c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

AIAA Modeling and SimulationTechnologies Conference and Exhibit6-9 August 2001 Montreal, Canada

AIAA-2001-4421

A01-37335

A Missile Flight Simulation Using Interdisciplinary Coupling

*Mel HumanNC A&T State Univ.

Greensboro, NC

Abstract

In this paper, we are concerned with inves-tigating an interdisciplinary problem which in-volves more than two disciplines. Specificallywe would like to include an example whichcontains the key factors involved in designingnext generation aircraft configurations, namelystructural-flow field coupling, propulsion, andcontrol issues. The goal was to select a modelthat is relatively tractable in terms of the ana-lyzing the individual disciplinary modules, thusallowing for the primary effort to be concernedwith the coupling methodology.

The example described here is a simplifiedsolid fuel ramjet missile. Each discipline as-sociated with the missiles functionally can beapproximated well with closed form and simplenumerical approaches. The flow field is approx-imated by Taylor- Maccoll conic flow which al-lows for a simple calculation for determiningthe inlet conditions for the engine combustionprocess. We establish a coupling mechanismbetween these three systems by assuming thatthe missile has very low radial stiffness. Ac-cordingly, we allow the structure's radius tofluctuate depending on the chamber pressure;

this in turn alters the cone angle which affectsthe inlet conditions.

Results to date have been completion of theexternal flow field calculation module, the dy-namics module which integrates the equationsof motion, and response of the structure to thechanging internal pressure.

BACKGROUNDDesign and engineering of future

aerospace/aeronautical systems will re-quire a keen attention to coupling interactionsamong traditionally non related disciplines.The area of multidisciplinary design andoptimization (MDO) is a rapidly developingset of procedures and techniques for handlingsuch complex problems.

There has been a good deal of work bya number of investigators involving two dis-cipline coupling, primarily structural-fluid in-teractions. However, there has been signifi-cantly less attempts on more than two disci-pline problems1'2'3, particularly those includingcontrol applications

°*Assoc. Prof. AIAA Sr. Member

ICopyright̂ 2001 by theAmerican Institute of American Institute of Aeronautics & AstronauticsAeronautics&AstronauticsInc. All rights reserved."

c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

A major impediment in realistic problems isthat the disciplinary analysis modules such asflow or stress analysis must be numerical for-mulations such as finite element calculations.It then may be rather cumbersome to couplethe modules numerically. The problem selectedhere is a simple enough so that closed form re-sults from three computations - structure, flow,and propulsion, may be used in the formula-tion. The problem provides an excellent modelfor estimating coupling influences, introductionof control methodology, and ultimately, the ap-plication of MDO procedures. Note that oncethe design methodology is established, the an-alytical modules may be replaced with higherfidelity computational disciplinary blocks.

MISSILE MODEL

The generic shape design of the airframe isshown in Figure 1. The dimensions are char-acteristic of a solid fuel ramjet missile designfor operation in the mid supersonic range atan altitude of about 20km. Relevant physicalparameters include:

L/:forebody length — 0.5mInduct length — 0.75mLc: combustion chamber length = 1.0m<5:initial cone angle = 20oii inner radius = O.lmr0 outer radius = 0.2mIgnition for the device is initiated at a speed

of Mach 2.Flow FieldThe flow upstream of the engine intake will

be assumed to be governed by Taylor-Maccollflow. Solutions for downstream shock condi-tions are tabulated in a number of sources.These solutions4 were represented in a two di-mensional table lookup where the free streamMach number and cone angle are the inputs.With the sound speed (a) relationship5

aQ(1)

we may determine state properties as to allowthe mass flow rate intake.

Upon entry into the inlet, we select an expo-nential profile duct variation in a diffuser sec-tion A(x) = A$e~Px. This variation allows de-termination of an exact solution to the ductflow equation.

dM M(l +dx 1-M2

4/ l_dAA dx

whose solution becomes

1-M2

(2)

(3)

where Ciis determined with M(x — 0) = MQ.Note that we repeat the use of this equationat the downstream side of the duct shock, us-ing M2 as the initial condition. Normal shockwave equations are used for determining statevariations across the discontinuity. In the ductarea, the stagnation pressure change is deter-mined by

dP_P

7M2

(4)

The present study calculates the friction co-efficient / in a simple fashion, employing therelation6

1fl/2 = 21og[JRe/1/2]-0.8 (5)

where the Reynolds number is based on aver-aged flow properties of the duct. A more accu-rate variable flow area correlation is currentlybeing implemented.

The position of the duct's normal shock isiteratively determined by satisfying final com-bustion chamber conditions to the requiredmass flow.

Propulsion

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Using the calculated mass flow, we employan erosive burning rate correlation7

f(mm/s) = 0.0066ra°-628(#/s) (6)The corresponding fuel burn rate and energy

release becomera/ =Q = irifHvwhere Lc is the combustion chamber length.

The combustion heating is then assumed to bea stagnation temperature enhancement track-ing a Rayleigh curve process. The increase inTO is merely

AT° =

where the specific heat is calculated with a com-mercial combustion code.

The end result in the calculation then be-comes essentially reservoir conditions POJ^Owhich then drives the gas through the nozzleexpansion. As we desire shockless flow through-out the nozzle, we determine the required exitarea Ae as a function of the pressure ratioThe thrust is computed from

.

~ (ma 4 (8)and TP are determinedwhere ue =

from exit conditions.Structural InfluenceA simplified coupling of the flow/propulsion

environment with the missiles frame is as fol-lows. We assume that the structure is radiallyflexible so that perturbations in the internalpressure lead to alterations in the outer radius.A simple way of doing this is the use of Lame'sformula8 for a thick shell

(9)Ewhere E is an effective radial modulus. Thecoupling occurs as the cone angle will beslightly altered by

cscAr (10)

This will change the flow results in subse-quent time steps.

SIMULATION

The set of calculations described in the pro-ceeding sections are performed in a given timestep. Outputs are acceleration, a new outerradius due to the internal pressure, new in-ner radius due to fuel erosion, and new coneangle from the pressure deformation. A newvelocity is calculated due to the acceleration;this includes updating the C^ value from theTaylor-Maccoll solution. The process is re-peated throughout the trajectory. A similarsimulation procedure is found in reference 9.

Another important calculation is the deter-mination of the exit throat area ratio as to pre-vent shock formation within the exhaust duct.This parameter represents the control functionas to achieve the isentropic duct requirement.

RESULTS AND FUTURE ENHANCEMENTS

Results to date are demonstrated in Figures2-6 where key model calculations are summa-rized. The five time steps in Figs. 2-4 are at 10second intervals while showing axial variations.

Some general performance characterisitcs areshowm in Fugures 2 & 3. Figure 2 shows theinternal Mach number values where the freestream value is decelerated through the obliquecone shock, the normal duct shock, and the fric-tion retardation. One can see the progressivenature of the duct shock where subsonic valuesare reached. Because of the heating effects, theflow becomes choked and sonic conditions re-sult at the end of the combustion process. Thestagnation pressures are shown respectively in3, where the latter is normalized to the free

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stream stagnation pressure (external pressurewill occur over a wide range of values for dif-ferent altitudes and velocities).

Figure 4 compares the model' trajectory withthat of a rigid body, performance throughoutthe missile's trajectory. (These curves were lin-early smoothed within ten second intervals).Clearly structural flexibility as captured by themodel degrades the acceleration performance.

Figures 5 through 8 demonstrate the sensi-tivity of the missile dynamics with respect todesign parameters: initial cone angle, materialdensity and modulus, and an operational para-meter - altitude.

The trend of better performance for smallerinitial cone angles may be explained by Equa-tion 10 which suggests smaller angles give riseto smaller radial perturbations throughout theflight. In Figure 6, a 10% above/below vari-ation in the material density shows the great-est effect at the higher value, a significant de-crease in performance; using a lighter materialincreases accelearation only marginally. Simi-larly, Figure 7 shows how a stiffer material ap-proches rigid body performance.

The allowance for flexure is probably doesnot influence altitude selection very much asdeduced from Fig. 8. We have the expectedresult of having poor acceleration at the lowerlevel of denser air which dominated the oppos-ing effect of lower ambient pressure supportinghigher flexure levels. However, the effect maybesignificant in non constant altitude flight.

The baseline model presented in this paper iscurrently undergoing significant improvementsand refinements. These include

• replacement of the analytical flow solutionwith a CFD generated numerical solution

• determination of pressure induced defor-mations with a finite element structuralmodel

• CFD computation of internal flow withinthe duct and combustion chamber

• enhanced combustion model using speciegeneration and enthalpy variations

• time series system identification of modeldynamics for control plant construction

REFERENCES1. Dovi, A.R., el at,"Multdisciplinary De-

sign Integration Methodology for a Super-sonic Transport Aircraft", Journal of Aircraft.March 1995.

2. Malone, B., & W.H. Mason, "Multi-disciplinary Optimization in Aircraft DesignUsing Analytical Technology Models," AIAAJournal of Aircraft, March, 1995.

3. Raney, D.L, J.D. McMinn, & A.S. Po-totzky, "Impact of Aeroelastic-Propusive Inter-actions on Flight Dynamics of a Hypersonic Ve-hicle," AIAA Journal of Aircraft, March, 1995.

4. Shapiro, A.H., Dynamics and Thermo-dynamics of Compressible Flow, v. 2, RonaldPress Company, 1954.

5. Plett, E.G., & R.A. Stowe, "NumericalSimulation of Solid Fuel Ramjet Missile Propul-sive Performance," Joint Propulsion Confer-ence and Exhibit, July, 1995, San Diego.

6. Schlicting, H., Boundary Layer Theory,6th ed., McGraw-Hill, N.Y. 1966.

7. Zandbergen, B.T., " Comparison of The-oretical and experimental Results of the SolidFuel Combustion Chamber Project," Proceed-ings of High Speed Air Breathing Propulsion,June 1991, The Netherlands.

8. Baumeister, T., & L.S. Marks, StandardHandbook for Mechanical Engineers, 7th ed.,McGraw-Hill, NY., 1967.

9. Bauer, A., el at, "Simulation of En-gine/Aircraft Dynamic Behavoir for Hyper-sonic Flight Vehicles," ICAS # 94-6.7.2, Sept.1994, Los Angeles, Ca.

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BOW SHOCK

NORMAL SHOCK

Figure 1 - Missile Schematic

internal Mach

*oQ_

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Figure 2Internal Pressure

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Figure 35

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-20 0 20 40 60 80 100 120 140t (sec.)Figure 4

3.5

3

2.5

2

1.5

1

Initial Cone Angle

-20 0 20 40 60 80 100 120 140

E Sensitivityi ' ' ' i

-20 0 20 40 6,0 80 100 120 140

Figure 5

Figure 6

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Added Mass3.

-20 0 20 40 60 80 100 120 140Figure 7

Altitude

-20 0 20 40 60 80 100 120 140

Figure 8

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