PART 1
2-D & 3-D MINIMUM LENGTH NOZZLE DESIGN
USING THE METHOD OF CHARACTERISTICS
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Summary of Features
1. Determine shapes and flow properties of 2-D Minimum Length Nozzles (MLN) given exit Mach number (Mdesign) and throat diameter (Dt).
2. NEW! Determine shapes and flow properties of 3-D Minimum Length Nozzles using an approximation procedure based on 2-D 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 (meter-newton-seccond), CGS (centimeter-dyne-second), FPS (foot-pound-second) and IPS (inch-pound-second).
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.

General Discussion
AeroSpike performs minimum length nozzle (MLN) design using the method of characteristics. A minimum length nozzle has the smallest possible throat-to-exit 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 throat-curve just downstream of the throat. For a nearly minimum length nozzle simply specify the streamline from the throat-curve so the curvature lines up with the nozzle wall shape generated by AeroSpike.

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 Prandtl-Meyer function using the Newton-Raphson 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 one-half the Prandtl-Meyer function, n(Mn) / 2, at the design exit Mach number. For a minimum length nozzle the maximum expansion angle is equal to one-half the Prandtl-Meyer 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 2-D Minimum Length Nozzle selection the "X" and "Y" coordinates of the nozzle contour represent the horizontal and vertical dimensions that define the 2-D characteristic mesh. Therefore, the Exit Area Ratio (Aexit/Athroat) = [2*Yexit*WIDTH] / [2*Ythroat*WIDTH] = Yexit/Ythroat because the flow is 2-Dimensional. Likewise, for the 3-D Minimum Length Nozzle selection the "X" and "Y" coordinates of the contour represent the horizontal and radial dimensions that define the 3-D axisymmetric mesh. Therefore, the Exit Area Ratio (Aexit/Athroat) = [p*Yexit^2] / [p*Ythroat^2] = (Yexit/Ythroat)^2 because the flow is 3-Dimensional and not 2-Dimensional. Finally, MLN Project files generated by previous versions of AeroSpike must be updated by adding "1" for 2-D flow or "2"  for 3-D 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 start-up 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 start-up 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 over-ride 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.


Main start-up screen for the Aerospike Nozzle and Minimum Length Nozzle (MLN) analyses.

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 start-up.
8. Return to the main start-up screen.

Input Variable Definitions
1. Ratio of specific heats (g): Selected from pull-down menu or user-defined.
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 2-D or 3-D by clicking either the 2-D characteristics or 3-D approximation option buttons.


Figure 1. Description of the inclination angle (q) from the sonic line (at throat) where Mn =1.0

AeroSpike MLN Results

Figure 2. Example 11.1, on page 282 from Modern Compressible Flow, With Historical Perspective, by John D. Anderson


Figure 3. MLN Characteristic Mesh For exit Mach number of 2.4


2-D and 3-D Minimum Length Nozzle Validation at M = 3.0
The following table compares 2-D and 3-D AeroSpike MLN results with data scaled from Figure 17.5, Gasdynamics: Theory and Applications. Two wall-points, one from the center and one at the end of the nozzle contour have been selected for comparison. Notice that 3-D Minimum Length Nozzles are substantially shorter than equivalent 2-D Minimum Length Nozzles that have identical Area Ratio (Aexit/Athroat). For comparison purposes all results are referenced to the curved sonic line analysis for 2-D and 3-D axisymmetric nozzles. Please reference Gasdynamics: Theory and Applications* page 325, Figure 17.15 where g = 1.4 and Mexit = 3. Finally, please note that AeroSpike MLN uses the straight sonic line method of characteristics analysis.

Figure 4. 2-D MLN Characteristic Mesh when g =1.4 and exit Mach number = 3


Figure 5. 3-D Axisymmetric MLN Characteristic Mesh when g =1.4 and exit Mach number = 3

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PART 2
AEROSPIKE NOZZLE DESIGN
ANNULAR (3-D) & LINEAR (2-D) 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 semi-log scale.
7. Units include, MKS (meter-newton-sec), CGS (centimeter-dyne-second), FPS (foot-pound-second) and IPS (inch-pound-second).
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 (3-D) or linear (2-D) aerospike nozzles.
12. Define the angle the sonic section of the thruster makes with the axis of the aerospike nozzle.
13. NEW! Added a hybrid rocket motor propellant having the following fuel and oxidizer to the list of combustion gases: 85% Nitrous Oxide, 15% HTPB.
14. NEW! 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. NEW! In the Aerospike Nozzle Data section added a display of Truncation as percent of total aerospike length.
16. NEW! In the Aerospike Nozzle Data section added a display of Distance from throat (origin) to end of thruster.
17. NEW! In the Aerospike Nozzle Data section added a display of Distance from end of thruster to end of ramp.

General Discussion
AeroSpike performs an expansion-wave analysis from the throat of the thruster nozzle, where Mn=1.0, to the thruster nozzle internal-exit as a series of simple wave expansions. Then, for the external ramp AeroSpike performs a series of Prandtl-Meyer 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 Prandtl-Meyer 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. Presently, the base pressure for a truncated aerospike is assumed to equal the atmospheric pressure (Pa) in which the nozzle is operating.

Procedure
From the menu on the top of the main start-up 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 start-up 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 over-ride 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 (Figure-2) and the secondary data entry area (Figure-3). 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 sonic-section angle (60 degrees to 120 degrees) is located in the secondary data entry area. The thruster sonic-section 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.
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 start-up.
8. Return to the main start-up 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 pull-down menu or user-defined.
5. Gas constant of exhaust (Rgas): Selected from a pull-down menu or user-defined.
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 Cross-Sectional Area and Pressure Ratio Relationship.


Figure 1. Aerospike Nozzle Displaying Basic Geometry and the External Expansion Fan.

AeroSpike Validation

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.

CF vs. PR Validation		Flow Field Validation
Figure 4. CF verses Pressure Ratio (Pc/Pa)  - Semi-Log plot, Maximum PR = 1000. AeroSpike CF verses PR compared to 20% (80% truncated) plug nozzle  Base Flow CFD analysis. Reference: AIAA 2001-1051, 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
		Figure 5. AeroSpike program nozzle flow field results compared to Base Flow CFD analyses for PR=9, PR=71 and PR=500. Reference: "Aerospike Nozzle Flow Fields", AIAA 2001-1051, by Takashi Ito. Contour plots used with permission from reference.

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System Requirements
(1) Screen resolution: 800 X 600
(2) System: Windows 98, 2000, XP, Vista, 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

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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 sonic-section (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 (11/26/06)
1) The gas Nitrogen Dioxide in the Gases pull-down 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 3-D Minimum Length Nozzles using an approximation procedure based on 2-D results.

For more information about AeroSpike please contact AeroRocket at aerocfd@aerorocket.com.
For more technical information about aerospike nozzles please refer to Takashi Ito's web site.

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