vacuum bonding of thin sheets of metal results in injector passages of stepwise
form with enhanced hydraulic losses, the propellant mixing efficiency is consider-
ably improved, especially for hypergolic propellants. Injectors with three or five
jets forming a narrow long spray (see Fig. 1) were placed on the injector assembly
as a means to increase the combustion zone length and thus to decrease its response
to pressure fluctuations.
Application of injectors with intersecting jets has practically ceased for high-
and medium-thrust engines with staged-combustion cycles, except for gas
generators in which high-frequency instabilities were not nearly so critical due
to the lower power intensity and longer operation processes. Injectors with inter-
secting coaxial sheets have found applications in so-called pintle injectors6 for
throttable LRE, as shown in Fig. 4. The disadvantages common to all of the
designs of combined slit injectors with variable passage cross sections are
stingy requirements on manufacturing accuracy. The generally permitted mis-
alignment ~0.03 mm is too large for this injector. For example, in the case of
nominal liquid-sheet thickness of 1 mm for a 10-fold decrease of thrust, the mis-
alignment amounts to ±30% of the average width of the injector passage cross
section. The ensuing deviation in the mixture ratio reduces the engine efficiency.
The presence of a movable element in the immediate vicinity of the heat release
zone poses another serious challenge in ensuring reliable operation of injector
assemblies, and consequently restricts the design to single-injector assemblies.

2. Swirl Injectors
Swirl injectors are predominantly used in Russian LRE gas generators and in
combustion chambers with pressurized feed systems or gas-generator cycles.

fuel distribution inlet (from

regeneration chamber)

diffusion bond

platelet assembly

16 face ring channels

redundant EB weld
ox manifold
304L S.S. body fuel manifold

Fig. 3 Space Shuttle OMS engine injector.

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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 27

r~ Oxidizer
Movable sleeve
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Primary
reaction
zone
Fig. 4 Pintle injectors of throttable LRE.

Monopropellant swirl injectors of screw-conveyer or tangential-channel type

were used in early LRE, as shown in Fig. 5. The former employs a screw con-
veyer as swirler and are predominantly used in gas generators with moderate
combustion temperatures. Each injector has a casing (1) brazed into the bottom
of the assembly. The casing has a nozzle (2) and an axisymmetric cavity (3) con-
nected to the liquid manifold through tangential passages (4). The bottom (5) is
flared out and welded in the casing (1) forming a vortex chamber (6). In recent
injector designs, the bottom (5) is made profiled to optimize the shape of the
passage cross section of the vortex chamber (6). In screw-conveyer injectors
(Fig. 5a), a screw-conveyer swirler (7) whose external passages (8) serve as tan-
gential passages (4) is fitted into the casing (1).
Important geometric parameters determining swirl-injector characteristics are:
1) nozzle radius Rn\
2) cross-sectional area of the inlet flow passage Ain;
3) swirling arm, i.e., the distance from the axis of the tangential passage to the
injector axis, Rm.
These parameters form a dimensionless number known as the geometric
characteristic parameter of a swirl injector:

(3)
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28 V. BAZAROV, V. YANG, P. PURI

6 r-\
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a)

b)
Fig. 5 Monopropellant swirl injectors with a) screw conveyer and b) tangential
passages.

It determines the injector flow coefficient //,, the nozzle filling coefficient 9, the
spray cone angle at the cylindrical nozzle exit, and other output parameters.
A table of expressions for calculating these parameters is given in Ref. 7.
In addition, there are some secondary parameters, which are of importance in
determining the liquid flow residence time and viscous losses in injectors.
These include:
1) the diameter and length of the vortex chamber;
2) the nozzle length and the convergence angle of the vortex-chamber wall
adjacent to the nozzle.
When fed through the tangential (4) or screw passages (8), the liquid is set in
rotary motion in the chamber (3), and forms a liquid vortex with a free internal
surface whose radius smoothly changes from the minimum at the bottom
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 29

(5) rmk = RN\/^ to rmx = P/vVl — 9 in the nozzle, where the flow area ratio
<p = AL/AN is the liquid-occupied fraction of the injector nozzle section, and
a = 2(1 — <f>)2/(2 — <p). At the nozzle exit, the spreading angle of the liquid
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sheet slightly increases due to the conversion of the centrifugal pressure produced
by the rotating liquid sheet to the axial velocity component, i.e., the Skobelkin
effect.2 The pressure on the liquid surface is equal to the pressure in the combus-
tion chamber. For an ideal fluid, the velocity at the liquid surface can be written as

U
u + Ut = COnSt = - AP
< (4)

where AP, is the pressure drop across the entire injector. Any change in the vel-
ocity components Ut is unequivocally associated with a change in the surface
radius rm . The liquid flows out of the nozzle exit as a thin sheet whose thickness
and spreading angle are independent of the pressure drop in a wide range of its
variations, and the initial section has the shape of a single-cavity hyperboloid.
The thickness of the sheet decreases farther downstream. It loses its stability
and disintegrates into droplets according to the mechanism described in Ref. 3.
It should be noted that under typical chamber conditions of a modern LRE, the
liquid sheet from a swirl injector does not have enough time to get thin and to
lose its stability since the aerodynamic effects of the surrounding high-pressure
turbulent flow tends to disintegrate the liquid sheet more effectively. Thus, one
has to analyze atomization for the startup and steady operating conditions of
the same injector using two different methods.
The following features of swirl injectors, which determine their predominant
applications in Russian LREs, are noteworthy:
1) For the same pressure drop and liquid flow rate, the average median diam-
eter of droplets is 2.2 to 2.5-fold smaller than that of jet injectors. This advantage
prevails for high flow rates and decreases when the counter pressure (i.e., the sum
of the combustion chamber pressure and the centrifugal pressure created by
liquid swirling motion) grows.
2) Compared with jet injectors, swirl injectors are not so sensitive to manufac-
turing errors such as deviation from prescribed diameter and surface
misalignment.
3) The flow passage areas of swirl injectors are much larger than those of jet
injectors with the same flow rates, and consequently they are less susceptible to
choking or cavitation.
4) The pressure drop across a swirl injector is shared between the tangential
channels APr and the vortex chamber APVC . Under steady-state conditions, the
relation between APT and APVC can be easily defined, with the latter much
higher than the former in most operational injectors. During the engine startup,
when the vortex chamber is initially empty, the entire pressure drop is applied
to the tangential channels and the liquid velocity is much higher than its
steady- state value. The vortex chamber begins to be filled with high-speed rotat-
ing liquid. The ensuing increases in the centrifugal pressure and viscous losses
then decrease the pressure drop across the inlet passage and subsequently the
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30 V. BAZAROV, V. YANG, P. PURI

mass flow rate prior to ignition. This self-tuning capability with variable flow
resistance under transient conditions improves the engine startup operation.
Swirl injectors also feature wide spray-cone angles (in the range of 36 and
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140 deg) and sharp non-uniformity of flow distribution. The liquid is practically
absent near the injector axis. These features can be considered as
a disadvantage when flow non-uniformity is detrimental for the process, for
example, irrigation of catalyst grains for hydrazine decomposition. They are,
however, advantageous for most applications in LRE since hot combustion
products are recirculated to the injection region to stabilize the flame. The forma-
tion of a thin uniform liquid sheet also protects the injector face plate against
excessive heat transfer from the high-temperature region.
Extensive effort has been applied to develop theories of swirl-injector hydrau-
lics in Russia since as early as the 18th century (e.g., Leonard Euler, a member
of the St. Petersburg Academy of Sciences). Among them, the principle of
maximum flow rate postulated by N. Abramovich8 in 1944 (Diploma of Discov-
ery No. 49) is most noteworthy, which essentially laid the foundation of modern
development of swirl injectors. The work was supplemented by the studies of
L. A. Klyachko2 and A. M. Prakhov,3 as well as empirical data of
Y. I. Khavkin9 and many other scientists. It now gives reasonably reliable cal-
culation results. Investigation of the dynamics of swirl injectors7'10 has
made it possible to design injectors with prescribed dynamic properties and
use them as a means of suppressing various mechanisms of high-frequency
instabilities. Work was conducted to study injectors with flow modulation capa-
bilities11 for modulating spray cone angles and flow rates with prescribed degrees
of atomization.
The disadvantages of swirl injectors are:
1) the internal-cavity volume is significantly larger than that of a jet injector
and has a longer startup transient time, which restricts their application in low-
thrust LREs with pulse operations;
2) complex configuration and heavy weight.
The types of swirl injectors used in LREs are diversified because of specific
requirements for different applications.

3. Monopropellant Swirl Injectors

Monopropellant injectors (see Fig. 5) were widely used in open-loop engines.
As a rule, they are arranged in assemblies in the chess-board or honeycomb pat-
terns with alternating fuel and oxidizer injectors. In welded and brazed assem-
blies, oxidizer injectors with elongated casings are attached to the assembly
bottoms and thus serve as structural elements. They often have two- to four-
run screw-conveyer swirlers with cylindrical configurations. Conic screw-
conveyer injectors, which are technologically less effective and widely used in
jet engines, are seldom used in LREs. They are used only in low-flow injectors
to minimize the vortex-chamber volume and to eliminate nonswirling liquid
leaking through the gap between the screw conveyer and casing.
Contemporary monopropellant swirl injectors have tangential passages. As a
rule, the number of passages is three or four. A smaller number of passages
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 31

increases the nonuniformity of flow distribution along the spray cone circle. Two
passages are used only in highly closed injectors or in very long injectors. One
passage is used for special purposes, when flow nonuniformity is required
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along the circumference, such as in near-wall regions. At present, monopropel-

lant swirl injectors are mainly used in gas generators to provide efficient propel-
lant delivery or as near-wall injectors in combustion chambers. Their relatively
low combustion efficiency (~97%) and high sensitivity of fuel and oxidizer
mixing increases the susceptibility to combustion-driven flow oscillations.

4. Bipropellant Swirl Injectors

Since the mid-1960s, bipropellant swirl injectors have been used most often in
Russian LREs of various types and applications. Figure 6a shows a typical design
of such injector elements. It has a hollow casing (1) with a nozzle (2). The
hollow insert (3) with the nozzle (4) and flared-out bottom (5) are brazed into

A-A
a)

b) B-B
C-C

Fig. 6 Two different designs of bipropellant swirl injectors; 1-casing; 2-casing

nozzle; 3-insert; 4-insert nozzle; 5-bottom; 6-central vortex chamber; 7-9
tangential passages; 8-peripheral vortex chamber.
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32 V. BAZAROV, V. YANG, P. PURI

it. The bottom and the insert form the vortex chamber (6), which is connected to
the propellant-delivery manifold through tangential passages (9). The insert (3)
and casing (1) form a peripheral vortex chamber (8) for the other propellant con-
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nected to its delivery manifold through tangential passages (7).

A more advanced and compact design of bipropellant injectors is given in
Fig. 6b. It is beam welded to the fire face. For hypergolic propellants, the edge
of the nozzle (4) is usually buried in the nozzle (2) of the peripheral injector
by 0.7-2.2 mm. The specific dimension depends on the propellant type and injec-
tor geometry. Oxidizer injectors can be used as peripheral injectors (so-called
direct scheme). Such an arrangement is more preferential from the standpoint
of mixing hydrodynamics since it will produce sheets of the same or approxi-
mately the same thickness. In more recent engines, however, the reverse
scheme with fuel delivery through the peripheral stage has found applications,
due to the ease of arrangement with the cooling jacket and the fire bottom of
the injector assembly. Injectors of the direct scheme have either propellant
mixing outside the injector (in this case, the spreading angle of the external
spray should be less than its counterpart of the internal spray), or mixing at the
external-nozzle edge. Injectors of the reverse scheme, whose peripheral fuel
stage have a considerably wider spray cone angle than that of the internal oxidizer
injector, may have propellant mixing at the nozzle edge alone. Internal mixing of
hypergolic propellants should be avoided in such injectors since they are not safe
against ingress of one of the propellants to the cavity of the other one. Cryogenic
propellants of the oxygen-kerosene type can be mixed directly in the vortex
chamber of the peripheral oxygen stage, in which the swirling liquid-oxygen
layer provides the cooling of the injector walls.

5. Combined Jet-Swirl Injectors

To increase the length of the combustion zone and to decrease the combustion
response to chamber flow fluctuations, combined jet and swirl injectors have been
widely used since the 1950s. Figure 7 shows sectors of the injector assemblies
equipped by such injector elements. Each or some of the screw swirlers inside
the vortex chambers are drilled to form a passage for a liquid jet inside the sur-
rounding conical spray. A combination of impinging jets and such a type of injec-
tor element was used in place of baffles in several Chinese booster LREs (see
Fig. 8) to suppress high frequency combustion instabilities. One should be
careful with this design because such an injector could be a source of LRE unstea-
diness if improperly designed,12 and may lead to operations in two different
manners, even at the same pressure drop. There may be a situation with separate
outflows of jet and swirling liquid or with a joint outflow. The central jet could
also disappear and the swirling spray could have a narrow spreading angle and
become poorly atomized. Figure 9 schematically shows a sector of a combustion
chamber injector assembly with alternating bipropellant swirl and impinged jet
injectors.

C. Gas-Liquid Injectors
Gas-liquid injectors utilize the kinetic energy of a gas flow for liquid atomiza-
tion and are mainly used in combustion chambers of closed-loop LREs and in
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 33

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a) b)
Fig. 7 Injector assemblies with alternating swirl and spray-swirl injectors.

some open-loop LREs having special operating conditions such as large thrust
control while using prevaporized, decomposed, or gasified propellant, with
preburning of a small amount of the propellant.
Figure 10 shows the main design schemes of gas injectors used in LREs. They
can be classified into three basic categories: injectors with peripheral liquid and
central gas delivery (a, d); injectors with peripheral gas and central liquid deliv-
ery (b, c, and e); and injectors with two-sided atomizing-gas delivery to the liquid
sheet being atomized (f).

1. Jet Injectors
The gas-liquid jet injector (Fig. lOa) is one of the designs most commonly
used in both hypergolic and hydrocarbon-oxygen LREs. It has a tubular casing
(1) with an axial gas passage (2) and holes (3) for liquid propellant delivery. Pas-
sages can be arranged along the length of the casing (1) in different ways, depend-
ing on the propellants used. Passages (3) are predominantly located in the vicinity
of the exhaust edge of the casing (1). The axes of passages (3) were made inter-
secting with the casing axis at an angle of 45 to 60 deg. Recently, it has become
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34 V. BAZAROV, V. YANG, P. PURI

first plate
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oxidizer

second plate

basic and high flowrate division Injector

Fig. 8 High flow rate and division injection elements for Chinese YF-1 engine.

preferential to make passages (3) chordal, which increases the uniformity of

propellant mixing.
The diameter of passage (2) usually varies between 6 and 18 mm, depending on
the engine thrust requirement. The length of passage (2) is chosen to maximize the

Fig. 9 Combustion chamber injector assembly with adjacent bipropellant jet and
swirl injectors.
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 35

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a)

Fig. 10 Pneumatic injectors used in main combustion chambers and gas generators
of LRE: a) jet-jet injector; b), c) jet-swirl injector; d), e) swirl-swirl injector; f) slit
injector with alternating liquid and gas injection; 1-casing; 2, 9 gas passage; 3-
liquid passage; 4-liquid manifold; 5-mixer; 6-bottom; 7-vortex chamber; 8-nozzle;
10-gas swirler.

removal of acoustic energy from the combustion chamber by treating the injector
as a half-wave acoustic resonator. Figure 11 shows a gas-liquid spray injector
assembly with a swirl fuel injector at the periphery, which protects the fire
bottom of the assembly against overheating. Liquid propellant jets injected into
the mixer (5) are atomized by the gas flow from passage (2). The combustion
process occurs immediately downstream of the injected jets since the flame is
stabilized on the walls of casing (1) behind holes (3). Simplicity of design and fab-
rication and high-quality mixing are the advantages of injectors of the type shown
in Fig. lOa. The disadvantage is the relatively high sensitivity to pressure pulsa-
tions, which, however, can be compensated by changing the injector geometry
to remove acoustic energy at prescribed frequencies from the combustion chamber.
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36 V. BAZAROV, V. YANG, P. PURI

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A-A
Fig. 11 Injector assembly with gas-liquid jet-jet injector equipped with a peripheral
liquid swirl stage.

2. Swirl Injectors
In the aforementioned case, if holes (3) in the casing wall are made tangential,
blocked from the entrance face, a swirl injector is formed with peripheral liquid
propellant delivery and central delivery of a hot swirl gas flow (Fig. lOd). This
design features a contraction between passages (2) and (3) and a shoulder in
front of passages (3) to ensure the formation of a swirling liquid sheet on the
internal walls of the mixer (5). The gaseous propellant arriving through the tan-
gential passages (2) forms a swirling flow in the internal cavity of casing (1),
travels around the liquid sheet running along the walls of the mixer (5), forms
surface waves on it, and finally blows away the crests and disperses the liquid
into droplets. Such injectors are widely used in high-power LREs with hypergolic
propellants. The main advantage is a wide stability margin for operation over a
broad range of mass flow rates. This feature may be attributed to the low sensi-
tivity of the atomization and mixing processes stabilized by the mixer wall to
pressure fluctuations and flow-velocity variations. Positioning liquid passages
(4) at the periphery of the mixer (5) allows cooling of the fire bottom of the
assembly by means of a cone-shaped screen formed by the evaporating-liquid
sheet.
High-quality atomization achieved by swirling gas and liquid flows allows
manufacturing of large injectors with diameters of the mixer (5) in a range of
50-60 mm and flow rates of several kilograms per second. To suppress high-
frequency instability in the combustion chamber, unique acoustic properties of
injectors such as acoustic impedance of the mixer (5) and acoustic resonance
of the gaseous-propellant vortex chamber are used. Important results are obtained
when the acoustic properties of the mixer and vortex chamber are combined.
The disadvantage of swirl injectors is the non-uniformity of flow intensity and
mixture composition along the spray-cone radius, which does not provide ade-
quate combustion efficiency. Figure 12 shows an example of a full-scale injector
with central delivery of swirling gas. The gas resonance cavity minimizing flow
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 37

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01.5 00.5

A-A C-C

Fig. 12 Injector with central injection of swirling gas and acoustically tuned
gas stage.

fluctuations is clearly visible. In low mass flow-rate injector elements with

exterior swirling liquid flows, the central gas stage could be made of a non-
swirling stepped channel, as shown in Fig. 13.

3. Coaxial Injectors with Central Liquid Stage

This type of design includes jet, jet-swirl, and swirl-swirl injectors, as shown
in Figs. lOb, c, and e. The injector has a tubular casing (1) with passages (2) for
gaseous propellant delivery [usually between the pylons having passages (3) for
liquid propellant delivery]. The pipe (6) with passage (7) and nozzle (8) is usually
buried in the tubular casing (1) and mounted rigidly on the pylons. When the
liquid stage (6) is made as a jet injector, passages (3) are radial and passage (7) is
elongated. In this case, injectors are used for mixing liquid oxygen and hydrogen-
enriched gas. When the central stage is made as a swirl injector, passages (3) are
of either the tangential or screw-conveyer type (the latter is mainly used in hydro-
gen-oxygen gas generators). In this case, a swirling liquid sheet flowing out of the
nozzle (8) is in the form of a cone-shaped film interacting with the coaxial gas
flow. The quality of propellant atomization and uniformity of mixing are signifi-
cantly enhanced, which allows manufacturing of larger injectors (with nozzle
diameters of 8-12.5 mm) with propellant flow rates of up to 2 kg/s for each
injector element. The presence of a hollow liquid vortex inside the passage (7)
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38 V. BAZAROV, V. YANG, P. PURI

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Fig. 13 Oxygen-kerosene injector with tangential injection of liquid oxygen.

makes it possible to use the liquid stage of the injector as a quarter-wave resona-
tor to absorb the acoustic energy in the combustion chamber, for which purpose
the passage (2) should be further elongated (see Fig. lOb), and in some cases it is
necessary to make additional holes in the walls of the acoustic damper (6) or to
equip the liquid stage of the injector with an additional Helmholtz transmission
resonator.
The main advantage of such injectors is high-quality atomization and mixing
of propellants. Their disadvantage lies in the complexity of manufacturing and
adjustment. The possibility of self-oscillations of the liquid sheet in the coaxial
gas flow in injectors of this type, which were first noted in Ref. 13, is an
additional problem. Testing of full-scale engines revealed the presence of self-
oscillations at the frequency of ~5500 Hz, leading to pressure pulsations in the
oxidizer passage with an amplitude of up to 0.9 MPa accompanied by subsequent
breakdown of pipe connections with no significant pressure fluctuations in the
combustion chamber.
Investigations of these injectors using models and actual propellants made it
possible to establish the mechanism of self-oscillations described in Ref. 14 and
to develop methods for their suppression. The presence of self-oscillations may
sharply intensify propellant atomization and mixing, especially for injectors with
low pressure drops. It has been proved both theoretically and experimentally that
such induced high-frequency fluctuations are not related to intrinsic oscillations
in the combustion chamber and are safe and even beneficial for combustion stability
and efficiency. The self-oscillation regime is sometimes displaced into the engine
operating regime requiring throttling (below 75% thrust), and injectors are equipped
with devices dampening pressure oscillations arising in the combustion chamber.
The principle of the swirl-swirl injector (see Fig. lOe) is similar to that of
the liquid-liquid injector shown in Fig. 6a. The only difference lies in the cross-
sectional area of the gaseous propellant passage. Because the gaseous propellant
comes through tangential passages (2) from the periphery of the casing (1), the
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 39

mixer is not made long. This, however, may lead to overheating and burnout of the
casing. Additional means of cooling are thus required to prevent overheating of the
fire bottom. Similar measures should also be introduced in jet injectors (see
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Fig. lOa), usually by putting an additional low-flow swirl injector at the periphery
of the mixer nozzle (see Fig. 11). The designs of coaxial swirl-swirl injectors are
highly diversified. As an example, Figs. 14 and 15 show swirl-swirl injectors of
gaseous oxygen-kerosene engines with gaseous oxygen-cooled walls. Figure 16
shows injectors of oxygen-hydrogen gas generators.

4. Slit Injectors
Figure lOf shows an example of the injector design to provide atomization of a
liquid sheet (or a vast number of fine jets) with the gas flow on both sides. This
design has found applications in both jet engines and LREs. It contains a hollow

C-C

B-B
A-A
Fig. 14 Swirl-swirl injectors of gaseous oxygen and liquid hydrocarbon.
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40 V. BAZAROV, V. YANG, P. PURI

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B-B
A-A

Fig. 15 Modified oxygen-kerosene injector with screw conveyer swirler.

casing (1) with an insert having axial gas passage (2) and slit passage (3) formed
by the insert, and a casing (1) for the liquid propellant. The passage (3) is con-
nected to the feed line (4). The casing (1) is mounted on an external yoke (9)
by means of a gas-flow swirler (10) and forms a vortex chamber (7) for the
liquid, which ends in a nozzle (8). The mixer (5) consists of the nozzle (8) and
yoke (9). The gas-flow swirler (11) or an additional low-flow liquid injector
(not shown in the figure) can be placed into the passage (2). Several alternating
liquid (3) and gas (2) circumferential passages can be made. The advantages of
this design are the most uniform and best-quality propellant atomization and
mixing. As a consequence, one engine with such a design has the combustion
efficiency of around 0.995. The disadvantages include increased production
complexity and cost and insufficient understanding of its dynamic characteristics,
which forbid its application in high-thrust engines.

5. Jet-Swirl Injectors
Injector assemblies using jet and swirl elements in the fuel or oxidizer lines are
shown schematically in Fig, 11. Application of a liquid swirl injector in combination
with a jet injector improves the cooling conditions of the fire bottom. The design has
allowed the development of tripropellant injectors (see Fig. 17), which represent in
essence a combination of well-developed designs shown in Figs. lOa and lOb. It
consists of a casing (1) with a cavity (2) and a chamber (3) with a nozzle (4) coaxi-
ally mounted in the casing, which forms a cavity (5) connected to the liquid propel-
lant delivery manifold by tangential or radial passages (6). Passages (7) are formed
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 41

O2.5
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A-A B-B

Fig. 16 Injector of oxygen-hydrogen gas generator.

between the chamber (3), the pylons with tangential passages (6) and casing (1). The
mixer (8) is formed between the nozzle (4) and the casing (1). Chordal (9) and radial
(10) passages for additional propellant delivery are made in the walls of the
casing (1). This injector is designed for two applications: reducing and oxidizing
generator gases. These versions differ in the dimensions of the passage cross sec-
tions alone. In case reducing gas is fed to the injector through passages (7), liquid
oxidizer comes to the chamber (3) through passages (6) (similar to the injector in
Fig. lOb), and additional fuel (hydrocarbon of the liquefied methane or propane
type) is fed through passages (10). In the case of operation according to the
scheme with afterburning of the oxidizing generator gas fed via passages (7), less
volatile liquid fuel (kerosene) is fed to the chamber (3) through passages (6) and
more volatile fuel (hydrogen) via passages (9) and (10). Under this condition,
there exists a possibility of throttling of kerosene passage (6) up to its complete dis-
engagement. The injector will operate in the regime characteristic of the injector in
Fig. lOb. The results of injector development and testing are given in Refs. 14 and 15.

D. Intensification of Propellant Atomization and Mixing in

Liquid Injectors
The efficiency of liquid injectors is relatively low because much energy is
expended for liquid acceleration rather than for atomization. Injectors possessing
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42 V. BAZAROV, V. YANG, P. PURI

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A-A B-B C-C

Fig. 17 Design of a tripropellant injector.

higher atomization and mixing characteristics have been developed. The

principal means of improving injector efficiency involve reducing viscous
losses of the liquid flow and applying nonstationary processes to intensify propel-
lant atomization and mixing. In jet injectors, the former can be accomplished by
increasing the quality of the liquid surface and reducing its area (e.g., by short-
ening injectors). Investigations of viscous-loss mechanisms in swirl injectors
have shown the following two contributing processes: interlayer liquid friction
in the vortex chamber and friction of the external liquid layers on the walls of
the vortex chamber and nozzle. To reduce viscous losses, it is advisable to
shorten the lengths of all of the elements, in particular the nozzle, and to
reduce the liquid flow velocity by increasing the swirling arm of the liquid,
Rin. Optimization of the injector shape to meet these requirements has led to
the development of short and flat platelet swirl injector assemblies consisting
of a stack of plates welded in vacuum with holes, grooves, and saw-cuts to
form nozzles, vortex chambers, and passages. Figure 18 shows an injector assem-
bly with stacked plate arrangement having two passages for propellant delivery.
The assembly has a casing (1) with an annular manifold (2) to deliver the first-
stage flow and disks (3) and (4) fitted in the casing with separation (5), tangential
passages (6), vortex chambers (7), and nozzles (8). The casing has an additional
manifold (9) for the second-stage propellant, and the disk (4) has additional
supply (10) and tangential (11) passages. To decrease the dimensions of the
assembly and the distance between the injectors, propellant distribution
between them is provided with disks (12) and (13) having grooves (14) and
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 43

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15

Fig. 18 Design of a stacked bottom injector assembly; 1-casing; 2, 9 propellant

manifolds; 3-fire bottom; 4-injector bottom; 5, 10, 14, 16 supply passages; 6, 11-
swirl passages; 7-injector vortex chambers; 8-injector nozzles; 12-13 separating
disks.

(15) for separation. The fire bottom (3) has grooves (16), which not only deliver
part of the propellant to the injectors but also cool it. When accommodated in the
casing, disks (3), (4), (12), and (13) are diffusion welded to it in vacuum, after
which propellant delivery pipes are welded to the casing.
When fed to the manifolds (2) and (9), the propellant enters tangential pas-
sages (6) and (11) through grooves (14), (15), and (16) and delivery passages
(5) and (10), gains rotating motion in the vortex chambers (7) and exits from
the nozzles (8) in the form of thin circumferential sheets. To decrease viscous
losses inside the vortex chamber when using viscous, adhesive or non-Newtonian
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44 V. BAZAROV, V. YANG, P. PURI

liquids such as jells and metal particle suspensions, the vortex chamber and
nozzle can be made of a porous material. Figure 19 schematically shows a
porous swirl injector in a mandrel for cold-flow tests. The injector has an
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insert mounted on a branch pipe (1). The insert is made by diffusion-welded

porous bottoms (2) and casing (3) fixed by a coupling nut (4). The casing has
a heat-resistant coating on the combustion-zone side. The annular manifold
formed by the casing and coupling nut (4) can be connected by holes to the
branch pipe, as shown in Fig. 18, or to an independent manifold, to deliver
low-viscous liquid that penetrates through the porous material of the casing
and forms a boundary layer that separates main propellant from the surfaces of
vortex chamber and the nozzle of the injector.16
When fed through the branch pipe, the liquid enters the casing cavity through
the tangential passages, forms a liquid vortex in the vortex chamber, and leaves
the nozzle as a circumferential liquid sheet. Simultaneously, part of the liquid
leaks through the porous bottom and casing and reaches the boundary layer
flowing around the internal surface of the vortex chamber, thereby decreasing

B-

A-A

Fig. 19 Swirl injector with porous ceramic-metal swirler; 1-mandrel; 2-bottom;

3-porous casing with tangential grooves; 4-coupling unit.
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 45

the gradient of the circumferential velocity and hence the viscous losses. In the
case of increased propellant viscosity, a small amount of low-viscosity liquid
such as kerosene is fed through the porous walls from an independent source.
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In liquid swirl injectors of the classical design, the exhaust process is stable
and not accompanied by regular self-oscillations. However, in the presence of
some destabilizing factors such as exhaust of a boiling liquid and bubbled
media, or with specially profiled vortex chambers, self-oscillations at frequencies
from several Hz to several hundred Hz can be produced and used to intensify the
process of mixture formation.17 As for vibrational liquid injectors with magneto-
striction or piezoelectric actuators, driven by additional sources of electrical
energy, they were used only in experimental designs for research purposes.18

E. Intensification of Propellant Atomization and Mixing in

Gas-Liquid Injectors
Because gas-liquid injectors use generator gas produced mainly by one of the
propellants, the engine is not in short supply of the atomizing flow. The problem
of improving the pneumatic atomization process is not so critical. The limiting
factors are length of the combustion zone permitted in chambers of limited
dimensions, high flow intensity, maximum flow uniformity and mixture ratio
(which is especially important for low-thrust systems), and sensitivity to acoustic
instability. In such chambers, methods for intensifying the mixture formation
process are:
1) decreasing injector dimensions;
2) increasing surface area wetted by the liquid and flown around by the
atomizing gas;
3) emulsification of the liquid, i.e., formation of gaseous bubbles in the liquid flow.

II. Theory and Design of Liquid Monopropellant Jet Injectors

A. Flow Characteristics
The mass flow rate of a jet (spray) injector, shown schematically in Fig. 20,
can be determined using Bernoulli's theorem:

PQI = Pi (5)

Pf

Fig. 20 Schematic diagram of jet injector.

 Purchased from American Institute of Aeronautics and Astronautics

46 V. BAZAROV, V. YANG, P. PURI

where kp\-2 represents the pressure losses in the injector passage and /?oi the
total pressure at the entrance. From mass continuity,
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m = pU\A\ = pU2A2 (6)

The flow area at the injection exit A2 is usually smaller than the injector cross-
sectional areaA n . Thus, A2 < An, and A 2 = sAn, with s (<1.0) being the coeffi-
cient of jet contraction. For continuous exhaust, A2 = An and e = 1.0. If the
propellant is delivered from a large manifold with A\ ^> A 2 , the inlet velocity
U\ can be neglected and p\ = p0i • The pressure drop across the injector Apj
can be obtained from Eq. (5):

AA =Pf -Pa = An -P2 = -(1 + £) (7)

where ^ — A/?i_2/(p£/f /2) is the hydraulic-loss coefficient. The subscripts / and

c refer to the conditions in the propellant manifold and combustion chamber,
respectively. The exit velocity U2 takes the form

(8)

Substitution of U2 from this expression into the continuity equation gives the
mass flow rate of the injector:

nit = -7^==An^/2pKpi (9)

V 1 i bz

For an ideal liquid flow with & = 0 and s= 1.0, we have

(10)

Comparison of Eqs. (9) and (10) leads to the expression of the flow coefficient of
an injector JLL:
8
m,id yrr£ "• (ID
Since the hydraulic loss coefficient £• is always positive, the injector flow coeffi-
cient fji is less than unity in all cases. In practical designs, £ and e are determined
experimentally, as functions of injector length // and cross-sectional area An. The
mass flow rate can be calculated from Eq. (9) for given pressure drop, fluid
density, and injector area.

B. Effect of Injector Configuration

Figure 21 schematically shows the liquid flow in a real injector passage. When
entering the injector, the liquid flow separates from the sharp edge at the entrance
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 47

0.4d
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Fig. 21 Schematic diagram of liquid flow in jet injector.

due to the inertia force, and contracts as it proceeds. The largest contraction takes
place at a distance of 0.4 dt from the inlet. The flow then expands and exhausts
from the injector. Two different modes of injector operation have been observed:
1) If the passage is short with the aspect ratio of lt/di less than 1.5, the expand-
ing jet is unable to reach the nozzle wall (see Fig. 21a) with the jet contraction
coefficient e less than unity, and an annular space is formed between the jet
and the injector wall, which is in communication with the combustion
chamber. The exhaust liquid flow is unstable and the mass flow rate is
reduced. For this reason, short injectors are seldom used in operational engines.
2) If the injector passage is sufficiently long with lt/di greater than 1.5, the
expanding jet reaches the injector wall (see Fig. 21b) and the contraction ratio
e equals to unity. A stagnant zone, however, forms between the flow boundary
and injector wall, and a vortex-type liquid flow reaches its steady state there.
At high flow velocities, cavitation may occur in the contraction section, and
the stagnant zone becomes a cavity. It is clear that each mode of injector oper-
ation affects the flow coefficient ju, in its own way.

C. Flow Coefficient
The flow coefficient JUL can be determined experimentally based on either the
measured mass flow rate or the hydraulic loss coefficient. The total pressure
losses Api_ 2 = ^pf/2/2 for a jet injector can be represented as follows:

= £ (12)

where A/?i_ c and A/?2-c are the losses in the 1 — c and c — 2 sections, respect-
ively, and Apfr the friction loss on the passage wall. Here,

(13)
VI
The coefficient ^\-c represents the energy loss associated with the vortex genera-
tion when the liquid flows into the injector passage. Figure 22 shows the
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48 V. BAZAROV, V. YANG, P. PURI

l.U

0.8 s
\
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0.6 ^
^
V

»„
0.4
"Xs

0.2 ^v^
"' ^^,
"^!-*. -".„..
0.0.
102 104 1(
Fig. 22 Effect of Reynolds number on coefficient ^!_c

measured &_ c vs the Reynolds number (Re = Uidi/v) on a logarithmic scale. It

decreases with increasing Reynolds number. The coefficient £ c _ 2 represents the
energy losses associated with the vortices formed when the flow expands after the
contraction at the c — c station. The explicit expression for £ c _ 2 can be derived
from the following equations:
Bernoulli's equation

(14)

where the subscript n denotes the flow properties for an ideal injector without
flow contraction.
Momentum conservation

= pU2A2(Un - U2) (15)

Mass continuity

pU2A2 = pUnAn (16)

A simple manipulation of Eqs. (14)-(16) gives the expression for the total-
pressure loss:

4-= (17)

which is also known as the Borda-Carnot theorem. It follows from this

equation that

Un
(18)
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 49

As is shown by experiments, this coefficient depends not only on the degree of flow
contraction but also on the Reynolds number in the narrow passage section.
However, for Re > 105, which, in fact, is always the case with LRE injectors,
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viscosity practically exerts no effect on £ c _ 2 . It is solely determined by the

inlet configuration that affects flow contraction, i.e., &_ 2 — fin- Figures 23 and
24 show such effects on £in for two different designs.19 As the inlet convergence
angle /3 increases, the value of £in initially decreases, reaches a minimum at
about 50 deg, and then increases. This optimum value decreases with an increase
in the ratio of /in/<^. Figure 25 shows the £in behavior when the injector passage has
sharp inlet edges and is tilted with respect to the inlet plane.19
The coefficient £fr represents the friction losses on the passage wall, and can be
expressed as follows:

fir = A//M (19)

where A is the drag coefficient. For hydraulically smooth pipes in the turbulent
regime (Re > 4 x 103), A can be expressed as A = 0.3164/fe~a25. A passage
is considered hydraulically smooth if the wall-surface roughness, which
depends on the quality of injector production, is less than 0.007d/.
The hydraulic-loss coefficient of a jet injector takes the form

D. Design Procedure
Once the theory behind the operation of jet injectors is established, the follow-
ing steps can be used to design a jet injector based on the equations derived in the
preceding section. These are eight equations and one experimental correlation for
11 variables (i.e., A/? f , m r , lit Re, Uh dit A, f fr , //,, f m , and fi_ c ). The system can
be easily solved using an iterative scheme if two of these parameters such as A/?,
and mi are known:
1) Specify initial data of A/?;, mi9 //, p, and v for the expected injector design.
2) Initiate the calculation with the flow coefficient JUL = l/^/l + fin as first
approximation.

80 120
v
(3 (degrees)
Fig. 23 Effect of inlet-edge contraction on coefficient £in in jet injectors.
 Purchased from American Institute of Aeronautics and Astronautics

50 V. BAZAROV, V. YANG, P. PURI

0.5
0.4
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0.3
0.2
0.1
0.0
0.5 1.0 1.5

Fig. 24 Effect of inlet-edge Founding on coefficient gin in jet injectors.

3) Calculate the diameter of the injector passage, dt = 0.95m?>5(^)~

4) Determine the velocity and Reynolds number based on the values obtained
in the previous steps with Ut = 1.273m/p~1(d/)~2 _ 025
5) Determine A and £fr from their respective relations, A = 0.3 l64Re ' and

6) Determine &_ c from Fig. 22 and calculate & = &- c + f m + &•

7) Update the flow coefficient with /JL = I/ VI + £• and repeat steps 3-7 until
the calculated coefficient /x converges.
The injector diameter dt usually falls in the range of 0.8-2.5 mm because a
small diameter is susceptible to clogging and a large diameter gives rise to
poor atomization quality and increased spray length. If dt falls outside the afore-
mentioned limits, some changes should be introduced into the mixture formation
process and the mass flow rate mt should either be decreased or increased.
The possibility of occurrence of cavitation in the injector passage should be
checked. Assuming that p2 = ps and £._2 = £ n , from Eq. (14) we nave

Ps-Pn=^\7£-Ll--
(21)

0.9 % /

0.8 X,
Xs
X
in 0.7 ss^
0.6
^
X,
X •y^

0.5-
30 40 50 60 70 80
a(degrees)

Fig. 25 Effect of tilt angle of injector passage on coefficient £in.

 Purchased from American Institute of Aeronautics and Astronautics

DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 51

Then,
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(22)
Ps-Pn

The condition for preventing cavitation becomes

Pn=Ps~ —— >Pv (23)

where pv is the vapor pressure of saturated propellant.

III. Theory and Design of Gaseous Monopropellant Jet Injectors

A. Flow Characteristics
The delivery of gaseous combustible mixtures produced in the gas generator
into the main combustion chamber is achieved with the aid of jet injectors.
The general theory of gas injectors is based on the conservation of mass and
energy and the equation of state. The mass flow rate of an injector is expressed as

ritt = W2U2A2 (24)

where JUL is the flow coefficient defined in Eq. (11). The gas density at the injector
exit is obtained by assuming an isentropic process through the flow passage:

i/k
(25)

The ideal exit velocity U2 becomes

(26)
\
where R denotes the gas constant and k the ratio of specific heats. The total press-
ure of the gas flow /?0i is the sum of the chamber pressure pc and the pressure drop
across the injector A/?;, i.e., /?01 = pc + A/?/. The total temperature T01 equals the
gas temperature in the manifold. The exit velocity reaches its maximum t/th, in
the limit of pc = 0. We define the velocity coefficient A as

1-1..Poi
^ (27)
 Purchased from American Institute of Aeronautics and Astronautics

52 V. BAZAROV, V. YANG, P. PURI

Substitution of Eq. (27) into Eq. (24) gives the injector mass flow rate in terms of A:

(pc + kpi)An
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m; = /x———————q(\ 2 ) (28)

where C* is the characteristic velocity20 defined as

= v< u» 01 ——
^[2/(fc+l)f +1) ^- 1)

The gasdynamic coefficient function q (A2) takes the form

The flow coefficient JJL of a gas injector for incompressible fluids can be written as

In evaluating the flow loss coefficient £-, the high permeability of the gas injector
assembly should be taken into account. The ratio of the area occupied by the gas
flow in the c — c cross section, Agas, to the total injector assembly area, Atotal, is
rather high compared with liquid injector assemblies. This highly packed situation
tends to straighten the flow at the injector entrance and reduce the subsequent flow
contraction in the injector passage. To account for the effect of neighboring injec-
tors, the following empirical relationship is used:

&-2 = 6 n ( l - « M ) (32)

where d\ is the diameter of the gas flow before entering the injector (i.e., at the 1 — 1
cross section). It is taken numerically to be the average distance between the axes
of the neighboring injectors. In addition, as a consequence of high-intensity turbu-
lence, viscosity exerts no influence on the local hydraulic losses, i.e., ^_ c = 0.
Thus,

€i = find - df/di) + Xli/di (33)

B. Design Procedure
Several parameters are usually specified at the engine system design stage.
These include the mass flow rate of the injector assembly and associated flow
conditions, diameter of the injector assembly, pressure in the combustion
chamber, and pressure drop across injectors. The type of injectors and their sur-
mised number are also provided. The problem can be formalized to determine the
injector passage diameter for the prescribed conditions. The numbers of unknown
parameters and equations used to close the formulation are the same as those for
 Purchased from American Institute of Aeronautics and Astronautics

DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 53

liquid injectors. There are, however, three additional unknowns, A 2 , C*, and
#(A2), which can be solved using Eqs. (27), (29), and (30), respectively.
The injector design proceeds in the following steps:
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1) Assume a value for the flow coefficient JJL.

2) Calculate p0i — PC + 4P/ an^ determine the velocity coefficient A2 from
Eq. (27).
3) Calculate <?(A2) using Eq. (30).
4) Calculate C* from Eq. (29).
5) Determine An = rhtC*/fipQiq(\2) and </,-
6) Calculate £• from Eq. (33) and determine /x = !/>/! +
7) Repeat steps 3-6 until the calculated dt converges.
If calculations show the lack of space to accommodate the intended number of
injectors with the calculated passage diameter dt on the injector assembly (e.g.,
the diameter dt is too large), changes should be made in the engine design by
increasing the pressure drop Apz since dt is inversely proportional to Ap/. If, on
the contrary, the injectors with the diameter di prove to underutilize the area of
the injector assembly (i.e., low permeability of the injector assembly due to
small df), then a lower value of A/?, should be implemented. The injector
design is considered to be completed when the parameters in Eq. (28) are corre-
lated not only with each other, but also with the engine parameters of the propul-
sion system and the design of the injector assembly.

IV. Theory and Design of Gas-Liquid Jet Injectors

The main objective of a gas-liquid mixing element is to provide a uniform
initial distribution of liquid propellant through the gas flow. This can be achieved
by introducing thin liquid jets into the gas flow. Numerous versions of injectors of
this type have been designed and fall into the three categories shown schemati-
cally in Fig. 26. Slit injectors with gas passages made in the form of concentric
slots in the injector bottom represent a derivative of the preceding designs. The
principle behind slit injectors and gas-liquid spray injectors is identical. As
shown by practice, slit injectors having high permeability are less susceptible
to high-frequency instability.
Keeping all of the other factors the same, injectors with external propellant
mixing (see Fig. 26a) lead to lower combustion efficiency and therefore have

b) c)
Fig. 26 Major designs of gas-liquid jet-jet injectors.
 Purchased from American Institute of Aeronautics and Astronautics

54 V. BAZAROV, V. YANG, P. PURI

not found wide applications. Liquid propellant jets can be introduced into the gas
flow over a range of injection angle of a = 0 to 90 deg. The value of a influences
the depth of liquid jet penetration into the gas flow, distribution of droplets, and
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eventually combustion efficiency. An injector in which the liquid-propellant jet is

introduced along the gas-flow axis (a = 0 deg) is referred to as coaxial (see
Fig. 26b). Injectors of this type have been implemented in liquid oxygen/hydro-
gen engines, where liquid oxygen is fed through the central passage and gasified
hydrogen is delivered through the circumferential passage. The gas-flow velocity
should far exceed the liquid-jet velocity to achieve effective atomization. The
recess distance between the central post and outer tube also plays an important
role in determining the efficiency and stability of injector operation.
As a specific example illustrating the injector design, we consider the follow-
ing injector geometry and operating conditions (the corresponding nomenclature
is given in Fig. 27):
1) injector geometry (taken based on recommendations)
dg = 20 mm d\ — 2.2 mm 8 = 4 mm
A/ — 15mm lm = 12mm // = 45mm
a = 45 deg /3 = 30deg t{ = 30mm
2) operating parameters (obtained from the engine design and preliminary
analysis of mixture formation)
mitg = 2.21 kg/s; mitl = 0.516 kg/s; pc = 150 • 105 N/m 2
(RT)g = 180,000 J/kg; Pl = 785 kg/m3
The objectives are
1) to construct the liquid jet trajectory in the injector passage;
2) to calculate the flow coefficients of the gas and liquid passages;
3) to evaluate the pressure drops of A/?^ and Ap/./, and the pressures of pg and
pi required to provide the prescribed gas and liquid mass flow rates through the
injector.

Fig. 27 Schematic diagram of gas-liquid injectors.

 Purchased from American Institute of Aeronautics and Astronautics

DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 55

Liquid jet trajectory. The analysis of the liquid trajectory in the gas flow is
based on the results of experimental and theoretical studies of the jet shape in a
cross flow. The gas flow in the passage tends to bend the injected liquid jet more
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severely as compared to the situation with an unbounded flow. The following

expression is usually used for the external boundary of the jet:

(34)

where Jcext = *ext/^/ is the longitudinal coordinate of the external jet boundary
reckoned from the 1-1 section, 3; = y/di, the radial coordinate from the internal
surface of the gas passage, and a/ the angle of the jet exhausted from the
passage. If 8/di > 1.0, the jet direction at the exit can be assumed to coincide
with the passage direction, i.e., a}- = a. As expected, the momentum ratio of
the gas to the liquid flow, pgU*/piUf, and the injection angle a determine the
jet trajectory.
Flow coefficient of gas passage. A gas-liquid injector is usually designed
using the results of cold-flow tests. The flow coefficient of the gas passage can
be written in the following form:

I
(35)

where £in(l — d^/lf) is the hydraulic-loss coefficient at the inlet to the gas passage
(see Fig. 23). It depends on the ratio between the gas and liquid heads, the angle
of the jet entering the flow, and the number and diameters of jets. Figure 28 plots
vs p ' ' and a for the case of four jets.

1.0

0.8

0.6
60°

0.4 45°
X
0.2

0.0
0.4 0.8 1.2 1.6 1.8

Fig. 28 Effect of momentum ratio on jjig for gas-liquid injectors.

 Purchased from American Institute of Aeronautics and Astronautics

56 V. BAZAROV, V. YANG, P. PURI

n=4
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Fig. 29 Effect of momentum ratio on ^ for gas-liquid injectors.

Flow coefficient of liquid passage. In calculating the flow coefficient for the
liquid passage /x/, an increase in the pressure loss due to tilted entry of the liquid
into the cylindrical gas passage should be taken into consideration. Furthermore, /x/
is also influenced by the conditions outside the passage, which depend on the
energy of the gas flow and the injection angle a. If we ignore these factors, fa
becomes

1
(36)
VI + £„ + 6.o
The coefficient £n represents the hydraulic loss at the inlet to the cylindrical tilting
passage with sharp edges (see Fig. 25), and £/.out is the coefficient characterizing the
additional losses when the tilted jet is introduced into the gas flow. Figure 29 shows
the effects of the momentum ratio and injection angle on £/.out> based on the data of
cold-flow tests. In calculating jjih the length of the tilted passage is assumed to be
greater than its diameter, i.e., d > dt. If the gas injector wall is thinner, the effect of
wall thickness on the flow coefficient should be taken into account.17'19

V. Theory and Design of Liquid Monopropellant Swirl Injectors

Although the fundamentals of swirling flow dynamics were established more
than 60 years ago by G. N. Abramovich8 in 1944 and independently by Taylor12
in 1947, the hydraulic characteristics of a liquid swirl injector remains a compli-
cated problem since fluid properties and injector geometric parameters have pro-
nounced and sometimes conflicting effects on injector characteristics. For
example, the mass flow through a swirl injector increases with an increase in
liquid viscosity, while the situation is reversed in jet injectors, despite the fact
that the general trend of the two types of injectors is identical for ideal fluids.
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 57

Publications on the effect of injector geometric and operating parameters on

spray characteristics often report contradictory results. Numerous and quite
different empirical expressions16 used to calculate the flow properties of swirl
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injectors do not allow evaluation of the effect of the primary parameters, and
can only be applied to the specific configuration of concern or those similar to it.

A. Flow Characteristics of Ideal Swirl Injector

To gain a fundamental understanding of the flow development in a swirl injector
and various underlying parameters, we first consider an ideal situation by ignoring
the flow non-uniformity associated with the discrete tangential passages. Figure 30
shows a typical swirl injector consisting of a hollow casing (1), an axisymmetric
vortex chamber (2), a nozzle (3), and tangential passages (4) connected upstream
with the propellant feed system. Liquid propellant enters the vortex chamber at the
velocity U-m, forms a circumferential swirling flow, exhausts at the axial velocity
Uan through the nozzle, and finally establish a near-conic sheet in the mixture-
formation zone. Ideally, the liquid sheet has the shape of a hyperboloid of revolu-
tion of one nappe. The spreading angle is determined by the ratio between the
circumferential and axial components of the liquid velocity near the injector exit.
In such a swirl injector, the pressure all over the internal surface of the liquid
vortex is equal to the pressure in the combustion chamber. The liquid potential
energy in the form the pressure drop across the injector is fully converted to
the kinetic energy. Thus, the liquid flow velocity on the surface becomes

(37)

Fig. 30 Schematic diagram of liquid flow in swirl injector; 1-injector casing;

2-vortex chamber; 3-nozzle passage; 4-tangential passages.
 Purchased from American Institute of Aeronautics and Astronautics

58 V. BAZAROV, V. YANG, P. PURI

On the injector wall Ur — 0, and

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1/2 = ul + Ul = UL + Uln = Uuk (38)

The subscripts k, s, and n denote the conditions at the injector head end, vortex
chamber, and nozzle, respectively. At the injector head end, Ua = 0. The circum-
ferential component of the liquid velocity Uuk is maximum and the radius of the
liquid-vortex surface, on the contrary, is minimum. In the vortex chamber, the
axial velocity Uas is positive and the circumferential velocity Uus is smaller
than Uuk, giving rms > rmk. In the nozzle, the smaller liquid passage area leads
to an increase of the axial velocity Uan and a decrease of Uun, giving
rmn > rms. Finally, at the nozzle exit, the centrifugal force arising from the swir-
ling motion acts as a velocity head, leading to an additional increase of the axial
velocity and subsequently an increase of the liquid-surf ace radius rme. The swir-
ling-liquid flow in the field of centrifugal force bears a resemblance to the liquid
flow through a dam in the field of gravitational force of the Earth. According to
N. E. Zhukovsky,21 the longitudinal velocity along the dam cannot exceed the
velocity of surface-wave propagation, much as the velocity of a gas flow in a
pipe of constant cross section cannot exceed the sound velocity. The concept
of a critical liquid flow in a swirl-injector nozzle results from the princi-
ple of maximum flow postulated by G. N. Abramovich8 and later proved by
L. A. Klyachko2 in 1962. It serves as the basis of modern theories of swirl
injectors. The whole theory of an ideal swirl injector is based on three principles,
namely, Bernoulli's equation, conservation of mass energy, and conservation of
angular momentum.
The swirl injector design involves 20 main parameters listed in Table 1. The
variable r with subscript stands for the radius of the liquid film, and R with
subscript the radial dimension of the injector.
All of these parameters can be related to each other through Bernoulli's
equation and conservation of mass, energy, and angular momentum. The
total velocity can be determined in terms of the pressure drop (Pf~pc) using
Bernoulli's equation:

Pc)lp (39)

where pf is the pressure in the propellant feed system for tangential channels and
pc the chamber pressure. The mass flow rate can be expressed in terms of the total
velocity U^ and the nozzle area An as well as the mass flow coefficient:

m = ^An^/2p(pf - p c) (40)

The total velocity is the vectorial sum of its three components:

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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 59

Table 1 Parameters involved in swirl injector design

Parameter Definition
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m Mass flow rate

pf Pressure in propellant feed system
pc Combustion chamber pressure
/?in Inlet pressure at tangential channel
a Spray cone angle
IJL Mass flow coefficient
<p Fractional area occupied by liquid in nozzle
h Liquid film thickness
C/2 Total velocity
Uun Swirl velocity in nozzle
Uan Axial velocity in nozzle
Urn Radial velocity in nozzle
Uu k Swirl velocity at head end of vortex chamber
Urk Radial velocity at head end of vortex chamber
t/in Velocity at entrance of vortex chamber
rmk Radius of liquid film at head end of vortex chamber
rmn Radius of liquid film in nozzle
An Area of nozzle
#in Radial location of center of tangential channel
A Geometrical characteristic parameter

Application of Bernoulli's equation at the tangential entry gives

(42)

where p-m is the tangential inlet pressure. It has been assumed in ideal injector
theory that the radial of velocity in the liquid film is zero in the vortex
chamber and nozzle:

U^ = Urn = 0 (43)

According to the conservation of angular momentum, the azimuthal velocity

satisfies the following relations:

Uumrm = Uukrmk = £/in#in = Urnrmn (44)

Since Ua and Ur are zero at the liquid surface at the head end of the vortex
chamber, the total velocity equals the circumferential velocity:

£/S = Uuk (45)

Equation (44) implies that the swirl velocity becomes infinity as the radius
approaches zero. Since the angular velocity cannot be infinite, a gas core must
be present, and the liquid will not fully occupy the entire injector. We can thus
define a parameter <p, known as the coefficient of passage fullness, that relates
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60 V. BAZAROV, V. YANG, P. PURI

the area filled by the liquid to the nozzle area:

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where Rn is the radius of the nozzle and rmn is the radius of liquid film in the
nozzle. Similarly, a non-dimensional parameter /x, known as the mass flow coef-
ficient, is defined that relates the actual flow rate to the maximum possible flow
rate through the nozzle. The two parameters can also be related to each other.

_ pUanAn<p _ Uan<p

where Uan is the axial velocity in the nozzle. Thus,

<^-*,.-*
Various geometrical parameters can be correlated to form a non-dimensional
geometrical characteristic parameter defined by

A=AnRin/AinRn (48)

where Rn is the nozzle radius, Rm the radial location of the center of the inlet
passage, and Ain is the total area of inlet passages. Finally, the spreading angle
of the liquid sheet at the nozzle exit, a, can be expressed in terms of the velocity
components:

tana = Uun/Uan (49)

Parameters such as m,pf,pc,p-m, and a are design specifications that are

chosen at the engine system design stage. Thus, there are in total 15 unknown par-
ameters involved in the injector analysis. Since there are only 14 equations, Eqs.
(39)-(49), an additional equation is required, which can be found using the prin-
ciple of maximum flow or an alternative differential volume approach as shown
next. Introducing Pt = Pf — Pc for this derivation and equating the pressure and
centrifugal forces on a liquid element of radius r with width dr, length rd<&, and
unit thickness, we have

U2
rd<&dP = dm—^ (50)
r

where Uu is the circumferential velocity at radius r inside the liquid film, and
dm = prdrdQ. The conservation of angular momentum,

Uu = Uumrm/r (51)
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 61

leads to

dP = pU2umr2m^ (52)
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where Uum is the circumferential velocity at the liquid film radius rm. Integrating
the preceding equation and applying P = 0 at r — rm, we have

P = ^(U2um-U2) (53)

From Bernoulli's equation,

?(U2a + U2u) + P = Pt (54)

Substituting the value of P from Eq. (53) into Eq. (54), and rearranging the result,
we have

(55)

The conservation of angular momentum gives

(56)

The total volumetric flow rate can be expressed as the product of the inlet passage
area and velocity £/in:
Q = n*miU-m (57)

where n is the number of tangential inlet passages and rin the radius of tangential
inlet passage. Substituting the value Uwn from Eq. (56) in Eq. (55) and replacing
the value of £/in from Eq. (56), we obtain

Riff-
(JO)

The definition of the coefficient of passage fullness, <p, gives

Q = <p(irR2nUa) (59)
With the preceding three equations, we may eliminate Ua and substitute for the
parameter A to obtain the following expression for the volumetric flow rate Q:

1-9 <p2
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62 V. BAZAROV, V. YANG, P. PURI

Since ^/2Pt/p represents the total velocity, 7rR*^/2Pt/p is the total volumetric
flow rate possible through the nozzle. By substituting the definition of the flow
coefficient /LL, Eq. (60) can be rearranged as follows:
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1
//, = - (61)

\-<p

The flow discharge coefficient depends on the injector geometric parameter A and
the coefficient of passage fullness (p. If <p is decreased, then the decrease in the
equivalent flow area is faster than the increase in the axial velocity, and so
the mass flow rate decreases. Similarly, for an increase in <p, the decrease in
the axial velocity is faster than the increase in the equivalent flow area, and so the
mass flow rate decreases. Thus there exists an optimum maximum mass flow rate
(or the discharge coefficient). Figure 31 shows the mass flow coefficient as a func-
tion of the <p for various values of the geometric characteristic A. A maxi-
mum value of iji for each value of A is observed, indicating the existence of
the maximum flow rate for a given value of A. Application of the condition
dfji/d<p — 0 gives the optimum value of the discharge coefficient:

(62)

Equation (62) is the final equation required to close the formulation for the injec-
tor analysis. The axial velocity in the cylindrical nozzle is

a 2
(63)

0.9
0.8 A=0.2
a

I 0.7
1 0.6
$ °-5
£ 0.4
_o
E
CO
0.3
CO
2
I °'
0.1
0.
8.0 0.2 0.4 0.6 0.8 1.0
Coefficient of Passage Fullness, <p
Fig. 31 Effect of coefficient of passage fullness on mass flow coefficient for various
geometric characteristic parameters.
 Purchased from American Institute of Aeronautics and Astronautics

DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 63

According to the flow continuity condition, the liquid mass flow rates in the
nozzle and tangential passages are equated:
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pin) (64)

We now determine the azimuthal velocity Uum at some arbitrary point of the
liquid vortex at the radius rm . The conservation of angular momentum gives

Uum = —— Uin = ^ t/S - —— /- (Pf ~ p in ) (65)

rm rm rm]jp

where U% is the idealized total liquid velocity. Application of Bernoulli's

theorem to the cylindrical part of the nozzle passage leads to

(66)

Substitution of Eqs. (63) and (65) into Eq. (66) gives

= ^ ^k (67)
C r
^ mn

A simple manipulation yields

Pf ~Pin = Affin __ 1 - ^2/92

(68)
Pf-Pc A/7/ (Rin/rmnf

With the aid of the principle of maximum flow, JJL = -\/<p3/(2 — 9), and
rmn = Rn^l — <p, Eq. (68) results in the ratio of the pressure drops across the
tangential passage and the injector as a whole:

The preceding equation is valid only for injectors having Rm/Rn > 1; otherwise,
it may give a physically unrealistic solution with A/?in/A/?/ > 1, which makes no
sense. Indirectly, this means that for such injectors the principle of flow
maximum does not hold.
We now normalize all of the radii with respect to Rn and all of the velocities
with respect to U% to express injector parameters in terms of three non-
dimensional parameters 9, JLL, and A. To simplify notation, all the non-
dimensional quantities are expressed with a bar over them. Thus the radius of
the liquid film at the head end of the vortex chamber becomes

rmk = 2(1 - <p)2/(2 - <p) =

 Purchased from American Institute of Aeronautics and Astronautics

64 V. BAZAROV, V. YANG, P. PURI

This quantity will be hereinafter included in many expressions. It is convenient to

use because of its sole dependence on the injector geometric characteristic par-
ameter A. Figure 32 shows the effects of the geometric characteristic parameter,
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A on various commonly used injector parameters. The azimuthal velocities Uun

and Uue increase with A, whereas the axial velocities Uan and Uae decrease
with increasing A. The coefficient of passage fullness <p and the mass flow coeffi-
cient JJL also show a decrease with an increase in A.
At the head of the vortex chamber, Ua = 0, and

2 (70)
Uuk Rm

With Eq. (70) in mind, the physical meaning of the parameter a = r^ is evident.
The ratio of the liquid film radius at the head end to that in the nozzle becomes

- <p) = 2(1 - - <p) - b (71)

Equations (70) and (71) are employed to derive the components of the steady-state
velocity in the injector element. Applying the conservation of angular momentum,
Uunrmn — U^mki we obtain the circumferential velocity in the nozzle:

Uun = rmk/rmn = (72)

The axial velocity in the nozzle becomes

UM = 1 - Um = x/1 - 2(1 - <p)/(2 - <p) = (73)

1.2

-8

3.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

Geometric Characteristic Parameter, A
Fig. 32 Effects of geometric characteristic parameter A on other injector design and
flow parameters.
 Purchased from American Institute of Aeronautics and Astronautics

DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 65

Combining Eqs. (72) and (73) determines the spreading angle of the liquid sheet at
the injector exit:
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an = tm-l(Uun/Uan) = im-\Um/Uan) = tan'1 ^2(1 - <p)/<p (74)

Owing to the effect of centrifugal force, the liquid surface radius at the nozzle exit
rme is larger than rmn, and the circumferential velocity decreases as well.2 Accord-
ing to Eq. (66), the circumferential velocity downstream of the nozzle exit is

Uue = rmk/Rn = V2d - <?)2/(2 - <p) = Ja (75)

The axial velocity becomes

UM = JT^~a (76)

Combining Eqs. (75) and (76) gives the spreading angle of the liquid sheet down-
stream of the injector exit:

ae = tan-1 y/a/(l - a) (77)

It should be noted from Eq. (76) that the axial velocity at the nozzle exit,

- 2<p)<p/(2 - <p) (78)

exceeds that in the nozzle section. The ratio of the two velocities depends solely on
the coefficient of passage fullness <p:

d)(2 - <p)/<p = ^3 - 29 (79)

In the limit of <p -> 0 (i.e., infinitesimally thin liquid film),

Uae = V3 Uan (80)

The liquid velocity exceeds the critical velocity at the nozzle exit. Hence, the
nozzle throat is offset by some distance from the exit. The relation between rme
and injector parameters can be expressed by the following transcendental
equation2:

= V 1 - M2A2 = rmeVrme - tfA* - y?A2 In _ ~ = (81)

A more accurate evaluation of the liquid spreading angle ae can be made by

substituting rme from Eq. (81) for a.
 Purchased from American Institute of Aeronautics and Astronautics

66 V. BAZAROV, V. YANG, P. PURI

The velocity in the tangential passage, £/in, can be determined from Eq. (44):
^in-*Mn ~~ 1"mk
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This implies
(82)

We determine the velocity components on the liquid surface in the vortex

chamber. In the root section of the vortex chamber, Uam is calculated from the
condition of flow continuity:

Uam = Uan(R2n - r2mn)/(R2m - r2J (83)

Note that ir(Rn — r2mn) is the flow area in the nozzle and 7r(Rm — r2m) is the flow
area in the vortex chamber. Equation (83) can be written as

U™ = Uan<p/(Rm - r ) = ^/(Rm - r) (84)

Applying the conservation of angular momentum, Umrm — U^rmk, and noting
Uln + U2um = U2 and Eq. (71), we obtain:

rl = al(\ - U2m) (85)

By substituting rm from Eq. (85) into Eq. (84), Uam and rm can be obtained by the
successive approximation method.

B. Flow Characteristics of Real Swirl Injectors

The flow process in a real swirl injector can be described by taking into account
viscous effects with the Navier-Stokes equations.14 No analytical solutions are
available for general cases, and the use of numerical calculations is inevitable.26
In practice, the real conditions can be approximately taken into account by intro-
ducing the hydraulic loss coefficient, £, which characterizes the total-pressure
loss in the injector, and the angular-momentum loss coefficient K:

Thus,

\p. - f.f/? _ JJ2 (gg)

where A/?,- — p/~ pc is the pressure drop across the injector. The actual mass flow
rate through the injector nozzle can be represented in the following form:

(87)
 Purchased from American Institute of Aeronautics and Astronautics

DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 67

where
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where ^represents the loss of angular momentum. Substitution of the definitions

of the geometric parameter A and the coefficient of nozzle opening, Rm = Rin/Rn,
gives

(88)

Kmt
- 9 ~9

Substitution of Eqs. (88) and (89) into Eq. (87) and rearrangement of the result
yields an explicit expression for the flow characteristic parameter A that accounts
for viscous losses:
(90)
where the mass flow coefficient ^ takes the form

(91)
A2

As is evident, the flow coefficient depends on the flow area ratio, (p, combination
of the geometric dimensions, A and /?in , hydraulic losses, £• , and angular momen-
tum losses, K. With the use of the principle of maximum flow,

(92)

we have

(93)
2-<p

The spray-cone angle is determined from the ratio between the circumferential
and total velocities in the nozzle exit section: sin a = Uun/U^n. Application of
Eqs. (88) and (90) gives

(94)
 Purchased from American Institute of Aeronautics and Astronautics

68 V. BAZAROV, V. YANG, P. PURI

and the total velocity becomes

———————- I————— A2 fc
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(95)

Equations (94) and (95) lead to the following equation for the liquid-sheet spread-
ing angle:

(96)

Equation (96) suggests that the spreading angle is different for liquid particles
located at various distances r from the axis (sin a ~ 1/r). In calculations, with
some inaccuracy assumed, the spray cone angle a corresponding to the average
radius is used:
? _L r P / ——————\
m
"^ " "/1 • ^ ^ (97)

Thus,
2 A K
(98)

For ideal liquid (^/ = 0 and K — 1), with the neglect of the radial component of the
liquid velocity, the spray cone angle 2a and flow parameters JUL and q> are deter-
mined by the injector geometric characteristics alone and can be calculated as a
function of A.

C. Effect of Viscosity on Injector Operation

Propellant viscosity and the ensuing friction losses affect the injector charac-
teristics in terms of JJL, a, and (p. The momentum loss measured by the coefficient
Kis first considered. For simplicity, the hydraulic losses are neglected with & = 0.
The injector performance can be conveniently evaluated by the equivalent-
injector characteristic parameter Aeq from Eq. (92):

Aeq =AK = ±———-

Consequently,
" , M<eq
; and jt,.= —— (99)
/nm
 Purchased from American Institute of Aeronautics and Astronautics

DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 69

1.0
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0.

Fig. 33 Effect of nozzle opening on flow parameters of swirl-injector vortex chamber.

This method is convenient because the numerical relations obtained for an

ideal injector and displayed graphically in Fig. 32 remain valid for the equivalent
injector by replacing A with A eq . Since K < 1.0 and Aeq < A for the same injector
under the effect of viscosity, it is evident from Fig. 33 that the momentum losses
lead to increase of the mass flow (fji&q) and passage fullness (<peq) coefficients and
a decrease of the spray cone angle 2aeq.
To evaluate Aeq for the suggested injector geometry (A = R-mRn/nrin) and to
find real values of ju, a, and q>, the following expression can be used23:

———— (100)
A
"2J
where

xO.25
and Rem — (101)

Equation (100) characterizes the behavior of an open-type injector (C =

R[n = 1.0) with R[n = Rn, Aeq = A. Unlike the momentum losses K, the total-
pressure losses £• decrease the flow coefficient. The main total-pressure losses
occur in the inlet passages. For most designs, we can assume that

f. = ^in + A— (102)

D. Design Procedure
When designing an injector, the mass flow rate m/, pressure drop A/?/, and pro-
pellant properties are usually known, and we need to evaluate the actual flow
 Purchased from American Institute of Aeronautics and Astronautics

70 V. BAZAROV, V. YANG, P. PURI

coefficient fjLt and the injector dimensions. The problem is reduced to correlating
the parameters in Eq. (99). The calculation proceeds as follows:
1) Prescribe the spray cone angle based on the injector operating conditions
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(usually between 90 and 120 deg, lower values may be used for special
cases). The geometric characteristic parameter A and the flow coefficient fa
are then determined from the plots in Fig. 32.
2) Determine the nozzle radius using

Rn= 0.475 / ' (103)

3) Specify the number of inlet passages (usually between two and four) and the
coefficient of injector opening, based on structural considerations. Then, the
radius of the inlet passage is obtained:

(104)

4) Revise the following injector parameters:

a) length of the tangential passages, usually /in = (3-6)rin;
b) nozzle length, /„ = (0.5-2)Rn, vortex-chamber length (ls>2Rin) and
vortex chamber radius (Rs = Rin + rin)
5) Find the Reynolds number in the inlet passages using

Rein = 0.637 ™{

and the friction coefficient using A = 0.3164/(/tein)0'25.

6) Determine Aeq using Eq. (100), and find ^ieq and aeq from the plots of Fig. 32.
7) Calculate the hydraulic-loss coefficient in the tangential passages using
£
b — t +A
_ bin ——
' A0
^'in
The coefficient £in is determined from the plots of Fig. 25 with allowance for the
tilting angle of the tangential passage relative to the external surface of the vortex
chamber, which can be calculated using the following formula:

a = 90 deg- tan'1 ~ (105)

An

8) Determine the actual flow coefficient ^ using Eq. (99).

9) Calculate the nozzle radius using the new approximation

Rn = 0.475
 Purchased from American Institute of Aeronautics and Astronautics

DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 71

10) Calculate the geometric parameter A in the new approximation

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11) Repeat steps 1-10 until the calculated injector parameters converge.
Another method of designing swirl injectors is also possible. It is based on the
results of model experiments shown in Figs. 34 and 35. The advantage of this
approach is its simplicity and adequate accuracy. The limited amount of experi-
mental data is the major limitation. The design procedure is as follows:
1) Prescribe the spray cone angle and ln/Dn from the plots in Fig. 34. Find the
value of A, and then obtain ^exp = Min, using the plots in Fig. 34.
2) Calculate the nozzle radius using Eq. (103).
3) Prescribe the number of inlet passages and the coefficient of injector
opening Rin based on structural considerations. Then, the radius of the inlet
passage is obtained:

4) Determine the Reynolds number Rein in the inlet passages from Eq. (101),
and use this result for design if Rem > 104.
5) Determine the other injector parameters such as /in, /„,/,, and Rs, find the
relative liquid vortex radius rm from the plots in Fig. 35, and then rm from

\ I
*W-
120
0.3
100 =d / =2.0 \i\V
\\
0.2

60
/ Ur=l-4 0.1
C=4
C=l

40 0.0
0 2 4 6 8 0 2 4 6 8
a) Geometric Characteristic parameter, A b) Geometric Characteristic parameter, A

Fig. 34 Experimental plots of a) spray cone angle 2a and b) flow coefficient as

functions of geometric parameter of swirl injector.
 Purchased from American Institute of Aeronautics and Astronautics

72 V. BAZAROV, V. YANG, P. PURI

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0 2 4 6 8
Geometric Characteristic parameter, A

Fig. 35 Experimental plot of relative liquid vortex radius in vortex chamber as

function of geometric parameter of swirl injector.

VI. Theory and Design of Liquid Bipropellant Swirl Injectors

In bipropellant injectors, the liquid-phase mixing occurs even before the liquid
sheet starts disintegrating. Both designs with internal and external mixing have
been implemented.

A. Injectors with External Mixing

Figure 36 shows three different configurations of injectors with external mixing.
Two swirl injectors are structurally connected in such a way that the nozzle of stage
1 is located concentrically inside the nozzle of stage 2. The exit sections of both
injectors are located at the same plane. Sometimes, to decrease the overall dimen-
sion of the injector, stage 2 or both stages are made completely open.
A basic design requirement for injectors with external mixing is that the spray-
cone angle of stage 1 should be larger than its counterpart of stage 2, such that the
fuel and oxidizer sheets intersect and mix outside the injector even before they start
disintegrating into droplets. The injector designs can be further classified into two
categories. If one of the nozzles is inside the other one, two injector designs are
possible. In version 1, the nozzle of stage 1 is accommodated by the gas vortex
of stage 2. Both injectors are hydraulically independent of each other and can be
designed using the procedure described in Section II. In version 2, the nozzle of
stage 1 is submerged in the liquid stream in stage 2. This design is usually associated
with the quest for increased flow capacity of stage 1 without increasing the pressure
drop A/?/ and decreasing the spray cone angle by increasing the nozzle dimension
(Ani = vrRli). The operation of stage 2, in particular its flow coefficient /^2,
depends on the ratio R = Rn2/Ri where Rn2 is the radius of the nozzle of stage 2
and RI is the external radius of the nozzle of stage 1. Figures 37 and 38 show the
theoretical results of the spray cone angle and flow coefficient of stage 2 as functions
of A/w and A 2 , respectively.

1. Design Procedure for Version 1

To initiate the injector design, the pressure drop A/?/, mass flow rate m/, and
propellant properties for each injector stage are prespecified. In addition, the
following parameters are provided from structural considerations:
 Purchased from American Institute of Aeronautics and Astronautics

DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 73

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Fig. 36 Versions of bipropellant swirl injectors with external mixing.

1) R-mi and #in2 are the coefficients of nozzle opening;

2) ln\ and In2 are the relative nozzle lengths ln = ln/2Rn\
3) HI and n2 are the number of inlet passages.
The design is based on the results of model experiments (see Section II) and is
carried out in accordance with the following procedure:
1) Prescribe the spray cone angles 2a2 and 2«i, according to the empirical
condition 2a\ — 2a2 = 10 to 15 deg based on injector operating conditions.
With these values and the correlation given in Fig. 34a, find the geometric charac-
teristic parameters, AI and A2. The flow coefficients of stages 1 and 2, JJLI and fji2,
are then determined from Fig. 34b.
2) Calculate the nozzle radii Rni and Rn2 from Eq. (103), and determine the
tangential-entry radii r-m\ and rin2 from Eq. (104).
3) Determine the Reynolds numbers Reini and Rein2 using Eq. (101). The
design is completed if Rein > 104, and the injector dimensions and flow
parameters are calculated.

0 1.2 1.4 1.6

Fig. 37 Spray cone angle as function of relative radial spacing between nozzles in
bipropellant swirl injectors.
 Purchased from American Institute of Aeronautics and Astronautics

74 V. BAZAROV, V. YANG, P. PURI

^
0.5
/
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0.4 —— -/'^2=1
•/ /

0.3
//
i
I/ /3
2

in
0.2
4
x/ ^
It
V-
0.1

y
.0 1.2 1.4 l.e

Fig. 38 Flow coefficient as function of relative radial spacing between nozzles in

bipropellant swirl injectors.

There are cases that the initial requirements are not satisfied. For example, the
nozzle of stage 1 is not accommodated inside the gas vortex of stage 2, a situation
frequently observed when oxidizer is fed through stage 1. In this case, version 2
of the injector design should be chosen, with the nozzle of stage 1 submerged in
the liquid stream in stage 2.

2. Calculation Procedure for Version 2

The initial injector requirements are the same as those in the previous version.
With the spray cone angles 2a2 and 2ai prescribed, the injector of stage 1 is
designed following the same procedure for version 1. The calculation proceeds
as follows:
1) Specify the thickness of the nozzle wall <5W and determine the external
radius of the nozzle of stage 1, RI = Rni + S w i.
2) Specify the spacing between the nozzles Ar (no less than 0.3 mm) and
calculate the nozzle radius Rn2 = RI + Ar of stage 2.
3) Determine the geometric parameter A2 from Fig. 37 and find /^ from Fig. 38.
4) Calculate the inlet-passage radius rin2 = V^m2^«2/«2^2-
5) Determine the required pressure drop across stage 2 following the standard
formula for the mass flow rate in terms of the injector pressure drop:

6) Repeat steps 1-5 using another until the calculated matches its
prespecified value.
 Purchased from American Institute of Aeronautics and Astronautics

DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 75

B. Injectors with Internal Mixing

Figure 39 shows three different versions of injectors with internal mixing. The
inner injector (stage 1) is recessed from the exit of stage 2, to achieve stable and
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efficient mixing of propellants on the internal surface of the nozzle of stage 2. This
part of the nozzle of stage 2 is referred to as the injector mixer, whose length can
be varied to provide the desired propellant flow residence time in the mixer, r/ . If
Ti is too long (e.g., 1.5-10 ms), burnouts and explosions may occur in injectors.
Conversely, if rz is too short (e.g., less than 0.1 ms), poor mixing of the propel-
lants may take place leading to degraded combustion efficiency. The optimal
value of Ti depends on propellant properties, injector flow rate, and several
factors whose effects are still not clearly understood. Provisionally, rz — 0.1 ms
is recommended for hypergolic propellants, and T/ = 0.2 ms for non-hypergolic
propellants with the total propellant flow rate ma + mi2 in the range of
0.2-1.0 kg/s. The final rt value (and hence, the recess length A/n) is determined
during the engine development.
The spray cone angle, when both stages operate simultaneously, depends on
many factors. It is generally assumed that the total angle 2a2 is 30-40 deg,
smaller than the spray cone angle of an isolated stage 2 without the inclusion
of stage 1. During the design of an injector, hydraulically independent operation
of each stage should be provided, namely,
1) the gas-column radius of stage 2 should exceed the external radius of the
nozzle of stage 1, with rm2 — rm\ = 0.2-0.3 mm;
2) the spray cone angle of stage 1 should be such that the propellant arrives at
the mixer wall 2-3 mm downstream of the tangential entries of stage 2.
The preceding conditions prevent the ingress of propellant from one of the stages
into the other.
To initiate the injector design, propellant properties, pressure drops, Ap;1 and
Ap /2 > and mass flow rates, m/i and m/ 2 , for both stages should be prespecified as
basic requirements. In addition, the following parameters should be given for
each stage based on structural considerations:
1) jjini and Rin2: coefficients of nozzle opening; = 3 for closed stages and
Rm = 0.1— 0.8 for open ones;
2) / w i and In2: relative nozzle lengths, ln\ = 1.0;
3) HI and n2: number of inlet passages (2-6);

Fig. 39 Versions of bipropellant swirl injectors with internal mixing.

 Purchased from American Institute of Aeronautics and Astronautics

76 V. BAZAROV, V. YANG, P. PURI

4) 2a\. spray cone angle of stage 1, 60-80 deg

The injector design proceeds in the following steps.
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1. Design of Stage 1
1) From the experimental correlations given in Fig. 34, determine the non-
dimensional parameter a, coefficient of passage fullness <p, geometric character-
istic parameter A x , and mass flow coefficient /^.
2) Calculate the nozzle radius Rni from the relation

Rni = 0.475^

radius /?inl using /?inl = 7?in^«i, and the radius of the inlet passages,

Inl = ^RmlRnl/

3) Determine the Reynolds number in the inlet passages from Eq. (101).
If Re > 104, consider the design of stage 1 completed and calculate the other
parameters of stage 1.
4) Calculate the length of the tangential passages using /inl = (3-4) r inl , the
length of the nozzle using ln\ — 2Rn\, and the length of the vortex chamber using
ti = (2-3)/? inl .
5) Calculate the external radius of the nozzle, RI = Rni + Sw, where the
nozzle wall thickness is 8W = 0.2-0.8 mm.
The relative vortex radius rm is found from Fig. 35, and the vortex radius rm\ is
calculated.

2. Design of Stage 2
1) Determine the permitted gas- vortex radius, rm2 = RI + 0.3 mm.
2) Assume rm2 = Rn2 to first approximation, and calculate JJL using ^(/) =
Q225mi2/(Rn2)2^/p2Api2, where the superscript (/) denotes the initially
guessed value of the mass flow coefficient.
3) Determine A2 from experimental correlations in Fig. 34, and then the
relative vortex radius rm2 from Fig. 35.
4) Determine the nozzle radius R^ — rm2/r(^2 based on the known values of
r ^2 and rm2, and calculate /4/7) using /4/7) = 0.225ro/2/(/^)VfcAP/2 with the
(

updated value of R^2, where the superscript (77) denotes the iteration step.
Repeat steps 3 and 4 until the calculated R^ converges, and update the values
of A2 , Rn2 , and rm2 .
5) Calculate Rin2 with /?in2 = Rm2Rn2 and r-m2 with rm2 — *jRm2
6) Determine Rein from Eq. (101). If Rem > 104, consider the design of stage 2
completed, and calculate the other parameters of stage 2. Determine the spray
cone angle 2a2 from Fig. 34 with stage 1 being idle and assume the total spray
cone angle of the injector 2a to be 2a2 - 35 deg. Using the prescribed value
of r/ = 0.1 -0.2ms, calculate the length of propellant mixing using the
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 77

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Fig. 40 Schematic diagram of liquid flow along swirl injector with divergent nozzle.

following equation:

where Km is the propellant ratio and <p\ and <p2 are the coefficients of stage
passage fullness (<p = I — r^).
7) Calculate the nozzle length In2 = 2ln2Rn2, compare it with /mix, and finally
obtain /mix + A/n = In2. Determine the final values of r/ and 2 a during the
experimental development.

VII. Modulation of Liquid Spray Characteristics of Swirl Injectors

Theoretically, the spray of a swirl injector with an ideal liquid resembles a
hyperboloid of revolution of one nappe. The spray cone angle is confined by
the asymptotes of the hyperbolas bounding it and is determined solely by the geo-
metric characteristic parameter A. For real injectors, as a result of viscous losses,
the spray shape varies from a tulip-like to a near-conic configuration, depending
on the pressure drop. In practice, there are often cases in which the spray cone
angle needs to be changed without affecting the geometric parameter and mass
flow of the injector. A notable example is the requirement for the spray of a
coaxial bipropellant liquid injector to intersect in the mixture formation zone,
especially when hypergolic propellants are used. Theoretically, it is impossible
to vary the spray cone angle without changing A. Designers are thus forced to
make the sheets intersect on the wall of the peripheral-injector nozzle,
as described in Section VI. The potential disadvantages of such propellant
mixing are:
1) nozzle erosion due to decreased distance between the combustion zone and
the injector; and
2) ingress of one propellant into the vortex chamber of the other and ensuing
explosions of the bipropellant mixture during engine restarts. The other important
 Purchased from American Institute of Aeronautics and Astronautics

78 V. BAZAROV, V. YANG, P. PURI

requirement imposed on flow-controlled injectors is minimum changes in the

spray cone angle with respect to variations of the geometric parameter.1
Methods of modulating the spray cone angle independently of the geometric
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parameter have been developed by Bazarov at the Moscow Aviation Institute.

The techniques do not require any moving parts in injectors. According to
classic theories of ideal swirl injectors,2 the spray cone angle of an injector is
unambiguously determined by its geometric characteristic A, which is related
to the flow coefficient ^t. However, as shown in Ref. 1, the flow coefficient ^
and the spray cone angle can be changed independently. One can either act
upon the liquid sheet in the injector nozzle once it has passed through the
throat section where the liquid axial velocity Ua equals its surface wave velocity
UW9 or use means leading to the violation of the principle of maximum flow.
Several methods may be used to control the spray cone angle independently of
the liquid flow, as shown in Fig. 40. The simplest method is profiling of the post-
throat section of a swirl injector nozzle. When the degree of nozzle opening
increases, the circumferential velocity decreases, thereby leading to an increase
in both the axial and radial components. The relationship between the liquid cir-
cumferential velocity and the nozzle radius is based on the conservation of
angular momentum:

Uuri = U^rmk (106)

where rt is the radius of the passage section of the injector nozzle; rmk is the radial
location of liquid film at the head end of the vortex chamber. The nozzle profile
determines the ratio between the axial and radial velocities:

Ur/Ua = drt/dz (107)

where z is the coordinate along the nozzle length and drt/dz represents the slope
at the nozzle.
According to gas-hydraulic analogy between the free-surface liquid flow and
gas flow through a pipe proposed by N. E. Zhukovsky, liquid velocity is critical in
the narrowest section and is equal to the velocity of disturbance (long waves)
propagation over the surface. This conclusion refers not only to spillways, but
also, as shown by L. A. Klyachko,2 to other potential liquid flows with free-
surface swirling motions in the field of gravitational forces in particular.
The inference about the impossibility of reaching a supercritical velocity,2
however, does not hold for swirling liquid flows in the divergent part of the
nozzle. It is reasonable to suggest that with the gas-hydraulic analogy proposed
by N. E. Zhukovsky, the free-surface liquid flow may become supercritical
in the divergent nozzle with its velocity exceeding the surface wave propagation
speed.
Consider an ideal swirling liquid flow along an axisymmetric divergent
nozzle (see Fig. 40). With all of the velocity components normalized by the
total velocity of the liquid exhausted from the nozzle, U^9 and the geometric
dimensions by Rn, Eqs. (41), (106), and (107) can be written in the dimensionless
 Purchased from American Institute of Aeronautics and Astronautics

DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 79

form:

Ul + Ut + U? = l ; Uu = rmk/fi'9 Ur/Ua = drf/dz = q (108)

Downloaded by PURDUE UNIVERSITY on January 27, 2019 | http://arc.aiaa.org | DOI: 10.2514/5.9781600866760.0019.0103 | Book DOI: 10.2514/4.866760

where q is the tangent of the nozzle-profile angle in reference to the injector axis,
and rmk is the liquid-vortex radius at the head end of the vortex chamber. From
the theory of ideal swirl injectors, the normalized circumferential velocity at the
nozzle throat is

C/M,th = rmk (109)

where rmk depends solely on the geometric characteristic parameter A and can be
expressed in terms of the coefficient of passage fullness (p. With r/ = J(z) available,
the velocity components in each nozzle section can be calculated from Eq. (108),
and hence the spray cone angle is obtained.
Substitution of Ur = qUa in Eq. (108), along with the use of Eq. (75) and the
conservation of angular momentum, gives

-q2) (110)

The spray cone angle becomes

tana = Ja(l+q2)/(f? - a) (111)

The desired spray angle can thus be achieved by varying q. For a cylindrical
nozzle exit q — 0 and

(r, 2 -a) (112)

For an axisymmetric divergent nozzle with A = 2, a — 1/3, rl•,= 2, and

tana % 0.301. The resultant spray cone angle becomes a = 16.7 deg, which is
12 deg less than that for a cylindrical nozzle with A = 2.
The liquid velocity along the wall is

Substitution of Uu from Eq. (108) into Eq. (113) gives

(114)

The value of UL in the divergent nozzle exit section (f/ > 1) exceeds the velocity
of wave propagation in the throat section given by Uw = Ue = \/l — a.
The ratio between the liquid velocity along the nozzle wall UL and the velocity
of wave propagation in the nozzle throat section [7th, Eq. (73), can be found in a
 Purchased from American Institute of Aeronautics and Astronautics

80 V. BAZAROV, V. YANG, P. PURI

manner similar to that for gas flow, and is denoted as A/:

A/ - UL/Uih = J(l - a/r- 2 )(2 - <p)/<p

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(115)

Substitution of the a value in terms of 9, Eq. (75), for the cylindrical nozzle exit
into Eq. (115) gives the expression for A/ in the cylindrical nozzle with allowance
for the Skobelkin effect described in Ref. 2:

(116)

Since <p lies between zero and unity, A, is always greater than unity.
The liquid velocity UL increases in the remaining length of the nozzle as rt
increases according to Eq. (115). As rt -> oo, A/. max = ^/(2 — <p)/<p. There
exists a limiting A, value for each geometric characteristic parameter A that
can be achieved by the liquid.
The coefficient of passage fullness at the exit of a profiled nozzle, <p/.exit> can
be determined as follows. Application of conservation of mass gives

fa (H7)

Substitution of Ua from Eq. (110) to Eq. (117) yields

V2-<pAfyi-a/r2

For a cylindrical nozzle (q = 0 and A/ = 1), the coefficient of passage fullness is

obtained by considering for the Skobelkin effect:

(118)

The liquid-vortex radius rv in the profiled nozzle is readily calculated

(119)

and

(120)

where q>t is the coefficient of passage fullness of the injector. Since rt — rv = h,

with h being the liquid-layer thickness, we have

(121)
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 81

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a) b) c)

Fig. 41 Liquid spray cone of swirl injector with Rs = 7 mm, rin = 1 mm, n = 2:
a) cylindrical nozzle with dt = 2 mm; b) divergent nozzle with duh = 2 mm and
^.exit = 6 mm; c) cylindrical nozzle with dt = 6 mm.

Equation (111) shows that a profiled divergent extension of a nozzle whose

length is of the order of its diameter is an effective means to provide spray inter-
section in coaxial bipropellant injectors. Figure 41 shows photographs of swirl-
injector sprays for various nozzle configurations. The divergent nozzle extension
decreases the spray cone angle and the range of its variations. It does not ensure a
constant spray cone angle. Furthermore, such a divergence piece increases fric-
tional losses at the wall and, consequently, leads to deterioration of atomization
quality.
A decrease in the flow velocity by increasing the injector area A/ causes a sim-
ultaneous decrease in the spray cone angle. On the other hand, a decrease in the
flow velocity by increasing ^ increases the spray cone angle. Thus, the spray
cone angle can be fixed within a prescribed narrow region by approximately
adjusting these two factors. Figure 42 schematically shows the liquid flow
along a combined dual-orifice injector with its spray cone angle varied using a
hydrolock. When fed to the chamber (3) through passages (5) and (6) and to
the nozzle (2) from the manifold through the slot (11), the liquid is swirled
and flows out from the nozzle edge as a hollow near-conic spray (12). The
liquid flow running out through the slot is bent by the main swirling flow and
forms a liquid bulkhead (13) in the nozzle, through which the main swirling
flow exhausts in a manner similar to a profiled nozzle. With the throttle valve
(10) open, the flow passage in the nozzle (2) is reduced, thereby decreasing the
main liquid flow and the spray cone angle. With the throttle valve (8) closed,
the flow decreases and the spray cone angle increases. Modulating the flow
areas of throttle valves (8) and (10) makes it possible to vary either the spray
cone angle or the flow rate, without affecting the other parameters. The pre-
viously described method of maintaining the prespecified spray cone angle can
also be used in combination with a bypass injector. To do this, passage (6) is con-
nected to the throttle valve (8) in line (4) and the liquid from the vortex chamber
is throttled.
The passage area of the nozzle can be changed using other methods. For dual-
orifice injectors, application of partition cowlings with nozzles between the
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82 V. BAZAROV, V. YANG, P. PURI

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9 13
10

Fig. 42 Liquid flows in combined dual-orifice injector with its spray cone angle
varied using a hydrolock; 1-casing; 2-nozzle; 3-vortex chamber; 4-feed line; 5, 6
tangential passages; 7, 8, 10 throttles; 9-ring collector; 11-ring slot; 12-liquid
spray; 13-liquid bulkhead; 14-liquid film.

high- and low-flow parts of the vortex chamber is very effective. If the internal-
nozzle diameter is smaller than the external-nozzle diameter, liquid exhaust
from such an injector is close to that along a divergent extension piece. The possi-
bility of controlling the spray cone angle by compressing the liquid flow in the
area between the vortex chamber and the nozzle of the swirl injector is also con-
sidered. The classic theory of swirl injectors neglects the radial velocity com-
ponent in the vortex chamber. However, as shown by A. M. Prakhov,3 the presence
of the radial velocity component in the area between the vortex chamber and
nozzle is responsible for the failure of the principle of maximum flow, since the
axial velocity in the nozzle is higher than the velocity of disturbance propagation
along the liquid surface. In other words, a "supercritical" liquid flow appears with
a corresponding decrease of the sheet thickness in the nozzle. The conditions of
"supercritical flow" migrate downstream from the vortex chamber to the nozzle.
Both the axial and the radial velocity components in the transition area between
the vortex chamber and the nozzle can be increased by introducing a central body
whose diameter is larger than the diameter of the liquid-vortex free surface into
the nozzle, or by means of axial contraction between the vortex chamber and
nozzle. Although the former measure is undesirable because of increased losses
in the kinetic energy of the liquid flow, it is often unavoidable in practice due
to structural considerations in the case of coaxial arrangement of multiple-
orifice injectors. On the other hand, axial contraction does not cause any increase
in losses and is applicable to swirl injectors of any type.
Consider the liquid flow under the condition of axial contraction. Part of the
pressure is consumed in the annular radial slot connecting the vortex chamber
with the passages. The resultant decrease in the liquid flow velocity reduces
the circumferential velocity of the liquid sheet velocity in the nozzle and the
spray cone angle. By changing the cross-sectional areas of the tangential inlet
 Purchased from American Institute of Aeronautics and Astronautics

DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 83

passages and the width of the radial slot, the prescribed value of the flow coeffi-
cient fa and a spray cone angle smaller than the one based on the principle of
maximum flow can be obtained.
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This contraction decreases the liquid velocity in the tangential passages and
can be expressed as U*n = ££/ino • In this case, rmk decreases by a factor of £
according to the conservation of angular momentum:

r
mk = (122)

Physically, this decrease of rmk is caused by the presence of the radial velocity Ur
in the annular slot and can be determined from the conservation of energy,

A/Vp + tf/2 + U2J2 = const (123)

and the mass continuity equation,

2irrlUr = U-mAin (124)

The spray cone angle for ideal liquid is determined from the formula14

(125)

Substitution of r^t from Eq. (122) into Eq. (125) gives

a* = 2 tan"1 (126)

Figure 43 shows schematically an experimental injector, and Fig. 44 presents

the experimental data of the spray cone angle as a function of the slot widths hi
and h2. The flow testing results agree well with the calculations. Axial contrac-
tion is an effective method of modulating the spray cone angle without changing
the injector geometric characteristic parameter A. The decrease in the flow rate
due to the velocity drop in the inlet passages Uin can be compensated by increas-
ing the passage flow area. Such a measure decreases the geometric characteristic
parameter A, and thus provides an additional decrease of the spray cone angle
because not only £ but also rm^ — (1 — <p)\/2/^2 — <p decrease in this case.

Fig. 43 Experimental swirl injector with axial contraction between vortex chamber
and nozzle.
 Purchased from American Institute of Aeronautics and Astronautics

84 V. BAZAROV, V. YANG, P. PURI

125
100
•&* .*— -;2nd1 1st
a •r *
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> 75
50
95
0.0 0.5 1.0 1.50
Slot Width
Fig. 44 Effect of slot width on spray cone angle for an experimental swirl injector
with axial contraction.

The spray cone angle of the injector with axial contraction can be determined
from the formula
1
tan a — (127)

where a0 is the spray cone angle without flow contraction.

Figure 2 shows a bipropellant swirl injector [A.C. 792023 (USSR)] with an
annular slot made in its peripheral region between the vortex chamber and the
nozzle, to provide guaranteed intersection between sprays 1 and 2. This design
is especially attractive when peripheral fuel delivery is used, and the sprays
can hardly be intersected in space using conventional methods due to the
lower propellant flows and greater geometric characteristic parameter of the per-
ipheral injector. As discussed in earlier sections, to provide contact for hypergolic
ignition, the central nozzle should be deepened into the peripheral one for propel-
lants to mix at the edge of the external nozzle. Such a design leads to propellant
ingress to the pre-injector cavity of the other propellant, and causes explosions
when the engine start cyclogram is disrupted.
To reduce the effect of flow compression by either decreasing the flow rate or
increasing the injector geometric characteristic parameter is a disadvantage of the
preceding method of modulating the spray cone angle as applied to LREs. There-
fore, spray flow compression in multimode injectors with fixed elements is
applied only to provide intersection of sprays in a prescribed range of modes.
In the process of thrust control, such injectors have a wider range of spray
cone angle variations than conventional ones. For swirl injectors with moving
elements, especially with moving screw conveyers, the radial clearance can be
changed simultaneously with the tangential passages area, and thus, to achieve
constant spray cone angle.
Consider the effect of liquid swirl Uu/Ua in the vortex chamber on the spray
cone angle. According to the conservation of momentum, a change in Uu leads to
a change in Uu/Ua. This can be achieved, for example, by turning the injection
flow passage to the extent that the liquid swirl ceases completely and the swirl
injector becomes a jet one. Consequently not only the spray cone angle is
decreased but also the flow rate is increased in accordance with the change in
the nozzle flow coefficient. In this design, with the spray cone angle controlled,
the flow rate can be maintained fixed by correspondingly decreasing the inlet
passage section area.1
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 85

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Fig. 45 Profiled nozzle of swirl injector with jump in liquid level.

Stabilization of the spray cone angle can be achieved by using dual-orifice

injectors for each propellant. As for the flow rate and spray cone angle, each
injector should operate independently and be designed to fulfill the prescribed
requirements. In the transition mode, the spray cone angle remains unchanged
and the atomization quality is ensured by intersecting the external throttled
spray with the central spray operating under excess pressure drop. Stabilization
of the spray cone angle can also be obtained by varying the degree of swirling
through the use of low flow rate swirling passages with smaller swirling radii.
Consider the possibility of stabilizing the spray cone angle by introdu-
cing a liquid jump in the nozzle. Following the gas-hydraulic analogy of
N. E. Zhukovsky,2 we obtain a liquid flow with a jump of the liquid level in a
swirl injector similar to the compression jump in a supersonic gas flow. As
shown schematically in Fig. 45, a supercritical liquid flow should first be
achieved by making a divergent section (2) in the nozzle (1) and a subsequent
contraction (3). The liquid travels along the nozzle and forms a standing
annular wave (4) whose radius is no less than rmk. Because of flow anisotropy
in the jump, the circumferential velocity component does not reach Uuk.
During subsequent decrease of the liquid level and the flow exhaust from the
nozzle to the mixing zone, the spray cone angle of a swirl injector with a profiled
groove in the nozzle becomes smaller than its theoretical counterpart for the case
without a jump. The effect of the groove decreases with the decreasing ratio
between its depth H and the liquid sheet thickness h, and completely disappears
with H/h < 0.3. This allows the use of a liquid jump in the nozzle to stabilize the
spray cone angle of a swirl injector with variable flow coefficient fa and coeffi-
cient of passage fullness (p. In the mode of maximum flow, a groove with depth
H < 0.3/z exerts little influence on the spray. As fa and <p decrease, the liquid
sheet thickness h decreases, and the effect of groove on the spray increases.
The increase of the spray cone angle associated with the decrease of fa is partially
or completely counterbalanced.

VIII. Design of Gas Swirl Injectors

A. Design Procedure
A gas injector, shown schematically in Fig. 46, can be designed by determin-
ing its basic dimensions Dn, din, ^in, and n, on which the prescribed flow and
spray properties are dependent. The design proceeds in the following steps:
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86 V. BAZAROV, V. YANG, P. PURI

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Fig. 46 Schematic diagram of gas swirl injector.

1) Choose the nozzle expansion ratio TT^ the geometric characteristic

parameter A, and the degree of nozzle opening Rin to accommodate the specific
features of the injector under consideration.
2) Determine the flow coefficient /m from Fig. 47. For intermediate Rin values
in the range between 0.2 and 1, /JL is calculated from the following formula:

= Mref M (128)

0.7

0.6

0.4

0.3

0.2
3.0
0.1
.05 1.10 1.15 1.25 1.20
Nozzle Expansion Ratio
Fig. 47 Flow coefficient jmref vs A and nozzle expansion ratio TT, for /?in = 0.75 of gas
swirl injector.
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 87

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0.0 0.5 1.0 1.5 2.0

Geometric Characteristic Parameter, A
Fig. 48 Correction of flow coefficient based on the degree of nozzle opening.

where ^f is the initial flow coefficient for R^n = 0.75 and jl is a correction for the
nozzle opening determined from Fig. 48. While making calculations, one must meet
the following conditions under which the experimental data given by Figs. 47 and
48 were obtained: Rein > 3000; /in = 1-1.2; /„ = 0.2-1 and ls = 0.2-0.3.
3) Calculate the Reynolds number for the inlet passages Re^n using

ein = 1413- (129)

where jjid is the dynamic viscosity of the gas, Pa - s.

4) Calculate the nozzle diameter in mm using

dn = 0.948 (130)

where 7\ is the gas temperature at the inlet to the injector, K; p2 is the gas pressure
downstream of the injector, MPa; R is the gas constant, J/kgK; m is the mass
flowrate through the injector, kg/s; and <p is calculated from the formula:

K+I (131)
- 1
\

where K is the polytropic exponent of gas expansion.

 Purchased from American Institute of Aeronautics and Astronautics

88 V. BAZAROV, V. YANG, P. PURI

5) Calculate the gas flow rate m using the prescribed nozzle diameter,
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m = 1.1 (132)

6) Calculate the total gas velocity at the nozzle exit, U^9 in m/s using

= 1.413- (133)
/ - 2 \ ——
(i-dmjTi* COS «exit

Calculate the axial velocity component using

Ua = U^ cos aexit (134)

and the tangential component using

Uu = U% sin aexit (135)

7) Calculate the inlet passage diameter Jin in mm using

din = dn -^L (136)

Determine the inlet passage diameter for a fully open injector using Fig. 49.

0.35

3.5 10.0 20.030.0

Geometric Characteristic Parameter, A

Fig. 49 Relative diameter of inlet passage din/Dn vs A and n for fully open injector
(Ds = Dn).
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 89

8) Calculate the inlet radius Rin and the vortex-chamber diameter Ds using the
following formulae:
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Dn
(137)

Ds = 2Rm + d-m (138)

B. Selection of Geometric Dimensions and Flow Parameters

This section summarizes the common practice used in the design of gas
injectors in Russia:
1) The relative length of the inlet passages /in is chosen between 1.0 and 1.5.
2) The relative length of the vortex chamber ls — ls/Ds is chosen between
0.1 and 0.3. If ls is greater than 0.3, JJL is calculated from the formula

(139)

where ^i can be found from Figs. 47 and 48 for prescribed values of A, R[n and TT/
and jjL2 for 77; = 1.05. The parameter b is determined form the empirical formula

b = 0.0225A(/?in - 0.3)(/, - 0.3) (140)

The formula is applicable for A < 3, Ri < 4, and ls < 4.

3) The number of inlet passages n is chosen based on the condition that
the required coefficient of gas-distribution non-uniformity K is obtained from
Figs. 50 and 51.
4) The convergence angle at the nozzle entrance M/1 is chosen between 90 and
120 deg.

§
'
to
5
Cfl

o
<+-(
o

0)
o
U
3.5 1.0 1.5 2.0 2.5
Geometric Characteristic Parameter, A
Fig. 50 Effect of geometric characteristic parameter A on coefficient of gas-
distribution non-uniformity in mixing layer for different Rin (n = 4).
 Purchased from American Institute of Aeronautics and Astronautics

90 V. BAZAROV, V. YANG, P. PURI

80

ef
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.
60

O =0.75
40

(U
O 20
U

^ 2 3 4 5 6 7
Number of Inlet Passages, n

Fig. 51 Effect of number of inlet passages on minimum coefficient of gas-

distribution non-uniformity in mixing layer.

5) With the relative nozzle length ln between 1 and 10, the gas flow coefficient
\ju remains constant.
6) The tolerances for the basic dimensions are chosen according to the man-
ufacturing specifications. The error for the injector flow coefficient should be
less than 10% for Dn < 10 mm and 16% for Dn > 10 mm. The final geometric
dimensions are determined in accordance with the results of injector flow tests.
7) The surface roughness should be Rz < 40 (Jim for the inlet passages,
Rz < 20 (Jim for the vortex chamber, and Rz < 2.2 jjum for the cylindrical and
end surfaces of the nozzle. No burrs are permitted at the nozzle edge and in
the inlet passages. The radius of the blunting chamber is 0.05-0.2 mm.

IX. Dynamics of Liquid Rocket Injectors

A liquid rocket engine, as shown schematically in Fig. 52, contains various
sources of intense pressure fluctuations caused by turbulent flows in the feed
line, fluttering of pump wheel blades, vibrations of control valves, and unsteady
motions in the combustion chamber and gas generator. As a consequence, the
actual process of mixture formation in injector elements typically occurs in the
presence of highly developed fluctuations, as the feedback coupling (loop 1 in
Fig. 53) affects the processes occurring in the combustion chamber and forms
a self-oscillating circuit.23'27 All conceivable mechanisms of intrachamber
instability are included here. Additionally, the chamber pressure fluctuation P'c
directly affects the liquid stage of the injector, L, forming another feedback
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 91

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Fig. 52 Schematic diagram of liquid rocket engine with staged combustion.

gas-liquid injector

P'

Fig. 53 Interactions of dynamic processes in liquid rocket engine with staged

combustion.
 Purchased from American Institute of Aeronautics and Astronautics

92 V. BAZAROV, V. YANG, P. PURI

coupling (loop 2). Acting as an oscillator in the feed system with the feedback
coupling 3, the injector excites pressure fluctuations P'L which then affect the
injector response through the direct coupling 4. The ensuing fluctuation in press-
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ure drop across the liquid injector, P'L, causes liquid-flow fluctuations m'L in the
nozzle exit. In parallel, the chamber pressure fluctuations P'c affect the gas
stage of the injector, G, by means of the feedback coupling 6, and consequently,
the gas generator (G.G.) through the feedback coupling 8. For a gas-liquid injec-
tor with internal mixing, the pressure fluctuations P'c affect the mixer through the
feedback coupling 7, and consequently, the liquid (12) and gas (13) stages of the
injectors. The gas generator responds to the disturbances of the flow rate m'G, temp-
erature T'G, composition K'G, exhaust velocity U'G, and pressure P'G of the generator
gas which, when passing through the mixer, results in fluctuations of droplet mass
and size distributions of the combustible mixture spray. The feedback couplings 12
and 13 can form their own self-oscillating circuits, causing fluctuations in the pro-
pellant flow at the injector exit as well as changes in the main mixture-formation
parameters, including the atomized droplet-size distribution, spray angle, and
uniformity of mixture composition.14
In LRE systems, injection is a key process since through it all feedback coup-
lings of the combustion chamber with other engine components are realized. In
addition to its main function of preparing a combustible mixture, an injector
acts as a sensitive element that may generate and modify flow oscillations.24
This section summarizes various important aspects of injector dynamics. The
mechanisms of driving self-pulsations in both liquid and gas-liquid injectors
are addressed systematically.

A. Linear Dynamics of Jet Injectors

For a short injector whose length is much less than the wavelength of oscil-
lation, the equation of motion for inviscid liquid takes the form

dt 2Li pLi pLi

where ~ stands for the instantaneous quantity and Lt for the injector length. Each
flow property may be decomposed into mean and fluctuating parts:

AP = AP + A/" and U = U + Ur (142)

For time-harmonic oscillations,

kP' = IAPV'"' and U' = \U' eia)t (143

Substitute Eq. (142) into Eq. (141) and linearize the result to get
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 93

The solution to the preceding equation is

A/"
Ur =
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(145)
pU + i

A transfer function relating the fluctuating velocity Uf and pressure drop AP' is
obtained as follows:

U'/U 1. 1 - iuLj/U _ 1
(.46,
AF/AP APj '2' l + (a>Li/U)2~2' 1 + Shf

where the overbar denotes a dimensionless quantity. The Strouhal number of the
jet injector, ShJ9 is defined as Shj = coLi/U.
Figure 54 shows the amplitude-phase diagram of the transfer function H/ for a
short jet injector. The normalized pressure-drop fluctuation APy is taken to be
unity, and the phase angle between £/. and AA is Oy. The locus is obtained by
increasing the Strouhal number (or oscillation frequency) in Eq. (146). For prac-
tical injector dimensions and oscillation frequencies commonly observed in
LREs, a jet injector can be considered as a single inertial element in which the
amplitude of flow oscillation U' decreases smoothly as the Strouhal number
increases, and the phase angle <f>7 increases asymptotically to Tr/2. The transfer
functions IT7 for step-shaped and other shapes of jet passages can be calculated
as a synthesis of several passages connected in series. In the case of long
liquid injectors and coaxial gas-liquid injectors, resonance at multiple frequen-
cies may occur when the injector length becomes comparable to the wavelength
of the fluctuation. The influence of injector length should be taken into account

^ Im

Sh=oo
1/2 Re EL

increasing Sh or CD

Fig. 54 Amplitude-phase diagram of response function of a short jet injector.

 Purchased from American Institute of Aeronautics and Astronautics

94 V. BAZAROV, V. YANG, P. PURI

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Fig. 55 Schematic of liquid swirl injector; 1-casing; 2-vortex chamber; 3-nozzle;

4-tangential passage.

for cryogenic liquids since the sound speed is relatively low due to the existence
of gas bubbles.

B. Linear Dynamics of Swirl Injectors

Figure 55 shows schematically a swirl injector with liquid flow. The liquid is
fed to the injector through tangential passages (station 4), and forms a liquid layer
in the vortex chamber (station 2) with a free internal surface shown by the dashed
line for the stationary case. The liquid is exhausted from the nozzle (station 3)
in the form of a thin, near-conical sheet that then breaks up into fine droplets.
Compared with a jet injector of the same flow rate, the flow passage of a swirl
injector is much larger, and as such any manufacturing inaccuracy exerts a
much weaker effect on its atomization characteristics. The resultant droplets
are finer and have higher uniformity, thereby motivating the predominant
application of swirl injectors in Russian LREs. From the dynamics standpoint,
a swirl injector is a much more complicated element than a jet injector. The
liquid residence time is longer than that of a jet injector; its axial velocity com-
ponent Ua is smaller for the same pressure drop, and the speed of disturbance
propagation is lower due to the existence of a central gas-filled cavity in the
liquid vortex. A swirl injector contains an inertial element (i.e., tangential
passage), an energy capacitor (i.e., vortex chamber partially filled with rotating
fluid), and a transport element (i.e., nozzle). Each of these elements can be
described with rather simple relationships.
The unsteady behavior of the tangential passage can be determined following
the same analysis as that for a jet injector, Eq. (146). The dynamics of the liquid
layer inside the vortex chamber and the nozzle can be modeled by means of a
wave equation that takes into account the disturbance propagation in the liquid
with centrifugal force. If we ignore the liquid-layer thickness and radial velocity,
 Purchased from American Institute of Aeronautics and Astronautics

DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 95

and follow the approach given in Ref. 2, a wave equation characterizing the flow
oscillation in the liquid layer is obtained:
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o,,21 ~~ A ^in"inl ^ / A9 (14/)

ar

Here £ denotes the fluctuation of the liquid-layer thickness, and rm the radius of
the liquid surface. The surface-wave propagation speed Uw is

The first parenthesized term in the square root represents the centrifugal accelera-
tion, and the second parenthesized term the effective thickness of the liquid layer.
The expression for the wave speed is analogous to that for shallow-water wave
propagation. The solution to Eq. (147) for a semi-infinite vortex is

f=[lei<a(t-z/u^ (149)

where fl represents the amplitude of the liquid surface wave. For an axisym-
metric rotating flow with a free interior surface, linearization of the equations
of motion leads to a relationship between the fluctuations of the liquid surface
and axial velocity:

(150)
The amplitude of the axial velocity fluctuation U'a is
3
m) (151)

In a non-dimensional form, the liquid surface-wave velocity inside the vortex

chamber can be determined from Eq. (148):

\
Here U% is the liquid velocity at _the head end of the vortex chamber. The
non-dimensional parameters a and Rvc are defined respectively as

(153)
 Purchased from American Institute of Aeronautics and Astronautics

96 V. BAZAROV, V. YANG, P. PURI

The subscript k denotes the head end of the vortex chamber, and A is the
geometric characteristic parameter.
In accordance with the principle of maximum flow,2 the axial velocity of the
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liquid flow inside the injector nozzle is the same as the surface wave speed, ana-
logous to the gas flow in a choked nozzle. Thus,

= (Ua)n = = (154)

where <p = Ai/An is the coefficient of passage fullness, representing the ratio of
the cross-sectional area occupied by the liquid to that of the entire nozzle.
As the wave propagation speed varies in the injector, the unsteady liquid flow
rate also changes with the flow. To quantify the dynamic characteristics of the
vortex chamber subject to flow disturbances, a reflection coefficient of the
surface wave at the nozzle entrance (or the exit of the vortex chamber), /3, is
defined according to the fluctuation of the liquid flow rate. A simple analysis
based on mass conservation leads to the following relation between the flow-
rate oscillations in the vortex chamber and the nozzle:

x^vc ^w,vc «"vc t*m,vc

After some straightforward manipulations, a reflection coefficient (3 characteriz-

ing the nozzle dynamics is obtained:

2
= 6vc ~ Q'n = l _

The amplitude of the surface wave in an infinitely long vortex chamber (i.e.
no wave reflection) is

(137)
rmk
A2(*

For a vortex chamber with zero length, the surface-wave amplitude can be
determined by the acoustic conductivity of its nozzle:

^in ^ (158)

which is much greater than lloo. For an intermediate case, the surface-wave
amplitude depends on the reflection coefficients at the head end and the exit of
the vortex chamber, as well as on liquid viscosity. The surface wave causes
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 97

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Fig. 56 Amplitude-phase diagram of response function of a swirl injector.

pulsations of the circumferential velocity U'u in the radial direction according to

the conservation of angular momentum (Uur = const) and consequently gives
rise to pulsations of centrifugal pressure. The non-dimensional amplitude of
centrifugal-pressure pulsations AP^C (defined as the pressure x difference
between the tangential entry and the liquid free surface) caused by surface
wave motions is not high and is equal to the non-dimensional amplitude of
surface waves in the swirl chamber.
The main difference between a swirl and a jet injector as a dynamic element
lies in the different mechanisms of disturbance propagation between the combus-
tion chamber and the feed system. For conventional injector dimensions that are
significantly smaller than disturbance wavelengths in the gas and liquid, pressure
oscillations arising in the combustion chamber propagate through the liquid
vortex layer almost instantaneously. This results in fluctuations of pressure
drop across the tangential entries, A/^, as shown in the amplitude-phase
diagram in Fig. 56, where AP'r is set to unity on the abscissa. Similar to a jet
injector, these fluctuations lead to oscillations of the liquid flow rate, Q'T,
which subsequently produce surface waves in the vortex chamber propagating
back and forth. Their amplitudes and phase angles i/rvc// with respect to the press-
ure oscillations depend on the resonance properties of the liquid vortex in the
injector, and can be determined by the reflection coefficients based on Eq.
(156). When disturbances occur, part of these waves pass through the injector
nozzle and cause fluctuations of the flow rate QfN and spray angle at the exit. Con-
currently, the fluctuation Q'T gives rise to oscillations of the circumferential
velocity U'u in the vortex chamber that propagate with the liquid flow and
produce centrifugal-pressure fluctuations kP'vcin on the vortex chamber wall.
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98 V. BAZAROV, V. YANG, P. PURI

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v*=1.0—-
-271

2 4 6 10 12
Shvc

71/2 -

-71/2
I I
2 4 6 10 12
Shvc

Fig. 57 Phase angle of pressure pulsation in vortex chamber as function of Strouhal

number.

This secondary fluctuation AP'VC/// can be vectorially summed with the original
pressure-drop fluctuation in the liquid vortex AP^C// under the action of the
surface wave to obtain the total pressure-drop fluctuation AP^C. Finally, the
vector sum of AP^C and AP'r forms the dynamic pressure drop in the injector
APJ relative to which the flow-rate fluctuation in the tangential entry is phase-
shifted by an angle if/T. If the tangential-velocity disturbance is not damped by
viscous losses, it will reach the injector nozzle exit considerably later than the
surface wave. The resultant fluctuations of the flow rate and other properties
must be determined by their vector sum.
Compared to the unsteady flow in the tangential inlet channel, the character-
istic time of circumferential-velocity pulsations in the liquid vortex layer is much
shorter, and as such any disturbance in the liquid layer is rapidly transmitted in
the radial direction. Measurements of centrifugal-pressure pulsations at the
outer wall of a typical vortex chamber at frequencies of hundreds of Hz reveal
that the liquid vortex layer responds in a quasi-steady-state manner to radial
disturbances. Their amplitudes are only a few percent lower than the quasi-
stationary variations for given changes of the circumferential velocity.
Figure 57 shows the phase angle between the pressure pulsations in the feed
line and the vortex chamber, where i/>vc is the phase difference between oscil-
lations at the head end and exit of the vortex chamber, and ij/ab between the
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 99

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Fig. 58 Amplitude of liquid surface wave in vortex chamber as function of Strouhal

number; 1-head end, 2-exit.

exit of the tangential entry and the liquid free surface at the vortex-chamber head
end. The Strouhal number Shvc is defined as wl^^jU^. Two different liquids with
dimensionless viscosities of v* = 0.1 and 1.0 are considered here. For reference,
p* = 0.08 for water at room conditions. Figure 58 shows the amplitude of the
liquid surface wave in the vortex chamber, where station 1 corresponds to the
head end and 2 to the exit.
Theoretical and experimental studies on the effect of the velocity pulsations in
the tangential entries U'in on the liquid swirling flow suggest at least two different
mechanisms of disturbance propagation in the vortex chamber. First, U'm pulsa-
tions cause fluctuations of the liquid free surface r'mk which then propagate at
the speed Uw according to Eq. (148). Second, t/[n pulsations result in an
energy disturbance in the form of circumferential velocity fluctuations that pro-
pagate throughout the entire liquid layer in both the radial and axial directions.
Analogous energy waves in gaseous flows are observed and generally referred
to as entropy waves.24 When propagating along the axis of the swirler, the
lengths of these waves decrease but the amplitudes grow in accordance with
the conservation of angular momentum.
The pressure variation in the radial direction is obtained by integrating the
centrifugal force across the liquid vortex layer:

-^dr (159)

In dimensionless form, Eq. (159) becomes

°sc =
2pUmUfin =3-^ (160)

where arg Uu = U'u/U-m is the deviation from the stationary dependence of Uu

per dimensionless radius r. As an example, for an infinitely long vortex
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100 V. BAZAROV, V. YANG, P. PURI

0.0
-0.2 x^s 0.2 0.4 0.6 0.8 1.0
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increasing co
classic swirl injector
- - - - vortex chamber with increased
-Imk •i, m radial velocity
Fig. 59 Amplitude-phase diagram of amplification coefficient of a typical liquid
vortex chamber.

chamber (i.e., no wave reflection from the nozzle), we have from Ref. 14

- \rr Rvc(l - in -,
arg Uu = tan - • ^———— - exp r--—v-(1-r)
c / 1
(161)
[2 Rvc -

Figure 59 shows the amplitude-phase diagram of the liquid swirier response to

incoming pressure pulsations for different values of liquid viscosity. The sub-
script /// stands for the results obtained from the third model. At zero frequency,
the amplification coefficient k (defined as k = U(n/&P'SC) has its stationary value
of unity. The coefficient decreases rapidly with increasing frequency, but the
phase angle grows from 0 to rr/2. Thus, the swirling flow movement is stable
if the amplification coefficient is placed in the fourth quadrant of this complex
plane.
For small disturbances, surface and entropy (or energy) waves behave inde-
pendently and the net effect can be represented by their vectorial sum. At high
frequencies, the entropy wave and its influence on centrifugal pressure can be
ignored due to the high inertia of the liquid vortex layer. In contrast, the influ-
ences of surface waves become negligible at low frequencies because of the
rapid decrease of their amplitudes. Entropy waves prevail in this situation. Cal-
culations have shown that centrifugal-pressure pulsations resulting from circum-
ferential velocity fluctuations may exist for several periods of pulsation, but at the
same time surface waves may propagate throughout the liquid almost instan-
taneously with their high wave speed. This effect, known as the memory effect
of a swirling flow, may suppress these fluctuations if they are out of phase.
The overall response function of a swirl injector IISW can be obtained in terms
of the transfer characteristics of each individual element of the injector. The
transfer function between the fluctuating velocity in the tangential inlet
passage and the pressure drop AP'r across the inlet is defined as

(162)
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 101

The centrifugal-pressure pulsations caused by liquid surface wave motions and

circumferential velocity fluctuations can be characterized by the following trans-
fer functions, respectively:
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AP;c///APr AP;c///APr
live// = 9/7/ / / 7—— = 9n/ /n —— U0.3)
2Uin/U-m 2QT/QT
AP;c////APr AP;c////APr
live/// = 9/7, , -—— = 9n/ / n—— (104)
2Uin/U-m 2QT/QT

Note that all of the variables in the preceding equations are complex to account
for the phase differences between the various processes of concern. The fluctu-
ation of the total pressure drop across the entire injector APJ is the vector sum
of AP'r and Pfvc , with the latter being PfvcII + P'vcIII. Thus,

AP; = Ap'r + AP;C = Ap'r + AP;C// + AP;C/// (165)

Substitution of Eqs. (162-164) into (165) and rearrangement of the result give
rise to the transfer function between Q'T and APJ:

_ QT/QT 0, /AP
i + 2 (nvc// + nvc///)nr A
The fluctuating flow rate at the exit of the vortex chamber (or the entrance of
nozzle) <2vC can be expressed as

n (167>
l= "S
Similarly, the fluctuating flow rate at the nozzle exit becomes

*^«
=n FT ^VC
(168)
/1£Q\

o"
\Ln "o~
5^vc

Since the mean flow rate through all elements of the injector is identical,

QT = Gvc - Qn (169)

Substituting from Eqs. (167) and (168) and using Eq. (166), we obtain the mass
transfer function for the entire injector.
The overall response function of a swirl injector Ilsw can be obtained by com-
bining Eqs. (166-169)

llsw
_ _ f vc (170)
- A/>; " AP, 2nrnvc + 1
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102 V. BAZAROV, V. YANG, P. PURI

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400
300

Fig. 60 Amplitude-phase diagram of response function of a typical liquid swirl

injector.

Here AP/(=P/ — Pc) denotes the pressure drop across the entire injector. The
pressure drop ratio can also be expressed in terms of the geometrical parameters
of the injector. The intricate dynamics in a swirl injector produce complicated
amplitude-phase characteristics of its overall response function, as shown in
Fig. 60 where A/^ is set to unity for simplicity. This diagram allows one,
under practical design limitations, to obtain any desired pulsation characteristic
by either suppressing or amplifying flow oscillations. Thus, it becomes possible
to control the engine combustion dynamics by changing the injector dynamics
alone without modifying the other parts of the combustion device.

Acknowledgments
The authors wish to express their sincere thanks to Piyush Thakre for helping
to prepare the figures.

References
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Mashinostroenie Pub., Moscow, Russia, 1985.
2
Dityakin, Y. F., Klyachko, L. A., and Jagodkin, Atomization of Liquids,
Mashinostroenie Pub., Moscow, Russia, 1977.
3
Pazhi, D. G., and Prakhov, A. M., Liquid Atomizers, Khimiya, Moscow, Russia, 1979.
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Rollbuhler, H. J., "Experimental Investigation of Reaction Control, Storable Bipro-
pellant Thrusters," NASA TND 4416, 1976.
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"Space Shuttle Orbital Maneuvering Subsystem Rocket Engine Design Features,"
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Elverum, G., Jr., Staudhammer, P., Miller, J., Hoffman, A., and Rockow, R., "The
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Bazarov, V. G., Dynamics of Liquid Injectors, Mashinostroenie Pub., Moscow,
Russia, 1979.
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Abrarnovich, G. N., Applied Gas Dynamics, Nauka, Moscow, Russia, 1976.
9
Khavkin, Y. L, Swirl Injectors, Mashinostroenie Pub., Moscow, Russia, 1976.
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DESIGN AND DYNAMICS OF JET AND SWIRL INJECTORS 103

10
Andreev, A. V., Bazarov, V. G., Marchukov, E. Y., and Zhdariov, V. L, "Conditions
of Hydrodynamic Instability Occurrence in Liquid Swirl Injectors," Energetika, 1985,
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Lyul'ka, L. A., and Bazarov, V. G., "Investigations of the Self-Oscillation Mode of
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Taylor, G., "The Mechanism of Swirl Atomizers," Proceedings of 7th International
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14
Andreev, A. V., Bazarov, V. G., Dushkin, A. L., Ggrigoriev, S. S., and Lul'ka, L. A.,
Dynamics of Gas-Liquid Injectors, Mashinostroenie Pub., Moscow, Russia, 1991.
15
Bazarov, V. G., "Injectors for Three-Propellant Liquid Rocket Engine with Smooth
Thrust Variation," IAF-95-S.1.03, Proceedings of 46th International Astronautical
Congress, Oslo, Norway, 1995.
16
Baywel, L., and Orzechovski, Z., Liquid Atomization, Taylor & Francis, Washington,
DC, 1993.
17
Bazarov, V. G., "The Effect of Injector Characteristics on Combustion Efficiency and
Stability," IAF Paper 92-0645, Proceedings of 43rd Space Conference, Washington,
DC, 1991.
18
Dressler, L., and Jackson, T., "Acoustically Driven Liquid Sheet Breakup," Proceed-
ings of 4th ILASS American Conference, Hartford, CT, Vol. 4, 1990, pp. 132-141.
19
Idelchik, L E., Handbook on Hydraulic Resistances, Mashinostroenie Pub., Moscow,
Russia, 1975.
20
Kudriavtzev, V. M., Basics of Theory and Design of Liquid Rocket Engines, 4th ed.,
Visshaya Shkola, Moscow, Russia, 1993.
21
Zhukovski, N. E., Hydraulics, Vol. 7, ONTI-NKTP, 1937 (in Russian).
22
Bazarov, V. G., "Self-Pulsations in Coaxial Injectors with Central Swirl Liquid
Stage," AIAA Paper 1995-2358, 1995.
23
Bazarov, V. G., Fluid Injectors Dynamics, Mashinostroenie Pub., Moscow,
Russia, 1979.
24
Glickman, B. F., Dynamics of Pneumo-hydraulic Liquid Rocket Engine Systems,
Mashinostroenie Pub., Moscow, Russia, 1983.
25
Ditiakin, Y., Kliatchko, L., Novikov, B., and Yagodicin, V., Atomization of Liquids,
3rd ed., Mashinostroenie Pub., Moscow, Russia, 1987.
26
Zong, N., and Yang, V., "Dynamics of Simplex Swirl Injectors for Cryogenic Propel-
lants at Supercritical Conditions," AIAA Paper 2004-1332, 2004.
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Yang, V., and Bazarov, V., "Propellant Rocket Engine Injector Dynamics," Journal
of Propulsion and Power, Vol. 14, No. 5, 1998, pp. 797-806.
Downloaded by PURDUE UNIVERSITY on January 27, 2019 | http://arc.aiaa.org | DOI: 10.2514/5.9781600866760.0019.0103 | Book DOI: 10.2514/4.866760

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