[american institute of aeronautics and astronautics 43rd aiaa/asme/asce/ahs/asc structures,...

AIAA-2002-1307

AN INTEGRATED PLATFORM FOR THE

SIMULATION OF FLUID-STRUCTURE-THERMAL

INTERACTION PROBLEMS

Hai TRAN� and Charbel FARHATy

Department of Aerospace Engineering and Sciences

and Center for Aerospace Structures

University of Colorado at Boulder

Boulder, CO 80309-0429, U.S.A.

Abstract

We propose an integrated uid-structure-thermalsolver for the investigation of aerodynamic heatingand aerothermoelastic problems. The underlyingfour-�eld formulation of aerothermoelasticity is anextension of the three-�eld formulation of aeroelas-ticity established by C. Farhat and co-workers. Wediscuss issues pertaining to the enforcement of ther-mal boundary conditions at the uid/structure in-terface in the presence of a k � " turbulence modelwith a wall law function and the transmission ofthermal information on meshes with non-matchinginterfaces. We propose solutions for these problems.We illustrate our solution methodologies in perform-ing an aerodynamic heating analysis of the F-16'stwo-dimensional wing section and an aerothermoe-lastic stability analysis of a at panel.

1 Introduction

In the design of supersonic and hypersonic vehicles,problems of aerodynamic heating need to be care-fully addressed in order to determine an accurateprediction of the aerothermal loads, structural tem-

�Research associateyProfessor, AIAA FellowCopyright c 2002 by the authorsPublished by the American Institute of Aeronautics andAstronautics, Inc. with permission

perature, deformations and stresses. Indeed, the se-lection of materials of construction depends on theseresults. The implications of aerodynamic heatingcan range from simple deformations of structuralcomponents such as buckling12, to the alterationof the aeroelastic behavior of an aircraft. The lat-ter behavior has been demonstrated by experimentsconducted by the NACA on the aeroelastic e�ects ofaerodynamic heating, and reported by Hugh L. Dry-den and John E. Duber in a 1955 AGARD confer-ence7. In this experiment, an aluminum alloy wingof multiweb construction was placed in an air ow ofMach = 2. In the �rst test, the airstream had a stag-nation temperature of 500ÆF (260ÆC) and a chord-wise \ ag waving" type of utter was observed. Inorder to attribute the utter to aerodynamic heat-ing, a second test was conducted with a stagnationtemperature of 100ÆF (37ÆC). In this second test, no utter was observed.Aerodynamic heating is a coupled phenomenon

between the ow �eld and the structure and has tobe analyzed as such in order to yield accurate pre-dictions. The �rst simulation of fully coupled uid-structure-thermal interaction problems has been re-ported by E. Thornton et al.13, and has been re-cently followed by R. Loehner et al.14.In this work, we present a four-�eld formula-

tion of aerothermoelasticity for the analysis of uid-structure-thermal interaction problems. Here, weonly consider a one-way thermal-mechanical cou-

1

American Institute of Aeronautics and Astronautics

43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con22-25 April 2002, Denver, Colorado

AIAA 2002-1307

Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

pling, which means that a change in temperatureonly causes stress and deformations, but not viceversa. A staggered procedure is used to obtainthe solutions of the coupled system. Since infor-mation need to be exchanged at the uid/structureinterface in a partitioned method, we address prob-lems concerning the transfer of thermal quantitiesfrom the structure to the ow at the interface fork� " turbulence models with a wall law function onone hand, and the transfer of the discretized heat uxes from the uid to the structural analyzer acrossmeshes with non-conforming interfaces on the otherhand. We then present applications to validate ourmethodology.

2 Four-�eld formulation ofaerothermoelasticity

To study uid-structure-thermal interaction phe-nomena, we adopt the following semi-discretizedfour-�eld formulation of aerothermoelasticity

@(JW )@t

j� + Jrx: (F(W )� _xW ) =1Re

Jrx:R(W )

M�u+ f int(u)�C� = fext(W;x)

Q _TS +HTS = gext(W )

~Kx = 0

(1)The system above is an extension of the three-�eld

formulation of aeroelasticity established by Farhatand co-workers9. The �rst equation of system (1)is the ALE nondimensional conservative form of theNavier-Stokes equations and describes viscous owson dynamic meshes. Here, t denotes time, a dotdesignates a derivative with respect to time, x(t) de-notes the time-dependent position of the uid gridpoint, � its position in a reference con�guration,J = det(dx=d�), W is the uid state vector usingthe conservative variables, Re is the Reynolds num-ber, F , R denote respectively the convective anddi�usive uxes. For turbulent ow computations,the time-averaged Navier-Stokes equations are cou-pled to the k � " turbulence model formulated inALE form1:

@JWt

@tj� +Jrx:F

ct (Wt; _x) = Jrx:Rt(Wt)+J(Wt)

(2)

In this equation,Wt is the vector of conservative tur-bulent variables, Fc

t the turbulent convective term,Rt the di�usive term, and contains the productionand dissipation terms. The closure of the system (2)is realized by an ALE formulation of Reichardt's walllaw1:

(v:n? � _x:n?)jÆ = uf [2:5 log(1 + �Æ+)

+7:8(1� e�Æ+

11 �Æ+

11e�0:33Æ

+

)]; � = 0:41 (3)

where v is the velocity of the uid, n? the tangent

to the wall boundary at a given point, Æ+ = Re�Æuf�

the nondimensional wall distance, and uf the frictionvelocity.The second equation of system (1) is the nonlinear

form of the structural equations of dynamic equilib-rium. Here,M is the FE lumped mass matrix, u thegeneralized displacement vector, f int the vector ofinternal forces, C the thermal coupling matrix, fext

the vector of external forces acting on the structure,� = TS � Tref , where TS is the vector of the struc-tural temperature and Tref is the temperature forzero thermal stress.The third equation governs heat transfer in the

structure. Here, Q and H are the capacity and con-ductivity matrices, gext is the vector of external heat uxes acting on the structure. The last equation de-scribes the motion of the uid mesh assimilated to aquasi-static pseudo-structural system. In this equa-tion, ~K is the �ctitious time-dependent sti�ness ma-trix.

3 Interface boundary conditions

Let �I be the uid/structure interface boundary andn the normal at a point on �I . Then, on �I the uidand structure equations are coupled by

� a velocity compatibility and equilibrium of trac-tions conditions

_u = v

�S n = �p n+� n(4)

where v is the velocity of the uid, p its pressure�eld, �S and �F are the structure stress tensorand the uid viscous stress tensor;

2


� a temperature continuity and a heat ux equi-librium condition

TS = TF

�SrTS n = ��FrTF n(5)

where TS and TF represent the structure and uid temperatures, �S and �F are the coeÆ-cients of conduction of the structure and the uid.

The equations governing the structure and dy-namic mesh motions are coupled by the continuityconditions

x = u

_x = _u(6)

4 Partitioned solution procedure

One popular method for solving the system of cou-pled equations (1) is the partitioned or staggeredprocedure. In such a method, the solution of the uid, structure and mesh dynamics are solved sep-arately in a serial manner. The interaction e�ectsare taken into account by the transfer of informa-tion between the di�erent computational domainsat the interface boundary (also called wet bound-ary of the structure) �I . The appealing feature ofthe staggered algorithm is its exibility to conservethe discretization and solution methods that are wellestablished within each discipline. An elementarystaggered algorithm for solving the coupled systemof equations (1) goes as follows

1. advance the uid solver under the �rst bound-ary condition given by the equations (4) and(5)

2. transfer the aerodynamic forces to the dynam-ical structure and heat uxes to the thermalstructure using the second of equations (4) and(5)

3. update the structural temperature under thenew heat ux supply

4. send the new temperature �eld to the dynamicalstructure

5. compute the structural displacement under thenew uid and thermal loads

6. update the uid mesh according to (6)

The above method, also referred to as the conven-tional serial staggered (CSS) procedure3, is graphi-cally depicted in Fig. 1.

Tn

n+1θ θn+2

n+2gn+1gTn+1T n+2

��

��

��

��

��

��

��

��

��

��

��

��

��

n+2n+1un+2

��

��

��

��

��

��

��

��

fu

��

��

��

��

��

��

��

��

��

��

��

W n n+1W n+2W

n+1f��

��

��

��

��

��

Heat

Fluid

un

��

��

S

��

��

��

��

S

Structure

S

��

��

n+1 n+2n

Figure 1: The basic CSS procedure for aerothermoe-lasticity

5 Computational issues

With the coupling of an additional thermal �eld, twocomputational issues need to be addressed. Theseissues are:

� how to impose the updated structural tempera-ture to the energy equation of the uid systemin case of a turbulent simulation, because the uid equations are integrated only up to a dis-tance Æ from the real wall?

� how to transfer heat uxes across meshes withnon-matching interfaces?

The following two paragraphs cope with these twopoints.

5.1 Temperature wall law

When the structure transfers its temperature TS tothe uid �eld, TS is imposed as a Dirichlet con-dition in the energy equation with E = �cvTS(t)maintained at each structural time step. It is clearthat this wall boundary condition doesn't constitutea problem as long as one solves the Navier-Stokessystem. However, it poses a problem when the k� �turbulence model with wall laws is used since theequations are integrated only up to a distance Æ fromthe surface of the wall. Therefore, one cannot im-pose TS , which is the temperature of the real wall,on the �ctitious boundary at Æ. Nevertheless, this

3


problem can be solved by imposing a weak-type Neu-mann condition at Æ. In this work, we propose toapply a logarithmic temperature wall law to retrievethe heat uxes at the wall. These uxes are thenweakly imposed in the uid's energy equation.If TÆ denotes the temperature at the distance Æ

and uf the friction velocity, then the heat uxes qwat the wall can be calculated from2:

� ufTS � TÆ

qw= T+ (7)

The temperature pro�le T+ depends on the regionin the turbulent boundary layer:

T+ =

8<:

PrÆ+ Æ+ < 13:2

13:2Æ+ +Prt�

lnÆ+

13:2Æ+ > 13:2

(8)

5.2 Conservation of energy for nonmatching meshes

As we have mentioned previously, the partitionedprocedure permits each �eld of a coupled system, beit aeroelastic or aerothermoelastic, to have its ownmathematical and numerical properties. When thecomputational domains of the uid and structurehave matching discrete uid/structure interface, thediscretization of Eqs. (4), (5) and (6) is straightfor-ward. However, in practical applications, the uidand the structure meshes are incompatible at the uid/structure interface. In aeroelastic applicationsfor example, the uid mesh is typically �ner thanthe structure mesh.In4, the authors addressed the problem of dis-

cretizing the uid/structure interface conditions andthe exchange of aerodynamic data between the owand structural solver. They considered a closed sys-tem de�ned by the union of the uid and structuresubsystems, and showed that if the uid and struc-ture solvers employ di�erent discretization methods,for example di�erent shape functions, most if notall interpolation based algorithms designed for con-verting the uid pressure and stress �elds at the uid/structure interface into a structural load arenon-conservative, since the sum of the loads on thewet surface of the structure are not exactly equal tothe sum of the uid loads on the uid side. Hence,they proposed a more robust and reliable algorithm,based on the conservation of energy, to convert the uid pressure and stress �elds at the uid/structureinterface into a structural load.

In this section, the new conservative method pre-sented by those authors is adopted to the discretiza-tion of the heat uxes (Eq. 5) at the uid/structureinterface. In order to ensure the conservation of en-ergy in terms of heat uxes at the interface, we de-rive the following computational strategy from4.We denote �F and �S the discrete uid/structure

interfaces on the uid and structure sides. For alladmissible weight function wF , we can write the dis-cretization of the uid heat uxes as:

QF =Z�F

�FrTFn wF d� (9)

The approximation of wF can be expressed as:

wF =j=nFXj=1

DjwFj (10)

where Dj is some function with a local or globalsupport on �F . nF is the number of uid verticesmatched with a structural element of nS nodes. Thediscrete value wFj at the uid node j can of coursebe chosen such that:

wFj =i=nSXi=1

cjiwSi (11)

where cji are constants that depend on the approx-imation method. Hence, the discretization of theheat uxes at �F can be formulated as

QF =j=nFXj=1

Z�F

(�FrTFn)DjwFjd�

=j=nFXj=1

Z�F

(�FrTFn)Djd�wFj

=j=nFXj=1

fjwFj

(12)

where fj is the numerical heat ux of the uid:

fj =Z�F

(�FrTn)Djd� (13)

Substituting equation (11) into equation (12) yields

QF =i=nSXi=1

0@j=nFX

j=1

fjcji

1AwSi (14)

4


On the other side, the heat uxes acting on the struc-ture on �S can be approximated by:

QS =i=nSXi=1

giwSi (15)

To conserve energy at the uid/structure interface,QF must be equal to QS, which leads to:

gi =j=nFXj=1

fjcji (16)

The expression of gi is independent of the discretiza-tion method of the structure. fj depends only on thedicretization method chosen to solve the uid prob-lem and cji depends on the approximation methodto evaluate (11). For example, if the heat transferequation is solved by �nite element methods, thencij corresponds to the shape functions �i(�j), where�j is the projection of the uid point, paired with astructural element, on �S (see �gure 2). Then,

gi =j=nFXj=1

fj�i(�j) (17)

In the example of �gure 2, four uid vertices arepaired with a structural �nite element composed oftwo nodes. In this instance, we have

g1 = �1(�1)f1 + �1(�2)f2 + �1(�3)f3 + �1(�4)f4

g2 = �2(�1)f1 + �2(�2)f2 + �2(�3)f3 + �2(�4)f4

ΓS

ΓF

23

4

1

f1f2

f3f4

Fluid cell

1 2

g1 g2

ξ2 ξ3ξ1 ξ4

Figure 2: Example of the conservative method

6 Flow, structure, thermal, andmotion solvers

The two-dimensional unsteady ow solver consid-ered in this paper operates on unstructured trian-gular elements. The ALE convective uxes are ap-proximated by a Roe upwinding scheme, whereasthe di�usive uxes are approximated by a Galerkinmethod using P1 shape functions. Second-order ac-curacy is achieved through the use of a piecewiselinear interpolation method that follows the princi-ple of the MUSCL (Monotonic Upwind Scheme forConservative Laws) procedure. The approximationof the ALE turbulence convective terms is based onLarrouturou's positivity preserving ux function8 tomaintain the positivity of the turbulent quantities.Time-integration is carried out by a second-order im-plicit backward di�erence scheme whose implemen-tation satis�es the discrete geometric conservationlaws (DGCL)9. This unstructured and unsteady im-plicit ow solver is parallelized using domain decom-position.Finite element methods are used for the solu-

tions of structural dynamics and heat transfer sys-tems. The �rst is time-integrated with the im-plicit second-order midpoint rule, the second is time-integrated with the implicit second-order central dif-ference scheme. The solution of uid-structure inter-action problems is achieved by a second-order accu-rate partitioned procedure10.The dynamic mesh motion relies on the torsional

springs analogy method proposed by C. Farhat et al.11.

7 Applications

In this section, we propose two numerical applica-tions in order to illustrate our solution methodology.The �rst example concerns the aerodynamic heatingof a F-16 �ghter wing section and the second con-cerns the thermal e�ects on the aeroelastic behaviorof a clamped at panel.

7.1 Aerodynamic heating of the 2DF-16 �ghter wing

The idealized 2D model of the F-16's multicell wingsection is shown in �gure 3. There are six cells in-side the wing. It is L = 1 m long and has a maxi-mum height of 4 cm. The skin is 4 mm thin and issupported by sti�eners that pass through the wing.

5


The thickness of the sti�eners is assumed to be 5mm. The entire structure is supposed to be uniform(no joints, thus no heat resistance) and of materialAl 2024-T4 (�S = 2800 kg=m3, �S = 170 W=mK,cpS = 875 J=kgK). The heat transfer domain ofthe structure is discretized into 4,144 vertices usingunstructured triangular �nite elements. The compu-tational ow domain consists of a structured meshof 14,367 vertices. The wall boundary is placed at adistance Æ = 2 � 10�4 from the physical surface ofthe structure.

Figure 3: Model of F-16's multicell wing section

Throughout this section, the free-stream Machnumber is set to M1 = 2, the free-stream density to�1 = 0:4 Kg/m3, the pressure to P1 = 25; 714 Pa.The Reynolds number of the turbulent ow is �xedat ReL = 5� 106.We investigate three cases in our simulations. In

the �rst case, we only consider the wing being aero-dynamically heated by the ow (�gure 3). In thesecond case, in addition to aerodynamic heating, wealso consider cooling by natural convection occurringinside the wing. For this simulation, the cells of thewing are �lled with air at temperature Ta = 224ÆKand heat is convected by ha(TS � Ta). The convec-tion coeÆcient ha for natural convection can be esti-mated by empirical laws. In the third case, we con-sider air inside the �rst and sixth cell and kerosenein the other four (�gure 4). For this case, the cellsare also discretized by triangular elements. Heat istransferred to air and kerosene in the cells by con-duction. The properties of kerosene are: � = 820Kg/m3, cp = 2; 000 J/kgK, � = 0:15 W/mK.The temperature of the structure is initially set to

350ÆK for all three cases. The simulation is carriedout until the sum of the heat uxes acting on thestructure reaches equilibrium, which means that thetemperature �eld has converged to a steady-state.In the solutions, we plot for each case the temper-

KA AKKK

Figure 4: Test case 3 (A = air, K = kerosene)

368

370

372

374

376

378

380

382

384

386

388

390

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tem

pera

ture

[K]

x [m]

t=100s

t=200s

t=400s

t=650s

Figure 5: Temperature distribution with time of F-16's skin (case 1)

360

365

370

375

380

385

390

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tem

pera

ture

[K]

x [m]

t=100s

t=200s

t=400s

t=650s


6


360

365

370

375

380

385

390

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tem

pera

ture

[K]

x [m]

t=100s

t=400s

t=600s

t=800s

t=3000s


ature distribution with time in the chordwise direc-tion of the wing. For all three cases, no temperaturegradients have been found in the transverse direc-tion of the wing. This is due to the small value ofthe Biot number. The stagnation temperature is notcorrectly computed because of the coarse structuralmesh at the leading edge. Hence, the heat uxestransmitted by the uid are inaccurate in this areaof the structural surface.The results on �gures 5, 6, 7 show how the skin

temperature can be over-predicted if the heat trans-fer model accounts for the structure only, as in case1. At the time station t = 100 s for example, thetemperature at x = 0:5 m is about 374ÆK for case 1,and 364ÆK for cases 2 and 3.Except for case 2 with cooling e�ects, the skin

temperature will eventually reach the steady-statetemperature predicted by Crocco's formula Taw =

T1�1 + 0:5Pr

13 ( � 1)M2

1

�= 384:5ÆK. However,

case 1 predicts that the time to attain this equilib-rium state is 650 s, while case 3 predicts a time of3000 s.

7.2 Aerothermoelastic analysis of a at panel

When an aircraft functions in a high-speed environ-ment, its aeroelastic behavior will be a�ected byaerodynamic heating. Indeed, experiments7 showedthat a heated structure is more susceptible to utterthan an unheated one. To illustrate this, we applyhere our methodology to the study of thermal e�ectson the transient nonlinear aeroelastic behavior of a

x

y

οο

��

��

οοM

οο

L = 0.5 m��

�� P

= 2

P

adiabatic supports

P q

Figure 8: Illustration of the panel problem

panel on two rigid supports. The panel is representa-tive, for example, of a wing's skin section supportedby sti�eners and spars.For this purpose, we consider a turbulent ow

over a at panel clamped at both extremities, asillustrated in �gure 8. The supports of the panelare assumed to be adiabatic. The panel, of ma-terial Al 2024-T4, has a length L = 0.5 m anda 4 � 4 mm cross-section. Its material proper-ties are: �s = 2; 800 kg=m3, �s = 170 W=mK,cps = 875 J=kgK, E = 7:3 � 1010 N=m2, � = 0:33,�s = 22:5� 10�61=K. It is modeled by 30 beam el-ements. We apply geometric nonlinear Euler beamtheory because of large deformations encountered inthe considered problems. The ow domain abovethe panel is discretized into 7,421 vertices.We have stated earlier that our interest is the tran-

sient response of the panel. Typically, the time con-stant of an aeroelastic response is much smaller thanthat of a thermal response. During the time frameof a transient aeroelastic computation, the temper-ature of the structure is quasi constant. Since thetemperature will not undergo any substantial varia-tion, our strategy of conducting an aerothermoelas-tic simulation is to �rst consider that the panel hasbeen aerodynamically heated to a certain temper-ature. This temperature will consequently be pro-vided as the initial temperature in the heat trans-fer equation of the structure. Then, we start anaerothermoelastic computation in which we perturbthe panel with a small initial displacement.During all the computations, we maintain the far-

�eld ow at M1 = 2, ReL = 5� 106 and �1 = 0:4Kg/m3. The reference temperature Tref in the ther-mal coupling term C� is taken as Tref = 298ÆK. Inorder to show the importance of aerodynamic heat-ing on the aeroelastic response of the panel, we pro-pose to contrast the aerothermoelastic results with

7


aeroelastic simulations. The structural time-step isset to �tS = 5 � 10�5s and to �tS = 10�4s foraeroelastic and aerothermoelastic computations re-spectively. The thermal time-step is set to follow thestructural time-step. The employed time-steppingstrategy corresponds to sampling a response periodin 82 time-steps for both cases. We �rst proceed toperform aeroelastic computations for a variety of P1to �nd the utter speed (u1 = M1

q P1�1

). Then,

we conduct several aerothermoelastic computationsto show that the structure enters utter at a lower ow speed than its aeroelastic counterpart.The results of aeroelastic computations, given in

�gure 9 show the displacement at the maximum lo-cation of de ection, i.e at x = 7=10 L. Flutter oc-curs at about P1 = 700; 000 Pa and the motion ofthe panel is retained in limit-cycle oscillations withbounded amplitudes.For the aerothermoelastic analysis, we place the

panel at a uniform temperature T = 325ÆK, whichis below its adiabatic temperature. At this temper-ature, the thermal forces exceed the Euler bucklingload. In �gure 10, we can see that the panel buck-les up and stays buckled for low pressures. How-ever, with increased pressure, it utters with limit-cycle oscillations, and the maximum deformation islocated at x = 7=10 L, see �gure 11. We also no-tice that the structure enters into utter regime at alower pressure and with more important amplitudescompared to the aeroelastic case (�gure 12). Thesebehaviors agree with the studies of Fung5 and Dow-ell6.

8 Conclusion

We proposed a four-�eld formulation of aerother-moelasticity for the analysis of uid-structure-thermal interaction phenomena. We presented so-lution methodologies to overcome computational is-sues pertaining to thermal boundary condition en-forcement in the presence of a turbulence model withwall law, and the transfer of thermal informationacross meshes of non-matching interfaces. We illus-trated our implementation by carrying out aerody-namic heating simulations of an idealized model ofthe F-16's multicell wing section and aerothermoe-lastic simulations of a at panel.

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

0 0.02 0.04 0.06 0.08 0.1

y-di

spla

cem

ent a

t 7/1

0 L

[m]

Time [s]

P = 800000 PaP = 700000 PaP = 625000 Pa

Figure 9: Aeroelastic responses (displacement)

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0 0.02 0.04 0.06 0.08 0.1

y-di

spla

cem

ent L

/2 [m

]

Time [s]

P = 100000 PaP = 25714 Pa

Figure 10: Aerothermoelastic responses (displace-ment) at low pressures

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0 0.02 0.04 0.06 0.08 0.1

y-di

spla

cem

ent 7

/10

L [m

]

Time [s]

P = 350000 PaP = 500000 Pa

Figure 11: Aerothermoelastic responses (displace-ment) at high pressures

8


-0.01

-0.005

0

0.005

0.01

0 0.02 0.04 0.06 0.08 0.1

y-di

spla

cem

ent a

t 7/1

0 L

[m]

Time [s]

Aerothermoelastic P = 350000 PaAeroelastic P = 800000 Pa

Figure 12: Displacement comparison between aeroe-lastic and aerothermoelastic responses

References

[1] H. Tran, B. Koobus, and C. Farhat, NumericalSimulation of Vortex Shedding Flows Past Mov-ing Obstacles Using the k-� Turbulence Modelon Unstructured Dynamic Meshes, La RevueEurop�eenne des El�ements Finis, Volume 6, No5/6, 1997

[2] Adrian Bejan, Convection Heat Transfer, JohnWiley & Sons, Inc, 1995

[3] C. Farhat, M. Lesoinne, Higher-Order Staggeredand Subiteration Free Algorithms for CoupledDynamic Aeroelasticity problems, AIAA paper,36th Aerospace Sciences Meeting and Exhibit,Jan. 12-15, 1998, Reno, NV

[4] C. Farhat, M. Lesoinne, P. LeTallec,Load and Motion Transfer Algorithms forFluid/Structure Interaction Problems withNon-matching Discrete Interfaces: Momentumand Energy Conservation, Optimal Discretiza-tion and Application to Aeroelasticity, Comput.Meths. Appl. Mech. Engrg., Vol 157, pp 95-114,1998

[5] Y.C. Fung, The Static Stability of a Two-dimensional Curved Panel in a Supersonic Flowwith a Application to Panel Flutter, Journal ofthe Aeronautical Sciences, Vol 21 No 8, pp556-565, 1954

[6] E. H. Dowell, Panel Flutter: A Review of theAeroelastic Stability of Plates and Shells, AIAAJournal, Vol 8, pp 385-399, 1970

[7] Hugh L. Dryden and Dr. John E. Duberg,Aeroelastic E�ects of Aerodynamic Heating,Proceedings of the Fifth AGARD General As-sembly, 15-16 June 1955, Canada

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[9] M. Lesoinne and C. Farhat, Geometric Conser-vation Laws for Aeroelastic Computations Us-ing Unstructured Dynamic Meshes, AIAA pa-per 95-1709, 12th AIAA Computational FluidDynamics Conference, San Diego, June 19-22,1995

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[12] A.R. Wieting, P. Dechaumphai, A.K. Pandey,Fluid-thermal-structural interaction of aerody-namically heated leading edges, AIAA PaperNo. 89-1227-CP, 30th Structures, StructuralDynamics and Materials Conference, April 3-5,1989

[13] Earl A. Thornton and Pramote Dechamphai,Coupled Flow, Thermal, and Structural Analy-sis of Aerodynamically Heated Panels, Journalof Aircraft, Vol 25, pp 1052-1059, 1988

[14] R. Loehner, C. Yang, J. Cebral et al., Fluid-Structure-Thermal Interaction using a LooseCoupling Algorithm and Adaptive UnstructuredGrids, 29th AIAA Fluid Dynamcis Conference,June 15-18, 1998 Albuquerque, NM

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