14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006-8087

Modeling and Simulation of a Generic HypersonicVehicle using Merged Aerodynamic Models

Shahriar Keshmiri * and Richard Colgren §

The University of KansasDepartment of Aerospace Engineering

2120 Learned Hall, Lawrence, KS 66045

Maj Mirmirani ¥

California State University, Los AngelesCollege of Engineering, Computer Science, and Technology

California State University, Los Angeles5151 State University Drive, Los Angeles, CA 90032

Abstract

This paper covers the development of a six degrees of freedom simulation of a generichypersonic vehicle (GHV) using a merged aerodynamic database. The experimentalinvestigation of the aerodynamic characteristics of the blunt body GHV configuration is used asthe core of the aerodynamic model. The gaps in the wind tunnel data have been filled using thebest available CFD results. The new aerodynamic model results in a more realistic simulation.Results for longitudinal flight conditions are shown in this paper. The simulation includes both air-breathing and rocket propulsion engine cycles. This work is developed to support conceptualhypersonic vehicle design studies and related aerospace vehicle technologies.

Nomenclature

CD Drag CoefficientCL Lift CoefficientDOF Degrees Of Freedomq The quaternion for the rotational tensor of the body frame w.r.t. the Earth frame

QBE The rotational tensor of the body frame w.r.t. the Earth frame

BEΩ The angular velocity quaternion of the body frame w.r.t. the Earth frame

[E] Identity matrixp Roll ratePLA Pilot Lever Angleq Pitch rater Yaw rateT Thrustw.r.t. With Respect To

Introduction

This paper presents the development of a six degrees of freedom simulation of a generichypersonic vehicle (GHV) [1] based on aerodynamic data from a variety of sources.

* PhD Student, Department of Aerospace Engineering, University of Kansas, Member AIAA.§ Associate Professor of Aerospace Engineering, University of Kansas, Associate Fellow AIAA.

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA 2006-8087

Copyright © 2006 by Shahriar Keshmiri, Richard Colgren and Maj Mirmirani. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006-8087

¥ Professor of Mechanical Engineering, Chair of the M. E. Department Cal. State University, LosAngeles, Member AIAA.

GHV steady state simulation results using an experimental investigation of the GHV’saerodynamic characteristics are compared with CFD code generated aerodynamic models. Thisapproach helps validate the CFD results, which have been generated using APAS and STARScodes. In order to operate through all Mach regimes, a combined-cycle propulsion system hasbeen developed. The engine model and the simulation include both air-breathing and rocketpropulsion modes.

Kinematics and Dynamic Equations for the GHV

Equation 1 gives the time derivative of the vectors as transformed from the A to the B coordinatesystem as

A B

d s d ss

d t d t

= + ×

ω (1)

where ω is the angular velocity between the B and A coordinate systems.

[ ] [ ] [ ]A AB Bs T s= × (2)

Taking derivative of (2) gives:

[ ] [ ]A AB B

B ABds dT dss T

dt dt dt = × + ×

(3)

Factoring out [ ]ABT from (3) yields:

[ ] [ ]A AB B

ABAB Bds dT dsT T s

dt dt dt

= × × + (4)

whereABAB 1T T− = . Also, [ ]

BAAB 1AB BAdT dT

T Tdt dt

− × = × . This results in:

[ ] [ ] [ ]BAA 1 B

AB BA Bds dT dsT T s

dt dt dt

− = × × + (5)

The angular velocity vector ω is:

[ ]BA1

BA dTT

dt

− = ×

ω (6)

Body Coordinate System

The system is constructed using the three standard Euler’s angle transformations: yaw, pitch and

roll (ϕ, θ, ψ).

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006-8087

[ ] [ ] [ ] [ ]XGYXBYBG TTTT )()()( ψθφ= (7)

These three rotations provide the transformation to the body axes representation.

[ ]

−++−

−=

φθφψφθψφψφθψφθφψφθψφψφθψ

θθψθψ

coscossincoscossinsinsinsincossincos

sincoscoscossinsinsincossinsinsincos

sincossincoscosBGT (8)

The Flat-Earth Equations of Motion

The equations for a flight vehicle subject to aerodynamic and proportional forces fa,p and thegravitational force mg as simulated are discussed next. Newton’s 2nd Law with respect to theinertial frame “I” states that the time rate of change of linear momentum equals the externallyapplied forces, which are the aerodynamic and proportional forces fa,p and the gravitational forcemg. This is written as in equation 9:

I IB a,pmD v f mg= + (9)

IBv is the velocity of center of mass with respect to the inertial reference frame I. The flat earth

assumption allows us to use the Earth coordinate frame ’E’ as an inertial frame. Therefore:

E EB a ,p

E E B E BE EB B B

B E BE EB B a ,p

mD v f mg

mD v m D v v

m D v v f mg

= +

= + Ω + Ω = +

(10)

where ΩBE is the angular velocity between the coordinate system]A and ]B. To generate theordinary time derivative, all terms should be express in the preferred coordinate system ]B.

[ ] [ ] [ ] [ ]BBpa

BEB

BBEVB gmfvm

dt

vdm +=Ω+

, (11)

The gravitational acceleration [ ]Bg is modeled in the flat Earth system as [ ] [ ]ggL

00= .

To transform the ]B coordinate system into ]L coordinate system just apply equation (1). The

transformation equation in matrix form becomes:

[ ] [ ] [ ] [ ] [ ]LBLBpa

BEB

BBEVB gTmfvm

dt

vdm +=Ω+

, (12)

which is written in coordinate form as:

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006-8087

[ ], 1

, 2

, 3

0 0

0 0

0

BBB B L

a pBL

a p

a p

dudt r q u f

dvm r p v f Tdtq p w f mgdw dt

− + − = + −

(13)

The equivalent differential equations are:

, 113

, 223

, 333

a p

a p

a p

fdurv qw t g

dt mfdv

pw ru t gdt m

fdwqu pv t g

dt m

= − + +

= − + +

= − + +

(14)

Euler’s law states that the time rate of change of angular momentum equals the externally applied

moments. This law governs the rotational degrees of freedom. E is picked as an inertial

reference frame for equation 15.

BBEB

BE MID =ω (15)

Transforming the rotational derivative into the body frame:

BBEB

BBEBEB

BB MIID =Ω+ ωω (16)

and then expanding the angular momentum vector results in the following:

( )B B BE B B BE BE B bB B bD I I D D I= +ω ω ω (17)

The term BB

BID is zero as we assumed that the airplane is adequately modeled as a rigid body.

BEBBB

BEBB

B DIID ωω =)( (18)

Equation 16 becomes:

BBEB

BBEBEBB

B MIDI =Ω+ ωω (19)

The body axis coordinate system is chosen as it expresses the moment of inertia tensor in a

constant form [4].

[ ] [ ] [ ] [ ] [ ]BB

BBEBBB

Bbe

BBEBB

B MIdt

dwI =Ω+

ω (20)

( ) [ ]( )1BBE

B B B B BB be B BEB B B

dwI I M

dt

− = − Ω +

ω (21)

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006-8087

Generic Hypersonic Vehicle Description

The GHV mass model is based on the assumption of a rigid vehicle structure. The equations ofmotion used in the simulation account for the time varying center of mass, center of gravity, andmoments of inertia experienced during the flight of such a vehicle. The total mass of the vehicle,its c.g. location, and the products of inertia vary as fuel is consumed. It is assumed that the c.g.moves only along the body x-axis as fuel consumed. Fuel slosh is not considered, and productsof inertia are assumed to be negligible. A sizing analysis discussed below of the generichypersonic vehicle studied yielded a full-scale gross weight of 300,000 lbs and an overallfuselage length of 200 ft. The three-view drawing of the vehicle is given in Figure 1. Deflectionsof the elevons are measured with respect to the hinge line (perpendicular to the fuselagecenterline). A fuselage-centerline-mounted vertical tail has a full span rudder with its hinge line at25 percent chord from the trailing edge. Deflections of the rudder are measured with respect toits hinge line. Positive deflections are trailing edge left. Small canards (65 A series airfoil) aredeployed at subsonic speeds for improved longitudinal stability and control [1].

Figure 1: The GHV 3D Model

Aerodynamic Model - APAS

For simulation of the aerodynamic forces and moments, the output data from asubsonic/supersonic/hypersonic analysis code developed jointly by NASA Langley and RockwellInternational Incorporated, the Aerodynamic Preliminary Analysis System (APAS), are used [1].The APAS code is often used in conceptual design studies, due to its short processing timerequirements and the relatively good results it provides. APAS is actually a front end to twoseparate analysis codes, Unified Distributed Panel (UDP) and Hypersonic Arbitrary BodyProgram (HABP). APAS uses UDP to analyze subsonic and supersonic runs, and HABP toanalyze hypersonic runs [1]. The drag rises dramatically as the vehicle nears the speed ofsound. This change is caused by the shock waves and the wave drag. One of the methods usedto reduce the wave drag for an aerospace vehicle configuration is known as transonic area ruling,first theorized by D. Hayes. Richard T. Whitcomb at the NASA Langley Aeronautical Laboratorydeveloped this theory [1]. Wallace and Whitcomb developed the qualitative method by which therule is actually used to find the wave drag. The transonic area rule states that the wave drag ofan aircraft is essentially the same as the wave drag of an equivalent body of revolution having thesame cross-sectional area distribution as the aircraft [1]. APAS analysis can be done relativelyquickly, allowing multiple design iterations. The results have been shown to be within twentypercent of actual values. APAS is good enough for conceptual design studies, and the speedwith which results can be generated allows the designer to include aerodynamic calculationswithin multi-disciplinary design optimization loops. This method works reasonably well intransonic flight regimes when slender body theory is applied to the equivalent body of revolution.This method is employed in APAS to calculate the drag coefficient in the transonic region. Usingthe APAS software, aerodynamic data are computed at Mach numbers of 0.3, 0.7, 4.0, 6.0, 10.0,15.0, 20.0, and 24.2 and at angles of attack of -1°, 0.0°, 2°, 4°, 6°, 8°, 10°, and 12°. For eachMach number and angle of attack combination, coefficients are generated for a range of

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006-8087

deflections of the right elevon, the left elevon, and the rudder. Each deflection is takenseparately. Deflections of -20 °, -10°, 0.0°, 10°, and 20° are used for each control surface. Attwo subsonic Mach numbers, 0.3, 0.7, and eight angles of attack, -1°, 0.0°, 2°, 4°, 6°, 8°, 10°, and12°, data are estimated for canard deflections of -10°, -5°, 0°,5°, and 10°.

Aerodynamic Model - Wind Tunnel

Experimental longitudinal and lateral-directional aerodynamics were obtained for the GHVconfiguration. Data were obtained at Mach numbers of from 0.6 to 20.0; Reynolds numbers,

based on model length, between6 62.5 l0 and 5.3 l0× × ; and angles of attack from -4 degrees to 20

degrees. As was previously stated, the proposed GHV is expected to take off from a conventionalrunway, perform an atmospheric acceleration using primarily airbreathing propulsion to achieve alow-Earth orbit, re-enter, and land on a runway. Extensive use of computational fluid dynamics(CFD) codes is anticipated for the pre- flight analyses of this class of vehicle at different speeds,altitudes, and flight path angles. This increased reliance on CFD codes for future spacetransportation systems does not reduce the importance of experimental investigations. Theprimary objective of the NASA Langley wind tunnel experiments was to provide timely force andmoment, flow visualization, and thermal mapping measurements across the Mach number rangefrom M = 0.6 to 20.0. Other objectives were to determine what, if any, modification to the testtechniques used would be required to perform such a fast-paced study, and to provide earlyexperimental data for comparison with engineering design codes such as HABP.

Development of the Aerodynamic Database

An overview of the aerodynamic characteristics, along with the process for developing anaerodynamic database, for the Generic Hypersonic Vehicle (GHV) is presented in this section.The experimental investigation of the aerodynamic characteristics for the GHV as presented inthe previous section is used as the core of the database. The gaps in the wind tunnel data havebeen filled using the best available CFD results. The database is generated and tabulated aslook-up tables for different speed regimes. In the control system design and trajectoryoptimization process, look-up tables are not the most suitable. A mathematical method is appliedto each look-up table to find the best analytical expression for each aerodynamic coefficient.Multi-variable curve fitting is used. Acquiring an accurate curve to match a set of highly nonlineardata can become a challenging and time consuming process. Because of this, interpolation orextrapolation of values from the look-up tables is a more commonly used method forimplementing these aerodynamic models.

Curve Fitting the Aerodynamic Data using Multiple Variables

For our simulation application each aerodynamic coefficient is a function of several independentvariables. Examples are the vehicle angle of attack, the Mach number, the sideslip angle, andthe deflection of the control surfaces. It is desired to find an approximate relationship of the form:

( )1 2 3 mx , x , x ,..., xy y=(22)

The curve fitting procedure cannot be performed unless a linear or nonlinear relationship betweenthe independent variables is used. An example is the linear relationship as follows:

0 1 1 2 2 3 3 m+ x x x ... xmy a a a a a= + + + + (23)

The coefficients a0, a1, a2, … , am are determined using the data points available from theaerodynamic look-up tables. In the linear case, the sum of the square errors (S) can beexpressed as:

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006-8087

( )2

0 1 1,i 2 2,i 3 3,i m,i1

- x x x ... xn

i mi

S y a a a a a=

= − + + + +∑(24)

To minimize the sum of square errors (S), the partial derivatives of S with respect to theindependent variables are set equal to zero. This is shown in equation 25.

1, 1, 1, 02

1, 1, 1, 2, 1, , 1 1,2

2, 1, 2, 2, 2, 2, 2 2,

2, 1, , 2, , , ,

i i i i

i i i i i m i i i

i i i i i i i i

m i i m i i m i m i m m i i

n x x x a y

x x x x x x a x y

x x x x x x a x y

x x x x x x a x y

=

∑ ∑ ∑ ∑∑ ∑ ∑ ∑ ∑∑ ∑ ∑ ∑ ∑

∑ ∑ ∑ ∑ ∑

K

K

K

M M M M M M M

K(20)

In this research, multiple relationships between the independent variables are investigated to findthe best possible fit to the tabular datasets. During the fitting process, it is required to determinethe quality of the fit. For fast and efficient calculation of the coefficients, a MATLAB program isdeveloped based on the sum of square errors (S). This statistic measures the total deviation ofthe response values from the fit. It is also called the summed square of the residuals. A valuecloser to zero indicates a better fit. Thus, the analytical expression for each aerodynamiccoefficient is generated using a MATLAB program for each speed regime. A completemathematical model of the GHV is developed applying this technique. It is used in the modelingand simulation of the vehicle. The aerodynamic model calculator function, fitternew.m, isdesigned to calculate the aerodynamic analytical expression for the clean vehicle, or for thevehicle with a control surface deflection.

Analytical Procedure

The aerodynamic modeling procedure is done for all coefficients and the analytical results arecollected in a MATLAB function (aero.m). This function works as a complete aerodynamic modelof the GHV. This approach makes each element of the simulation a separate module. For anyother hypersonic vehicle with different configuration, the aerodynamic module can be generatedby applying the MATLAB code fitternew.m to the aerodynamic look-up tables. This keeps themodeling and simulation routine robust and flexible.

Engine

Horizontal take-off and landing vehicles continue to be of great interest for future space launchmissions. For a hypersonic vehicle to operate over all Mach regimes, a combined-cyclepropulsion system is the most promising concept. In this research, a ramjet/scramjet systemconsisting of a dual-cycle airbreathing core with a variable geometry inlet is discussed. Thiscombined cycle engine model can be used within any hypersonic vehicle conceptual designframework. The propulsion model for this configuration study is developed using a twodimensional forebody. A one dimensional combustor model is used. This engine model isimplemented within MATLAB. The effect of both flight path angle and angle of attack areinvestigated.

Turbojet Engine: 0.00 < Mach < 2.0

Ramjet-Scramjet Engine: 2.00 < Mach < 6.0

5 1 -4 2 -10 3 3 3Thrust PLA (2.99 10 - 1.00 10 (alt) 1.33 10 (alt) - 6.48 10 (alt) 3.75 10 (Mach) )= × × × × + × × × × + × ×

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006-8087

Liquid Rocket Engine: 6.00 < Mach < 10.0 (altitudes greater than 60,000 ft)

Trajectory

Using the method discussed in reference [2], the flight trajectory of the GHV is chosen within arange of dynamic pressures of from 500 to 2260 psf. The variation in dynamic pressure atsupersonic and hypersonic speeds is shown in Figure 2. Keeping the dynamic pressure in thisrange helps avoid large forces and drag values. As airbreathing engines generate thrust in directproportion to the mass flow rate, if the GHV flies faster at a constant dynamic pressure theavailable free stream mass flow rate per unit area is reduced. When the trajectory of the vehicleis designed to maintain the maximum mass flow rate while the Mach number is increasing, it isalso designed to fly at a constantly increasing dynamic pressure. This is shown in Figure 3.

Figure 2: Variation in Dynamic Pressure

Altitude versus Flight Mach Number for ConstantFreestream Mass Flow per Unit Area

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

0 5 10 15 20 25 30

Flight Mach Number

Alt

itu

de

(ft)

mass f l ow r at e=1 psf / sec

mass f l ow r at e=10 psf / sec

mass f l ow r at e=100 psf / sec

q0=500 psf

q0=1000 psf

q0=2000 psf

Figure 3: Mass Flow Rate Constraint

( ) ( ) ( )2 33 3 2 2Thrust PLA (-1.8585 10 + 2.6294 10 Mach - 9.5423 10 Mach +1.0834 10 Mach )= × × × × × × × ×

( ) ( ) ( ) ( )4 -1 5 -1Thrust -5.43 10 + 6.64 10 alt + 3.24 10 PLA + 3.74 10 ( alt PLA )= × × × × × × × ×

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006-8087

All the previous considerations are brought into account when generating the trajectory for theGHV. The result is shown in Figure 4. The ascent trajectory is designed to requirements andconstraints different from the descent trajectory. The main concern in descent is the temperatureseen on the body of the vehicle. In ascent, the effects of regenerative cooling, the fuel air ratio,and total the body temperature are all included as design factors.

Figure 4: Flight Trajectory

Steady-State or Trim Simulation Results

Following figures show the trim results for the GHV using a combination of CFD and experimentalaerodynamic data. As shown in Figure 5, the highest values for CD occur in the transonic region.The greatest thrust is needed in this flight regime to overcome the resulting drag force. The liftcoefficient decreases at supersonic speeds. This effect on the required thrust is shown in Figure6.

Figure 4: Variation in the Drag Coefficient

Trajectory

0

20000

40000

60000

80000

100000

120000

0 5 10 15 20Flight Mach Number

Alti

tude

(ft) Ascent

Level flight

Descent

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006-8087

Figure 5: Thrust versus Flight Mach Number

A large variation is seen in the trimmed angle of attack through the transonic region. Thetrimmed angle of attack increases at hypersonic speeds as the flight path angle decreases (seeFigure 7). Figure 8 shows the decreasing trend in the pitch angle with increasing Mach number.

Figure 6: Variation in Angle of Attack with Flight Mach Number

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006-8087

Figure 8: Variation in Pitch Angle with Flight Mach Number

Summary

This paper covered the development of a six degrees of freedom simulation of a GenericHypersonic Vehicle (GHV). The model and simulation are developed to support conceptualdesign studies of hypersonic vehicles with multiple cycle engines. The aerodynamic database forthe Generic Hypersonic Vehicle uses experimental data and multiple CFD codes. The simulationincludes both air breathing and rocket propulsion modes. Air breathing modes include turbojet,ramjet, and scramjet cycles.

References

[1] W. Pelham Philips, Gregory J. Brauckmann and William C. Woods, “ExperimentalInvestigation of the Aerodynamic Characteristics for a Winged-Cone Concept,” AIAA 87-49098, 1987.

[2] Shahriar Keshmiri, Richard Colgren and Maj Mirmirani, “Development of an AerodynamicDatabase for a Generic Hypersonic Air Vehicle,” AIAA 2005-35352, August 2005.

[3] Sang Bum Choi, “Investigation of the Aerodynamic Characteristics of the Generic HypersonicVehicle, Winged-Cone Configuration, by STARS CFD Codes,” MFDCLAB, Los Angeles,California, 2005.

[4] Peter H. Zipfel, “Modeling and Simulation of Aerospace Vehicle Dynamics,” AIAA EducationalSeries, 2000.

[5] Frank L. Lewis and Brian L. Stevens, “Aircraft Control and Simulation,” Wiley, 1992.

[6] Jan Roskam, “Airplane Flight Dynamics and Automatic Flight Control Part I,” DARCorporation, 1997.

[7] E. T. Curran and S. N. B. Murthy, “Scramjet Propulsion,” Department of the Air Force (Editor),Purdue University.

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA-2006-8087

[8] “Conceptual Design of the OREAD EXPRESS: TransAtmospheric Cargo (TAC) Vehicle,” TheUniversity of Kansas Propulsion Design Team, 1991/1992 AIAA Air Breathing PropulsionCompetition, June 1992.

[9] John H. Blakelock, “Automatic Control of Aircraft and Missiles,” Wiley, 1991.

[10] Paul Dierckx, “Curve and Surface Fitting with Splines,” Oxford University, 1995.

[11] Philip George, “Numerical Methods of Curve Fitting,” Cambridge [Eng.] University Press,1961.

[12] Shahriar Keshmiri, Maj D. Mirmirani and Richard Colgren, “Six-DOF Modeling andSimulation of a Generic Hypersonic Vehicle for Conceptual Design Studies,” AIAA-2004-4805, August 2004.