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Design and Numerical Analysis of Aerospike Nozzles with Different Plug
Shapes to Compare their Performance with a Conventional Nozzle
Conference Paper · March 2005
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AIAC-11 Eleventh Australian International Aerospace Congress
WC0023
Design and Numerical Analysis of Aerospike Nozzles
with Different Plug Shapes to Compare their
Performance with a Conventional Nozzle
Mehdi Nazarinia 1, Arash Naghib-Lahouti 2 and Elhaum Tolouei 1
1
Senior Research Engineer
2
Faculty Member
Aerospace Research Institute, P.O. Box 14665-834, Tehran, Iran
Tel: +98-21-8362010, Fax: +98-21-8362011
E-mail: mnazarinia@ari.ac.ir
Summary: The objective of the present study is to demonstrate thrust advantage of a small
axisymmetric aerospike nozzle in comparison with the equivalent conventional (convergentdivergent) nozzle through numerical simulation, and to study the effects of geometrical
parameters on the aerospike’s performance. Internal and external flow of the aerospike nozzle
and the equivalent conventional nozzle has been numerically analyzed in different ratios of
ambient pressure to design exhaust pressure (Patm/Pdes) representing optimum, under- and
over-expansion working conditions. Results indicate that the aerospike nozzle can deliver as
much as 5% more thrust in over-expansion conditions. To address practical aspects of using
an aerospike nozzle, effects of geometric parameters including base curvature, and different
values of plug truncation ranging from 0 to 75%, on thrust and base temperature distribution
of the nozzle have been studied. Results show that higher values of base truncation are more
appropriate for flight at altitudes higher than design altitude (under-expansion condition),
while lower values are more appropriate for flight below design altitude. It has also been
shown that adding curvature to the base of a truncated plug can improve local temperature
distribution without any significant effect on thrust.
Keywords: nozzle, aerospike, truncation, design, thrust, CFD.
Nomenclature
CF
CP
F
h
Isp
L
&
m
M
M
P
PR
ℜ
T
r
x
thrust coefficient
specific heat at constant pressure (J/kgK)
thrust (N)
altitude (m)
specific impulse (s)
length (m)
mass flow rate (kg/s)
Mach number
molecular weight (g/mol)
absolute static pressure (N/m2)
Pressure Ratio (= Patm / Pdes)
Universal gas constant, 8314 J/kg.mol.K
absolute static temperature (K)
radial coordinate
axial coordinate
Eleventh Australian International Aerospace Congress
Sunday 13 – Thursday 17 March 2005
Fifteenth National
Space
Engineering
Melbourne,
Victoria,
Australia Symposium
Greek Symbols
γ
ratio of specific heats, Cp/Cv
Subscripts
1
stagnation
atm
atmosphere (external flow)
b
base
des
design
e
exhaust
i
inlet
mom momentum
p
pressure
t
throat
Introduction
Expansion and discharge of a gas in different propulsion systems, e.g. jet engines and rockets,
is always accomplished by a nozzle. Thrust of a conventional nozzle with fixed geometry,
discharging in atmosphere can be expressed by the following simple relation:
& Ve + (Pe − Patm )A e
F=m
(1)
This relation indicates that for a nozzle designed to have a constant value of Pe (also known as
design exhaust pressure, Pdes), thrust is affected by change of altitude. At the design altitude,
where Patm = Pdes, the second term of the above relation (known as pressure thrust) is zero, and
the nozzle is said to be working in “optimum condition”. At altitudes lower than the design
altitude, where Patm > Pdes, pressure thrust assumes a negative value, and loss of thrust is
inevitable. These conditions, which occur at altitudes ranging from ground level to the design
altitude, are known as “over-expansion” conditions. Beside the inherent loss of thrust, the
conventional nozzle might suffer problems including shock waves and flow separation in the
divergent section, thrust oscillation, and flow asymmetry in over-expansion conditions.
Ever since jet and rocket propulsion systems have emerged, researchers have invented and
implemented many types of nozzles, mainly to increase the thrust performance of nozzles in
off-design working conditions. Among these various designs, features of the aerospike nozzle
have attracted researchers since mid-1950s [1]. Many theoretical studies of the aerospike
nozzle have been carried out in 1960s. Berman and Crimp’s work [2] is an example of these
studies, in which issues such as analytical design methods, thrust vectoring, and integration
with solid- and liquid-propellant systems have been addressed. Rao [3] has presented a more
accurate method based on calculus of variations for design of the plug in 1961, and Lee and
Thompson [4] have developed the first computer program for plug nozzle design based on
Rao’s work in 1964. In early 1970s, thermal and strength problems of the aerospike nozzle
and development of more efficient methods for fabrication of conventional nozzles led to a
decline in research activities in this field.
In 1990s, NASA initiated the SSTO (Single Stage to Orbit) project, which required a
propulsion system with maximum efficiency in a broad range of working altitudes. The
aerospike propulsion system was selected for this purpose, and extensive research and
development led to successful testing of the RS2200 aerospike engine [5]. Cancellation of this
project in 2001 has led to another decline in research activities in this field in the United
States in the recent years, but the aerospike nozzle is still a live research topic in Europe and
Japan. Hageman et al. [6, 7] have proposed application of a large scale aerospike nozzle in the
post-Ariane 5 launch vehicles to DLR. Tomita et al. [8] and Sakamoto et al. [9] have carried
out experimental studies of axisymmetric and linear aerospike nozzles, and Fujii and Ito [10]
have studied many aspects of aerospike nozzles numerically.
Most recently, two different groups in the United States have applied the axisymmetric
aerospike nozzle in propulsion system of sounding rockets, and demonstrated considerable
gain in the rockets’ performance [11, 12].
The aerospike nozzle is considered to have better overall performance compared to the
conventional bell nozzle since expansion of the jet is not bounded by a wall and the exhaust
flow can adjust itself to the environment by changing the jet boundary. In addition, the nozzle
performance is considered not to be influenced by the cutting off the nozzle because the base
pressure compensates the loss of thrust force. The base pressure can also be increased by
injecting a secondary flow at the base which may be tapped off the exhaust flow and injected
at the base [13]. For detailed discussion on the off-design performance comparisons of
aerospike and conventional nozzles, the reader is kindly referred to reference [14].
Despite its remarkable advantage over a conventional nozzle, practical application of an ideal
aerospike nozzle has structural and thermal problems, especially because of the sharp end of
the plug. These problems can be averted to a great extent by truncating the base of the plug.
Base truncation reduces weight and length of an aerospike nozzle, and increases its strength,
thus facilitating practical application of this kind of nozzle. In this paper, effect of different
values of base truncation on thrust performance of an aerospike nozzle has been studied in
various working conditions. To further understand the mechanism by which base flow affects
performance of the nozzle, variation of pressure and thrust in this area has been studied in
more detail. Effects of rounding the base of a truncated plug on flow pattern, thrust
performance, and temperature distribution are also studied.
Numerical flow simulation
This section describes numerical modeling and analysis of internal and external flow of the
aerospike nozzle with different plug shapes, and the equivalent convergent-divergent nozzle
using a commercial CFD code. The objective of the analysis is comparison of thrust produced
by aerospike nozzles with different plug shapes with a conventional nozzle in off-design
working conditions.
Geometry
Two methods, namely the method of characteristics [15] and an approximate method based on
ideal rocket assumptions [16], have been used to design the central plug of the aerospike
nozzle, which have resulted in almost identical plug shapes. Schematic of the axisymmetric
aerospike nozzle, displaying the characteristic lines representing Prandtl-Meyer expansion
waves is shown in Fig. 1. The two methods are implemented to design an aerospike nozzle in 4-5
kN thrust class, with specifications originally introduced in [11], assuming the following design
parameters:
Chamber Pressure:
Design Altitude:
Optimum Thrust:
Specific Impulse:
Mass Flow:
Propellant:
P1 = 2067857 N/m2
h = 3657.6 m
F = 4443.9 N
Isp = 235 s
& = 3.25758 kg/s
m
Ethanol-Oxygen, γ = 1.21
The following specifications are also calculated for the nozzle using the relations derived in
the approximate method [14]:
Throat Angle:
Throat Area:
Throat Radius:
Exhaust Area:
Exhaust Radius:
Exhaust Mach number:
θt = 57.1665°
At = 1.438569×10-3 m2
rt = 0.045221 m
Ae = 7.04612×10-3 m2
re = 0.047888 m
Me = 2.80434
Fig.1: Schematic of the axisymmetric aerospike nozzle, displaying the
characteristic lines representing Prandtl-Meyer expansion waves.
Details of the design process by both methods can be found in [14]. Results of the two
methods show a maximum difference of 5% in radius. In the following sections, results of the
approximate method are used as the baseline for numerical flow simulation due to their better
agreement with the original plug shape. Fig. 2 shows geometry of the aerospike and the
conventional nozzle and the preceding convergent sections.
(a) Aerospike nozzle (inset: plug contour)
(b) Equivalent conventional convergent-divergent nozzle
Fig. 2: Nozzle Geometry
Two methods were also examined to define the truncated plug shapes: implementation of the
relations of the approximate method to design a whole new truncated plug, and cutting off the
ideal aerospike nozzle at different stations of the central plug without altering the remaining
portion.
Exhaust and throat area of the aerospike nozzle is calculated using the following relations in
the approximate method (Fig. 1):
2
2
A e = p re − rb
(2)
At
(
p (r
=
− rt
Cos? t
e
2
2
)
)
(3)
Considering Eqns 2 and 3, it can be concluded that for constant and specified values of throat
area and expansion ratio (e=Ae/At), increasing base radius causes the exhaust radius to
increase, thereby increasing the throat radius. Fig. 3 shows the mentioned effect for an
aerospike nozzle with different values of plug truncation of 0% (ideal), 75%, 50%, and 25%.
It is clearly seen in the figure that for low values of truncation (25% and 50%) the increase in
radius of the aerospike nozzle is more pronounced. Considering restraints imposed by
combustion chamber and the vehicle’s radius, the application of such truncated aerospike
nozzles may not be practical.
Fig. 3: Trend of aerospike nozzle radius increase for different plug shapes
with constant and specified values of At and e
Therefore, in this study in order to determine the profile of truncated aerospike nozzles with
lengths of 75%, 50%, and 25% of the ideal aerospike nozzle, 25%, 50%, and 75% of the ideal
plug were cut off respectively in a way similar to [10]. As for the remaining portions, instead
of redesigning the aerospike nozzle contour, these parts remained unchanged to prevent
changes in exhaust and throat radius. Geometric specifications of the truncated nozzles are
presented in Table 1.
Table 1: Geometric specifications of truncated aerospike nozzles
% of
plug length
100
75
50
25
Plug length
(m)
0.127530
0.0942975
0.0628650
0.0314325
Base radius
(m)
0.004363
0.011551
0.020910
Unlike a conventional nozzle, in which the throat (where M=1) remains at the same
longitudinal position in all working conditions (optimum, over-, and under-expansion), the
throat of an aerospike nozzle covers a different angular domain for each ratio of Patm/Pdes.
Therefore, in modeling of an aerospike nozzle, the convergent section should also be included
in the solution domain, and the throat should be allowed to establish by itself, rather than
imposed at a fixed location. To achieve this objective, the solution domains for both nozzles
analyzed herein include a convergent section with a length of L=0.2m and the following inlet
properties:
Ai = 7.212855×10-3 m2 (Ai/At = 5.01384)
Mi = 0.13
Pi = 0.98984 P1 = 2045430 N/m2
Authors’ experience [17] has also shown that to satisfy the convergence criteria, far-field
boundaries of the solution domain should have a distance of at least 40re from the nozzle’s
exhaust surface. Considering axial symmetry of the problem, numerical solution is carried out
in half of the entire domain, bounded by a boundary with axial symmetry condition.
Grid
The solution domain has been divided into 4 regions (labeled Fluid 1-4 in Fig. 4), to facilitate
definition of different initial conditions in each region. In case of the conventional nozzle,
regions 1-3 (which include convergent and divergent sections of the nozzle) are discretized
using a structured grid of quadrilateral cells, while in region 4 (the surrounding domain) an
unstructured grid of triangular cells has been used for discretization. The resulting grid has
50767 cells (Fig. 5a). In case of the aerospike nozzle, geometric variations in regions 1 and 2
make it impossible to generate a structured grid with acceptable quality. Therefore, the entire
solution domain is discretized by an unstructured grid of triangular cells. The total number of
cells for the ideal aerospike nozzle a finer grid with 340544 cells has also been generated to
study the effect of grid size on results (Fig. 6), which is shown in Figure 5b, is 85136.
Different geometries of the truncated aerospike nozzles cause region 2 of the solution domain
(Fig. 5) to have different number of cells. Total number of grid cells for the 75%, 50%, and
25% cases are 90582, 93334, and 95892, respectively, which are comparable with the grid
cells of the ideal aerospike nozzle (85136).
As it was mentioned previously, adding curvature to the base of a truncated plug can improve
local temperature distribution without any significant effect on thrust. To analyze this effect, a
similar aerospike nozzle with a 50% truncated plug with rounded base was analyzed in the
same working conditions as the three other cases. In this case, the straight line defining the
base was replaced with a circular arc tangent with the contour of the remaining portion of the
plug (Fig. 7). Fig. 5f shows the grid used for the 50% plug rounded end with the total number
of grid cells of 95374.
Fig. 4: Solution domain and boundary conditions
Fluid properties
Average properties of combustion products resulting from chemical reaction of ethanol and
oxygen have been assumed for the fluids in all regions. These properties, which satisfy the
following relation:
CP =
are as follows:
γ = 1.21, CP = 1286.68 J/kgK, M = 37.23 g/mol.
?ℜ
? (? − 1)
(4)
(a) Conventional nozzle
(b) Ideal Aerospike nozzle, 100% plug length
(c) Truncated aerospike nozzle, 75% plug length
(d) Truncated aerospike nozzle, 50% plug length
(e) Truncated aerospike nozzle, 25% plug length
(f) Truncated aerospike nozzle, 50% plug
with rounded base
Fig. 5: Grid for analysis of conventional and aerospike nozzles with different plug shapes,
displayed in vicinity of the plug
Fig. 6: Comparison of the results of numerical analysis using
the original and refined grids for the ideal aerospike nozzle
Fig. 7: Geometry of the 50% truncated aerospike nozzle
with rounded base
Boundary conditions
The condition implied at the inlet of the convergent section is inlet with specified mass flow,
with the following boundary values:
& = 3.25757 kg/s
m
T0 = 1577.826 K
P = 2045430 N/m2
Seven different cases of atmospheric conditions (Patm/Pdes) have been analyzed, which
correspond to different working conditions (including over-expansion, optimum and underexpansion). Boundary values imposed at pressure farfield and pressure outlet boundaries in
the 7 cases are presented in Table 2. Apart from case 4, which corresponds to the nozzles’
optimum working condition as introduced in [11], the other cases have been chosen
hypothetically to represent under-expansion (cases 1-3) and over-expansion (cases 5-7)
conditions. Supersonic Mach numbers of external flow have been selected for all cases in
order to facilitate satisfaction of the convergence criteria.
Table 2. Values imposed at farfield boundaries in
different analysis cases
M
T (K)
T0 (K)
&DVH Patm/Pdes P (N/m2)
1
4.00
257736
1.5
288.15
356.04
2
3.10
200000
1.5
288.15
356.04
3
1.57
101325
1.5
288.15
356.04
4
1.00
64434
2.80434 264.378 482.689
5
0.41
26415.4
2.5
223.12
369.54
6
0.19
12037.1
3.0
216.61
421.31
7
0.10
6410
3.0
216.61
421.31
Initial conditions
Numerical values of flow variables vary greatly in different regions of the solution domain in
analysis of a nozzle with external flow. For example, while values close to stagnation
properties prevail at the inlet of the convergent section, the external flow might involve
substantially lower pressures and higher (even supersonic) Mach numbers. In such
circumstances, initialization of a flow variable with a constant value throughout the entire
domain can make convergence difficult or sometimes impossible. To deal with this problem,
properties at nozzle inlet, throat and exhaust surface (which can be roughly estimated using
one dimensional isentropic flow relations [18]) have been used to define custom field
functions describing initial values of axial velocity, pressure and temperature in Fluid 1-3
regions. The initial state of the solution domain for the 75% truncated aerospike nozzle, which
is generated using the above-mentioned functions, is displayed in Fig. 8, as an example.
Fig. 8: Contours of Mach number at the initial state of the solution domain
for the 75% truncated aerospike nozzle
Analysis features
Combustion products have been assumed to behave as a viscous (laminar) and compressible
ideal gas (P=ρRT). The coupled implicit method described in [19] has been used for solution
of the four governing equations (continuity, conservation of momentum in longitudinal and
radial directions, and conservation of energy), considering severe compressibility effects
existing in the solution domain. Fluxes of convected variables at cell walls are approximated
by the first order upwind scheme. Courant’s number has been set to 0.7 for cases 5, 6 and 7,
and to 2.0 for the other cases. Two criteria have been posed for convergence. One is reduction
of the global residual of solution of all governing equations to the order of 10-5, and the other
is establishment of mass balance between inlet, far-field and outlet boundaries, which is
checked by integration of mass flow through the mentioned boundaries at each iteration. In all
cases, the solution process has been continued until both criteria have been satisfied.
Results and discussion
In this section, flow pattern of the ideal aerospike nozzle in different working conditions is
compared to that of the truncated aerospike and conventional nozzles in this section.
Performance of all the nozzles in off-design conditions is also compared by calculation of
thrust for each case.
Qualitative study of the results
Exhaust flow of the aerospike nozzle is characterized by formation of a series of expansion
waves, which originate from the upper lip of the convergent section (Fig. 1). Since the exhaust
flow is not bounded by a solid wall, these expansion waves can adjust their intensity and
domain to match the exhaust flow with the external flow.
For the ideal aerospike, in over-expansion conditions, the domain covered by these waves
ends before the end of the plug. At that station, flow properties are close to those of the
optimum condition, which usually involve a higher Mach number and a lower pressure
compared to the external flow. From this station onwards, flow encounters reflection of the
expansion waves in form of a series of compression waves, which increase the pressure and
reduce Mach number to a value close to that of the external flow. The above-mentioned
process can be noticed in Fig. 9a in form of contours of static pressure for case 1, where
Patm/Pdes = 4.00. Effect of these expansion and compression waves on pathlines (Fig. 9b) is so
that the exhaust flow leaves the exhaust surface straight and without further contraction and
residual radial velocity.
But for the truncated aerospike nozzles, the situation is such that the expansion waves originated
from the upper lip of the convergent section will face the truncated portion of the plug, while in
the ideal case these expansion waves meet the plug surface. The flow facing the truncation first
encounters a sharp expansion, then by continuing its way to the centre of the plug base a
compression, stagnating exactly at the centre of the plug base. This phenomenon is due to the
formation of two symmetric vortices in the base of the plug (Fig. 10), which counteract the effect
of each other at two locations, one of which is located at the centre of the plug base, where flow
conditions change into the stagnation conditions. It should be pointed out that regardless of the
amount of truncation and the extent of the plug base area, the flow parameter distribution pattern
is the same. Figs. 9c, 9e, and 9g clearly show the above-mentioned process in form of contours
of static pressure for different plug shapes. Effect of these vortices and also the expansion and
compression waves on pathlines is also seen in Figs. 9d, 9f, and 9h.
The exhaust flow of a conventional nozzle, on contrary, does not have the chance to adapt
itself to the condition prevailing outside the nozzle before leaving the nozzle. After leaving
the nozzle, flow is compressed through a series of compression waves originating from the edge
of the exhaust surface, which resemble converging shock waves. These compression waves
contract the flow, and impose a radial velocity component, which contributes to loss of thrust in
over-expansion conditions [14].
(a) Contours of static pressure, 100% plug length
(b) Pathlines, 100% plug length
(c) Contours of static pressure, 75% plug length
(d) Pathlines, 75% plug length
(e) Contours of static pressure, 50% plug length
(f) Pathlines, 50% plug length
(g) Contours of static pressure, 25% plug length
(h) Pathlines, 25% plug length
Fig. 9: Flow pattern of the ideal and truncated aerospike nozzles in
over-expansion conditions (Patm/Pdes = 4.00)
Fig. 10: Flow pattern at the base of a truncated nozzle
For an aerospike nozzle in optimum condition, the domain covered by expansion waves ends
right at the end of the plug. At that station, flow properties match those of the external flow,
and no considerable further expansion or compression are encountered (Fig. 11a). Pathlines
also indicate that the flow leaves the exhaust surface without any radial component (Fig. 11b).
But for the truncated aerospike nozzles, once again the exhaust flow will face the truncated
portion of the plug before expanding to optimum values. Flow pattern in the optimum
condition is the same as over-expansion condition, explained in the preceding paragraphs.
Figs. 11c, 11e, and 11g show the expansion process in form of contours of static pressure for
different plug shapes. Effect of the base vortices and also the expansion and compression waves
on pathlines is also seen in Figs. 11d, 11f, and 11h. It should be noted that in spite of existence
of rotational flow at the base area, pathlines will continue their way parallel to the axis of the
plug after the longitudinal position corresponding to the end of a virtual ideal plug, even for
the 25% truncated aerospike nozzle (Fig. 11h).
For the conventional nozzle in optimum condition, expansion waves inside the divergent
section affect the flow so that properties at the intersection of nozzle axis and the exhaust
surface match those of the external flow [14].
(a) Contours of static pressure, 100% plug length
(b) Pathlines, 100% plug length
(c) Contours of static pressure, 75% plug length
(d) Pathlines, 75% plug length
(e) Contours of static pressure, 50% plug length
(f) Pathlines, 50% plug length
(g) Contours of static pressure, 25% plug length
(h) Pathlines, 25% plug length
Fig. 11: Flow pattern of the ideal and truncated aerospike nozzles in
optimum conditions (Patm/Pdes = 1.00)
For an aerospike nozzle in under-expansion conditions, the domain of the expansion waves
covers an area larger than the plug itself, and the flow continues to expand after the plug.
Effect of these waves on the exhaust flow can be seen in Fig. 12a in form of contours of static
pressure for case 7, where Patm/Pdes = 0.10. As seen in Fig. 12b, pathlines also keep diverging for
some distance after the plug.
For truncated plug nozzles in under-expansion, flow pattern at the plug base and pathlines are
same as those seen in over-expansion and optimum conditions. It should be mentioned that parts
of the expansion waves encounter the rotational base flow, hence drifting the base vortices,
especially the second stagnation point seen in Fig. 9, away from the nozzle axis. Figs. 12c, 12e,
and 12g show the process in form of contours of static pressure for different plug shapes. Effect
of base vortices and also the expansion and compression waves on pathlines is also seen in Figs.
12d, 12f, and 12h.
In case of the conventional nozzle in under-expansion conditions, expansion of the flow is
continued outside the nozzle through a series of expansion waves originating from the edge of
the exhaust surface. This expansion causes pathlines to diverge before adapting to the external
flow, in a manner similar to the aerospike nozzle [14].
(a) Contours of static pressure, 100% plug length
(b) Pathlines, 100% plug length
(c) Contours of static pressure, 75% plug length
(d) Pathlines, 75% plug length
(e) Contours of static pressure, 50% plug length
(f) Pathlines, 50% plug length
(g) Contours of static pressure, 25% plug length
(h) Pathlines, 25% plug length
Fig. 12: Flow pattern of the ideal and truncated aerospike nozzles in
under-expansion conditions (Patm/Pdes = 0.10)
Effect of aerospike base truncation on thrust performance
Thrust is the best measure of performance for a nozzle intended for use in a propulsion
system. Thrust of an axisymmetric convergent-divergent nozzle is calculated by the following
integral on the exhaust surface:
F = ∫ ?Vx2 dA + ∫ (P − Patm )dA = Fmom + FP
(5)
It is clearly seen that the second term, known as pressure thrust, will be negative in overexpansion conditions, causing a considerable loss of thrust [14].
For an axisymmetric aerospike nozzle, thrust is calculated using the following relation:
F = ∫ ( ρVdA)Vx + ∫ (P − Patm )Cos? t dA + ∫ (P − Patm )Sin? t dA = F1 + F2 + F3
(6)
The first two terms are integrated on the throat of the convergent section, while the third term
is integrated on the entire plug surface. This relation shows that the pressure component of
thrust of the aerospike nozzle never assumes a negative value in over-expansion conditions.
Therefore, loss of thrust of an aerospike nozzle is always less than the conventional nozzle in
similar conditions. Figure 13 shows that the ideal aerospike nozzle can deliver as much as 5%
more thrust in over-expansion conditions in comparison with the conventional nozzle [14].
Effect of base truncation can also be seen in Fig. 13. The thrust calculation procedure is the
same as that of the ideal aerospike nozzle, noting that the term F3, also contains the thrust
produced by the base as well. It should be pointed out that according to Fig. 13, base
truncation, contrary to what may be expected, does not have any significant effect on thrust in
under-expansion conditions. This graph shows that for under-expansion conditions, thrust of
all nozzles are similar to each other. But in over-expansion conditions the 25% plug has thrust
equal or less than the conventional nozzle. In these conditions the 75% and 50% plugs have
less thrust compared to the ideal aerospike nozzle, but present better overall performance than
the conventional nozzle.
Fig. 13: Variation of thrust delivered by different nozzles studied herein
with external flow pressure
In order to understand the reason why thrust delivered by different truncated aerospike nozzles
remains unchanged in under-expansion conditions but differs in over-expansion conditions, it
is necessary to approach the thrust components differently. Dividing thrust into the three
following components, explains this phenomenon more clearly:
1) Thrust produced by the nozzle convergent section (F1+F2)
2) Thrust produced by the plug surface (part of F3)
3) Thrust produced by the plug base (other part of F3)
Fig. 14 shows the contribution of the above-mentioned components to the total value of thrust
for under- and over-expansion conditions. In under-expansion conditions, when the plug is
truncated, its lateral area decreases. Therefore the pressure thrust produced by the plug
reduces. On the other hand, the thrust generated by the base region increases because of the
increase of the base area. These two effects compensate each other, and the total nozzle thrust
becomes almost the same for different nozzle truncation. This effect can be seen most clearly
for the 25% plug.
But in over-expansion conditions, the situation is totally different. In these conditions, as the
nozzle length becomes shorter, hence decreasing the plug area, thrust produced by the plug
still decreases, while as the atmosphere pressure is higher than the exhaust pressure thrust
produced by the base pressure would have a negative value. So by increasing truncation, the
negative value of base thrust will increase, hence decreasing total thrust in over-expansion
conditions. Fig. 14b shows that for the 25% plug, total thrust is lowest.
Conv+Plug+Base
Conv
Conv+Base
0.7
Plug
Base
Conv
1
0.9
0.6
0.8
0.5
F / Fdes
0.7
F/Fdes
0.6
0.4
0.5
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0
25
50
75
100
0
25
50
75
%Plug
%Plug
(a) Patm / Pdes = 0.10
(b) Patm / Pdes = 4.00
100
Fig. 14: Proportion of the thrust for each truncated nozzle
To describe the reason with more detail, average base pressure for the 75, 50 and 25% plug
nozzle is plotted versus external flow pressure in Fig. 15. The solid line represents the
atmospheric pressure which decreases as the pressure ratio (Patm/Pdes) decreases or the altitude
becomes higher. At low altitudes (i.e., over-expansion conditions), base pressure linearly
increases as atmospheric pressure increases. Therefore, the pressure thrust which is produced
by the pressure difference between the atmosphere and the base becomes small and even
negative in most cases at low altitudes.
On the other hand, at high altitudes, pressure at the base remains constant despite variation of
altitude. As altitude is increased, atmospheric pressure decreases and the difference between
base pressure and atmospheric pressure increases, hence increasing the base thrust.
10
6
Base Pressure
Plug-75%Base P
Plug-50%Base P
Plug-25%Base P
Patm
10
5
10
4
103
1
2
3
4
P atm/P des
Fig. 15: Variation of the averaged base pressure with the external flow pressure
To compare behavior of the portion of thrust produced by the plug’s base pressure (base
thrust), the non-dimentionalized base thrust coefficient is defined as follows:
FBase
=
CF
Base 1 ?P M 2 S
e Base
2 des
(7)
This coefficient is plotted against Patm/Pdes for different values of plug truncation in Fig. 16.
This figure indicates that base thrust, which has a positive value in under-expansion
conditions, decreases as atmospheric pressure is increased. This decrease continues in overexpansion condition for the 25% plug, explaining its lower thrust performance in these
conditions. For the 50% and 75% plugs, base thrust begins to increase again beyond a certain
value of atmospheric pressure, and even assumes a positive value in case of the 75% plug.
This behavior explains the superior performance of plugs with lower amount of truncation in
over-expansion conditions.
0.5
Plug-75%CFBase
Plug-50%CFBase
Plug-25%CFBase
0.4
0.3
0.2
CFBase
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
1
2
3
4
Patm/P des
Fig. 16: Variation of the base thrust coefficient with the external flow pressure
Effect of base curvature of a truncated plug on thrust, and temperature distribution
As it was previously mentioned, adding curvature at the base of a truncated plug can improve
local temperature distribution without any significant effect on thrust. To show this, the
rounded base nozzle introduced in Figs. 5f and 7 was analysed in the same working conditions
as the other nozzles.
Fig. 17 shows, thrust of the 50% rounded base and truncated plug nozzles against external
flow pressure. It can be seen that there is no difference in thrust performance of a rounded
base and a truncated plug nozzle. Therefore, adding curvature to the base has a negligible
effect on thrust of the nozzle.
1.1
1.05
F/Fdes-50%
F/Fdes-50%-Rounded
1
F/Fdes
0.95
0.9
0.85
0.8
0.75
0.7
0
1
2
3
4
P atm/P des
Fig. 17: Comparison of the variation of thrust with external flow pressure
for a 50% rounded and truncated plug
However, this modification can have a positive effect on base temperature distribution. Table
3 compares average values of base temperature for the 50% rounded base and truncated plugs,
calculated by integration of base surface temperature distribution. This table shows that
average base temperature of the rounded base plug is considerably lower than that of the
truncated plug, hence making the former more attractive from fabrication and structural point
of view.
Table 3. The averaged base temperature of the 50% rounded
and truncated plug
Plug
Patm/Pdes
0.10
4.00
Rounded
Truncated
945.38 K
1128.80 K
1377.91K
1435.01 K
Conclusion
Flow structure and the performance of truncated aerospike nozzles in off-design conditions
has been compared with conventional convergent-divergent and ideal aerospike nozzles with
the same design values, by numerical analysis. The results clearly indicate that the aerospike
nozzle is capable of generating as much as 5% more thrust in over-expansion conditions. This
amount of excess thrust is particularly valuable for a launch vehicle, where it can contribute to
a remarkable advantage in the vehicle’s mass performance. As the base pressure thrust
compensates the loss of thrust in under-expansion conditions, plug truncation has minor effect
on the loss of thrust in these conditions. But in over-expansion conditions, thrust loss will
increase with the increase of truncation.
Base pressure thrust is closely related to variation of base pressure with atmospheric pressure.
Base pressure is constant in under-expansion conditions, but increases with the increase of the
atmospheric pressure in over-expansion conditions. Adding curvature to the base of a
truncated plug does not influence its thrust, but significantly improves local temperature
distribution at the base of the plug.
Based on the observed behavior of the thrust delivered by aerospike nozzles with different
amounts of plug truncation, it can be concluded that selection of the amount of plug truncation
depends on the flight regime of the vehicle which will use this propulsion system. If most of
the thrusted flight phase is going to be at altitudes lower than the design altitude (overexpansion conditions) it is recommended to use lower values of truncation, but if most of the
thrusted flight phase is going to be at altitudes higher than the design altitude (underexpansion conditions) it is recommended to use higher values of truncation
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