PART 1:
AEROSPIKE NOZZLE DESIGN
ANNULAR (3D) & LINEAR (2D) CONTOURS
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Summary of Features
1. Determine the shape of an annular or linear aerospike nozzle
given the thruster exit area ratio (Aei/At), projected area expansion
ratio (Ae/At), pressure ratio (Pc/Pa), thruster internal radius
(Ra), radius to lip of cowl (Re), total nozzle length from origin
(Lnozzle), chamber temperature (Tc), chamber pressure (Pc), ratio
of specific heats (g) and gas constant.
2. Click the UpDown command button to move a locator to
one of seven points in the flow field.
3. All important flow properties are displayed in real time as
the locator moves from point to point in the flow field described
by the characteristic mesh of the aerospike.
4. Generate color contour plots of Mach number (Mn), Pressure
(P/Pc), Temperature (T/Tc) and density (R/Rc) with a single click.
5. Plot Mn, P/Pc, T/Tc, R/Rc, CF, CFvac, ISP, ISPvac as a function
of aerospike nozzle axial location at a particular Pc/Pa.
6. Plot CF, Thrust, CFvac, ISP, and ISPvac verses pressure ratio
(Pc/Pa) on a semilog scale.
7. Units include, MKS (meternewtonsec), CGS (centimeterdynesecond),
FPS (footpoundsecond) and IPS (inchpoundsecond).
8. Graphically display the outer flow boundary for under expanded
flow, over expanded flow and the angle of the outer boundary
flow.
9. Graphically display the initial shock wave formed at the lip
of the cowl for over expanded flow (Pa/Pc > Pe/Pc) and the
shock angle from the lip.
10. Define gas properties for inert gases, liquid propellant
gases and solid fuel propellant gases or insert your own values.
11. Define the analysis for annular (3D) or linear (2D) aerospike
nozzles.
12. Define the angle the sonic section of the thruster makes
with the axis of the aerospike nozzle.
13. Added a hybrid rocket
motor propellant having the following fuel and oxidizer to the list of
combustion gases: 85% Nitrous Oxide, 15% HTPB.
14.
Added the ability to
save F(x) verses PR (Pressure Ratio) and F(x)
verses x to a CSV file for use with Notepad or Excel.
15.
In the Aerospike Nozzle Data section added a
display of Truncation as percent of total aerospike length.
16.
In the Aerospike Nozzle Data section added a
display of Distance from throat (origin) to end of thruster.
17.
In the Aerospike Nozzle Data section added a
display of Distance from
end of thruster to end of
ramp.
18. NEW! Added the ability to include base
thrust for truncated aerospike nozzles.
Propellant Gases Available
Inert Gases 
Dry Air 
Hydrogen 
Helium 
Water Vapor 
Argon 
Carbon Dioxide 
Carbon Monoxide 
Nitrogen 
Oxygen 
Nitrogen Monoxide 
Nitrous Oxide 
Chlorine 
Methane 





Liquid Fuel Propellant Gases 
Oxygen, 75% Ethyl Alcohol(1.43) 
Oxygen, Hydrazine(.09) 
Oxygen, Hydrogen(4.02) 
Oxygen, RP1(2.56) 
Oxygen, UDMH(1.65) 
Fluorine, Hydrazine(2.3) 
Fluorine, Hydrogen(7.60) 
Nitrogen Tetroxide, Hydrazine(1.34) 
Nitrogen Tetroxide, 50% UDMH, 50%
Hydrazine(2.0) 
Nitric Acid, RP1(4.8) 
Nitric Acid, 50% UDMH, 50% Hydrazine(2.20) 

Solid Fuel Propellant Gases 
Ammonium Nitrate, 11% Binder, 420% Mg 
Ammonium Perchlorate, 18% Binder, 420% Al 
Ammonium Perchlorate, 12% Binder, 420% Al 
Hybrid Rocket Motor Propellant Gases 
85% Nitrous Oxide, 15% HTPB 


UserDefined Gases 
Specify custom or userdefined gases by inserting Ratio of specific heats for
exhaust (g) and Gas constant of
exhaust (Rgas) in the Aerospike Nozzle Data section. 
General Discussion
AeroSpike performs an expansionwave
analysis from the throat of the thruster nozzle, where Mn = 1.0,
to the thruster nozzle internalexit as a series of simple wave
expansions. Then, for the external ramp AeroSpike performs a
series of PrandtlMeyer expansions from the lip of the cowl,
where R=Re, to the entire length of the external ramp of the
aerospike nozzle. The ideal contour or shape of the external
ramp of the aerospike nozzle is determined using isentropic supersonic
flow theory. Then, depending on whether the flow is underexpanded
or if the flow is overexpanded AeroSpike performs either a PrandtlMeyer
expansion analysis or an oblique shock wave analysis to determine
the angle of the outer flow boundary from the lip of the cowl.
As a by product of the oblique shock wave analysis AeroSpike
determines the shock wave angle for overexpanded flow and plots both the
outer boundary contour and the initial shock wave from the lip of the cowl. If
the Check to Include base thrust
check box is not checked then base pressure is assumed
equal to atmospheric pressure (Pb = Patm) which means base thrust is zero and
only the centerbody and thrusters contribute to total aerospike
thrust. However, if the Check to Include base
thrust check box is checked then base pressure and atmospheric pressure are
nonequal
resulting in the following aerospike nozzle total thrust equations, F_{total} = F_{centerbody}
+ F_{base} + F_{thruster}. and CF = F_{total} / (At
* Pc).
Procedure
From the menu on the top of the main startup screen,
select units (MNS, CGS, FPS or IPS) from the Units menu and then
the propellant gas from the Gases menu. A number of inert gases,
liquid fuel propellants and solid fuel propellants are available.
The value for the ratio of specific heats (g) are determined from the Units and Gases menus and
are passed on to the Aerospike Nozzle program after clicking
the Aerospike Nozzle command button on the main startup screen.
The ratio of specific heats, gas constant (Rgas), chamber pressure
(Pc), and pressure ratio (PR) are required for the Aerospike
Nozzle analysis. Additionally, the pressure ratio (PR) represents
the maximum value for Pc/Pa that AeroSpike will use to plot CF,
Thrust, CFvac , ISP and ISPvac as a function of PR. The chamber
pressure is computed based on the atmospheric pressure (Pa) and
the pressure ratio (Pc/Pa). These values are automatically passed
to the Aerospike Nozzle analysis when the command button is clicked.
However, the user
can override any input value by inserting new data directly
into each data entry box on the Aerospike Nozzle Design screen.
Each time the user changes any data entry the results are automatically
updated and displayed. The user only needs to click the Plot
button to see a new contour plot of the results or the UpDown
button to see flow results at any of the characteristic mesh
points.
Toolbar Operations
1. Click [X] to switch between the main data entry area (Figure2)
and the secondary data entry area (Figure3). The main data entry
area is displayed by default. Specify either annular aerospike
geometry or linear aerospike geometry by clicking one of two
option buttons in the secondary data entry area. In addition,
the thruster sonicsection angle (60 degrees to 120 degrees)
is located in the secondary data entry area. The thruster sonicsection
angle is measured from the axis of the aerospike nozzle to the
section that defines the throat of the thruster (where Mach number
= 1). Default = 90 degrees. Finally, check the Check to Include base
thrust check box to
include base thrust for the determination of total thrust and thrust coefficient
(CF) for truncated aerospike nozzles.
2. Send all flow properties (X, Y, Mn etc.) at each characteristic
mesh point to the printer.
3. Send an image of the screen to the printer.
4. Save all flow properties (X,Y, Mn etc) at each characteristic
mesh point to a data file.
5. Read the nozzle description file from a previous session.
6. Save the nozzle description file from a previous session.
7. Refresh the displayed analysis to the default analysis seen
during startup.
8. Return to the main startup screen.
Input Variable Definitions
1. Thruster exit area ratio
(Aei/At): Ratio of thruster internal exit area (Aei) to thruster
throat area (At). Equation 1 is inverted to find Pc/Pei from
Aei/At.
2. Thruster pressure ratio (Pc/Pei): Ratio of chamber pressure
to thruster exit pressure. Found by interation of Equation 1
and displayed in lower data region.
3. Aerospike expansion ratio (Ae/At): The projected area of the
aerospike nozzle (Ae = p * Re^2)
divided by the total thruster throat area.
4. Ratio of specific heats (g):
Selected from a pulldown menu or userdefined.
5. Gas constant of exhaust (Rgas): Selected from a pulldown
menu or userdefined.
6. Aerospike pressure ratio (Pc/Pa): Ratio of the chamber pressure
(Pc) to the atmospheric pressure (Pa).
7. Thruster internal circular radius (Ra): Radius of the internal
portion of the thruster duct from point 1 (throat) to point 2
(Ra).
8. Radius to lip of cowl (Re): Radius that defines the projected
area of the aerospike nozzle (Re).
9. Aerospike length from origin (Lnozzle): Total length of the
aerospike nozzle from the origin (throat) of the thruster to the end of
the ramp.
10. Chamber temperature (Tc): Chamber temperature in either degrees
Rankine or degrees Kelvin depending on the units selected.
11. Chamber pressure (Pc): Chamber pressure whose units depend
on the units selected.
12. Width of ramp for linear aerospike nozzles (Lramp).
.
Equation 1: Thruster CrossSectional Area and Pressure Ratio
Relationship.
Figure 1. Aerospike Nozzle Displaying Basic Geometry and the
External Expansion Fan.
AeroSpike Validation1
EXTERNAL FLOW FIELD FOR AN ANNULAR AEROSPIKE NOZZLE
Figure 2.
Aerospike Nozzle  Optimum Expansion (PR = 71.5) and 20% plug nozzle
configuration
Figure 3. Aerospike
Nozzle  Secondary Input Data Entry Area for Annular/Linear nozzle
and thruster angle inputs.
Truncated Aerospike Nozzle Base Pressure, Total
Thrust and Thrust Coefficient
To determine truncated aerospike
nozzle base pressure (Pb) for
computing total thrust and pressure coefficient (CF) simply check the Check to Include base thrust
check
box in the Annular or Linear Ramp Selection and Thruster Angle data entry
area. When checked this check box activates base pressure computation for truncated aerospike nozzles
where base
pressure is included for determination of total aerospike thrust and thrust
coefficient. Truncated aerospike nozzle base thrust is determined using two
curvefit relationships for computing base thrust. The first relationship is atmospheric pressure (Patm)
verses pressure ratio (PR=Pc/Pa) and the second relationship is base pressure
verses percent truncation. Where, X% truncation refers to an aerospike nozzle
where (100X)% of the expansion ramp has been removed leaving a blunt base
region. The plots in Figure 4 and Figure 5 illustrate the
relationship between Patm verses PR and Pb verses Percent Truncation for the
computation of base thrust, total thrust and thrust coefficient. The curvefit
for base
pressure verses percent truncation displayed in Figure 5
was developed using several Computational Fluid Dynamics (CFD) analyses using aerospike nozzles having 20%, 30%, 40% and 50% truncation.
Aerospike
nozzle total thrust is
computed using the following equations knowing PR and percent
truncation.
F_{base}= (Pb  Patm) * A_{base}, F_{total} = F_{centerbody}
+ F_{base} + F_{thruster }and CF = F_{total} / (At * Pc)
where Patm = fn(PR) and Pb = fn(% Truncation)
Figure 4. Atmospheric pressure (Pa)
verses pressure ratio (PR=Pc/Patm) 

Figure 5. Base pressure (Pb) verses percent truncation curvefit using CFD
analyses of aerospike nozzles having 20%, 30%, 40% and 50%
truncation.

CF vs.
PR Validation 

Flow Field Validation 
Figure 6. CF verses Pressure Ratio (Pc/Pa)  SemiLog
plot, Maximum PR = 1000. AeroSpike CF verses PR compared to 20% (80% truncated) plug nozzle Base Flow CFD analysis
includes base pressure coefficient.
Reference: AIAA 20011051, T. Ito, K. Fujii, Flow Field Analysis of the Base
Region
of Axisymmetric Aerospike Nozzles.
"I used (AeroSpike) to design several types of nozzle(s) and
found your software is really useful".
Takashi Ito, JAXA 

Reference:
"Aerospike Nozzle Flow Fields", AIAA 20011051,
by Takashi Ito. Contour plots used with permission
from reference. 

Figure 7. AeroSpike program nozzle flow field
results compared to Base Flow CFD analyses for PR=9, PR=71 and
PR=500.

Aerospike Rocket Motor Design Example 



AeroCFD 5.2 Analysis by
John Cipolla

Annular aerospike
designed to produce 225,000 pounds of thrust. 

AeroCFD analysis of an aerospike rocket. M =
1.5. 
The annular aerospike rocket motor pictured above
(left) is a
design developed and rendered by
Richard Caldwell
(Rocket Nut) intended to produce 225,000 pounds of thrust at sea level.
This design concept was developed using AeroSpike 2.6 software to
specify the internal thruster and external ramp geometries for efficient
operation from sea level to orbital altitude. Rocket Nut's design uses an untruncated Annular
Aerospike ramp as illustrated in Figure3. Please click
here for
more information.
AeroSpike
Validation2
X33 XRS2200 AEROSPIKE ROCKET ENGINE ANALYSIS EXAMPLE
In
this example
the Linear Ramp or 2D option is used to determine sea level
and vacuum thrust for
the X33 XRS2200 aerospike rocket engine. Click the Hide or Show Aerospike
Nozzle Data (X in the toolbar) to select the Linear Aerospike
option and then insert 90 for the 2D Ramp width in the space
provided. The data from Figure9 were used as input for the aerospike
analysis illustrated in Figure10 where vacuum thrust and specific impulse
(Isp) are predicted to be 264,600 lbf and 454.7 sec where Pc/Pa = 100000.
Dimensional data for this analysis are based on Boeing's results for
the XRS2200 linear aerospike rocket engine
where vacuum thrust and Isp are 266,230 lbf and 436.5 sec
for a 0.6% variation in thrust and 4.2% variation in Isp. Due
to unavailable thruster dimensions and conflicting dimensional and performance information provided by NASA and
relevant technical papers this aerospike rocket engine analysis is an
approximation.
NASA Description of the X33 Spaceplane: The X33 was to have been a
wedgedshaped subscale technology demonstrator prototype of a potential
future Reusable Launch Vehicle (RLV) that Lockheed Martin dubbed VentureStar.
The company hoped to develop VentureStar early this century. Through
demonstration flight and ground research, NASA's X33 program was to have
provide the information needed for industry representatives such as Lockheed
Martin to decide whether to proceed with the development of a fullscale,
commercial RLV program. The X33 design was based on a lifting body shape
with two revolutionary linear aerospike rocket engines and a rugged
metallic thermal protection system. The vehicle also was to have had
lightweight components and fuel tanks built to conform to the vehicle's
outer shape. Time between X33 flights was planned to normally be seven
days, but the program hoped to demonstrate a twoday turnaround between
flights during the flighttest phase of the program. The X33 was to have
been an unpiloted vehicle that took off vertically like a rocket and landed
horizontally like an airplane. It was planned to reach altitudes of up to 50
miles and high hypersonic speeds. The X33 Program was managed by the
Marshall Space flight Center and was planned to have been launched at a
special launch site on Edwards Air Force Base. Technical problems with the
composite liquid hydrogen tank resulted in the program being cancelled in
February 2001.

XRS2200 Engine 
5K ft 
Vacuum 
Thrust, lbf 
204,420 
266,230 
Specific Impulse, sec 
339 
436.5 
Propellants 
Oxygen, Hydrogen 
Mixture Ratio (O/H) 
5.5 
Chamber Pressure, psia 
857 
Cycle 
Gas Generator 
Area Ratio (Ae/At) 
58 
Throttling, Percent
Thrust 
50  100 
Dimensions, inches
Forward End
Aft End
Forward to Aft 

134 wide x
90 long
42 wide x 90 long
90 

Figure8 aerospike engine side view. 
Figure9, XRS2200 aerospike rocket engine description data
repeated in Figure10. 
Figure10, XRS2200 vacuum
(Pc/Pa=100,000) analysis. Input data for the results in Figure10 based on
data from Figure9 using AeroSpike version 2.6.0.5.
Aerospike Engine
Results
Compared 
5K ft (Pc/Pa =
70.0621) 
Vacuum (Pc/Pa =
100,000) 
Thrust, lbf 
Isp, sec 
Thrust, lbf 
Isp, sec 
AeroSpike 2.6.0.x 
155,500 
267 
264,600 
454.7 
Boeing XRS2200 Results 
204,420 
339 
266,230 
436.5 
BOUNDARY SHAPE: The outer
boundary angle from the lip of the thruster cowl to the end of the
external ramp increases as altitude increases. The pressure ratio
(Pc/Pa) defines the extent to which the outer boundary expands as
altitude and pressure ratio increase. For pressure ratio, Pc is the
thruster chamber pressure and Pa is the local atmospheric pressure.
This section compares the XRS2200 expansion boundary shapes
at 5K feet (Pc/Pa =
70.0621) and 50K feet (Pc/Pa = 509.21)
determined by Navier Stokes
and AeroSpike. 

BACK TO
TOP
PART 2:
2D & 3D MINIMUM LENGTH NOZZLE DESIGN
USING
THE METHOD OF CHARACTERISTICS (FREE BONUS ADDITION)
BACK TO TOP
Summary of Features
1. Determine shapes and flow properties of 2D Minimum Length Nozzles (MLN) given
exit Mach number (Mdesign) and throat diameter (Dt).
2. Determine shapes and flow
properties of 3D Minimum Length Nozzles using an
approximation procedure based on 2D results.
3. Click the UpDown command button to move a locator from
point to point in the flow field.
4. All important flow properties are displayed in real time as
the locator moves from point to point in the flow field described
by the characteristic mesh.
5. Generate color contour plots of Mach number (Mn), Pressure
(P/Pc), Temperature (T/Tc) and density (R/Rc) with a single click.
6. Units include, MKS (meternewtonsecond), CGS (centimeterdynesecond),
FPS (footpoundsecond) and IPS (inchpoundsecond).
7. Define gas properties for inert gases, liquid propellant gases
and solid fuel propellant gases or insert your own values.
8. Output all flow variables to the printer or text file for use with
spreadsheet applications.
Propellant Gases Available
Inert Gases 
Dry Air 
Hydrogen 
Helium 
Water Vapor 
Argon 
Carbon Dioxide 
Carbon Monoxide 
Nitrogen 
Oxygen 
Nitrogen Monoxide 
Nitrous Oxide 
Chlorine 
Methane 





Liquid Fuel Propellant Gases 
Oxygen, 75% Ethyl Alcohol(1.43) 
Oxygen, Hydrazine(.09) 
Oxygen, Hydrogen(4.02) 
Oxygen, RP1(2.56) 
Oxygen, UDMH(1.65) 
Fluorine, Hydrazine(2.3) 
Fluorine, Hydrogen(7.60) 
Nitrogen Tetroxide, Hydrazine(1.34) 
Nitrogen Tetroxide, 50% UDMH, 50%
Hydrazine(2.0) 
Nitric Acid, RP1(4.8) 
Nitric Acid, 50% UDMH, 50% Hydrazine(2.20) 

Solid Fuel Propellant Gases 
Ammonium Nitrate, 11% Binder, 420% Mg 
Ammonium Perchlorate, 18% Binder, 420% Al 
Ammonium Perchlorate, 12% Binder, 420% Al 
Hybrid Rocket Motor Propellant Gases 
85% Nitrous Oxide, 15% HTPB 


UserDefined Gases 
Specify custom or userdefined gases by inserting
Ratio of specific heats (g) in the
Minimum Length Nozzle Data section. 
General Discussion
The
Minimum
Length
Nozzle
routine
performs a minimum length
nozzle (MLN) design using the method of characteristics. A minimum
length nozzle has the smallest possible throattoexit length
that is still capable of maintaining uniform supersonic flow
at the exit. Strictly speaking a minimum length nozzle requires
a sharp corner at the throat. However, sometimes a sharp corner
at the throat may be impractical. A nearly minimum length nozzle
may be generated by specifying a very small but finite radius
of curvature at the throat with the inflection point of the throatcurve
just downstream of the throat. For a nearly minimum length nozzle
simply specify the streamline from the throatcurve so the curvature
lines up with the nozzle wall shape generated by MLN.
A straight sonic line is assumed to occur at the throat of the
minimum length nozzle. For the example presented in Figure 2
and Figure 3, where the exit Mach number is 2.4, the first characteristic
(C_ ) propagating from the corner of the throat is inclined by
a small amount (q
= 0.375 deg)
from the normal sonic line. The slope of the first characteristic
is dy/dx = (q
 m) = 73.725
deg. See Figure 1 below. The remaining expansion fan is divided
into six increments. The Mach number at each point in the flow
is determined from the PrandtlMeyer function using the NewtonRaphson
iteration method and the unit processes dictated by the method
of characteristics. The nozzle contour is drawn by starting at
the throat corner where the maximum expansion angle of the wall,
qw_max is equal to onehalf
the PrandtlMeyer function, n(Mn)
/ 2, at the design exit
Mach number. For a minimum length nozzle the maximum expansion
angle is equal to onehalf the PrandtlMeyer function for the
design exit Mach number. For other nozzles the maximum expansion
angle must be less than n(Mn=Mdesign)
/ 2. For a detailed discussion
of the method of characteristics please refer to the reference,
Modern Compressible Flow, With Historical Perspective, by John
D. Anderson, pages 260 to 282.
PLEASE NOTE: For the 2D Minimum Length Nozzle selection the
"X" and "Y" coordinates of the nozzle contour represent the horizontal and
vertical dimensions that define the 2D characteristic mesh. Therefore, the
Exit Area Ratio (Aexit/Athroat) = [2*Yexit*WIDTH] / [2*Ythroat*WIDTH] =
Yexit/Ythroat because the flow is 2Dimensional. Likewise, for the 3D
Minimum Length Nozzle selection the "X" and "Y" coordinates of the contour
represent the horizontal and radial dimensions that define the 3D axisymmetric
mesh. Therefore, the Exit Area Ratio (Aexit/Athroat) = [p*Yexit^2] / [p*Ythroat^2] = (Yexit/Ythroat)^2 because the flow is
3Dimensional and not 2Dimensional. Finally, MLN Project files generated by
previous versions of MLN must be updated by adding "1" for 2D flow or "2"
for 3D flow at the bottom of the MLN file. Do not forget to save each Project
file with an MLN extension when updating older Project files for
use with AeroSpike 2.5 or higher.
Procedure
From the menu on the top of the main startup screen,
select units (MNS, CGS, FPS or IPS) from the Units menu and then
the propellant gas from the Gases menu. A number of inert gases,
liquid propellants and solid fuel propellants are available.
The value for the ratio of specific heats (g) are determined from the Units and Gases menus and
are passed on to the MLN program after clicking the Minimum Length
Nozzle command button on the main startup screen. Only the ratio
of specific heats are required for the MLN analysis. The other
values including the gas constant (Rgas), chamber pressure (Pc)
and pressure ratio (PR) are not required for the MLN analysis.
When performing an MLN analysis the only values required are
the Inclination angle from the sonic line (see above), Design
Mach number (Mdesign), and throat diameter (Dt). The ratio of
specific heats has already been specified from the main screen
selection. However, the user can override the inserted ratio
of specific heats by simply inserting his own ratio of specific
heats in the data entry box. Each time the user changes any data
entry the results are automatically updated and displayed. The
user only needs to click the Plot button to see a new contour
plot of the results or the UpDown button to see flow results
at any of the characteristic mesh points.
Toolbar Operations
1. Show or hide the main data window from view or from being
printed.
2. Send all flow propeties (X, Y, Mn etc) at each characteristic
mesh point to the printer.
3. Send an image of the screen to the printer.
4. Save all flow properties (X,Y, Mn etc) at each characteristic
mesh point to a data file.
5. Read the nozzle description file from a previous session.
6. Save the nozzle description file from a previous session.
7. Refresh the displayed analysis to the default analysis seen
during startup.
8. Return to the main startup screen.
Input Variable Definitions
1. Ratio of specific heats (g):
Selected from pulldown menu or userdefined.
2. Inclination angle from sonic line: This angle is used to compute
the slope of the first characteristic from the edge of the throat.
3. Design Mach number (Mdesign): The Mach number at the exit
of the nozzle where the flow is uniform.
4. Throat diameter (Dt): The entrance to the minimum length nozzle
where Mn = 1.0
5. Area ratio (Aexit/Athroat): The resulting exit area ratio of
the nozzle determined by the method of characteristics.
6. Specify whether the nozzle is 2D or 3D by clicking either the 2D
characteristics or 3D approximation option buttons.
Figure 1. Description of the inclination angle
(q) from the sonic line (at throat) where Mn =1.0
Minimum
Length
Nozzle
Validation1
Example 11.1, on page 282
from Modern Compressible Flow, With Historical Perspective, by John D. Anderson
Figure
2. Example 11.1, on page 282 from Modern Compressible Flow, With
Historical Perspective, by John D. Anderson
NOTE ABOUT MLN ANALYSIS ACCURACY:
The Minimum Length Nozzle (MLN) analysis illustrated in Figure2 uses exact
input data from Modern Compressible Flow With Historical Perspective. For
maximum accuracy simply insert 0.0 degrees in the Inclination angle from
sonic line input block. When this simple modification is performed the exit
Area ratio (AR) becomes 2.43
which compares to AR = 2.403 for exact 2D isentropic flow and represents a
1.124% difference from isentropic theory.
Figure 3. MLN Characteristic Mesh For exit Mach number of 2.4
where Inclination angle from
sonic line = 0.0 degrees produces AR = 2.43.
Minimum
Length
Nozzle
Validation2
Figure 17.5, Gasdynamics:
Theory and Applications,
2D and Approximate 3D MLN
Validation at M = 3.0
The following table compares 2D and 3D
MLN results with data scaled from Figure 17.5, Gasdynamics: Theory and
Applications. Two wallpoints, one from the center and one at the end of the
nozzle contour have been selected for comparison. Notice that 3D Minimum Length
Nozzles are substantially shorter than equivalent 2D Minimum Length Nozzles
that have identical
Area Ratio (Aexit/Athroat).
For comparison purposes all results are referenced to the curved sonic
line analysis for 2D and 3D axisymmetric nozzles. Please reference
Gasdynamics: Theory and Applications*
page 325, Figure 17.15 where g
= 1.4 and Mexit = 3. Finally, please note that MLN uses the
straight sonic line method of characteristics analysis.
2D MLN Analysis 
X_Coordinate 

Y_Coordinate 
Difference 
X_Coordinate 
Difference 
Y_Coordinate 
Difference 
AeroSpike 2.6 
6.522 

3.331 
10.7% 
17.43 
4.79% 
4.354 
6.8% 
Straight Sonic
Line* 
6.522 

3.30 
11.5% 
16.83 
0.53% 
4.198 
10.1% 
Curved Sonic
Line* 
6.522 

3.73 
 
16.92 
 
4.670 
 









3D MLN Analysis 
X_Coordinate 

Y_Coordinate 
Difference 
X_Coordinate 
Difference 
Y_Coordinate 
Difference 
AeroSpike
2.6 
3.126 

1.603 
0.06% 
8.353 
2.68% 
2.087 
2.9% 
Axisymmetric
Curved Sonic Line* 
3.126 

1.604 
 
8.59 
 
2.028 
 
Figure 4. 2D MLN Characteristic Mesh when
g
=1.4 and exit Mach number = 3
Figure 5. Approximate 3D Axisymmetric MLN Characteristic Mesh when
g
=1.4 and exit Mach number = 3
BACK TO TOP
AeroSpike System Requirements
(1) Screen resolution: 800 X 600
(2) System: Windows 98, XP, Vista, Windows 7 (32 bit and 64 bit), NT or Mac with emulation
(3) Processor Speed: Pentium 3 or 4
(4) Memory: 64 MB RAM
(5)
English (United States) Language
(6)
256 colors
Please note this web page requires your
browser to have
Symbol fonts to properly display Greek letters (a,
m, p,
∂
and w)
ADDITIONAL REQUIREMENT: Input data for all AeroRocket programs must use a period (.)
and not a comma (,) and the computer must be set to the English (United States)
language. For example, gas constant should be
written as Rgas = 355.4 (J / kg*K = m^2 / sec^2*K)
and not Rgas = 355,4. The English (United States)
language is set in the
Control Panel by clicking Date, Time, Language and
Regional Options then Regional and Language Options
and finally by selecting English (United States). If periods are not used in all inputs
and outputs the
results will not be correct.
AEROSPIKE REVISIONS
AeroSpike 2.3 Features
and Error Fixes
1. Fixed plot resolution problem that occurred for some high
aspect ratio aerospike nozzles.
2. Fixed the ratio of specific heats (g) manual entry error that would not accept gamma
(g) = 2 and some other
minor errors.
3. Fixed the incorrect gas constant (Rgas) value for hydrogen.
For hydrogen, Rgas = 4122.11 m^2/(sec^2*K).
4. Added the ability to specify thruster sonicsection (throat)
angle. Throat angle can vary from 60 to 120 degrees, the default
is 90 degrees.
AeroSpike 2.4.1 Features
and Error Fixes
1. Added a hybrid rocket
motor propellant having the following fuel and oxidizer to the list of
combustion gases: 85% Nitrous Oxide, 15% HTPB.
2. Added the ability to
save F(x) verses PR (Pressure Ratio) and F(x) verses
x to a CSV file for use with Notepad or Excel.
3.
In the Aerospike Nozzle Data section added a
display of Truncation as percent of total aerospike length.
4. In the Aerospike Nozzle Data section added a display of Distance from
throat (origin) to end of thruster.
5. In the Aerospike Nozzle Data section added a display of Distance from
end of thruster to end of
ramp.
6. Corrected a few Status Bar display errors for plots of F(x) verses x.
AeroSpike 2.4.2 Error Fix
(11/26/2006)
1) The gas Nitrogen Dioxide in the Gases pulldown menu should be
labeled Nitrous Oxide (N2O). (Fixed)
AeroSpike 2.5.0 Features
(01/23/2007)
1. Added ability to determine shapes and flow properties of 3D Minimum Length
Nozzles using an approximation procedure based on 2D results.
AeroSpike 2.6.0.1 Features
(09/21/2008)
1. Added the ability to include base thrust of truncated aerospike nozzles
to determine total thrust and thrust coefficient.
2. To modify or change units in previous versions of AeroSpike the user
needed to close the main aerospike nozzle analysis screen and then redefine
units in the startup screen. However, starting with this new version the user
goes directly to the startup screen without closing the aerospike nozzle
analysis screen to modify pressure ratio, units, gases and altitude to
instantly included those changes on the main aerospike nozzle analysis
screen.
3. Darkened all data display boxes to prevent confusion with white data
entry boxes for the MLN and aerospike analyses.
AeroSpike 2.6.0.2 Features (09/14/2009)
1) For AeroSpike, fixed all input data text boxes for 32 bit and 64 bit
Windows Vista. When operating earlier versions of AeroSpike in Windows Vista the input data
text boxes failed to show their borders making it difficult to separate each
input data field from adjacent input data fields.
This simple change did not alter
any computational result.
AeroSpike 2.6.0.3 Features (04/27/2011)
1) For the Minimum Length Nozzle routine corrected the 3D axisymmetric
meshdata display.
This simple change did not alter
any computational result.
AeroSpike 2.6.0.4 Features (09/21/2011)
1) For AeroSpike the color contour plots and aerospike nozzle shape data are now displayed to scale.
In previous versions aerospike color contour plots and aerospike nozzle shape displays were not to scale
to allow full use of the available display area. However, recent work
indicates it is more useful to display scaled aerospike geometry than to
fill the entire plot area. The factor 0.5390 was used to properly scale the
Xcoordinates of the X, R plot data. This simple change did not alter any
computational result.
AeroSpike 2.6.0.5 Features (09/25/2011)
1) For AeroSpike the Truncation as percent of total aerospike length
data field displayed incorrect
values intermittently based on UNITs selection. This simple change did not alter any
other computational result.
For more information about
AeroSpike please
contact AeroRocket.
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