PART 1
SUBSONIC PANEL METHOD
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SUBSONIC PANEL METHOD FEATURES
AeroEuler uses 3-D source panels for the body to rapidly solve the frictionless potential flow equations in minutes to approximate subsonic compressible flow up to Mach 0.80. A complete set of graphical tools such as velocity and pressure contour plots are provided to allow the user to visualize 3-D flow around the model. The subsonic portion of AeroEuler computes the pressure (form) drag which is the integral of the normal pressure forces acting parallel (CX) to the axis of the model and perpendicular (CY or CN) to the axis of the model.

1. Determine pressure coefficient (Cp), static pressure, dynamic pressure and velocity around three-dimensional bodies in subsonic flow (M < 0.8).
2. Compute axial drag coefficient (CX) and normal drag coefficient (CN).
3. Generate models using built-in Sphere-Cone, Elliptical, Conical and Ogive shapes.
4. Generate models by importing shapes having up to 1,000 X, R points.
5) Automatically incorporates flow compressibility using the Prandtl-Glauret rule for flight analyses to Mach 0.80.
6. Generate color contour plots of pressure coefficient (Cp), static pressure, dynamic pressure and velocity with a single click.
7. Units include, MKS (meter-newton-second), CGS (centimeter-dyne-second), FPS (foot-pound-second) and IPS (inch-pound-second).
8. Define atmospheric properties up to 150,000 feet.
9. Send tabulated surface data to a text file in the following format: AXIAL LOCATION (I), MERIDIAN (K), PSTATIC, PDYNAMIC, CP, VELOCITY.
10. Send all plots directly to a color printer.
11. Operating instructions always available, up to date and located on-line.
12. Maximum mesh dimensions of 150 X 75 grids.
13. A series of ten project files (input data) are available. Simply unzip Projects.zip and transfer to the EulerCFD directory.

GENERAL PROCEDURE
A) BASIC UNITS
From the menu on top of the main start-up screen, select units (MNS, CGS, FPS or IPS) from the Units menu.

B) MODEL GEOMETRY
Begin model definition by clicking the GEOMETRY icon in the toolbar. First, in the GEOMETRY definition section the user must define the Outer Surface Shape of the solution region (exterior region of flow away from the body) by selecting either the CIRCULAR or ELLIPTICAL options in the Outer Surface Shape pull-down menu. CIRCULAR and ELLIPTICAL outer shapes are intended for subsonic flow and the SUPERSONIC outer shape is intended for supersonic flow. The CIRCULAR and ELLIPTICAL outer boundary shapes cannot be used to define supersonic outer boundary shapes because the MacCormack space-marching procedure requires the mesh in the marching direction to be vertical, parallel and separated by constant intervals from the initial plane of data to the end of the computational region. Therefore, to perform a subsonic analysis please select either the CIRCULAR or ELLIPTICAL Outer Surface shape.

Second, the shape of the model must be defined. The model or surface shapes available will depend on whether the outer boundary shape is CIRCULAR, ELLIPTICAL or SUPERSONIC. For the subsonic Outer Surface Shapes of CIRCULAR and ELLIPTICAL the model shapes available are SPHERE-CONE, ELLIPTICAL, CONICAL and IMPORTED. The user can import up to 1,000 X-R airframe geometry points from a text file. To create a model based on imported geometry, define the project geometry in NotePad and then Import the shape and finally save the Project file as usual. To read the Project file, open the Project file as usual and then Import the shape. The Import feature is located in the File command as Import Shape. The data has the following format. First line: Total number of X-R point locations. Second and subsequent lines: X, R airframe locations separated by commas.

Third, depending on which Outer Surface Shape is selected and which Body Shape is chosen various geometry definitions appear. The following matrix of inputs are defined for each Outer Surface selected in relation to each Model Shape. These geometry inputs are automatically displayed when the Body Shape and the associated Outer Surface is selected.

CIRCULAR and ELLIPTICAL Outer Shapes
SPHERE-CONE Body Shape
Number of grids in axial direction (NI). Please limit the number of grids to 150.
Number of Grids in vertical direction (NJ). Please limit the number of grids to 75.
Height of exit flow field above body surface (H)
Distance before nose stagnation point (DS) [Not required for CIRCULAR Outer Shape. Required for ELLIPTICAL Outer Shape]
Spherical nose radius (RNOSE)
Total sphere-conical body length (LBODY)
Angle of conical after-body (THETA)
Initial spacing off body in vertical direction (S1)
Vertical grid distribution as either LINEAR or TANH (Hyperbolic Tangent)
Number of grids on circumference of model (NM)

ELLIPTICAL Body Shape
Number of grids in axial direction (NI). Please limit the number of grids to 150.
Number of Grids in vertical direction (NJ). Please limit the number of grids to 75.
Height of exit flow field above body surface (H)
Distance before nose stagnation point (DS) [Not required for CIRCULAR Outer Shape. Required for ELLIPTICAL Outer Shape]
Body radius at base of model (RBODY)
Total body length (LBODY)
Elliptical body clustering in the axial direction (1 < Beta < 100)
Initial spacing off body in vertical direction (S1)
Vertical grid distribution as either LINEAR or TANH (Hyperbolic Tangent)
Number of grids on circumference of model (NM)

CONICAL and IMPORTED Body Shapes
Number of grids in axial direction (NI). Please limit the number of grids to 150.
Number of Grids in vertical direction (NJ). Please limit the number of grids to 75.
Height of exit flow field above body surface (H)
Distance before nose stagnation point (DS) [Not required for CIRCULAR Outer Shape. Required for ELLIPTICAL Outer Shape]
Body radius at base of model (RBODY)
Total body length (LBODY)
Initial spacing off body in vertical direction (S1)
Vertical grid distribution as either LINEAR or TANH (Hyperbolic Tangent)
Number of grids on circumference of model (NM)

C) CFD ANALYSIS
PANEL METHOD CFD ANALYSIS FOR SUBSONIC FLOW
Begin the CFD analysis by clicking the PANEL CFD icon for subsonic flow in the toolbar. Then enter the following.
1) Define altitude effects on pressure and density in the Operational Altitude pull-down menu. Select from Sea Level to 150,000 feet.
2) Define velocity in the Free field velocity data entry box.
3) Define angle of attack in the Angle of attack data entry box in degrees.
4) Specify the velocity in FT/SEC, MPH, M/SEC or Mach number and see the velocity in the Basic Units displayed just above.
5) Perform a subsonic CFD analysis and view results by clicking the Solve icon (below) in the 3-D PANEL ANALYSIS (SUBSONIC) section on the main screen. Then, after the solution is found view filled-contour plots and property surface plots for pressure coefficient (Cp), static pressure, dynamic pressure and velocity. Toggle between various properties such as Static Pressure, Dynamic Pressure, Cp and velocity by clicking the option buttons. Select Plot contours or surface plots by clicking either the Plot Contours or Plot Surface Curves check boxes. Note: line-contour plots are not available for subsonic flow results, but line-contour plots are available for supersonic flow results.

Toolbar Operations

1) Exit the AeroEuler computer program.
2) Reset the analysis to the default start-up CFD model (sphere-cone).
3) Save all flow properties (X,Y, Velocity etc) at each panel control point to a data file.
4) Show/Hide the Model Geometry definition section.
5) Show/Hide the 3D Panel Analysis (Subsonic) section.
6) Show/Hide the Euler CFD Analysis (Supersonic) section.
7) Show/Hide the Contour Plots and surface plot section.

VALIDATION RESULTS #1: 3:1 ELLIPTIC BODY, SUBSONIC PRESSURE DISTRIBUTION

Figure-1: Main AeroEuler screen - Model Geometry. Grid generation inputs.

Figure-2: Main AeroEuler Screen. 3-D PANEL ANALYSIS (SUBSONIC), CFD performed..


Figure-3: Main AeroEuler screen. CONTOUR PLOTS, Plot surface curves.


Figure-4: Main AeroEuler screen. CONTOUR PLOTS, Cp filled contours.


Figure-5: AeroEuler Subsonic Results Validation. AeroEuler results and data from the paper, Low-Speed Pressure Distribution on Axisymmetric Elliptic-Nosed Bodies (red dots). From Journal of Aircraft, page 969, October 1988. Data from Figure 3, Pressure coefficient distribution on semi-infinite axisymmetric bodies with elliptical noses.
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PART 2
SUPERSONIC SPACE MARCHING
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SUPERSONIC SPACE-MARCHING FEATURES
AeroEuler uses a space-marching approach based on the MacCormack method to solve the hyperbolic Euler equations for supersonic flow (Mn > 1). When the flow is entirely supersonic the equations of fluid motion are hyperbolic and any space-marching technique will be appropriate. The space-marching method will not work if the flow is locally subsonic and elliptic. The supersonic blunt body problem, where the flow is locally subsonic near the stagnation point, is an example where the space-marching methods do not work. However, this analysis assumes the flow is totally supersonic and that no subsonic regions in the flow exist and that the flow is entirely hyperbolic. The following list of features apply to the space-marching supersonic analysis.

1. Determine pressure ratio (P/Pinf), temperature ratio (T/Tinf), density ratio (R/Rinf) and Mach number around 2-D and axisymmetric bodies and ramps for M > 1.
2. Compute axial drag coefficient (CX) and normal drag coefficient (CN).
3. Generate models using built-in Conical and Ogive shapes.
4. Generate arbitrary axisymmetric models by importing shapes having up to 1,000 X, R points.
5. Generate filled-contour and line-contour color plots of pressure ratio (P/Pinf), temperature ratio (T/Tinf), density ratio (R/Rinf) and Mach number with a single click.
6. Generate surface plots of pressure ratio (P/Pinf), temperature ratio (T/Tinf), density ratio (R/Rinf) and Mach number .
7. View tabulated listings of station data in RichText format.
8. Send tabulated data to a text file in the following format: I,J,X,Y,P/PINF,R/RINF,T/TINF,M
9. Send all plots directly to a color printer.
10. Operating instructions always available, up to date and located on line.
11. Maximum mesh dimensions of 500 X 500 grids.
12. Improve stability and the odds of a solution by using artificial viscosity to dampen numerical oscillation around shocks.
13. A series of ten project files (input data) are available. Simply unzip Projects.zip and transfer to the EulerCFD directory.

GENERAL PROCEDURE
A) UNITS
From the menu on top of the main start-up screen, select units (MNS, CGS, FPS or IPS) from the Units menu. Not required because all values are inviscid and are normalized by the free stream conditions.

B) MODEL GEOMETRY
Begin model definition by clicking the GEOMETRY icon in the toolbar. First, in the GEOMETRY definition section that appears, the user must define the Outer Surface Shape of the solution region (exterior region of flow away from the body) by selecting either the CIRCULAR, ELLIPTICAL or SUPERSONIC options in the Outer Surface Shape pull-down menu. CIRCULAR and ELLIPTICAL outer shapes are intended for subsonic flow and the SUPERSONIC outer shape is intended for supersonic flow. Therefore, to perform a supersonic analysis please select the SUPERSONIC Outer Surface shape.

Second, the shape of the model must be defined. The model or surface shapes available will depend on whether the outer boundary shape is CIRCULAR, ELLIPTICAL or SUPERSONIC. However, for the SUPERSONIC outer boundary shape the available model shapes are CONICAL, OGIVE and IMPORTED. The user can import up to 1,000 X-R airframe geometry points from a text file. To create a model based on imported geometry, define the project geometry in NotePad and then Import the shape and finally save the Project file as usual. To read the Project file, open the Project file as usual and then Import the shape. The Import feature is located in the File command as Import Shape. The data has the following format. First line: Total number of X-R point locations. Second and subsequent lines: X, R airframe locations separated by commas.

Third, depending on which Outer Surface Shape is selected and which Body Shape is chosen various geometry definitions appear. The following matrix of inputs are defined for the SUPERSONIC Outer Surface and the CONICAL, OGIVE and IMPORTED Body Shapes. These geometry inputs are automatically displayed when the Body Shape and the associated Outer Surface is selected.

SUPERSONIC Outer Shape
CONICAL, OGIVE, IMPORTED Body Shapes
Number of grids in axial direction (NI), Please limit the number of grids to 500.
Number of Grids in vertical direction (NJ). Please limit the number of grids to 500.
Height of exit flow field from centerline (H)
Distance before nose stagnation point (DS)
Nose cone base radius (RNOSE)
Nose cone length (LNOSE)
Angle of conical after-body (THETA)
Conical after-body length (LBODY)
Number of grids on circumference of model (NM)

C) CFD ANALYSIS
EULER CFD ANALYSIS FOR SUPERSONIC FLOW
Begin the CFD analysis by clicking the EULER CFD icon for supersonic flow in the toolbar. Then enter the following.
1) CFL stability criterion, normally in the range of 0.0 to 1, but practically the value of CFL should be limited to 0.5 to 1.
2) Artificial viscosity. Sometimes improves the stability of a solution across shock waves by reducing the oscillatory response of a solution. Please see page 391 in Computational Fuid Dynamics, by John D. Anderson for further explanation of the concept of using artificial viscosity to improve solution stability. Artificial viscosity should be limited to the range of 0 to 0.9.
3) X-tabular increment ( Limit to 0 to NI or the number of grids in the axial direction) for printing to a scrolling RichText box. A value other than 0 allows AeroEuler to output the data in tabular form to a text box. A small non-zero value of this value makes the program output each marched plane of data to the scrollable text box. Printing large amounts of data to a scrollable text box is very memory intensive and may cause the program to crash. It is recommended the value for this input be NI or NI/2 at first and then reduce this value as needed.
4) Y-tabular increment (Limit to1 to NJ-1) for limiting the vertical data output at each data plane. This value allows the user to limit the data output to the RichText file along each plane of data. Greatly reduces memory requirement for storage of data.
5) Perform a supersonic CFD analysis and view results by clicking the Solve icon (below) in the EULER CFD (SUPERSONIC) section on the main screen. Then, after the solution is found view vertical property plots, property surface plots and see the tabulated results for pressure ratio (P/Pinf), density ratio (R/Rinf), Mach number (U/a), X-Velocity ratio (Ux/a) and Y-Velocity ratio (Uy/a). The pull-bar locates the vertical property plots at each station in the marching direction. Surface property plots are generated when the option button is selected for the property on the surface in the marching direction. The maximum and minimum values for the entire flow are located in the boxes above Max Value and Min Value, respectively. The maximum value for the plots may be altered by modifying the Max Value box entry. The Max Value entry will be used to define the maximum plot value for contour plots on the main screen. Line-contour plots and filled contour plots may be generated on the main screen. Finally, the maximum and minimum for each vertical plot location (station number) are labeled, Maximum curve value and minimum curve value, respectively.

Note: the height (H) of the exit flow field must enclose the shock wave as it forms. Many times a solution will fail because a smaller value than necessary is used for H. Simply increasing H in many cases resolves solution problems for supersonic flow.

Toolbar Operations

1) Exit the AeroEuler computer program.
2) Reset the analysis to the default start-up CFD model (sphere-cone).
3) Save all flow properties (X,Y, Velocity etc) at each panel control point to a data file.
4) Show/Hide the Model Geometry definition section.
5) Show/Hide the 3D Panel Analysis (Subsonic) section.
6) Show/Hide the Euler CFD Analysis (Supersonic) section.
7) Show/Hide the Contour Plots and surface plot section.

VALIDATION RESULTS #2: 10 DEGREE, 2-D CONE-CYLINDER AT M = 2

Figure-6:Main AeroEuler screen. MODEL GEOMETRY - SUPERSONIC FLOW.

Figure-7: Figure-6:Main AeroEuler screen. EULER CFD (SUPERSONIC) results

Figure-8:EULER CFD results at station 2.2, where 2-D cone-cylinder is 10 degrees and P/Pinf=1.7

Figure-9: EULER CFD results along surface along 2-D cone-cylinder

Figure-10: EULER CFD results at station 5.16, where 2-D cone-cylinder is 0.0 degrees

Figure-11: EULER CFD results at station .62, where 2-D cone-cylinder is 10 degrees and P/Pinf=1.7 at surface

Figure-12: P/Pinf 10 degree cone-cylinder line-contour plot on main screen.

Figure-13: P/Pinf 10 degree cone-cylinder filled-contour plot on main screen.

Figure-14: AeroEuler Supersonic Results Validation. NACA 1135 results for a 2-D, 10 degree cone-cylinder at Mach = 2 indicate P/Pinf = 1.70 and shock angle equal to 39.3 degrees. AeroEuler space-marching results on the 10 degree cone are: P/Pinf = 1.70 and shock angle approximately 39 degrees from plot of P/Pinf in Figure-14 (above).
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VALIDATION RESULTS #3: 5.352 DEGREE EXPANSION, M = 2, Pages 374 - 414, Computational Fluid Dynamics

Figure-15, AeroEuler Supersonic Results Validation. AeroEuler results compared to a 5.352 degree Prandtl-Meyer expansion wave example from the book, Computational Fluid Dynamics by John D. Anderson, pages 374 - 414 where the forward Mach line of the expansion fan should be 30 degrees and the rearward Mach line of the expansion fan should be 27.04 degrees. The green region on the AeroEuler filled-contour plot indicates a forward Mach line set at approximately 30 degrees and the rearward Mach line set at approximately 25 degrees. For the methodology used to compute the properties of a Prandtl-Meyer expansion, please see page 113 of the book Modern Compressible Flow with Historical Perspective.


Figure-16: AeroEuler Supersonic Results Validation. AeroEuler results for an expansion corner compared to a 5.352 degree Prandtl-Meyer expansion wave example from the book, Computational Fluid Dynamics by John D. Anderson, pages 374 - 414 where the exact results are stated to be: M 2= 2.2, P2/Pinf = .732, R2/Rinf = .80 and T2/Tinf = .916. AeroEuler results are: M2 = 2.11, P2/Pinf = .730, R2/Rinf = .80 and T2/Tinf = .910.
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Euler Code Theory Report (EulerCode.pdf)
Table of Contents

Two Dimensional Planar and Axisymmetric Euler Equations MacCormack Method Space-Marching Theory EULER90 Program Listing Program Definitions Shock Capturing Analysis AeroEuler References Program Listing Derivation of the MacCormack Finite Difference Equations Free-Field Points Derivation Wall Points Derivation Upper-Boundary Points Derivation Note: After installation the report is located at: c:/Program Files/EulerCFD/EulerCode.pdf

1 2 3-10 3 3 4 5-10 11-16 11-12 13-14 15-16

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AeroEuler Project Files (Projects.zip)

STANDARD GEOMETRY TEST CASES 5.352 DEG 2D EXPANSION - Pages 374 - 414, Computational Fluid Dynamics 10 DEG 2D CONE-CYLINDER - NACA 1135 Test case 10 DEG CONE 2D - NACA 1135 Test Case 10 DEG CONE 3D - NACA 1135 Test Case 15 DEG CONE 2D - NACA 1135 Test Case 6 DEG CONE-CYLINDER 3D - NACA 1135 Test Case OGIVE NOSE - SUPERSONIC ELLIPTICAL BODY - Low-Speed Pressure Distribution on Axisymmetric Elliptic-Nosed Bodies IMPORTED GEOMETRY TEST CASES CONE-CYLINDER IMPORT SHAPE - 10 DEG CONE-CYLINDER 2D CONE-CYLINDER PROJECT - SUBSONIC CONE - CYLINDER PROJECT -SUPERSONIC - NACA 1135 Test case Note: After installation the project files and code listing is located at: c:/Program Files/EulerCFD/Projects.zip

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System Requirements
Screen resolution: 1024 X 768 (800 X 600 is marginal)
System: Windows 98, 2000, XP, Vista, NT or Mac with emulation
Processor Speed: Pentium 3 or 4
Memory: 100 MB RAM
256 colors

PROGRAM REVISIONS
AeroEuler 2.0.2
1) Corrected the output error that occurred when attempting to save subsonic surface data by clicking SAVE DATA. By clicking SAVE DATA the user now correctly saves static pressure, dynamic pressure, Cp (pressure coefficient) and velocity along the axial length of the body and at each meridian or circumferential location. The subsonic results use the following format. Axial Location (I), Meridian Location (K), PSTATIC, PDYNAMIC, CP, VELOCITY.

For more information about AeroEuler please contact AeroRocket at aerocfd@aerorocket.com.

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