Proceedings of 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, andMaterials Conference

April 23-26, 2012, Honolulu, HI, USA

AEROELASTIC SIMULATION OF STRUCTURES IN HYPERSONIC FLOW

Robert L. Brown∗, Kaushik Das†, John D. Whitcomb‡, and Paul G. A. Cizmas‡

Department of Aerospace Engineering, Texas A&M UniversityCollege Station, Texas 77843-3141

ABSTRACTThis paper presents recent developments of a high-

fidelity nonlinear aerothermoelastic analysis code used forevaluating the effectiveness of novel materials in mitigat-ing the effects of hypersonic flow. This aerothermoelasticmodel consists of: (1) an aerodynamics model based on theReynolds-averaged Navier-Stokes equations that capturescompressible flow effects and shock/boundary layer inter-action, (2) a thermoelastic finite element structural model,and (3) a solution methodology that couples the nonlin-ear structure and fluid flow, including a consistent geo-metric interface between the deforming structure and theflow field. The parallel computation algorithm and the gridgenerator of the flow solver were developed concurrentlyto improve the efficiency of parallel computation. Resultsare presented for three configurations: a two-dimensionalwedge, a fin and a generic wing of a hypersonic vehicle.

NomenclatureD Elasticity constantD airfoil/wing structural domaind displacementE Young modulus or total energye infinitesimal strain tensor~F flux vector

∗Graduate Research Assistant†Postdoctoral Research Associate‡Professor

f viscous stress or body force~G source term vectorH total enthalpyK heat conductivity of structurek heat conductivity of gasM Mach number~Q state vectorq heat fluxRe Reynolds numberT temperaturet timeu,v,w velocity componentsV free-stream velocityx position vectorα coefficient of thermal expansion or

thermal diffusivityκ turbulence kinetic energyρ densityσ stress tensorΩ cell volume or airfoil/wing domainω specific dissipation rate

1 Copyright c© 2012 by authors

Sub- and superscripts˙ time derivative0 reference value or initial valueconv convective termsf finalvis viscous termsx x-direction componenty y-direction componentz z-direction component∞ free-stream value

1 IntroductionOne of the challenges of developing hypersonic

reusable launch vehicles is the mitigation of the effects ofthe thermo-vibro-acoustics loads and fatigue on the struc-ture [1]. It is expected that these hypersonic vehicles willoperate up to a Mach number of 15 and that they must flyin the atmosphere for extended periods of time. Conse-quently, there is significant aerodynamic heating of the ve-hicle, which adversely alter the structure by increasing ther-mal stresses and by degrading material properties. There-fore, the design and analysis tools for hypersonic vehi-cles require a tight coupling between the structural, aero-dynamic, propulsion and control systems. The proper, cou-pled modeling of the multidisciplinary effects is needed fordeveloping reliable numerical simulation capabilities. Thisneed is exacerbated by the fact that testing scaled mul-tidisciplinary models in the wind tunnel is very difficultto achieve because of the conflict between the free-streamMach numberM∞, Reynolds numberRe∞, aeroelastic pa-rameterρ∞V 2/E0, heat conduction parameterk∞/K0 andthermal expansion parameterα0T0 [2].

The development of hypersonic vehicles reliesheavily on the development of novel multifunctionalceramic/metal/polymer hybrid composites for high-temperature environments. An accurate representation ofstructural boundary conditions is necessary for determin-ing the requirements for new materials. In the case of anaerospace vehicle, these conditions are the aerodynamicand thermal loads created by the surrounding fluid. Foraccurate prediction of the entire system and evaluationof new materials within a relevant context, both thestructure and fluid flow must be modeled. Because theaccurate calculation of aerodynamic and thermal loads isintrinsically tied to the current position of the structure,

a coupled model that includes both the structural andaerodynamic models must be implemented. This couplingmust be implemented such that the loads be accuratelytransferred between the two models.

For the correct prediction of unsteady aerodynamics,computational fluid dynamics (CFD) models are needed toproperly capture the flow physics, such as shock/boundarylayer interaction and flow separation. This is necessaryto predict the occurrence of detrimental aerothermoelasticphenomena (such as limit cycle oscillation, flutter, buffet-ing) and acoustic loads.

This paper presents recent developments on the simula-tion of aerothermoelastic loads for elements of a hypersonicvehicle. The next section presents the governing equationsof the aeroelastic model. This is followed by a descrip-tion of the numerical method used for the aeroelastic solver.The results section presents the loads predicted for an air-foil, a fin and wing of a hypersonic vehicle.

2 Aeroelastic ModelThis section presents the governing equations of the

aerodynamic and structural models, and the coupling be-tween them. The first part of the section briefly de-scribes the aerodynamic model that was based on theReynolds-averaged Navier-Stokes equations. The secondpart presents the structural model that approximated the finand wing as plates. The last part of this section describesthe coupling of the aerodynamic and structural models.

2.1 Aerodynamic ModelThe unsteady, compressible flow around a deforming

hypersonic fin and wing was modeled by solving the mass,momentum and energy conservation equations. Theseequations, collectively known as the Reynolds-averagedNavier-Stokes equations, are

∂∂t

Z

Ω

~Q ·dΩ+

I

S

(

~Fconv −~Fvis

)

·dS =

Z

Ω

~G ·dΩ, (1)

where~Q is the state vector of conservative variables,~Fconv

is the vector of convective fluxes,~Fvis is the vector of vis-cous fluxes, and~G is the vector of source terms. The statevector of conservative variables is~Q = (ρ,ρu,ρv,ρw,ρE)T ,whereρ is the density of the fluid,u, v andw are thex-, y-

2 Copyright c© 2012 by authors

and z-components of the velocity, andE is the total en-ergy of the fluid particle. The vector of convective fluxesis ~Fconv = (ρV,ρuV + nx p, ρvV + ny p,ρwV + nz p,ρHV )T ,whereV is the contravariant velocity andH is the total en-thalpy. The vector of viscous fluxes is~Fvis = (0, fx, fy, fz,uk fk −qn)

T , where fx, fy and fz are the viscous stresses inthex-, y- andz-directions, andqn is the heat conduction inthe fluid.

The effects of turbulence were modeled by the two-equation eddy-viscosity shear stress transport model [3].The time-dependent integral form of these equations waswritten in a vectorial form similar to (1). The state vec-tor of turbulent conservative variables was~QT = (ρκ,ρω)T

whereκ was the turbulence kinetic energy andω was thespecific dissipation rate.

2.2 Structural ModelThe structural model is three-dimensional finite ele-

ment based for both the heat transfer and thermoelastic be-havior. The finite element formulations are based on theweak form of energy conservation for the heat transfer andconservation of momentum for the thermoelastic behavior.A mix of tetrahedral and hexahedral elements was used inthe meshes. The mesh refinement requirements are differ-ent depending on whether one is concerned with the inter-action with the CFD model or trying to predict local failure.Prediction of local failure requires a much more refinedmesh. A multi-scale approach was used for the model. Arelatively coarse grid was used for the model that is coupledwith the CFD model. It is much less refined than the CFDgrid. Hence, mapping procedures are used in conveyinginformation between the models. Since the time steppingrequirements are also more stringent for the CFD model,different time integration schemes were used for the CFDand the thermal and thermoelastic models.

The balance of linear momentum of the airfoil/wingdomainD is given by:

σi j, j(x, t)+ fi(x, t) = ρdi(x, t), in D×(0, t f ), i = 1,2,3 (2)

where(0, t f ) is the time duration of interest andx is theposition vector of a point in the domainD. Index notationis being used andi = 1, 2, 3 are the three directions of aCartesian coordinate system.σi j is the stress tensor,fi isthe body force, andui is the displacement. A superimposed

dot represents derivative with respect to time. Across theinterface between the airfoil/wing and the fluid domain, thetractionσi jn j = hi(x, t) and velocities ˙ui(x, t) are continu-ous. The initial condition of the airfoil/wing is

di(x, t = 0) = d0i (x) anddi(x, t = 0) = ν0

i (x) on D (3)

We assume that the material of the body is an elastic mate-rial and its constitutive relation is given by

σi j(x, t) = Di j klekl (4)

whereDi j kl are the elasticity constants and the infinitesi-mal strain tensorekl is related to the displacements by

ei j(x, t) = (di, j + d j,i)/2 (5)

The equation of heat transfer of the airfoil/wingD is givenby:

T (x, t)−αT,ii(x, t) = 0, in D × (0, t f ), i = 1,2,3 (6)

whereT is the temperature, andα is the thermal diffusiv-ity. Across the interface between the airfoil/wing and thefluid domain, the heat fluxαT,ini and temperatureT (x, t)are continuous. The initial condition of the airfoil/wing is

T (x, t = 0) = T 0(x) (7)

2.3 Coupling between Aerodynamic and StructuralModels

The coupling between the aerodynamic and structuralmodels was done using a tightly-coupled solution that al-lowed the two models to communicate with a certain fre-quency. For the aeroelastic simulation the communicationoccurred at every time step [4]. For the aero-thermal cou-pling results presented herein, the communication occurredafter every 100 steps of the flow simulation.

The aerodynamic loads were generated by integratingthe pressure and shear stresses over the wing surface. Thethermal loads were generated by integrating the area spe-cific heat flux. The aerodynamic and thermal loads were

3 Copyright c© 2012 by authors

then passed to the structural solver and were used to calcu-late the deformation and temperature of the structure.

The coordinates of the new position of the structurewere passed to the grid deformation algorithm which up-dated the grid of the flow solver. The temperature of thesurface was directly passed to the boundary cells of the flowsolver. The flow solution was then calculated and the aero-dynamic and thermal loads generated for the new positionof the structure.

3 Numerical MethodThis section briefly presents the numerical methods

used to solve the unsteady flow and the grid deformation.The methodology utilized to solve the Reynolds-averagedNavier-Stokes equations is presented first, followed by thefinite element approach to solve the wing deformation.

3.1 Reynolds-averaged Navier-Stokes SolverThis section presents the spatial and temporal dis-

cretization of the Reynolds-averaged Navier-Stokes equa-tions, the parallelization strategy and the grid generationand deformation algorithm.

3.1.1 Spatial Discretization A finite volumemethod was used to solve the Reynolds-averagedNavier-Stokes equations. The governing equations werediscretized using an unstructured grid with mesh duals ascontrol volumes [5]. The median dual-mesh was adoptedherein because of its flexibility of handling unstructuredmixed meshes [6]. The cell-averaged variables were storedat the nodes of the grid, that is, the vertices of the cells.

The surface integral of the inviscid and viscous fluxeswas approximated as a sum over the faces of each controlvolume. The source terms were calculated using the con-trol volume-averaged solution and the derivatives of flowvariables at the cell centroid. Equation (1) was rewritten ina semi-discrete form as

∂(Qi ·Ωi)

∂t=

∂Qi

∂tΩi +

∂Ωi

∂tQi =−

I

Si

~F ·~n ·dS+Gi ·Ωi (8)

where Qi denotes the volume-averaged flow variableQover the volumeΩi.

The surface integral term on the right-hand side of thisequation was approximated as

I

S

~F ·~n ·dS = ∑j=k(i)

(

~Finv −~Fvis

)

·Si j (9)

where k(i) is the set of vertices adjacent to nodei,(

~Finv −~Fvis

)

is the flux normal to the dual-mesh cell face

andSi j is the cell face surface area. Both the surface areaand the flux were associated with the dual mesh face thatwas associated with the edge(i, j).

The Navier-Stokes equations (8) and the turbulencemodel equations were written at grid nodei asΩi∂Qi/∂t =Ri, where the residual was

Ri = −

I

Si

~F ·~n ·dS + Gi ·Ωi −Qi∂Ωi

∂t. (10)

The upwind method with the Roe’s approximate Riemannsolver and Harten’s entropy fix were used to compute theinviscid fluxes. The least-squares method with QR de-composition was used to compute the gradients of the vis-cous fluxes at vertices. Direction derivatives were used toimprove the accuracy of the averaged gradients at edges.Piecewise linear reconstruction was used to obtain second-order spatial accuracy.

3.1.2 Temporal Discretization The unsteady solu-tion was computed using a multi-stage explicit time inte-gration algorithm. A residual smoothing algorithm wasused to increase the integration time step [7]. Implicitresidual smoothing added an implicit flavor to the explicitmethod by blending the residuals of the neighboring nodes.Residual smoothing allowed the use of larger time steps.

The time evolution of the cell volume∂Ωi/∂t in (10),which depends on grid deformation, was calculated usingthe geometric conservation law. The geometric conserva-tion law [8] relates the time derivative of the volume tothe motion of the faces of the cell∂Ωi/∂t =

H

Si

~VS ·~ndS =

∑j=k(i)

~VS ·~ni jSi j, where~VS denotes the velocity vector corre-

sponding to the center of the faceSi j whose unitary normalvector is~ni j.

4 Copyright c© 2012 by authors

3.1.3 Grid Generation and Deformation AlgorithmThe computational domains around the airfoil, fin and wingwere discretized using a hybrid grid consisting of hexa-hedra and triangular prisms, as shown in Figs 1 to 3. Itshould be noted that although the flow over the airfoil istwo-dimensional, a three-dimensional model was used thathad only one layer in the spanwise direction. Due to theparticular configuration of the wing, it was possible to di-vide the computational domain into layers that were topo-logically identical, as shown in Figs. 2 and 3. Triangularprisms or hexahedra volume elements were generated byconnecting the nodes that were topologically identical onadjacent layers. As a result, a 2.5-D grid could be used todiscretize the 3-D domain.

Each layer had a structured O-grid around the bound-ary of the structure, and an unstructured grid at the exteriorof the O-grid. The O-grid allowed good control over themesh size near the structure and permitted clustering cellsin the direction normal to the structure to properly captureboundary layer effects. The unstructured grid was flexiblein filling the rest of the domain.

The computational mesh deformed according to thestructure displacement. While the computational grid de-formed as the wing bent and twisted under aerodynamicloads, the grid connectivity remained unchanged. To main-tain a good quality grid as the wing deformed, the mesh de-forming algorithm was required to generate a moving gridthat was both perpendicular to the wing surface and to theouter boundary of the computational domain [9].

3.1.4 Parallelization Algorithm The RANS flowsolver was parallelized to reduce the turnaround time. Adomain decomposition approach was used that relied onthe mesh splitting on topologically identical layers, whichsimplified parallel communication. In addition, the parallelcommunication benefited from the fact that mesh deforma-tion did not change node connectivity while the wing wasdisplaced. Further details on the parallelization algorithmare provided in [4].

4 ResultsThis section presents results for three configurations:

a double-wedge airfoil, a fin and a wing of a generic hy-personic vehicle. The double-wedge airfoil is identical tothe root airfoil of the fin and wing. The use of the double-

Figure 1. Mesh of double-wedge airfoil.

Figure 2. Mesh of fin.

Figure 3. Wing mesh.

wedge airfoil, which had a smaller grid size compared tothe wing, allowed us to do a parametric study that couldalso be relevant for the wing. A number of boundary con-ditions were explored, representing different heat flux ratesin the interior of the structure and different flight speeds.

5 Copyright c© 2012 by authors

The double-wedge airfoil is shown in Fig. 1. Thefin, shown in Fig. 2, had a double wedge shaped airfoil,with a simple planform. The wing could be used for afull aerothermoelastic simulation but the results presentedherein included only a preliminary aeroelastic analysis.Due to the computational cost of this simulation, a fullparametric study was not performed yet.

4.1 Double-wedge Airfoil Aerothermodynamic Analy-sis

Aerodynamic analysis was performed on the double-wedge airfoil to determine heating rates for the structureunder various flight conditions. This analysis provides abasis for determining the feasibility of different skin con-figurations based on required cooling rates and maximumtemperatures throughout the structure.

4.1.1 Grid Convergence To assess the effect of themesh size on the solution, several grids were generated forthe double-wedge airfoil. Since the double-wedge airfoilwas modeled as one three-dimensional layer, the grids hadtwo identical sides. The medium grid had 120 nodes onthe airfoil surface per layer. Each layer had a total of 5422nodes, and the entire grid had 10844 nodes. The fine gridhad 400 nodes on the airfoil surface per layer. Each layerhad a total of 20963 nodes, and the entire grid had 41926nodes. The medium and fine meshes of the fin are shownin Fig. 4.

The heat flux variation along the airfoil, shown in Fig. 5for the medium and fine grids, indicates that the grid sizeaffects the results for the first half of the airfoil. This canbeattributed in part to the mesh size and in part to a carbuncleeffect [10] at the airfoil leading edge.

Temperature and Mach number contours are shown inFigs. 6 and 7 for different Mach numbers. In these threecases, the heat flux at the interior of the airfoil was constantand equal to 6.6 kW/m2.

Figure 8 shows the temperature variation in the struc-ture near the leading edge for Mach 4 and an interior heatflux of 6.6 kW/m2. Note that the airfoil had an angle ofattach of 0.6 degrees and therefore there is a difference be-tween the values on the pressure side and the suction sideof the airfoil.

Figure 9 shows the surface temperature variation as afunction of Mach number and interior heat flux. As ex-

(a)

(b)

Figure 4. Fin meshes: (a) medium, (b) fine.

pected, the largest temperature occurs at the largest Machflight number and the smallest interior heat flux. It shouldbe noted that the surface temperature drops at the spar lo-cations. These locations are shown in Fig. 10.

4.2 Rigid Fin Aerothermodynamic AnalysisAerodynamic analysis was performed on the fin to de-

termine approximate stagnation temperatures on the sur-face of the wing. This analysis allows for the determina-tion of necessary structural materials for the plate modeland also provides a starting point for any aeroelastic analy-sis.

Two computational grids of the fin are shown inFig. 11. The medium grid had a total of 36 layers along

6 Copyright c© 2012 by authors

Chordwise Position [m]

Hea

t F

lux

[W/m

^2]

-1 0 1 2 3 4 5 6103

104

105

106

Medium GridFine Grid

Figure 5. Heat flux variation along the airfoil at Mach 5 as a function of

grid size.

the span, with 128 nodes on the airfoil surface per layer.Each layer had a total of 6337 nodes, and the entire gridhad 304,436 nodes. The fine grid had 60 layers along thespan, with 256 nodes on the airfoil surface per layer. Eachlayer had a total of 18,839 nodes, and the entire grid had1,813,021 nodes.

The maximum temperature for fin flying at Mach 5.0with an adiabatic wall boundary condition was calculatedto be 2100 K. The surface temperature variation is shownin Fig. 12. Flying at Mach 7.0 under the same conditions,the maximum temperature on the surface of the wing wascalculated to be 3850 K. Both of these temperatures aretoo high for conventional materials, so some level of activecooling must be used.

4.3 Wing Aeroelastic AnalysisA preliminary aeroelastic analysis was performed on

the wing shown in Fig. 3 for various Mach numbers and dy-namic pressures. The Mach number contours on the suctionand pressure side of the wing flying at Mach 3 are shown inFig. 13. Note that in this simulation the flow was assumedto be adiabatic. The effects of aerodynamic heating on theaeroelastic response will be later included in these calcu-lations, allowing the properties of the structure to change

z

y

0.1 0.11 0.12 0.13 0.14

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02T[K]

79076073070067064061058055052049046043040037034031028025022019016013010070

(a)

z

y

0.1 0.11 0.12 0.13 0.14

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02T[K]

79076073070067064061058055052049046043040037034031028025022019016013010070

(b)

z

y

0.1 0.11 0.12 0.13 0.14

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02T[K]

79076073070067064061058055052049046043040037034031028025022019016013010070

(c)

Figure 6. Temperature contours for an interior heat flux of 6.6 kW/m2:

(a) Mach=3, (b) Mach=4 and (c) Mach=5.

7 Copyright c© 2012 by authors

z

y

0.1 0.11 0.12 0.13 0.14

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02Mach

5.004.754.504.254.003.753.503.253.002.752.502.252.001.751.501.251.000.750.500.250.00

(a)

z

y

0.1 0.11 0.12 0.13 0.14

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02Mach

5.004.754.504.254.003.753.503.253.002.752.502.252.001.751.501.251.000.750.500.250.00

(b)

z

y

0.1 0.11 0.12 0.13 0.14

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02Mach

5.004.754.504.254.003.753.503.253.002.752.502.252.001.751.501.251.000.750.500.250.00

(c)

Figure 7. Mach contours for an interior heat flux of 6.6 kW/m2: (a)

Mach=3, (b) Mach=4 and (c) Mach=5.

ature: 500 600 700 800 900 1000 1100 1200 1300 1400 1500

Temperature [K]

(a)

z

y

0 0.05 0.1 0.15

0

0.05

0.1

0.15

Temperature

15001450140013501300125012001150110010501000950900850800750700650600550500450400350300

[K]

(b)

Figure 8. Temperature contours in the structure at Mach=4 for an interior

heat flux of 6.6 kW/m2.

with time and temperature. Figure 14 shows the aeroelasticdeformation of the wing at Mach.

5 ConclusionsThe aerothermodynamic simulation of elements of a

hypersonic vehicle presented herein illustrated the impor-tance of thermal effects. At high Mach numbers, the ac-curacy of aerodynamic simulation was jeopardized by thecarbuncle effect, which consequently affected the accuracyof the heat transfer. A judicious griding at the leading edgeand choice of the flux computation strategy alleviated someof the carbuncle effect issues. The preliminary aeroelastic

8 Copyright c© 2012 by authors

Chordwise Location [m]

Tem

per

atu

re [

K]

-1 0 1 2 3 4 5 6200

400

600

800

1000

1200

1400

1600Mach 4.0, 3.0 kW/m^2Mach 5.0, 3.0 kW/m^2Mach 4.0, 1.5 kW/m^2Mach 5.0, 1.5 kW/m^2

(a)

Chordwise Location [m]

Tem

per

atu

re [

K]

-0.1 0 0.1 0.2 0.3 0.4 0.5

400

600

800

1000

1200

1400

1600Mach 4.0, 3.0 kW/m^2Mach 5.0, 3.0 kW/m^2Mach 4.0, 1.5 kW/m^2Mach 5.0, 1.5 kW/m^2

(b)

Figure 9. Surface temperature as a function of Mach number and interior

heat flux: (a) entire airfoil and (b) leading edge region.

simulation, in the absence of the thermal effects, was notaffected by local nonphysical results that originated at thewing leading edge.

Position along chord [m]

y

0 2

-0.5

0

0.5

Figure 10. Structural model.

(a)

(b)

Figure 11. Fin meshes: (a) medium, (b) fine.

6 AcknowledgmentsThis work was supported by the Multidisciplinary Uni-

versity Research Initiative grant FA9550-09-1-0686 fromthe Air Force Office of Scientific Research to the TexasA&M University with Dr. David Stargel as the programmanager. The authors also acknowledge the Texas A&MSupercomputing Facility (http://sc.tamu.edu/) for provid-

9 Copyright c© 2012 by authors

Figure 12. Surface temperature profile of fin at Mach 5.

ing computing resources useful in conducting the researchreported in this paper.

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11 Copyright c© 2012 by authors