Rocket nozzles: 75 years of research and development
SHIVANG KHARE1 and UJJWAL K SAHA2,*
1Department of Energy and Process Engineering, Norwegian University of Science and Technology,
7491 Trondheim, Norway2Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India
e-mail: [email protected]; [email protected]
MS received 28 August 2020; revised 20 December 2020; accepted 28 January 2021
Abstract. The nozzle forms a large segment of the rocket engine structure, and as a whole, the performance of
a rocket largely depends upon its aerodynamic design. The principal parameters in this context are the shape of
the nozzle contour and the nozzle area expansion ratio. A careful shaping of the nozzle contour can lead to a
high gain in its performance. As a consequence of intensive research, the design and the shape of rocket nozzles
have undergone a series of development over the last several decades. The notable among them are conical, bell,
plug, expansion-deflection and dual bell nozzles, besides the recently developed multi nozzle grid. However, to
the best of authors’ knowledge, no article has reviewed the entire group of nozzles in a systematic and
comprehensive manner. This paper aims to review and bring all such development in one single frame. The
article mainly focuses on the aerodynamic aspects of all the rocket nozzles developed till date and summarizes
the major findings covering their design, development, utilization, benefits and limitations. At the end, the future
possibilities of development are also recommended.
Keywords. Rocket nozzle; aerodynamics; thrust; expansion ratio; nozzle contour; shock wave; method of
characteristics; efficiency.
1. Introduction
The nozzles were invented with a primary motive to change
the flow characteristics such as velocity and pressure. In
1890, Carl Gustaf Patrik de Laval developed a convergent-
divergent (CD) nozzle that had the ability to increase a steam
jet to a supersonic state [1, 2]. A typical CD nozzle and the
variation of velocity, temperature, pressure across the length
of nozzle is shown in figure 1. This nozzle was termed as a de
Laval nozzle and was later used for rocket propulsion. An
American engineer Robert Goddard was the first to integrate
a de Laval nozzle with a combustion chamber, thereby
increasing the rocket efficiency and attaining the supersonic
velocities in the region of Mach 7 [2, 3].
For space propulsion, the rocket [4, 5] is the main system
that stores its own propellant mass and ejects this mass at
high speed in order to provide thrust. A rocket engine [6–9]
generates this thrust by accelerating the exhaust gases to the
desired speed and direction. In simple words, the nozzle
utilizes the pressure generated inside the combustion
chamber to enhance the magnitude of thrust by accelerating
the combustion products to a high supersonic velocity. The
nozzle exit velocity can be controlled by the nozzle
expansion ratio or the area ratio (i.e., the exit area of nozzle
divided by the area of throat). Lesser design complexity and
weight, maximum performance, and ease of manufacture
are some of the main desirable features of a rocket nozzle.
The development of novel rocket nozzles [10] for launch
vehicles faces a challenging design problem. In order to
meet the performance of nozzle at higher altitudes, the
nozzles are designed with high area ratios. However, this
would generate over-expanded flow conditions when oper-
ated at sea level. A nozzle is said to be an over-expanded
one when its exit pressure is less than the ambient pressure.
These conditions lead to an unsteady internal flow separa-
tion resulting in the generation of side loads which may
cause damage to the whole launch system. The generation of
high magnitude side loads inside the nozzles is one of the
most important issues under consideration in designing the
reusable, robust, and efficient launch vehicles [11–13].
1.1 Brief overview
Rocket nozzles comes in a variety of configurations like
ideal, conical, bell, plug, expansion-deflection (E-D) and
dual bell besides the recently developed multi nozzle grid
(MNG). An ideal nozzle is defined as the one that produces
an isentropic flow (i.e., without internal shocks) and gives a
uniform velocity at the exit. The contour of such a nozzle
can be designed with the help of method of characteristics*For correspondence
Sådhanå (2021) 46:76 � Indian Academy of Sciences
https://doi.org/10.1007/s12046-021-01584-6Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)
[14–16] (detailed in section 2). The conical type has his-
torically been the most common contour for rocket nozzles
because of its design simplicity and ease of manufacture
[16, 17]. In a conical nozzle, the exit velocity is essentially
one-dimensional (1D) corresponding to the area ratio, but
the flow is not in an axial direction across the outlet area
leading to performance loss due to flow divergence
[17, 18]. In the late 1930s and early 1940s, German sci-
entists performed extensive nozzle research [16, 19] by
considering all aspects of designing. They opined that there
was no major benefit in using contours with high com-
plexities. However, this was applicable only for low area
ratio nozzles like V-2 rocket [16, 17]. Because of its high
divergence losses, the conical type short nozzles find their
application in small thrusters and solid rocket boosters,
where simple fabrication is desired over aerodynamic per-
formance [17]. On increasing the cone angle, the thrust loss
of a conical nozzle gets enhanced due to flow divergence.
This thrust loss can be minimized by contouring the nozzle
wall and this type is referred to as the bell nozzle. This is
because, by doing this, the flow can be made to turn closer
to the axial direction [18]. Usually, the calculus of varia-
tions is the simple and direct approach for designing the
nozzle contours [14, 18]. Guderley and Hantsch [20]
investigated the problem of finding the nozzle exit area and
its contour to generate the optimum thrust for given values
of ambient pressure and nozzle length. However, the
method was not accepted widely until a simplified tech-
nique, as detailed later, was proposed by Rao [14]. In
Ariane 5 Vulcan or Space Shuttle Main Engine (SSME),
the conventional bell-type nozzle was used to expand the
propellant products [21]. On the other hand, the plug nozzle
is an altitude-compensating type rocket nozzle, where a
traditional CD nozzle expands the flow to a fixed area ratio
regardless of the freestream conditions. The free jet
boundary that acts as a virtual outer wall on a plug nozzle
expands to match the freestream ambient pressure [22, 23].
The first conceptual analysis of plug nozzles was conducted
in 1950s [21]. Though the performance benefits were
claimed in most of the literatures, however, the plug noz-
zles did not gain the hardware flight status. In the future,
this might change as the rocket engine having a linear plug
nozzle is foreseen as the propulsion system for the RLV
X-33 concept of the Lockheed Martin Corporation
[21, 24, 25]. In the E-D nozzle, the flow from the chamber
is directed radially outward and away from the axis of
nozzle. The flow is diverted towards the curved contour of
the outer diverging nozzle wall [26]. The hot gas flow
moving out of the chamber expands around a central plug.
The E-D nozzle concept had been the subject of various
experimental and analytical investigations. These studies
revealed the poor altitude-compensation capabilities of E-D
nozzle and were in fact poorer than the plug nozzles
because of over-expansion and aspiration losses. For noz-
zles with high expansion-ratio and comparatively smaller
length, an E-D nozzle performs superior than a comparable
bell nozzle of equal length. This is due to the lesser
divergence losses as compared to the bell nozzle [24]. In a
dual bell nozzle, two shortened bell type of nozzles are
joined into one with an inflection point between them. In
1949, Cowles and Foster were the first to introduce the dual
bell concept, and it was patented by Rocketdyne in 1960s
[27–29]. It is still in the conceptual stage but seems to be a
strong candidate for future rocket engines [30]. In recent
times, a newer concept of Multi Nozzle Grid (MNG) came
into the limelight where a thin and lightweight plate with
multiple small nozzles can be used instead of a lengthy and
heavy single nozzle. The saving in length is in direct pro-
portion to the square root of the number of small nozzles
(nozzlettes) in MNG (i.e., MNG with hundred nozzlettes is
ten times smaller than an equivalent single nozzle) [31–33].
1.2 Present objective
As evident from above, various shapes of rocket nozzles
have been evolved over the past 75 years, however, there is
not a single article that gives a comprehensive review of all
the types of nozzles developed till date. The present article
mainly deals with the aerodynamic features of ideal, coni-
cal, bell, plug, expansion-deflection, dual bell, and multi
nozzle grid type rocket nozzles. The review summarizes all
the facts and figures in a systematic way incorporating
several features of rocket nozzles such as design, devel-
opment, utilization, benefits and limitations along with
recommendations.
2. Ideal nozzle
When there is a parallel uniform flow with the exit pressure
matching with the ambient pressure at the nozzle exit, the
nozzle thrust becomes maximum. Such type of nozzle is
Figure 1. Variation of velocity, temperature and pressure across
the length of a De Laval nozzle [2].
76 Page 2 of 22 Sådhanå (2021) 46:76
termed as an ideal nozzle. The ratio of area at exit Ae to
area at throat At of such a nozzle [18] can be expressed by
Eq. (1).
Ae
At¼ c� 1
2
� �1=22
cþ 1
� � cþ1ð Þ=2 c�1ð Þ paPc
� �� 1=cð Þ1� pa
Pc
� � c�1ð Þ=c" #�1=2
ð1Þwhere c is the isentropic exponent. The ratio pa/pc is veryless for the rocket engines working at high altitudes, and
hence more expansion ratios for the ideal nozzles is needed.
A rocket engine working over a wide range of altitudes will
give optimum performance when the nozzle expansion ratio
is variable. Because of very high temperatures involved, the
mechanical variation of expansion ratio is not easy. Thus,
an appropriate fixed expansion ratio is selected by consid-
ering the performance requirements over the complete path
of the rocket. The performance of nozzle divergent section
can be calculated (Eq. (2)) in terms of its vacuum thrust
coefficient Cfv [18] and can be expressed as:
Cfv ¼ Fv
pcAtð2Þ
where Fv is thrust when discharging to vaccum, pcand At
are the combustion chamber pressure and area of the nozzle
throat, respectively. The vacuum thrust coefficient of an
ideal nozzle is a function of isentropic exponent c and the
nozzle area expansion ratio Ae/At.
The thrust coefficient of an ideal nozzle [18] working in
an ambient pressure pa, is given by
Cf ¼ Cfvi � paPc
:Ae
Atð3Þ
The above equation is known as the 1D thrust coefficient
equation.
The uniform exit flow can be obtained when the nozzle
wall configuration is designed by the method of charac-
teristics [14–16, 34]. With the help of this method, the
system of partial differential equations can be converted to
ordinary differential equations that are valid along the
characteristic curves that represent the path of propagation
of disturbances in supersonic flows. A characteristic may be
defined as a curve along which the governing partial dif-
ferential equations reduce to an ’interior operator’s that is a
total differential equation known as the compatibility
equation. Thus, along a characteristic, the dependant vari-
ables may not be specified arbitrarily being compelled to
satisfy the compatibility equation [35–39]. High-speed
computing machines are required for the wall contour
computation. Foelsch [40–42] gave an approximation
through which the nozzle contour can be obtained by
manual computation. The length of an ideal nozzle can be
decreased by permitting all the expansion to occur just at
the throat downstream and then constructing the nozzle
contour to turn the flow such that it can attain an axial
uniform flow at the exit. To attain a uniform flow at exit,
the minimum length of the nozzle required for various
values of expansion ratio (computed for c = 1.23) is shown
in figure 2 [3]. In computing these lengths of the nozzle, it
was found that at the downstream of throat, a sharp corner
should be eluded. Hence, a wall contour of a radius of
curvature equal to 0.4 times the radius of throat was used
[18, 43]. From figure 2, it is clear that the ideal nozzle that
gives the maximum thrust performance is heavy and
lengthy. The following sections deal with the ways of
decreasing the length of nozzles without much decrement
in the performance.
3. Conical nozzle
The application of a conical nozzle was very common in
early rocket engines. The principal appeal of the conical
nozzle is that it is easy to manufacture and it has the
flexibility of converting an existing design to lower or
higher area ratio without much redesign. A typical conical
nozzle configuration is depicted in figure 3. On the basis of
length, performance and weight, the best compromised
diverging angle (a) for a conical nozzle is 15� [17, 36, 44].The thrust coefficient of a 15� conical nozzle is only 1.7%
lesser than the ideal nozzle and changes slightly with alti-
tude [18]. Thus, the performances and lengths of newer
nozzles are often compared with a 15� conical nozzle. Thelimitation of conical nozzle being heavier and longer can be
mitigated by increasing the divergent cone half-angle (a).In a conical nozzle, the performance loss takes place at a
lower altitude as higher ambient pressure leads to flow
separation and over-expansion. The major disadvantage of
conical nozzles includes the trade-off between the diver-
gence angle and the nozzle length [45].
The flow divergence in the conical nozzle leads to a
reduction in thrust as the flow direction is not completely
Figure 2. Length (calculated) comparison of several types of
nozzle (MNG: Multi Nozzle Grid, E-D: Expansion-Deflection
nozzle) [3].
Sådhanå (2021) 46:76 Page 3 of 22 76
axial at the exit. Malina [46] showed that the momentum at
the exit was equal to the value calculated from 1D theory
multiplied by a correction factor, k [16, 17].
k ¼ 1þ cosa2
ð4Þ
The magnitude of correction factor, k decreases as aincreases as shown in figure 4 [46].
The conical nozzle thrust coefficient working into the
vacuum (Cfvc) is given below [18]
Cfvc ¼ pepc
:Ae
Atþ 1þ cos /
2
� �qeVe
2Ae
pcAtð5Þ
where qe, Ve, pe are the 1D density, velocity and pressure at
the nozzle exit.
Darwell and Badham [47] revealed the possibility of
shock formation within the nozzle by the method of char-
acteristics. They found that by modifying the wall contour
near the junction of the throat profile and the cone, the
shock formation could be eliminated. Khan and Shemb-
harkar [48] explained that one can capture the flow in a CD
nozzle in an overexpanded flow regime by employing a
computational fluid dynamics (CFD) code. They computed
the flow features like after-shocks, location of shocks and
lambda shocks in the CD nozzle. The centreline static
pressure plot (figure 5) for different nozzle pressure ratio
(NPR) shows that shock location moves towards the nozzle
exit as the NPR increases. Balabel et al [49] studied the
turbulent gas flow dynamics in a two-dimensional (2D) CD
nozzle and the associated physical phenomena were anal-
ysed for different operating conditions. Results showed
that, in predicting the point of separation and the position of
shock wave, the realizable v2–f and the SST (Shear StressTransport) k–x models gave better results in comparison to
other models. Hoffman and Lorenc [50] investigated 2D
gas particle flow effects in conical nozzles. Wehofer and
Moger [51] developed an analytical method to predict the
inviscid transonic flow fields linked with CD and conver-
gent conical nozzles. Jia et al [52] analysed numerically the
influence of the fire-in-the-hole staging event on the flow
separation at the start-up and side loads of a conical nozzle
during flight. Jia et al [53] further investigated numerically
the three-dimensional (3D) side loads and the associated
flow physics of a conical nozzle during the fire-in-the-hole
staging event of a multistage rocket. Zmijanovic et al [54]studied the transverse secondary gas injection into the
supersonic flow of a CD nozzle to explain the influence of
fluidic thrust vectoring within the framework of a small
satellite launcher. The results demonstrated the possibility
of attaining pertinent vector side forces with moderate
secondary to primary mass flow rate ratios ranging around
5%. They also revealed that the geometry and injector
positioning had a high influence on the nozzle performance
and shock vector control system. Figure 6 shows the
injection port and the scheme of separation alongside
nozzle wall. Zhang et al [55] performed the computational
investigation of convergent conical nozzles. The influences
of base drag and the freestream Mach number on the cal-
culated velocity coefficient were analysed. Sunley and
Ferriman [56] carried out tests to study the jet separation in
conical nozzles, and demonstrated that the pressure at
which the gas separates was neither constant nor
Figure 3. Typical configuration of a conical nozzle.
Figure 4. Variation of k with a in graphical form [46].Figure 5. Centreline static pressure (measured) plot for different
nozzle pressure ratio (NPR) [48].
76 Page 4 of 22 Sådhanå (2021) 46:76
independent of the nozzle length. Migdal and Kosson [57]
studied the shock predictions in conical nozzles. Migdal
and Landis [58] investigated the performance of conical
nozzles by the application of method of characteristics.
Table 1 summarizes the various numerical, experimental
and analytical studies in the area of conical nozzles
addressing the aerodynamic aspects such as flow separation
and the shock formation.
4. Bell nozzle
The bell type nozzle is the most commonly used shape in
rocket engines. This category of nozzle provides significant
benefits in terms of performance and size over the conical
type nozzle. It has a high angle expansion section (20� to
50�) right behind the nozzle throat. This is followed by a
gradual reversal of nozzle contour slope so that at the
nozzle exit the divergence angle is small, preferably below
10� half angle. It is possible to have large divergence anglesimmediately behind the throat (20�–50�) because the high
relative pressure, the large pressure gradient, and the rapid
expansion of the working fluid that do not permit separation
in this portion unless there are discontinuities in the nozzle
contour [3]. The length of bell nozzle is generally given as
a fraction of the reference conical nozzle length. As seen in
figure 7, an 80% bell nozzle configuration that has the same
expansion ratio (e) as 15� half angle conical nozzle is 20%
smaller in length. The exit angle of 80% and 60% bell
nozzles are 8.5� and 11�, respectively [3].
A near optimum contour of the bell nozzle can be
designed by using a simple parabolic approximation pro-
cedure given by Rao [14, 59, 60]. The parabolic approxi-
mation of the bell nozzle design is shown in figure 8 [36],
where at the upstream of throat T, the nozzle contour is a
circular arc whose radius is 1.5Rt. From T to the point N,the divergent portion of the nozzle contour is made up of a
circular entrance section of radius 0.382Rt and from N to
the exit E, it is a parabola [36].
The required data for the design of a specific bell nozzle
are: diameter of throat Dt, initial wall angle of the parabola
hn, nozzle exit wall angle he, area ratio e and nozzle axial
length from throat to exit plane Ln (or the desired fractional
length Lf, based on a 15� conical nozzle) [36]. The varia-
tions of wall angles hn and he with the expansion ratio at
different values of Lf are shown in figures 9 and 10,
respectively. It is evident in figure 9 that by increasing the
expansion ratio of the bell nozzle, the value of hn increases.For a particular expansion ratio, the magnitude of hn is
larger for lower fractional nozzle length. By choosing
proper inputs, the optimum nozzle contours can be
approximated.
The thrust loss can be minimized by contouring the
nozzle wall, as by doing this, the flow can be made to turn
closer to the axial direction [18]. For contouring the nozzle
walls, various methods have been suggested in the litera-
ture. Dillaway [61] computed nozzle contours by gradually
reducing the slope of nozzle wall as one approaches
towards the nozzle exit. In a contoured nozzle, there is a
high dependency of exit flow on the nozzle contour and no
simple relation such as Eq. (5) can be derived. The
Figure 6. Scheme of the separation alongside nozzle wall [54].
Table 1. Major studies conducted on conical nozzle.
Investigators Year Nature of study Focus of study
Migdal and Landis [58] 1962 Numerical Performance of conical nozzles
Darwell and Badham [47] 1963 Numerical Shock formation
Sunley and Ferriman [56] 1964 Experimental Jet separation
Migdal and Kosson [57] 1965 Numerical Shock predictions
Hoffman and Lorenc [50] 1965 Numerical Gas-particle flow effects
Wehofer and Moger [51] 1970 Analytical Inviscid transonic flow fields
Khan and Shembharkar [48] 2008 Numerical Location of shock
Balabel et al [49] 2011 Numerical Turbulent gas flow dynamics
Zmijanovic et al [54] 2014 Experimental and Numerical Fluidic thrust vectoring
Zhang et al [55] 2015 Numerical Convergent conical nozzles
Jia et al [52] 2015 Numerical Flow separation and side loads
Jia et al [53] 2016 Numerical Side loads
Sådhanå (2021) 46:76 Page 5 of 22 76
contoured nozzle vacuum thrust coefficient can be evalu-
ated as [18]
Cfv ¼Z
Ae
p
pcAt2prdr þ
ZAe
qV2coshpcAt
2prdr ð6Þ
where the integration is performed throughout the whole
area Ae of the exit plane, which is perpendicular to the axis
of nozzle. The velocity (V), direction (h) density (q), andpressure (p) are calculated at a distance r from the axis of
nozzle in the exit plane. According to Landsbaum [62], one
can analyse the various truncated ideal nozzles of different
expansion ratios and select the one which gives the best
performance [63]. Farley and Campbell [64] investigated
such truncated ideal nozzles experimentally and the results
were found to be very close to the theoretical values.
Ahlberg et al [65] presented the graphical technique for
choosing the optimum nozzle contours from a family of
truncated perfect nozzles. Rao [14] developed a method by
using the calculus of variations to design the wall contour
of the optimum thrust nozzle. For the same performance as
that of 15� conical nozzle, the length required for the bell
nozzle is shown in figure 2. Allman and Hoffman [66]
examined a technique for the design of maximum thrust
nozzle contours using direct optimization methods. The
contour of the nozzle was given by second-order
polynomial:
yðxÞ ¼ Aw þ Bwxþ Cwx2 ð7Þ
where the coefficients Aw, Bw, and Cw are determined by
mentioning the attachment angle, the exit radius and by
requiring that the polynomial contour attach continuously
to the circular-arc initial expansion contour. The authors
compared thrusts developed by the calculus of variation
contours (Rao’s method) with the thrusts generated by
polynomial contours. It was concluded that both the
methods predict essentially the same maximum thrust (i.e.,
for zero ambient pressure, the agreement was within 0.2%)
justifying the proposed technique.
Frey et al [67] presented a new nozzle contouring tech-
nique called TICTOP (figure 11), merging both Truncated
Ideal Contour (TIC) and Thrust Optimized Parabolic (TOP)
designs. The nozzle obtained is shock-free as the TIC
design and does not induce restricted shock separation
Figure 7. Comparison of a 15�conical nozzle (reference nozzle)
with 80% and 60% bell nozzles, all at an expansion ratio e of 25[3].
Figure 10. Variation of wall angles he (calculated) with the
expansion ratio e at different values of Lf (Lf = desired fractional
length (m) [36].
Figure 8. Parabolic approximation bell nozzle design configu-
ration [36].
Figure 9. Variation of wall angles hn with the expansion ratio eat different values of Lf [36].
76 Page 6 of 22 Sådhanå (2021) 46:76
(RSS) which leads to high side loads. Simultaneously, it
permits a higher nozzle wall pressures at exit giving a better
margin of separation than the TOP design. Pilinski and
Nebbache [68] analysed the separated flow numerically at
various NPRs in a TIC nozzle and the pattern of a free
shock separation (FSS) was obtained for very low and high
pressure ratios. Between these two ranges of pressure
ratios, an uncommon cap shock pattern without reattach-
ment of the boundary layer appeared. Verma et al [69]
conducted tests campaign on a sub-scale TOP nozzle in
order to investigate the link between unsteady characteris-
tics of separation and reattachment shocks and the source of
side loads in rocket nozzles. Lawrence and Weynand [70]
studied the separated flow in 2D and axisymmetric nozzles
having various wall contours. Nguyen et al [71] investi-
gated the turbulent flow separation in an overexpanded
TOC nozzle. Nebbache and Pilinski [72] carried out com-
putational investigation of the flow in a TOC nozzle and
analysed the structures of flow separation at various pres-
sure ratios. Potter and Carden [73] described a method to
design the nozzles for low-density, high-speed flows with a
focus on treatment of very thick laminar boundary layers.
Baloni et al [74] performed 2D axisymmetric flow inves-
tigation within the bell type nozzle at off-design and design
conditions with the help of CFD software Fluent 6.3.26 and
GAMBIT 2.4.6. Numerical simulation was conducted
separately for two different flow conditions, i.e., hot and
cold.
Verma et al [75] carried out an experimental study to
identify the source of several flow conditions that led to the
generation of side load in a TIC nozzle. The main con-
tributors were found to be the transition in flow conditions,
i.e., the change in the circumferential shape of recirculation
portion inside the nozzle from a cylindrically dominated
regime to a conical one and the end-effect regime that
initiated a highly unsteady condition of flow in the sepa-
ration region preceding these transitions. Stark and Wagner
[76] studied the separation of boundary layer and the
related flow field in a TIC nozzle. Verma [77] performed
experiments to investigate the separation of flow in a TIC
nozzle and reported that the off-design over-expanded
nozzle flow was dominated by shock-induced boundary
layer separation that demonstrated fluctuating characteris-
tics. The separation shock fluctuates back and forth, the
magnitude of which showed a high dependence on the
contour of nozzle. Asymmetry in flow separation was also
observed at particular pressure ratios. Hagemann et al [78]tested two different types of nozzle, a thrust optimized and
a truncated ideal nozzles, for the same performance data to
explore the origin of different side loads. Results demon-
strated the highest side load in the thrust optimized nozzle
when the separation pattern changed from free to restricted
shock separation. The side load measured in the truncated
ideal nozzle was only about 33.33% of side load in the
thrust optimized nozzle. The comparison of both the nozzle
contours is shown in figure 12. Stark and Wagner [79]
performed tests to analyse the TIC nozzle flow field at low
NPR. For NPR less than 10, a convex and for NPR greater
than 20, a slight concave shaped Mach disks were found.
Terhardt et al [80] studied the side-loads and flow separa-
tion in a TIC type nozzle experimentally and analytically.
Zhang et al [12] studied numerically the aeroelastic sta-
bility for a rocket nozzle at the start-up. It was reported that
the aeroelastic behaviour of rocket nozzles were strongly
dependent upon the wall thickness and the material prop-
erties of the wall. A comparison of different nozzle con-
tours is given in table 2, while the summary of major
studies on bell nozzle is shown in table 3.
5. Plug nozzle
The plug nozzle is an advanced rocket nozzle that consists
of a primary nozzle whose shape is somewhat conventional
and a plug that helps an external expansion. The main
characteristics of this nozzle is its interaction with the
external ambient that can avoid the separation of flow
phenomena that affects a conventional bell nozzle. Such
benefits arise from the generation of expansion fan at the
primary nozzle lip and its influence on the pressure beha-
viour along the plug wall [36].
The use of plug-type nozzles has been analysed inten-
sively in recent years because of their various attractive
performance and design features. The basic concept of a
plug nozzle is not new. The manufacturers of propulsion
systems and the Lewis Research Centre of NASA have
investigated their characteristics for several years [81]. The
early turbojets of Germany installed them to attain variation
Figure 11. The TICTOP design [67]. Figure 12. Comparison of Thrust optimized contour and Trun-
cated ideal contour type nozzles [78].
Sådhanå (2021) 46:76 Page 7 of 22 76
of throat area [82]. The main attraction of a plug nozzle is
that it is altitude compensating. In other words, where a
conventional CD nozzle expands the flow to a fixed area
ratio regardless of the freestream conditions, the free jet
boundary that acts as a virtual outer wall on a plug nozzle
expands to match the freestream ambient pressure. This
feature is useful for rockets used in launch vehicles, where
the NPR can range from low magnitude at launch to infinity
in the vacuum of space. The schematic of the flow field near
a plug nozzle at varying NPR is shown in figure 13 [83, 84].
At or above design NPR, the plug nozzle behaves like a
conventional CD nozzle ejecting the exhaust gases straight
back axially (at design), or, resulting in a typical under-
expanded flow condition. For NPRs lesser than the design
value, the jet boundary is pulled nearer to the plug by a
series of compression waves and expansion fans that occur
naturally in an attempt to match the ambient pressure [84].
Plug nozzles refer both to a full spike and a truncated
spike nozzles. The contoured, full-length spike nozzle is
normally referred to as a spike; while a conical, full-length
spike nozzle is defined as plug nozzle. The aerospike comes
from the idea of introducing an additional flow into the base
region of the truncated spike, forming an ‘‘aerodynamic
spike’’ with the base flow. Basically, the base flow helps fill
in the area underneath the base and supports make up for
the performance loss from truncating the nozzle [3, 36, 85].
The plug nozzle throat is situated as an annulus at the
outer diameter (not all plug nozzles utilize an annular
throat; some use engines clustered on a shared plug) with
the exhaust gases flowing in an inward direction. At the
annulus outer edge or cowl-lip, the exhaust gases expand
suddenly to the ambient pressure. The expansion waves
generating from the cowl-lip controls the flow of exhaust
gases and the flow turning is affected by the surface of the
Table 2. Comparison of various nozzle contours.
Nozzle type Key features
Conical Straight walls from throat to exit, incomplete flow turning
Truncated ideal contour (TIC) Curved walls near throat transition to nearly straight walls near the exit, virtually
complete flow turning, shortened version of the method of characteristics
Thrust optimized contour (TOC) Curved walls near throat transition to nearly straight, walls near the exit, more
sudden transition than TIC, virtually complete flow turning
Thrust optimized parabola (TOP) Parabolic approximation of TIC, higher wall pressure at exit reduces risk of side
loads
Table 3. Major studies conducted on bell nozzle.
Investigators Year Nature of study Focus of study
Dillaway [61] 1957 Analytical 3D analysis of supersonic contoured nozzles
Rao [14] 1958 Analytical Method to optimize the wall contour of nozzle
Landsbaum [62] 1960 Numerical Design of bell-shaped nozzle contours
Farley and Campbell [64] 1960 Experimental Cut off portions of ideal nozzles
Ahlberg et al [65] 1961 Experimental and Numerical Optimization of truncated portions of ideal nozzles
Lawrence and Weynand [70] 1968 Experimental Separated flow in 2D and axisymmetric nozzles
having various wall contours
Potter and Carden [73] 1968 Experimental and Numerical Design of nozzles for low-density and high-speed flows
Terhardt et al [80] 2001 Experimental and Analytical Flow separation and side-loads in TIC nozzles
Verma [77] 2002 Experimental Flow separation phenomena in a TIC nozzle
Hagemann et al [78] 2002 Experimental Origin of side load in TIC and TOP
Nguyen et al [71] 2003 Experimental Turbulent flow separation in TOC nozzles
Pilinski and Nebbache [68] 2004 Numerical Separated flow in a TIC nozzle
Verma et al [69] 2006 Experimental TOP nozzle
Nebbache and Pilinski [72] 2006 Numerical Flow in an axisymmetric overexpanded TOC nozzle
Stark and Wagner [79] 2006 Experimental Analysis of flow field in a TIC nozzle at low NPR
Stark and Wagner [76] 2009 Experimental Boundary layer separation in a TIC nozzle
Frey et al [67] 2017 Numerical TIC and TOP design
Baloni et al [74] 2017 Numerical 2D axisymmetric flow analysis within the bell type nozzle
Verma et al [75] 2017 Experimental Origin of side-load generation in TIC nozzle
Zhang et al [12] 2017 Numerical Aeroelastic stability
76 Page 8 of 22 Sådhanå (2021) 46:76
plug. The design of the plug surface is such that the
expansion of gases from a chamber pressure pc to an
ambient pressure pa can take place smoothly producing a
uniform flow at exit parallel to the axis of nozzle. The
external diameter of such an ideal plug is same as the exit
diameter of a uniform flow CD nozzle expanding the gases
to the same ambient pressure pa. However, the plug nozzle
is much smaller than an equivalent CD nozzle [18].
Krase [82] suggested simple approximate methods to
design ideal plug nozzle contours by manual calculations.
Berman and Crimp [81] studied the performance compar-
ison between the conventional and the plug nozzles. As
seen in figure 14, the plug nozzle shows a thrust benefit
over a conventional nozzle when it is working at below
design pressure ratio as the nature of flow in this case is
self-adjusting [81, 86, 87].
Rommel et al [21] performed computational investiga-
tion of a plug nozzle and gave an understanding of the
development of flow field at various ambient pressures.
Figure 15 shows a typical conventional nozzle, a truncated
and a full length plug nozzle. Ito et al [88] investigated the
flow fields of a plug nozzle with the help of a numerical
simulation. They designed the plug contour by the method
of characteristics and various types of plug nozzles were
considered by truncating the nozzle length at various
positions. Results showed an increase of thrust performance
of the contoured plug nozzle by about 5 to 6% than the
conical plug nozzle, and the pressure distribution on the
nozzle surface was unaffected by the external flow for the
pressure ratios more than the designed point. Chutkey et al[89] analysed the flow fields associated with truncated
annular plug nozzles of varying lengths. On the other hand,
Shahrokhi and Noori [90] studied the flow properties of
various aerospike nozzle shapes. They used the k-e turbu-
lence model and Navier-Stokes equations for the simulation
of flow field. A uniform cubic B-spline curve was used to
generate the various plug shapes, and the best configuration
was determined by considering the total thrust force as a
performance merit. Kumar et al [91] discussed the proce-
dure to design a plug nozzle and the parameters governing
its design. Johnson et al [92] presented an optimization
analysis for plug nozzles with variable inlet geometry.
Besnard et al [93] presented the design, manufacturing and
tests of a 1000 lbf thrust ablative annular aerospike engine.
Results showed that variations in c with temperature led to
small, albeit not null, differences in thrust and plume
characteristics. Further, the nozzle was found to be very
efficient. Lahouti and Tolouei [94] carried out numerical
modelling of external and internal flows of a truncated plug
nozzle with several amounts of base bleed in under-ex-
pansion, optimum and over-expansion working conditions
Figure 13. Flow field of a plug nozzle with full length at different pressure ratios pc/pa, off-design [pc/pa\ (pc/pa)design] left, [pc/pa[(pc/pa)design] right and design pressure ratio (centre) [83, 84].
Figure 14. Comparison of theoretical performances (calculated)
of conventional and plug nozzles [81, 86, 87].
Figure 15. Typical (a) conventional nozzle, (b) truncated nozzle
and (c) full length plug nozzle [21].
Sådhanå (2021) 46:76 Page 9 of 22 76
to obtain the nozzle’s base thrust, base pressure distribution
and flow pattern. A variation of the non-dimensional total
base thrust (CFbase) with the pressure ratio (pa/pdes) for
different amount of base bleed is shown in figure 16.
Shanmuganathan et al [95] carried out the study of linear,
annular aerospike nozzles and the characteristics of its flow
field. They opined that the annular aerospike nozzle was
better than the linear aerospike nozzle.
The central plug body truncation is beneficial because it
eliminates the huge ideal length and heavy structural mass
of the well contoured body, and this truncation results in a
different flow and performance behaviour as compared to
the full length plug [3, 96]. At lesser pressure ratios, an open
wake flow is established with a pressure level equal to the
ambient pressure. At a certain pressure ratio close to the
design pressure ratio of the full length plug nozzle, the base
flow suddenly changes its character and turns over to the
closed form, characterized by a constant base pressure that
is no longer affected by the ambient pressure. Analyses
indicated that smaller plug bodies with higher truncations
leads to an earlier change in wake flow at pressure ratios
below the design pressure ratio [24]. At the point of tran-
sition, the pressure inside the wake reaches below the
ambient pressure, and the full base area generates a negative
thrust. This loss of thrust depends on the percentage of
truncation and the total size of the base area. The results
demonstrated an increase in the thrust loss for smaller plug
bodies as the total base area increases. Beyond the pressure
ratio at transition, the pressure inside the closed wake
remains constant. A further decrease of the ambient pressure
leads to a positive thrust contribution of the total base area
[24, 83]. An experimental investigation by Ruf and
McConnaughey [97] stated that a 50% truncation of the plug
nozzle resulted in only a 0.5% reduction in its performance.
In order to avoid a flat base at the vertex of plug after the
truncation, a larger cone angle could be used at the plug end.
Berman and Crimp [81] showed the performance decrement
of only 1% by using cone half angles, even up to 30� at the
plug end. The approach that is used to optimize the contour
of conventional nozzles can also be used to optimize the
plug contour. The solution of such problems was discussed
by Rao [98], where he presented typical optimum plug
contours. By the use of a plug, the reduction in length of the
nozzle can be obtained as indicated in figure 15.
As compared to other types, the major disadvantage of a
plug nozzle lies with its high cooling requirements because
of higher heat fluxes and greater surface areas to be cooled
[21, 36]. The central plug of a spike nozzle if heated to
higher values (greater than the material limitations) would
require a secondary cooling system to be installed to cool
the plug and prevent failures. Overcoming this problem
using secondary active cooling system greatly affects the
mass of vehicle. The summary of major numerical and
experimental studies of plug nozzles is shown in table 4. In
case of plug nozzles, the key areas that have drawn atten-
tion to the researchers are the optimization of the nozzles’
contour and their truncation.
6. Expansion-deflection nozzle
The expansion-deflection (E-D) nozzle is an altitude com-
pensating type rocket nozzle where the altitude compen-
sation is attained through the interaction of exhaust gas
with the atmosphere. It looks similar to a bell nozzle, but a
’pintle’ or ’centre body’ is placed at the throat and because
of that the flow is diverted towards the wall [26]. The
benefit with the centre body is that it makes the exhaust gas
to flow in a more outward direction than in bell nozzles.
Due to this, the size of the nozzle decreases considerably as
compared to the conventional nozzles of the same expan-
sion ratio. This nozzle works in closed and open wake
modes. In closed wake mode, the entire exit area of nozzle
is filled by the exhaust gases. The ambient pressure at
which the transition of wake takes place from open to
closed modes is known as the design pressure. If the
ambient pressure is further decreased, then the remaining
expansion would occur outside the nozzle just like a bell
nozzle and in that case there is no benefit of altitude
compensation. In open wake mode, the ambient pressure
controls the exit area of nozzle and the exhaust gas does not
fill the complete nozzle. As the area ratio is controlled by
the ambient pressure, the altitude compensation is achieved
up to the design pressure [99, 100].
When there is a high chamber to ambient pressure ratio
(pc/pa), the performance of the E-D nozzle is similar to that
of a plug nozzle [18]. However, when the pressure ratio is
lesser, the flow in a plug nozzle adjusts itself to the pressure
at the cowl-lip. On the other hand, in an E-D nozzle, the
flow adjusts itself to the pressure occurring behind the
central plug. The combustion chamber of the E-D nozzle is
small in size, and it has thus some benefits regarding
cooling requirements and weight. When a nozzle with veryFigure 16. Variation of total base thrust (calculated) with
ambient pressure for different base bleeds [94].
76 Page 10 of 22 Sådhanå (2021) 46:76
high expansion ratio is required, the best choice is an E-D
nozzle because of its compact sized combustion chamber
and smaller length [18]. Rao [101] studied the flow between
the walls of the combustion chamber and the plug by using
a series expansion and described the method of designing
the optimum nozzle wall contour. Results showed that for
the same performance as that of bell nozzle, the length of
an E-D nozzle required was only 50% to that of bell nozzle
length. The variation of the various nozzles thrust coeffi-
cient with the altitude is shown in figure 17 [101].
Schomberg et al [100] studied the dependency of thrust
coefficient on the E-D nozzle geometry when the flow
conditions were highly over-expanded. Results show that
the nozzle thrust depends upon the rate of change of area
through the transonic region and the linear E-D type nozzle
has shown a higher thrust as compared to conventional
nozzle when the flow conditions are highly over-expanded.The flow behaviour in open and closed modes in the E-D
nozzle is shown in figure 18 [100]. Schomberg and Olsen
[102] designed and tested the E-D nozzle at different
pressure ratios to represent operation over a range of the-
oretical altitudes (figures 19 and 20). They reported the
efficiency of E-D nozzle to be more than the conventional
CD nozzle. Wasko [103] studied the performance of plug
and E-D nozzles where the influence of base bleed were
also analysed. Results showed that the performance of full
length plug nozzle was better, whereas the performance of
the E-D nozzle was only comparable to that of a conical CD
nozzle. Schomberg et al [26] compared the linear variant of
an E-D nozzle to a conventional CD nozzle through an
experimental investigation, and demonstrated that the thrust
coefficient in the E-D type nozzle was 25–100% more than
the CD nozzle across the tested range of pressure ratios.
Schomberg et al [104] carried out numerical analysis of
E-D and CD nozzles. Results were noted over a range of
theoretical altitude and were compared to experimental
static pressure readings and schlieren images. The corre-
lation between experimental and numerical static pressure
values were found to be high for all operating conditions.
Currao et al [105] reported the use of Pressure-Sensitive
Paint (PSP) and infrared camera to a small linear asym-
metric nozzle. Their project aimed to validate numerical
results through the PSP measurements. Taylor et al [106]studied the evacuation effect in E-D nozzles. Mueller and
Hall [107] analysed the region of separated flow within a
E-D type nozzle. Taylor and Hempsell [108] carried out the
optimization of E-D nozzles for vacuum thrust. The major
experimental and numerical studies in the area of E-D
nozzles are summarized in table 5. In most of the cases,
they are related to the design of optimum wall contour of
nozzle, flow separation, and evacuation effects.
Table 4. Major studies conducted on plug nozzle.
Investigators Year Nature of study Focus of study
Krase [82] 1959 Numerical Design of ideal plug nozzle contours
Berman and Crimp [81] 1961 Analytical Modification of plug end
Rao [98] 1961 Numerical Optimization of plug contours
Johnson et al [92] 1974 Numerical Optimization for axisymmetric plug nozzles
Rommel et al [21] 1997 Numerical Flow field development at different ambient pressures
Ruf and McConnaughey [97] 1997 Numerical Truncation of plug nozzle
Ito et al [88] 2002 Numerical Designing of plug contour
Besnard et al [93] 2002 Experimental Design and testing of thrust ablative annular aerospike engine
Lahouti and Tolouei [94] 2006 Numerical External and internal flows of a truncated plug nozzle with base bleeds
Shahrokhi and Noori [90] 2010 Numerical Flow properties of aerospike nozzle shapes
Chutkey et al [89] 2014 Numerical
and Experimental
Flow fields in truncated annular plug nozzles of varying lengths
Shanmuganathan et al [95] 2015 Numerical Flow field study of linear and annular aerospike nozzles
Kumar et al [91] 2017 Numerical
and Experimental
Design procedure of aerospike nozzle
Figure 17. Variation of various nozzles thrust Performance
(calculated) with altitude (E-D: Expansion-Deflection nozzle)
[101].
Sådhanå (2021) 46:76 Page 11 of 22 76
7. Dual bell nozzle
Dual bell nozzles have been seen as a solution to maximize
the efficiency at high altitudes while eluding hazardous side
loads at lower altitudes. As can be seen in figure 21, a dual
bell nozzle consists of two different contours as opposed to
one between the throat and exit. It is composed of a base
contour that is separated from the extension contour by an
inflection in the wall. [110, 111]
In 1949, Cowles and Foster first introduced the concept
of a dual bell nozzle, and the design was patented by
Rocketdyne in the 1960s [27–29]. As the CFD capabilities
developed with time, the research activities gained attention
in the 1990s. The confirmation of the feasibility of this
nozzle was done by Horn and Fisher [27] by conducting
tests at Rocketdyne and in Europe by the Future European
Space Transportation Investigations Programme (FESTIP)
at the European Space Agency (ESA) [27]. They investi-
gated four contour combinations to find the extension
contour that provided the most favourable flow transition
characteristics and high altitude performance as compared
to the performance of two baseline contours. The perfor-
mance of the dual bell nozzles was demonstrated to be
lesser than the theoretical optimum because of the losses
linked with aspiration drag during low-altitude mode and
non-optimum contour in high-altitude mode. Horn and
Fisher found that even with such losses, a dual bell nozzle
could deliver sufficient thrust to carry 12.1% additional
payload as compared to a conventional CD nozzle having
the identical expansion ratio. The first numerical analysis of
dual bell nozzles was done by Goel and Jensen, which was
published in 1995 [112]. Throughout the 2000s, in Europe
and in the United States, various experimental and
numerical investigations of dual bell nozzles were per-
formed [24]. Modern studies focus on optimizing particular
design variables of the dual bell nozzle such as the ideal
contour and relative length of the extended section.
With the help of nozzle wall inflection, this nozzle gives
an altitude adaptation. Controlled and symmetrical sepa-
ration of flow takes place at this wall inflection at low
altitudes which leads to a less effective expansion ratio. The
flow in the nozzle is attached to wall at higher altitudes
until the full geometrical expansion ratio is utilized. An
improved vacuum performance is attained due to the higher
expansion ratio [110, 113]. However, some additional
performance losses are induced in dual bell nozzles. The
main benefits of this nozzle lie with the control of flow
Figure 18. E-D nozzle flow behaviour in a open mode and b closed mode [109].
Figure 19. Pressure distribution at a 450 kPa inlet [102].
Figure 20. Pressure distribution at a 550 kPa inlet [102].
76 Page 12 of 22 Sådhanå (2021) 46:76
separation and its design simplicity. While other altitude
compensating nozzle concepts have the limitations of
mechanical complexity, cooling difficulty, and more weight
and price tag. The dual bell nozzle provides an exceptional
blend of simplicity, lesser weight, performance and ease of
cooling [24].
Kbab et al [114] designed the profile of a dual bell nozzleand studied the flow behaviour using the method of char-
acteristics. Genin et al [115] conducted experimental and
numerical studies on a dual bell nozzle for the evaluation of
heat flux distribution. For both the operation modes (sea
level and altitude), the heat flux value was found to increase
in the region of the contour inflection. The heat flux value
increased by about 40% in altitude mode. In sea level mode
and during the sneak transition, the flow separation in the
region of the inflection increases this phenomenon. The two
operation modes of a dual bell nozzle are depicted in fig-
ure 22. Under sea level conditions, the flow separates at the
contour inflection in a controlled and symmetrical way
(figure 22a). The generation of side load is continued to
decrease and the thrust enhances due to the small area ratio.
During flight, the ambient pressure reduces, which results
in an increase in the NPR. At a particular height, the
transition NPR is reached and the separation point leaves
the contour inflection and moves rapidly to the nozzle exit.
The extension then flows full (figure 22b). The thrust is
enhanced due to the greater area ratio.
The variation of specific impulse with altitude for a dual
bell and two conventional nozzles is shown in figure 23
[115]. Schneider and Genin [116] analysed the effect of
various turbulence models and feeding pressure gradients
on the dual bell flow transition behaviour. They found
improved results for the Reynolds stress turbulence and
Spalart–Allmaras model. Genin et al [117] tested a planar
dual bell nozzle model under several test conditions in cold
and hot flows. The analysis of the shock in the region of the
contour inflection gave an idea of the shape and position of
the separation front. In sea level mode, the numerical and
experimental results were in good agreement; and for
higher NPR values, the calculated separation position was
situated further upstream than measured in the experiments.
Table 5. Major studies conducted on E-D nozzle.
Investigators Year Nature of study Focus of study
Rao [101] 1960 Numerical
and Experimental
Method of designing nozzle wall contour to yield optimum thrust
Mueller and Hall [107] 1968 Experimental Region of separated flow
Wasko [103] 1968 Experimental Effects of base bleed
Taylor and Hempsell [108] 2004 Numerical Optimization of E-D nozzles for vacuum thrust
Taylor et al [106] 2010 Experimental Analysis of the evacuation effect in E-D nozzles
Schomberg and Olsen [102] 2012 Experimental Designing and testing of E-D nozzles at different pressure ratios
Schomberg et al [100] 2014 Numerical
and Experimental
Effect of E-D nozzle geometry on calculated thrust coefficient
Schomberg et al [26] 2014 Experimental Comparison of linear variant of the E-D nozzle to a conventional
CD nozzle
Schomberg et al [104] 2014 Numerical Analysis of annular CD and E-D nozzles
Currao et al [105] 2014 Experimental Application of PSP and Infrared camera to a small linear asymmetric
nozzle
Figure 21. A typical dual bell nozzle.
Figure 22. Two operation modes of dual bell nozzle: a sea level
and b altitude mode [115].
Sådhanå (2021) 46:76 Page 13 of 22 76
Stark and Genin [118] studied the characteristics of side
load in a dual bell nozzle. Verma et al [119] did experi-
mental investigation to study the Reynolds number effect
on transition behaviour of dual bell nozzle for tests inside a
high altitude simulation chamber. Verma et al [120] furtherconducted experiments to investigate the dependency of
transition behaviour on ambient pressure fluctuations in
dual bell nozzle. Toufik et al [121] studied the design of
dual bell nozzles and evaluated several wall parameters and
performances with the help of the method of characteristics.
Davis et al [28] developed the design procedure of dual bell
nozzle contour and used it to analyse the nozzle that could
be installed on a nanosatellite launcher or a sounding
rocket. Verma et al [111] studied the unsteady flow con-
ditions occurred during sneak transition by conducting
cold-gas test on a subscale dual bell nozzle working under
sea level conditions. The results showed that the flow
during sneak transition was highly unsteady and was the
major source of side load generation. Genin et al [122]
performed the numerical and experimental studies to opti-
mize the transitional behaviour through the variation in
extension geometry. Hagemann et al [123] had done the
analytical and experimental work to explore the aerody-
namic characteristics of various dual bell nozzles. Frey and
Hagemann [112] investigated various aspects of design for
the wall inflection and nozzle extension with focus on the
dependency of transition behaviour on the type of nozzle
extension. Results showed that two different types of noz-
zle extensions (overturned and constant pressure) might
offer the sudden flow transition. Genin et al [124] investi-gated computationally the flow behaviour in a dual bell
nozzle and analysed the two operation modes and the
transition from one mode to the other by changing the NPR.
Table 6 summarizes the experimental, analytical and
numerical studies of the dual bell nozzles.
7.1 Extendable nozzles
Extendible nozzles were used as a substitute to dual bell
nozzles earlier. This nozzle is commonly known as an
extendible exit cone (EEC). It has a fixed low-area-ratio
section, which is enlarged to a higher area ratio by attaching
one or more nozzle cone extension parts. The specific
impulse of such nozzles increases by doubling or tripling
the initial area ratio. This configuration permits very high
area ratio nozzles to be bundled in a relatively short length
that helps in decreasing vehicle inert mass. [125]
8. Multi nozzle grid
The Multi-Disciplinary Optimization (MDO) procedure
[31] creates a thin and lightweight Multi Nozzle Grid
(MNG) plate (figure 24) instead of a lengthy and heavy
single nozzle. The length saving is in direct proportion to
the square root of the number of small nozzles (nozzlettes)
in the MNG (i.e., MNG with hundred nozzlettes is ten times
smaller than an equivalent single nozzle) [31–33]. Past
study reveals the overall improvement in performance via
mass saving and nozzle efficiency. The mass saving results
in substantial improvement of structural mass ratio, while
improvement in nozzle efficiency is attained by using
strategies that minimize thermodynamic losses yet increase
area ratio. Since the beginning of rocketry, the nozzle
designers mainly focused on how to increase the perfor-
mance of a nozzle without enhancing its weight and length.
The same task is just as important today with the devel-
opment of advanced materials. MNG is the only nozzle
configuration where the nozzle length (i.e., plate thickness)
to throat diameter can be less than one, yet is capable of
providing extremely high area ratio [31]. Chasman et al[31] have studied the MDO to design an innovative multi-
nozzle configuration. Later on, Chasman et al [32] have
carried out the hot-fire tests of 91 nozzlettes MNG con-
figuration characterized by exceedingly high nozzle erosion
at an average rate of 0.5 lb./sec. The high erosion of copper
infiltrated tungsten, traditionally considered being ‘‘a well-
defined refractory material’’ underlines the need to match
the structural material of the nozzle with the environment
of propellant by-products. Material experts are in the
opinion that the excessive erosion observed, occurred due
to the sensitivity of tungsten to flow of hot carbon
monoxide through the nozzlettes. Such sensitivity is inde-
pendent of nozzle configuration and would also prevail in
conventional single nozzle. Chasman et al [33] further
studied the viscous losses of MNG in hybrid motor tests.
The viscous losses in the flow through the MNG dropped
the efficiency by 3% as compared to that of an Equivalent
Single Nozzle (ESN). It can be mentioned here that a single
nozzle that shares the same scaled nozzle contour and
overall throat area as well as exit area is called an ESN. But
when compared to a conventional single nozzle (CSN) thatFigure 23. Variation of specific impulse with altitude for a dual
bell and two conventional nozzles [115].
76 Page 14 of 22 Sådhanå (2021) 46:76
are commonly used in air vehicles, the MNG system may
improve the performance by more than 11% because of its
mass and length savings. The major experimental studies in
the area of MNG are shown in table 7.
In summary, a list of rocket engines stating their appli-
cations and the type of nozzles used along with their cor-
responding nozzle area ratio are shown in table 8, while the
characteristics of all the rocket nozzles starting from the
ideal to the MNG types are shown in table 9.
The performance comparison of various nozzles is
shown in figure 25. The nozzle thrust coefficient, CF for an
ideal nozzle (in this figure, the ideal nozzle refers to a
variable area-ratio nozzle having the optimum expansion
for each chamber pressure to ambient pressure ratio, pc/pa)is shown together with those of the high area-ratio aero-
dynamic spike and bell nozzles. As is evident, the CF curve
of the aerodynamic spike follows the ideal nozzle perfor-
mance (altitude-compensation), rather than dropping off
rapidly like the bell design at low pc/pa (i.e., sea level)
operating points. All the nozzles have a higher CF at a high
pc/pa (i.e., vacuum).
9. Losses in rocket nozzles
The two important losses encountered in rocket nozzles are
summarized below:
Table 6. Major studies conducted on dual bell nozzle.
Investigators Year Nature of study Focus of study
Horn and Fisher [27] 1994 Experimental Performance characteristics of dual bell nozzles
Frey and Hagemann [112] 1999 Analytical
and Experimental
Dependence of the transition behaviour on
the type of nozzle extension
Hagemann et al [123] 2002 Analytical
and Experimental
Aerodynamic characteristics
Stark and Genin [118] 2010 Experimental Characteristics of side load
Genin et al [124] 2012 Numerical Flow behaviour in a dual bell nozzle
Genin et al [122] 2013 Numerical
and Experimental
Optimization of the transitional behaviour through variation
of extension geometry
Genin et al [115] 2013 Numerical
and Experimental
Determination of heat flux distribution.
Genin et al [117] 2013 Numerical
and Experimental
Planar dual bell nozzle under several test conditions
Verma et al [119] 2013 Experimental Reynolds number influence on dual bell transition behaviour
Verma et al [120] 2014 Experimental Influence of ambient pressure fluctuations on the dual bell transition
behaviour
Davis et al [28] 2015 Numerical
and Experimental
Dual bell nozzle contour design procedure
Verma et al [111] 2015 Experimental Unsteady flow conditions encountered during sneak transition
Schneider and Genin [116] 2016 Numerical Effect of various turbulence models and feeding pressure gradients on
dual bell flow transition behaviour
Toufik et al [121] 2016 Numerical Design of dual bell nozzle
Kbab et al [114] 2017 Numerical Design of dual bell nozzle profile
Figure 24. Schematic of Multi Nozzle Grid (MNG) [31].
Table 7. Major studies conducted on Multi Nozzle Grid (MNG).
Investigators Year Nature of study Focus of study
Chasman et al [31] 2005 Experimental MDO method for MNG design
Chasman et al [32] 2005 Experimental Nozzle erosion in a MNG
Chasman et al [33] 2012 Experimental Viscous losses of MNG in hybrid motor tests
Sådhanå (2021) 46:76 Page 15 of 22 76
Table 8. Key features of few rocket engines used in the past [43].
Rocket engines Application Nozzle used Nozzle area ratio
RS-1801 Lunar excursion module ascent engine 72% Bell 45.6
AJ10-138 Titan III transtage Bell 40
YLR99-RM-1 X15 Aircraft; Pioneer spacecraft Conical 9.8
YLR113-AJ-1 Rocket sled Conical 5.7
RL10A-3-3 Centaur upper stage Bell 57
AJ10-137 Apollo service module Bell 62.5
LR91-AJ-5 Titan II stage Bell 49.2
LR89-NA-7 Atlas MA-5 Booster engine 100% Bell 8
J-2 S-II and S-IV stages of Saturn V Bell 27.5
F-1 S-1C stage of Saturn V Bell 16
Model 8093 Centaur ACS and ullage orientation Conical 15
M2A Comsat positioning and orientation Conical 40
SE-7 Gemini attitude control 80% Bell (Scarfed) 40.2
PD6000179 Titan III transtage ACS Optimized Bell 49
SE-7-1 Saturn SIV-B ullage 80% Bell (Scarfed) 40
Table 9. Characteristics of various rocket nozzles.
Types of nozzle Characteristics
Ideal Give maximum thrust performance
Long and heavy
Conical Easier manufacturing
Easier optimization
Flow at exit is not completely axial
Requires a trade-off between divergence angle and nozzle length
Pressure thrust is not optimum at all altitudes
Bell Lesser length as compared to conical nozzle
Performance is higher as compared to conical nozzle
Manufacturing is difficult
Pressure thrust is not optimum at all altitudes
Optimization is difficult
Plug Flow separation can be avoided
Allow near optimum expansion at all altitudesu
Less noisy
Lesser size as compared to conventional nozzle
Cooling of central plug is difficult
E-D Allow near optimum expansion at all altitudes
Size is lesser as compared to conventional nozzle
Easier cooling
Combustion chamber is compact
Performance is low as compared to plug nozzle
Higher throat heat fluxes relative to a conventional bell nozzle with an
equal throat area
Dual bell Can carry more payload as compared to conventional nozzles
Controlled flow separation
Easier cooling
Higher performance and reliability
Difficult to optimize the wall contour
Lesser side loads
Lower weight
Difficult to manufacture
MNG Better performance as compared to other types of nozzles
Lesser length
76 Page 16 of 22 Sådhanå (2021) 46:76
9.1 Two-phase flow loss
Solid rocket motors (SRMs) carrying evenly mixed ener-
getic solid propellants have several benefits such as high
reliability, structural simplicity, and ease of maintenance.
Hence, such SRMs are extensively used as boosters for
spacecraft/launch vehicles. However, the co-existence of
high-energy metal additives and the oxides generated in the
chamber during the working process may lead to loss of
two-phase flow resulting in poor performance of the SRMs.
9.2 Impingement Loss
The impingement of a thruster exhaust plume on spacecraft
structures lead to loss of thrust depending on the configu-
ration of the surfaces affected. Mitigating the effects of the
interaction between rocket exhaust plumes and neighboring
structures is a perpetual problem in the spacecraft propul-
sion system design. The plume shear against the interior
cylindrical surface results in an effective thrust loss, while
the plume impingement against the cylinder base generally
has a negligible effect on the net axial thrust. [3, 127]
10. Thrust vector control in rocket nozzles
Rocket propulsion systems deliver a push in the direction of
desired end point, but such systems can also provide tor-
ques in order to rotate the vehicle in conjunction with the
propulsive force. The thrust vectors direction can be con-
trolled using the mechanisms described below. These
mechanisms can influence a vehicle’s yaw, pitch, and roll
rotations. [128–131]. The thrust vector control mechanisms
can be divided into four groups:
a. Mechanical deflection of the main nozzle or thrust
chamber [128].
b. Insertion of heat-resistant movable bodies into the
exhaust jet; these experience aerodynamic forces and
cause deflection of part(s) of the exhaust gas flow [129].
c. Injection of fluid into a side portion of the diverging
nozzle section causing an asymmetrical distortion of the
supersonic exhaust flow [130].
d. Separate thrust-producing unit that is not a part of the
main propulsion system providing flow through one or
more of its own nozzles [131].
11. Conclusion and recommendation
Rocket nozzle designers are always being challenged to
search for the resolution to use a smaller size nozzle in
harvesting a higher specific impulse while maximizing the
cost saving and simplifying the structural complexity.
Lesser weight, maximum performance, and ease of manu-
facture are some of the main desirable features of a rocket
nozzle. This review paper attempts to analyse the major
research work done in the area of rocket nozzles during the
last 75 years. Some of the salient points from the present
review work are summarized below.
• The thrust of a rocket nozzle becomes optimum when
there is a parallel uniform flow at the exit and having
the exit pressure equal to the ambient pressure, but this
makes the nozzle longer and heavier.
• A conical nozzle is easier to manufacture and has the
flexibility of converting the existing design to lower or
higher area ratio without much redesign. The thrust
coefficient of a 15� conical nozzle is only 1.7% lesser
than the ideal nozzle.
• The shock formation in a conical nozzle can be
eliminated by modifying the wall contour near the
junction of throat profile and cone. The numerical
results show that the realizable v2–f and the SST k–xmodels give better results as compared to other models
in predicting the position of separation point and the
shock wave in the conical nozzle. The major disad-
vantage lies with the trade-off between the divergence
angle and the nozzle length.
• Bell type nozzle is the most commonly used shape in
rocket engines. This type of nozzle offers significant
benefits in terms of size and performance over the
conical nozzle. The thrust loss experienced in a conical
nozzle due to the flow divergence can be decreased to
some extent by the bell nozzle. The side load measured
in the truncated ideal nozzle is about 33.33% of side
load in the thrust optimized nozzle. The use of the
parabolic approximation procedures seems to be the
easier method to optimize the bell nozzle contour.
• The length of a plug nozzle is significantly lesser than
an equivalent CD nozzle. These types of nozzles are
meant for altitude compensating. The main property of
a plug nozzle is its interaction with the external
ambient that is capable to eliminate the flow separa-
tion. Plug nozzle shows a thrust benefit over a
Figure 25. Nozzle performance comparison (calculated) [126].
Sådhanå (2021) 46:76 Page 17 of 22 76
conventional CD nozzle when it is working below
design pressure ratio. The thrust performance of a
contoured plug nozzle is around 5–6% more than the
conical plug nozzle. A 50% truncation in plug nozzle
results in only a 0.5% loss in its performance. The
major disadvantage of such nozzles lies with its
relatively high cooling requirements.
• The length of an E-D nozzle is approximately the same
to that of a plug nozzle for an equivalent thrust
performance. For the same performance as that of a
bell nozzle, the length of an E-D nozzle required is
only 50% to that of a bell nozzle. The thrust coefficient
of an E-D nozzle is 25-100% more than the CD nozzle.
As the combustion chamber of a rocket engine with an
E-D nozzle is compact in size, it has certain benefits
regarding cooling requirements and weight. When
there is a need of a very high expansion ratio, the best
choice is the E-D type nozzle.
• Dual bell nozzles have the advantage of maximum
efficiency at high altitudes while no side loads at low
altitudes. This nozzle provides sufficient thrust to carry
12.1% more payload as compared to a conventional
CD nozzle having the same expansion ratio. The main
benefits of a dual bell nozzle are its ability to control
flow separation, besides its design simplicity and high
reliability. The heat flux value in a dual bell nozzle
increases in the region of the contour inflection for
both the operation modes. The flow during sneak
transition is highly unsteady and is the major origin of
side-load generation in the dual bell nozzle preceding
the final transition. Other altitude compensating noz-
zles have the limitations of mechanical complexity,
cooling difficulty, and more weight and price tag. The
dual bell nozzle provides an exceptional amalgam of
performance, simplicity, lesser weight, and ease of
cooling.
• In MNG, the length saving is in direct proportion to the
square root of the number of nozzlettes. This is the
only nozzle configuration where the nozzle length (i.e.,
plate thickness) to throat diameter can be less than one,
yet is capable of providing extremely high area ratio.
Viscous losses in the MNG drop the efficiency by 3%
as compared to that of the ESN. However, as compared
to a CSN, the MNG system may improve the
performance by more than 11% because of mass and
length savings.
As evident from the above comprehensive literature
review, each design has its specific strengths and weak-
nesses, and there is no design that stands out as clearly
superior to the others. Rather, the selection of a particular
design results from a careful trade-off between different
factors such as application, performance, reliability,
mechanical complexity, manufacturability, weight, and
cost. The ‘winning’ concept from such a trade-off is further
influenced by how these factors are ranked in order of
importance. Based on the results presented in this review
and a look at which designs have received greatest attention
till date, the dual bell and MNG nozzles seems to be the
two most attractive candidates; the dual bell nozzle because
of its simplicity, and the MNG nozzle due to its lesser
length.
Acknowledgements
The authors wish to place on record their sincere gratitude
to all the authors of classical and popular papers/reports/
theses which formed the framework of the present review
work. Appreciation is extended to all the sources of
figures and data used in this work. A list of these sources
has been included in the references, and the authors
apologize if due recognition to any source is left out
inadvertently.
List of symbolsAe Nozzle exit area (m2)
At Throat area (m2)
Cfv Vacuum thrust coefficient
Cfvc Conical nozzle vacuum thrust coefficient
Cfvi Ideal nozzle vaccum thrust coefficient
CFbase Total base thrust
CF Thrust coefficient
CF,X Axial thrust coefficient
CFi Ideal thrust coefficient
Dt Throat diameter (m)
Fv Thrust when discharging to vaccum
HN Nozzle projected height (m)
Isp Specific impulse (sec)
L Nozzle length (m)
Ln Nozzle axial length from throat to exit plane (m)
LN Nozzle projected length (m)
Lf Desired fractional length (m)
pa Ambient pressure (Pa)
pc Chamber pressure (Pa)
pdes Design pressure (Pa)
pe Pressure at the exit (Pa)
pt Stagnation pressure (Pa)
pw Wall pressure (Pa)
R Radius of circular arc (m)
Rt Throat radius (m)
Ve Nozzle exit velocity (m/s
Greek symbolsa Divergent cone half-angle (deg.)
qe Density at the exit of nozzle (kg/m3)
c Isentropic exponent
e Nozzle expansion ratio or area ratio
h Nozzle convergent cone section half-angle (deg.)
he Nozzle exit wall angle (deg.)
hn Initial wall angle of the parabola (deg.)
76 Page 18 of 22 Sådhanå (2021) 46:76
AbbreviationsAR Area ratio
CD Converging-diverging
CFD Computational fluid dynamics
CSN Conventional single nozzle
E-D Expansion-deflection
EEC Extendable exit cone
ESA European space agency
ESN Equivalent single nozzle
FESTIP Future European space transportation
investigations programme
FSS Free shock separation
MDO Multi-disciplinary optimization
MNG Multi nozzle grid
MOC Method of characteristics
NASA National aeronautics and space administration
NPR Nozzle pressure ratio
PSP Pressure-sensitive paint
RSS Restricted shock separation
SRM Solid rocket motor
SSME Space shuttle main engine
SST Shear stress transport
TIC Truncated ideal contour
TOC Thrust optimized contour
TOP Thrust optimized parabola
1D One dimensional
2D Two dimensional
3D Three dimensional
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