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Aero-Thermodynamics for Conceptual Design

David J. Kinney*

NASA Ames Research CenterMountain View, CA 94035-1000

A software tool for the prediction of the aero-thermodynamic environments of conceptual aerospace configurations is presented. The vehicle geometry is defined using unstructured, triangulated surface meshes. For subsonic Mach numbers a fast, unstructured, multi-pole panel code is coupled with a streamline tracing formulation to define the viscous surface solution. For supersonic and hypersonic Mach numbers, various independent panel methods are coupled with the streamline tracing formulation, an attachment line detection method, and stagnation-attachment line heating models to define the viscous aero-thermal environment.

IntroductionThe prediction of the aero-thermodynamic

environments of launch, orbiter, and crewed space vehicle configurations early in the conceptual and preliminary design stages is often driven by the need for rapid turn-around and the ability to handle complex and unconventional geometries. NASA’s on going programs, such as NGLT (Next Generation Launch Technologies) and OSP (Orbital Space Plane), require the evaluation of the aero-thermodynamic environments of configurations ranging from stacked two-stage to orbit reusable systems, to expendable launch vehicles with crew transfer vehicles mounted atop.

During the 2nd Generation Launch Vehicle program, the development of the Advanced Engineering Environment (AEE) was begun. AEE includes the full set of conceptual to preliminary level analysis tools, including geometry, aero-thermodynamics, structures, weights, trajectory, thermal protection, cost, operations, and safety. In support of the AEE environment, the aero-thermodynamic tools are required to supply both integrated forces and moments for trajectory analyses, as well as distributed heating and loads information for thermal and structural analyses.

To adequately define the aero-thermal environment, a typical matrix of Mach, dynamic pressure, and angle of attack (M-Q-α) might span 16 x 12 x 16 = 3072 cases. The use of high fidelity Computational Fluid Dynamics (CFD) for all but a few of these cases, across the Mach number (0.3 to 26), dynamic pressure (0.01 to 1000+ psf), and angle of attack (0 to 40+ degrees) range, is impractical.

*Research Scientist, Senior Member AIAA

The large size of the aero-thermal databases, and the impracticality of running high fidelity (finite rate chemistry, Navier Stokes) CFD across the entire M-Q-αrange lead to the need for engineering level tools for the prediction of both subsonic and super-hypersonic aero-thermodynamics. The Configuration Based Aerodynamics tool (CBAERO), developed under NASA’s 2nd Generation Launch Vehicle program provides such a capability.

CBAERO combines the use of unstructured, triangulated surface grids with proven and understood,engineering level analysis, for both subsonic and super-hypersonic Mach numbers, across the angle of attack and dynamic pressure ranges. The use of unstructured, triangulated grids allows the re-use of the same software tools used to define surface grids for the many unstructured Euler solvers available, such as NASA Ames Research Center’s CART3D.

Geometry DefinitionCBAERO makes use of an unstructured surface

grid of triangles to define the Outer Mold Line (OML)of the vehicle configuration. No volume mesh is required or used. The use of triangulated surface grids was driven by many factors: 1) the ability to easily, and automatically, mesh complex configurations, 2) the ability to re-use the various unstructured mesh generation tools already in existence, and 3) the ability to compare, on the same mesh, with various unstructured Euler solvers (CART3D, AIRPLANE) which are regularly used during the conceptual and preliminary design stages.

Within the AEE, the geometry definition is defined by using PTC’s ProE, and surface meshes are generated using ProMesh. However, an unstructured triangulated mesh from any source can be accepted, once correctly

42nd AIAA Aerospace Sciences Meeting and Exhibit5 - 8 January 2004, Reno, Nevada

AIAA 2004-31

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

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formatted. Figure 1 shows a typical surface grid of the NASA X-33 configuration. Figure 2 illustrates a generic 2nd generation launch vehicle concept generated and meshed using the AEE geometry tool.

Fig. 1 Unstructured Triangulated Surface for X-33,containing 150,000 triangles

Fig. 2 Generic 2nd Gen Concept Generated Using AEE Geometry Tool, containing 12,000 triangles

Subsonic Mach NumbersFor the subsonic Mach number range CBAERO

makes use of a fast, multi-pole, unstructured panel code formulation. The linear, integral formulation for subsonic, incompressible flow is solved:

dSr

dSr SWSSWS

p ∫∫∫∫∞∞ ++++

−

∇=Φ 1

4

11

4

1 σπµπWhere µ is the doublet strength and σ the source

strength. The above integral formulation is solved discretely by assuming constant source and doublet distributions across each surface triangle. Enforcing tangency on each triangle panel (via a Dirichlet formulation) results in a large, and fully dense, linear

system for the unknown doublet strengths. The formulation is similar to the one used in references 8 and 10. A simple Prandtl-Glauert affine transformation is used to account, in a linear sense, for the compressibility affects of low Mach number flows.

The size of the linear system and the solution time will both grow as N^2, where N is the number of triangles. This quadratic complexity is the largest detractor of all such linear panel formulations. To circumvent such quadratic solution complexities, CBAERO make use of well known fast multi-pole methods, sometimes generically referred to as fast tree methods (reference 2).

The largest cost in implementing a typical panel method, either based on structured or unstructured grids, is the evaluation of the linear system of equations and its solution. To speed up this process, CBAERO uses an octree data structure to define the spatial distribution of the surface triangles (panels), and their relative proximity to one another. With the octree information, approximate, multi-pole, expansions can be built for entire regions of the of the solution domain. The net result is that the generation and solution of the linear system of equations can be reduced to a complexity proportionate to N*Log N, where N is again the number of surface triangles (panels), [references 2, 3, and 9].

Fig. 3 Octree data structure for the HL-20

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Fig. 4 Octree data structure for the HL-20

Figures 3 and 4 provide views of a typical octreedata structure that was automatically built for the NASA HL-20 crew transfer vehicle. The use of the fast-multi-pole formulation makes solutions on meshes with 20k to 100k triangles routine. Figure 5 presents a subsonic, Mach 0.3 solution about the NASA HL-20. The combined surface and wake mesh contains nearly 14k triangles, and required 82 seconds of CPU time on a 2.6 Ghz Pentium 4 desktop PC.

Fig. 5 Mach 0.3, 15 Degrees Angle of Attack,pressure coefficient for HL-20

Super-Hypersonic Mach NumbersFor super-hypersonic Mach numbers the inviscid

solution is based on typical independent panel methods, such as Modified-Newtonian, Tangent Cone, and Tangent Wedge formulations. Figure 6 depicts the surface pressure contours, using a Modified-Newtonian formulation, on the X-33 configuration.

Fig 6. Mach 6, 30 degrees Angle of Attack,pressure coefficient for X-33

The Modified Newtonian method provides the pressure at each surface triangle. The known entropy after the normal shock, the surface pressure provided by the Modified Newtonian, and the assumption of an isentropic process from the shock to the surface defines the thermodynamic state on the triangle.

For the Tangent Cone and Tangent Wedge methods two options exist. First, one may assume a fully attached conical or two-dimensional shock. The Taylor-Maccoll equations (conical flow), or the oblique shock relations then provide the post (conical or oblique) properties for pressure as well as entropy. Conversely, the Tangent Cone and Tangent Wedge methods may be used to provide only the pressure, and the post normal shock entropy can be used to fix the thermodynamic state at each triangle. With any of these methods, the equilibrium air model is used to define all the thermodynamic properties and relationships.

Unlike the subsonic, potential formulation, the various independent panel methods do not provide any information on the surface velocity pattern, or direction. The independent panel methods only provide the scalar thermodynamic properties, including the velocity magnitude. To provide an estimate of the surface velocity vectors, the very simple approximation:

nVnVsurface ˆˆ ××= ∞rr

is used. Here n̂ is the local surface normal, and ∞Vr

is

the free stream velocity vector.

Streamline Tracing and Attachment Line DetectionGiven an inviscid flow field, generated using either

the subsonic fast panel method, or the super-hypersonic independent panel methods, the standard ordinary differential equations (ODEs) for a three dimensional streamline, constrained to the vehicle surface, are solved.

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The present implementation assumes a linear variation of the surface velocity field across each triangle. The 3 equations for the x, y, and z coordinatesof the streamline paths are integrated on a given surface triangle until a triangle edge is reached. A triangle to edge, and an edge to triangle data structure is used to move to the adjacent triangle where the integration process is continued. The streamline ODEs are integrated in reverse, starting at a point near the base of the vehicle, and ending at the stagnation point.

To efficiently cover the surface with as few streamlines as possible, the surface nodes are sorted in increasing x, from nose to tail. A simple greedy algorithm is used to choose the next starting node. Using this approach, a limited number of streamlines are generated such that every node is adjacent to at least one streamline. Figure 7 shows a typical surface streamline pattern produced using the simple hypersonic surface velocity model on the X-33 configuration.Figure 8 shows the surface streamline pattern on the HL-20 configuration based on a Mach 0.3, 15 degrees angle of attack solution generated using the fast panel formulation.

Fig. 7 Streamlines based on hypersonic approximation to surface velocity field for the X-33

Fig. 8 Streamlines based on subsonic fast panel surface solution at Mach 0.3, 15 degrees Angle of Attack, for the HL-20

Using the same inviscid surface velocity field, the method of reference 1 is used to locate the attachment lines. This method makes use of topology considerations to locate attachment and separation lines. Here, only the attachments lines are of interest. Figures 9 and 10 show the detected attachment line locations for the X-33 configuration. The method successfully detects the chine attachment lines, as well as those along the fin leading edges.

Fig. 9 Detected attachment line locations and streamline pattern

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Fig. 10 Detected attachment line locations super-imposed on surface grid.

Running Lengths and Acreage Heating ModelsOnce the inviscid surface flow field, streamline

patterns, stagnation point, and attachment lines aredefined the viscous solution over the acreage of the vehicle can be easily estimated. The running lengths,from either the stagnation point or attachment lines, are calculated using the streamline patterns. The running lengths, the inviscid solution, and either a laminar or turbulent flat plate reference enthalpy method (Eckert),are used to solve for the skin friction coefficient. Using Reynold’s analogy, the Stanton number is calculated and the convective heat rate is set equal to the radiation heat rate, where a non-conducting wall is assumed.

4wallconvective Tq σε=&

This non-linear equation is solved for wallT . Once

again, all thermodynamic properties and relationships are based on an equilibrium air model.

The choice of laminar or turbulent flow is based on

a user supplied edgeM/Reθ criterion for transition. The

transition length is set equal to the laminar run length, and a linear transition from laminar to turbulent flow is assumed over the transition region.

Stagnation and Attachment Line HeatingFor super-hypersonic Mach numbers, a Fay-Riddell

analysis with a Lee’s distribution is used to define the stagnation and off-stagnation heating. The typical Fay-Riddell analysis requires the specification of the stagnation point radius of curvature. However, this requirement is actually driven by the assumption of a local Newtonian flow field. The Fay-Riddell formulation truly requires the local surface velocity gradient, which, for a Newtonian flow field, is inversely

proportional the square root of the local surface curvature.

Rds

duq

1∝∝&The present formulation does away with any

estimate of the local surface curvature and instead estimates the velocity gradient at the stagnation point directly from the local surface velocity magnitudes known at each triangle. With the stagnation velocity gradient known, the stagnation heating can be estimated.

Along the attachment lines, the same stagnation type heating analysis is performed. However, now the surface velocity gradients are evaluated at each surface triangle identified to lie on an attachment line. Once the stagnation and attachment line heating rates have been evaluated, a running length offset is calculated for the stagnation point and for each identified attachment triangle.

lineattachmentplateflat qqSo

__&& =

Here lineattachmentq _& is known, calculated using the Fay-

Riddell like analysis. So is an unknown running length offset. So is solved for, and used to offset the running lengths. This formulation guarantees that the stagnation or attachment line heating, and the acreage heating models are zero order continuous. Figure 11 presents a laminar, Mach 6, 30 degrees angle of attack solution on the X-33 configuration.

Fig. 11 Temperature Contours for a Mach 6, 30 degrees Angle of Attack, laminar condition.

Base DragThe base drag model is a simple semi-empirical

correlation of base pressure coefficient with Mach number. For hypersonic Mach numbers the base pressure varies inversely with the Mach number squared.

Control Surfaces

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CBAERO allows for an arbitrary number of user defined control surfaces. Control surfaces are interactively defined by selecting individual surfaces or groups of triangles using the CBSETUP tool provided with the CBAERO package. For subsonic Mach numbers the deflection of the local geometry about the user defined hinge line is approximated using linearized boundary conditions and small angle assumptions. For super-hypersonic Mach numbers the surface triangles are truly rotated about the hinge line as no requirement for surface continuity is required when using the independent panel methods. Streamline patterns and thermal solutions are based on the un-deflectedgeometry.

ImplementationCBAERO is coded in ANSI C++, using an object

oriented approach. The code runs in parallel on shared memory systems using a simple C/C++ forking approach where a matrix of M-Q-α cases is efficiently distributed over a user specified number of processors. The cases are distributed over angle of attack first, then Mach number, and finally dynamic pressure to reduce the overhead of calculating quantities that are only functions of angle of attack, or Mach, or dynamic pressure. For instance, the super-hypersonic surface flow field approximation is only dependent on angle of attack and not the Mach number or dynamic pressure. Near linear speed up for 64 processors is typically observed.

ResultsResults for a Mach 6, 30 degrees angle of attack,

laminar condition are presented in figures 12-17 for the X-33 configuration. The surface grid contains approximately 150k triangles, and a single solution requires approximately 152 seconds of CPU time on a 2.6 GHz Pentium 4 desktop PC. The heat transfer coefficient, scaled by the stagnation value, is plotted for various longitudinal stations and down the vehicle centerline. The experimental data [reference 4] covers only the wind-ward surface of the vehicle, while CBAERO results for both the windward and leeward sides of the vehicle are shown. The chine heating is clearly visible in figures 12 and 13. The increased heating on the fin leading edge is visible in Figures 14 through 16. Figure 17 shows the distribution down the centerline, starting at the stagnation point and extending to the aero-spike nozzle.

Figures 18-24 compare the present methods predictions with experimental data [reference 5] for the integrated lift, drag, and moment coefficients for the HL-20 configuration for subsonic through hypersonic Mach numbers. The surface grid contains approximately 10k triangles. A typical subsonic

solution takes approximately 80 seconds on a 2.6 GHz Pentium 4 desktop PC. A typical supersonic/hypersonic solution takes approximately 6 seconds on a 2.6 GHz Pentium 4 desktop PC. At Mach 0.3 the fast panel formulation compares nicely with the experimental data for lift and drag, but is off by nearly a constant amount in the moment. As the Mach number is increased to 0.6 and 0.9 the fast panel formulation continues to compare nicely with the experimental data for lift and drag, however the prediction of the moment becomes progressively worse.

As the Mach number is increased beyond Mach 1, the formulation switches to the independent panel methods. While Mach 1.2 is well outside the ‘appropriate’ range for the independent panel methods, the results here (Tangent Cone) compare surprisingly well with the experimental data. As the Mach number is increased, the comparison with the experimental data continues to improve.

X/L = 0.3, Mach 6, 30 degress AoA, Re = 3.6e6

0

0.1

0.2

0.3

0.4

0.5

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2Y/L

H/H

ref

Wind Tunnel DataCBAERO

Fig. 12 Heating comparison, Mach 6, 30 degrees Angle of Attack, Laminar, X/L = 0.3

X/L = 0.5, Mach 6, 30 degrees AoA, Re = 3.6e6

0

0.1

0.2

0.3

0.4

0.5

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Y/L

H/H

ref

Wind TunnelCBAERO

Fig. 13 Heating comparison, Mach 6, 30 degrees Angle of Attack, Laminar, X/L = 0.5

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X/L = 0.7, Mach 6, 30 degrees AoA, Re = 3.6e6

0

0.2

0.4

0.6

0.8

1

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

Y/L

h/h

ref

Wind TunnelCBAERO

Fig. 14 Heating comparison, Mach 6, 30 degrees Angle of Attack, Laminar, X/L = 0.7

X/L = 0.8, Mach 6, 30 degrees AoA, Re = 3.6e6

0

0.2

0.4

0.6

0.8

1

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Y/L

H/H

ref

Wind TunnelCBAERO

Fig. 15 Heating comparison, Mach 6, 30 degrees Angle of Attack, Laminar, X/L = 0.8

X/L = 0.9, Mach 6, 30 degrees AoA, Re = 3.6e6

0

0.2

0.4

0.6

0.8

1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

Y/L

H/H

ref

Wind TunnelCBAERO

Fig. 16 Heating comparison, Mach 6, 30 degrees Angle of Attack, Laminar, X/L = 0.9

Centerline, Mach 6, 30 degrees AoA, Re = 3.6e6

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

X/L

H/H

ref

Wind Tunnel DataCBAERO

Fig. 17 Heating comparison, Mach 6, 30 degrees Angle of Attack, Laminar, Centerline

Fig. 18 HL-20, Mach 0.3

CLMach 0.3

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-5 0 5 10 15 20 25 30 35

Alpha, deg

CL

Wind TunnelCBAERO

L/D vs CLMach 0.3

-3

-2

-1

0

1

2

3

4

5

-0.2 0 0.2 0.4 0.6 0.8 1

CL

L/D

Wind TunnelCBAERO

CL vs CDMach 0.3

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5CD

CL

Wind TunnelCBAERO

Cm vs AlphaMach 0.3

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

-5 0 5 10 15 20 25 30 35

Alpha, deg

Cm

Wind TunnelCBAERO

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Fig. 19 HL-20, Mach 0.6

Fig. 20 HL-20, Mach 0.9

CLMach 0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-5 0 5 10 15 20 25 30 35

Alpha, deg

CL

Wind TunnelCBAERO

L/D vs CLMach 0.6

-4

-3

-2

-1

0

1

2

3

4

-0.2 0 0.2 0.4 0.6 0.8 1

CL

L/D

Wind TunnelCBAERO

CL vs CDMach 0.6

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

CDC

L

Wind TunnelCBAERO

Cm vs AlphaMach 0.6

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-5 5 15 25 35

Alpha, deg

Cm

Wind TunnelCBAERO

CLMach 0.9

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-5 0 5 10 15 20 25 30 35

Alpha, deg

CL

Wind TunnelCBAERO

L/D vs CLMach 0.9

-4

-3

-2

-1

0

1

2

3

4

-0.2 0 0.2 0.4 0.6 0.8 1

CL

L/D

Wind TunnelCBAERO

CL vs CDMach 0.9

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

CD

CL

Wind TunnelCBAERO

Cm vs AlphaMach 0.9

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

-5 5 15 25 35

Alpha

Cm

Wind TunnelCBAERO

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Fig. 21 HL-20, Mach 1.2

Fig. 22 HL-20, Mach 2.0

CLMach 1.2

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-10 -5 0 5 10 15 20 25 30 35

Alpha, deg

CL

Wind TunnelCBAERO

L/D vs CLMach 1.2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

CL

L/D

Wind TunnelCBAERO

CL vs CDMach 1.2

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8

CDC

L

Wind TunnelCBAERO

Cm vs AlphaMach 1.2

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-5 5 15 25 35

Alpha

Cm

Wind TunnelCBAERO

CLMach 2.0

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-5 0 5 10 15 20 25

Alpha, deg

CL

Wind TunnelCBAERO

Cm vs AlphaMach 2.0

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

-5 5 15 25 35

Alpha

Cm

Wind TunnelCBAERO

L/D vs CLMach 2.0

-1

-0.5

0

0.5

1

1.5

2

-0.2 0 0.2 0.4 0.6 0.8

CL

L/D

Wind TunnelCBAERO

CL vs CDMach 2.0

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4

CD

CL

Wind TunnelCBAERO

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Fig. 23 HL-20, Mach 4.5

Fig. 24 HL-20, Mach 20.

CLMach 4.5

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-5 0 5 10 15 20 25 30 35

Alpha, deg

CL

Wind TunnelCBAERO

L/D vs CLMach 4.5

-1

-0.5

0

0.5

1

1.5

2

-0.2 0 0.2 0.4 0.6

CL

L/D

Wind TunnelCBAERO

CL vs CDMach 4.5

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5

CDC

L

Wind TunnelCBAERO

Cm vs AlphaMach 4.5

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-5 5 15 25 35

Alpha

Cm

Wind TunnelCBAERO

CLMach 20.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30 35

Alpha, deg

CL

Wind TunnelCBAERO

L/D vs CLMach 20.

-1

-0.5

0

0.5

1

1.5

2

-0.2 0 0.2 0.4 0.6

CL

L/D

Wind TunnelCBAERO

CL vs CDMach 20.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5

CD

CL

Wind TunnelCBAERO

Cm vs AlphaMach 20.

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

-5 0 5 10 15 20 25 30 35

Alpha

Cm

Wind TunnelCBAERO

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Concluding RemarksA new software package for predicting the aero-

thermal environment of conceptual aerospace vehicles has been presented. The present method makes use of unstructured surface triangulations to define the geometry, which greatly eases the geometry setup time and allows the automation of the grid generation process using one of the many commercially available surface grid generation packages. The current contains a fast, unstructured, multi-pole panel method for prediction of subsonic flow fields. For super-hypersonic Mach numbers various independent panel methods are used. For both the subsonic and super-hypersonic Mach numbers a streamline tracing method, in coordination with an automatic attachment line detection method is used to define the surface streamline pattern and the subsequent viscous surface solution based on laminar and turbulent flat plate methods. Results for both integrated force and moment data and distributed heating data are compared to available experimental data.

References[1] David N. Kenwright, “Feature Extraction of

Separation and Attachment Lines”, IEEE Transactions on Visualization and Computer Graphics, Vol. 5, No. 2, April-June 1999

[2] J.E., Barnes, “N-Body Models of Collisionless Systems”, volume 1, chapter 8, Springer-Verlag, 1996

[3] J.E., Barnes, and Hut, P., “A hierarchical O(N Log N) force calculation algorithm”, Nature, 324(4):446-449, December 1986

[4] H. Harris Hamilton II, K. James Weilmuenster, Thomas J. Horvath, and Scott A. Berry, “Computational/Experimental Aeroheating Predictions for X-33 Phase II Vehicle”, AIAA 98-0869

[5] Cruz, C., and Ware, G. “Predicted Aerodynamic Characteristics of HL-20 Lifting Body Using Aerodynamic Preliminary Analysis System (APS)”, AIAA-92-3941, July 1992

[6] Tauber, Michael E., “A Review of High-Speed Convective, Heat-Transfer Computation Methods”, NASA TP 2914, 1989

[7] Bonner, E., Clever, W., and Dunn, K., “Aerodynamic Preliminary Analysis System, II”, NASA-CR-165627, April 1981

[8] Brian Maskew, “Program VSAERO Theory Document – A Computer Program for Calculating Nonlinear Aerodynamic

Characteristics of Arbitrary Configurations”, NASA CR-4023, September 1987

[9] J.L. Hess, and A.M.O. Smith, “Calculation of Potential Flow About Arbitrary Bodies”, Progress in Aeronautical Sciences, Vol. 8, 1967, Pages 1-138

[10] B. Maskew, B.M. Rao, and F.A. Dvorak, “Prediction of Aerodynamic Characteristics For Wings with Extensive Separations”, Paper No. 31 in ‘Computation of Viscous-Inviscid Interactions’, AGARD CP-291, February 1981

[11] Harry W. Carlson, Marcus O. McElroy, Wendy B. Lessard, and L. Arnold Mc Cullers, “Improved Method for Prediction of Attainable Wing Leading-Edge Thrust”,NASA TP 3557, April 1996