Read Fluid-Thermal-Structural Modeling and Analysis of Hypersonic Structures under Combined Loading text version

52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference<BR> 19th 4 - 7 April 2011, Denver, Colorado

AIAA 2011-1965

Fluid-Thermal-Structural Modeling and Analysis of Hypersonic Structures under Combined Loading

Adam J. Culler

Air Force Research Laboratory, Wright-Patterson AFB, Ohio 45433

Jack J. McNamara

The Ohio State University, Columbus, Ohio 43210

Accurate predictions of structural response and life in extreme environments are necessary to achieve the United States Air Force' goals of affordable, reusable platforms capable of sustained hypersonic flight and responsive access to space. However, the predictive capability of current commercial software is limited for combined aerothermal and aeropressure loading due in part to the inability to seamlessly address multi-coupled, multi-scale fluid-thermal-structural interactions. This study aims to quantify the significance of a frequently neglected interaction, namely: the mutual (2-way) coupling of structural deformation and aerodynamic heating, on response prediction in hypersonic flow. In order to accomplish this objective, an additional focus is on the use of partitioned solution procedures to couple separate: fluid, thermal, and structural models. The response of a carbon-carbon hypersonic skin panel in Mach 12 flow is investigated. It is determined that the 2-way coupled quasi-static solution is converged for O(10) thermal time steps and O(10) deformation updates in aerodynamic heating computations within the characteristic thermal response time. Subsequently, it is shown that including the dependence of aerodynamic heating on structural deformation results in O(20%) increase in peak skin temperature and O(200%) increase in surface ply failure index for relatively modest peak displacement, O(2%) of panel length. Dynamic aeroelastic stability of the quasi-static response predictions is verified through the use of short-duration, transient dynamic response tests that use subiterations to converge the fluid-structural response. Additionally, a long-duration, staggered dynamic solution procedure is investigated. It is determined that the use of sequential cold restarts of the dynamic structural solution results in numerical errors that can alter the predicted response.

Nomenclature

B CF L c cf E EF S EP Fo Fply Fv f1 G H h hc

Assistant

= ratio of the characteristic thermal to structural response times = Courant-Friedrichs-Lewy number = specific heat = local skin friction coefficient = elastic modulus = fluid-structure convergence criteria = load convergence criteria (MSC.Nastran) = Fourier number = surface ply failure index = view factor = lowest natural frequency = shear modulus = enthalpy = skin thickness = convection heat transfer coefficient

Research Engineer, Universal Technology Corporation. Member AIAA. Professor, Department of Mechanical and Aerospace Engineering. Senior Member AIAA.

Postdoctoral

Copyright © 2011 by Culler, A. J. and McNamara, J. J. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

k L M Pr p Qaero Qrad q Rex r St T Tinitial Tstruct Tv t tF , tT , tS U w, w x x3 y z Subscripts 0 11 12 22 3 4 aw e J j M N n P w, surf x, y, z

= thermal conductivity = characteristic length scale = Mach number = Prandtl number = pressure = aerodynamic heat flux = radiation heat flux = U 2 /2, dynamic pressure = local Reynolds number = recovery factor = Stanton number = temperature = initial uniform panel temperature = panel structure temperature = viewed surface temperature = time = characteristic fluid, thermal, and structural response times, respectively = velocity, air = panel displacement and velocity, respectively, z-direction = chordwise direction, flow direction over undeformed panel = distance from transition to turbulence to leading edge of panel = spanwise direction = transverse direction, normal to undeformed panel surface = thermal expansion coefficient = oblique shock angle relative to freestream = ratio of specific heats, air = increment = emissivity = viscosity, air = undeformed panel surface inclination angle to freestream = Poisson's ratio = density = Stefan-Boltzmann constant, 3.302 × 10-15 BT U/sec/in2 /R4

= total condition = longitudinal direction, composite ply = lateral with respect to longitudinal, composite ply = lateral direction, composite ply = flow at leading edge of panel = point of interest along panel = adiabatic wall = edge of boundary layer = number of surface elements = surface element index = time step offset for application of a steady thermal load = number of time steps performed in each call to MSC.Nastran = time step index = total number of time steps performed in a dynamic analysis = wall, aerodynamic surface = x-direction, y-direction, z-direction = freestream

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Superscripts i = subiteration index = evaluated at the reference enthalpy

I.

Introduction

FFORDABLE , reusable platforms capable of sustained hypersonic flight and responsive access to space are needed to meet at least two of the Focused Long Term Challenges of the U.S. Air Force Research Laboratory, namely:1 Persistent & Responsive Precision Engagement and On-demand Force Projection, Anywhere. The current, widespread focus2­6 by the U.S. Department of Defense and the National Aeronautics and Space Administration on the development and demonstration of hypersonic technologies that will enable these types of prompt global reach objectives is indicative of the numerous technology gaps that must be addressed in order to develop these platforms. The five most critical enabling technologies (in addition to propulsion) were listed in order of priority in a 1998 review of the Air Force Hypersonic Technology (HyTech) Program:7 "1) airframe and engine thermostructural systems; 2) vehicle integration; 3) stability, guidance and control, navigation, and communication systems; 4) terminal guidance and sensors; and 5) tailored munitions." Over a decade later, these broad challenges still remain. An important requirement for the design of hypersonic airframe and engine thermostructural systems is accurate prediction of the response to simultaneous, aerodynamic heating and fluctuating pressures over long durations.8­15 This presents many significant computational challenges, including: coupling of the fluid-thermal-structural response (i.e., aerothermoelasticity).13, 16 Consider Fig. 1, which illustrates the degree of coupling between the different disciplines that compose the field of aerothermoelasticity.17 Typically, the aerothermoelastic problem is simplified by neglecting "weak" couplings, as well as the effect of aerodynamic pressure on aerodynamic heating. In such an approach, the aerothermal solution is obtained first, using a reference geometry of the structure, over the entire range of relevant operating conditions. Subsequently, the aeroelastic analysis is carried out using an updated structure based on the resulting temperature distribution. This simplification of the aerothermoelastic problem is denoted here as 1-way coupling, and relies on three important assumptions:17­20 1) thermodynamic coupling between heat generation and elastic deformation is negligible; 2) dynamic aeroelastic coupling is small, i.e., the characteristic time of the aerothermal system is large relative to the time periods of the natural modes of the aeroelastic system; and 3) static aeroelastic coupling (static elastic deflections due to steady-state pressure and thermal loading) is insufficient to alter the temperature distribution from the reference condition. Under conditions where these assumptions fail, feedback from the aeroelastic solution to the aerothermal solution is required in order to update the aerodynamic heating conditions based on structural deformation. This procedure is denoted here as 2-way coupling. Part of the focus of this paper is to investigate the significance of 2-way coupling for response prediction in hypersonic flow.

A

Figure 1. Degree of coupling in aerothermoelasticity, Ref. [17].

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Despite its potential importance, only a limited number of studies have considered mutual coupling of aerodynamic heating and structural deformation (2-way coupling) for response prediction in hypersonic flow. These studies have focused exclusively on panel structures and leading edges. Note that numerous hypersonic panel flutter studies have considered thermal effects by prescribing a steady-state temperature distribution.21, 22 Thus, these studies further simplified the aerothermoelastic problem by neglecting the transient aerothermal analysis in addition to 2-way coupling. The impact of 2-way coupling on quasi-static response was investigated for aerodynamically heated panels23 and leading edges24, 25 using a staggered procedure to couple finite element flow, thermal, and structural models. The solution sequence alternated between flow-thermal analysis (flow analysis with updated, nonuniform surface temperature) and thermal analysis of the structure. At select times, the quasi-static structural deformation due to "steady-state" pressure and thermal loads was updated in the flow-thermal analysis. Thornton and Dechaumphai23 found that heating rate distributions were altered significantly by only modest panel deformations (whether concave or convex). Dechaumphai et al.24, 25 concluded in the latter study25 that leading edge deformations due to shock-shock interference heating can alter the aerodynamic heating distribution and must be taken into account. Note that the impact of 2-way coupling on deformation and stress was not assessed in these studies.23­25 Two-way coupling was also considered for the quasi-static response of metallic thermal protection panels using a two-dimensional boundary element method26 and the finite element method27 for thermal and structural models, each loosely-coupled to a hypersonic computational fluid dynamics algorithm. Kontinos26 found that aerodynamic heating distributions were significantly altered by both concave and convex panel bowing. In Ref. [27], bowing of the metallic panels into the flow was modeled using an iterative procedure to compute the quasi-static deformation under transient heating. Kontinos and Palmer27 reported that the impact of coupling on deformation was negligible, however, the coupled analysis resulted in significant surface temperature variations and increased thermal stresses. Results of these studies23­27 demonstrated a significant dependence of aerodynamic heating on structural deformation. However, since these studies were limited in scope and/or modeling, more work is needed to fully understand the importance of 2-way coupling for response prediction. The authors' recent investigation28 into the effects of using different types of fluid-thermal-structural coupling for hypersonic aerothermoelasticity was undertaken to address these shortcomings in a comprehensive manner. This study used analytical modeling techniques to investigate a simply-supported, thermally-insulated metallic panel undergoing cylindrical bending. It was found28 that 2-way coupling can impact flutter boundary predictions and nonlinear flutter response, in addition to aerodynamic heating and transient temperature distributions. Furthermore, it was demonstrated that simplified temporal coupling procedures offer substantial reductions in computational expense, with negligible loss of accuracy, for aerothermoelastic analysis over long-duration hypersonic trajectories. More recently,29 the 2-way coupled modeling approach28 was used to investigate the response of a carbon-carbon hypersonic skin panel using finite element thermal and structural models. It was found29 that the significance of 2-way coupling for quasi-static structural response prediction depends largely on the in-plane boundary conditions, since increasing resistance to thermal expansion results in larger deformations. Including these deformations in the aerodynamic heating analysis resulted in nonuniform skin temperatures, asymmetric deformation, and elevated stresses. For the panel and trajectories investigated,29 2-way coupling resulted in low to moderate increases (7 ­ 23%) in peak skin temperature rise and large increases (100 ­ 200%) in the surface ply failure index for peak displacements in the range of 1 ­ 4 skin thicknesses (0.5 ­ 2.2% of panel length). The current effort is an extension of the authors' recent work29 on the topic and is aimed at: 1) investigating convergence and dynamic stability of quasi-static response predictions, and 2) developing a dynamic solution procedure for long-duration response prediction. Note in the previous effort,29 convergence studies were performed on the separate models. Thus, part of the current focus is on verifying convergence and stability of the coupled models. The specific objectives of this paper are: 1. Investigate convergence of the 2-way coupled quasi-static solution procedure with respect to: thermal time step, deformation feedback for 2-way coupling, and deformation-induced pressure load. Subsequently, demonstrate the effect of 2-way coupling on quasi-static response prediction. 2. Assess the dynamic aeroelastic stability of quasi-static response predictions through the use of shortduration, dynamic response tests that use subiterations to converge the fluid-structural response.

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3. Develop a long-duration, dynamic solution procedure in order to study the impact of 2-way coupling under transient, combined aerothermal and aeropressure loading. Fulfilling these objectives will make a significant contribution towards understanding the importance of fluid-thermal-structural coupling for response prediction in hypersonic flow. In addition, this work furthers the development of a comprehensive suite of tools for combined physics problems associated with extreme environment structures.

II.

Fluid-Thermal-Structural Model

The fluid-thermal-structural model used in this investigation is illustrated in Fig. 2. The blocks labeled: Aerodynamic Pressure, Aerodynamic Heating, Thermal, and Structural represent separate fluid, thermal, and structural models. These individual models are coupled by the parameters that label the input/output arrows. Note that Aerodynamic Pressure includes self-induced fluctuating pressures due to structural deformation, and Structural includes both elastic and inertial forces depicted in Fig. 1. Thus, the fluid-thermalstructural model includes all of the "strong" couplings depicted in Fig. 1, while neglecting the "weak" couplings. This results in 2-way coupling between the aerothermal (aerodynamic heating + thermal) and aeroelastic (aerodynamic pressure + structural) models, as depicted by the arrows labeled 1 and 2. Accordingly, the capability to study the impact of 2-way coupling (includes both arrows 1 and 2) versus 1-way coupling (neglects arrow 2) is provided.

Figure 2. Fluid-thermal-structural model.

II.A.

Representative Hypersonic Skin Panel

In this study, the fluid-thermal-structural modeling approach is applied to a representative carbon-carbon panel structure of an airbreathing hypersonic vehicle. The panel illustrated in Fig. 3 corresponds to a portion of the stiffened-skin located on the inlet ramp of the blended wing body airbreathing hypersonic vehicle concept11, 12 shown in Fig. 4. The inlet ramp forms a compression surface for the scramjet engine. This region of the vehicle is exposed to severe heating and pressure loading,11, 12 therefore skin panels on the inlet ramp are an interesting test case for coupled analysis. Blevins et al.11, 12 performed decoupled flow, thermal, static, and dynamic analyses of a larger section of the inlet ramp (shown inverted in Fig. 5) for transatmospheric flight trajectories. Details of the geometry and carbon-carbon construction of this larger panel structure (Fig. 5) are used in the present work to develop thermal and structural models of a representative hypersonic panel (Fig. 3).

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Figure 3. Representative hypersonic skin panel.

Figure 4. Blended wing body airbreathing hypersonic vehicle concept, Refs. [11, 12].

Figure 5. Inlet ramp panel section considered in Refs. [11, 12].

II.B. II.B.1.

Thermal and Structural Models Geometry

Thermal and structural finite element models of the representative hypersonic skin panel (Fig. 3) are developed using MSC.Patran and MSC.Nastran. Skin thickness (0.065 in), stiffener height (1.25 in), and stiffener spacing (10 in) of the representative panel are equal to the ramp panel (Fig. 5) investigated in Ref. [11], whereas, the representative stiffener thickness (0.0325 in) is half due to symmetry of the larger ramp panel.11 Additionally, in order to develop a generic representation, the substructure attachment flanges at the leading and trailing edges of the ramp panel (Fig. 5) are eliminated. Therefore, the length (12 in) of the representative panel is set to match the frequency of the first skin bending mode using clamped skin edges (approximately 250 Hz under uniform 1.25 psi load).11 The first six mode shapes and frequencies of the representative panel under no load are shown in Fig. 6. II.B.2. Material Properties

The carbon-carbon composite laminates are composed of internal 0/90 plies of carbon-carbon fabric, sandwiched between two outer plies of fabric oriented at 45 deg.11 A five-ply layup is used for the skin and a four-ply layup for the stiffeners.11 Thermal properties of the composite laminates at room temperature (70 F) are listed in Table 1. Temperature dependent specific heat and in-plane thermal conductivity (Table 2) are used. Density and emissivity,11 as well as through-thickness thermal conductivity30 are not temperature

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(a) Mode 1: 199 Hz.

(b) Mode 2: 414 Hz.

(c) Mode 3: 445 Hz.

(d) Mode 4: 667 Hz.

(e) Mode 5: 722 Hz.

(f) Mode 6: 798 Hz.

Figure 6. Mode shapes and frequencies of the representative panel under no load.

dependent in this model. Structural properties of the carbon-carbon plies are listed in Table 3. These properties (except density) are not provided in Ref. [11], therefore the carbon-carbon fabric is modeled as advanced carbon-carbon 4 (ACC-4).31­33 Temperature dependent thermal expansion coefficients (Table 4) are employed using a bi-liner fit of the data plotted in Fig. 1 of Ref. [33]. The remaining structural properties are not temperature dependent in this model. Note that carbon-carbon composites retain room temperature mechanical properties up to at least 4000 F in non-oxidizing environments (in oxidizing environments, coatings limit maximum use temperatures to approximately 3000 F).34 Strengths31 of the carbon-carbon plies used for static failure prediction are provided in Table 5. In this work, structural damping is modeled as equivalent viscous damping correlated at the first natural frequency, with a damping coefficient of 1.6% of critical based on Ref. [35].

Table 1. Thermal properties of the carbon-carbon laminates at 70 F.

c kx ky kz

0.065 0.180 18.6 18.6 3.0 0.8

lbm/in3 BT U/lbm/ F BT U/hr/f t/ F BT U/hr/f t/ F BT U/hr/f t/ F

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Table 2. Temperature dependence of specific heat and in-plane thermal conductivity of the carbon-carbon laminates, Ref. [11].

T ( F) 0 200 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000

c (BT U/lbm/ F ) 0.170 -- 0.242 -- 0.295 -- 0.330 -- 0.360 -- 0.390 -- 0.420

k x = ky (BT U/hr/f t/ F ) 17.5 20.9 23.6 24.2 24.2 23.9 23.3 23.1 22.5 21.9 21.4 20.9 20.3

Table 3. Structural properties of the carbon-carbon plies at 70 F.

E11 E22 G12 12 11 22

0.065 15 × 106 15 × 106 2.5 × 106 0.3 0.58 × 10-6 0.58 × 10-6

lbm/in3 psi psi psi 1/ F 1/ F

Table 4. Temperature dependence of thermal expansion coefficients of the carbon-carbon plies, Ref. [33].

T ( F ) 30 2190 2540

11 = 22 (10-6 1/ F ) 0.556 2.111 2.111

Table 5. Strengths of the carbon-carbon plies, Ref. [31].

Tension Compression In-Plane Shear Interlaminar Shear

40 × 103 24 × 103 6.0 × 103 1.5 × 103

psi psi psi psi

II.B.3.

Elements

The thermal model uses continuum (CHEXA) elements for the skin (one layer) and shell (CQUAD4) elements for the stiffeners, whereas, the structural model consists of shell (CQUAD4) elements (one layer) for

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both skin and stiffeners. Continuum elements are used in the thermal model to compute through-thickness temperature gradients in the skin. However, shell elements are required (in MSC.Nastran) in the structural model to compute stresses in each ply of the composite laminate, since continuum elements use "equivalent" (i.e., smeared) structural properties. Independent thermal and structural grid convergence studies were performed36 using uniform, square grids (0.25 x 0.25 in and 0.125 x 0.125 in). The grid selected (0.25 x 0.25 in) for subsequent investigations results in 2400 elements for each model; the thermal model contains 4508 nodes, and the structural model has 2499 (14,994 degrees of freedom). The difference in the number of nodes results from using 8-node CHEXA elements for the thermal skin and 4-node CQUAD4 elements for the structural skin. The selected grid (0.25 x 0.25 in) showed excellent agreement (generally less than 1% difference) with the 0.125 x 0.125 in grid for skin temperatures, displacements, and frequencies, and reasonable agreement (less than 10% difference) in peak stress. II.B.4. Loads

Thermal loads are applied to the structural model by passing the temperature distribution from the thermal model. The use of matching grids simplifies this operation, but is not required. Note that thermal bending moments in the skin are included by prescribing the through-thickness temperature gradients from the continuum element thermal skin onto the shell element structural skin. Aerodynamic pressure and heating loads from the fluid models (discussed in the following sections) are applied to the skin of the structural and thermal models, respectively. The internal panel pressure is set equal to the undeformed external panel pressure, thus the differential pressure load is deformationinduced. The deformation-induced pressure (p4 - p3 ) is given by Eq. (2). Aerodynamic heating is applied using a convection heating boundary condition, Eq. (11), where the convection heat transfer coefficient and adiabatic wall temperature are the input (see Fig. 2). Thermal radiation is modeled by considering the upper skin surface to be diffuse, gray, opaque, and enclosed (Fv = 1) by a low temperature environment (Tv = 0 R).37, 38 Thus, heat rejected by thermal radiation from the upper skin surface is given by Eq. (1), all other panel surfaces are adiabatic. Note that radiation from the upper skin surface is modeled within the MSC.Nastran thermal model. Also, the adiabatic boundary condition applied to the remaining panel surfaces is based on the insulated configuration described in Ref. [11].

4 4 Qrad = Fv (Tw - Tv )

(1)

II.C.

Representative Hypersonic Vehicle Configuration

A representative hypersonic vehicle configuration is used to generalize the study of fluid-thermal-structural interactions of panel structures in hypersonic flow. The representative configuration is a wedge-shaped body, such that a panel located on the surface is inclined to the freestream flow, as depicted in Fig. 7. Note that the panel can deform due to pressure and thermal loads, whereas, the wedge is considered to be flat and rigid. This configuration has been used previously in experimental investigations of the aerodynamic heating of panels in a hypersonic wind tunnel,39, 40 and in computational studies of panel response in hypersonic flow.23, 41 Additionally, this configuration is representative of airbreathing hypersonic vehicles (e.g., NASA X-43A, Fig. 8) that have lifting body designs with large flat surfaces. An advantage of this configuration (Fig. 7) is that approximate hypersonic aerodynamic theories and semi-empirical methods provide reasonably accurate pressure and heating loads. In the authors' previous work,28, 42 it was shown that the present fluid models (described in the following sections) produce heat flux predictions that compare well with experimental data40 for Mach 7 flow over spherical dome protuberances, and with Reynolds Averaged Navier-Stokes solutions for 2-D Mach 8 flow over deformed panels. In addition, the present fluid models were shown29 to predict similar mean pressure and heating values for the inlet ramp section of the conceptual hypersonic vehicle illustrated in Fig. 4 compared to Parabolized Navier-Stokes solutions11 for 2-D Mach 6 to 15 flow.

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Figure 7. Representative hypersonic flight vehicle configuration.

Figure 8. NASA X-43A Hyper-X airbreathing hypersonic test vehicle.

II.D. II.D.1.

Fluid Models Aerodynamic Pressure

In the representative configuration (Fig. 7), the freestream flow (location 1) is hypersonic, resulting in an attached oblique shock at the leading edge of the wedge-shaped body.43, 44 Inclined surfaces before and after the panel are flat and rigid, thus the inviscid flow properties at the leading edge of the panel (location 3) are the same as those directly behind the oblique shock wave (location 2). These properties are computed using the oblique shock relations,41, 43 and the undeformed configuration of the panel is parallel to this flow. Panel deformations (location 4) alter the inviscid flow properties. The unsteady pressure over the deformed panel is computed using piston theory;45, 46 temperature and Mach number of the inviscid flow are computed using this pressure in conjunction with isentropic flow relations.43 Note that the pressure, temperature, and Mach number of the inviscid flow over the deformed panel are the boundary layer edge properties used in the computation of aerodynamic heating (see Fig. 2). Thus, deformation of the panel impacts both the aerodynamic pressure and the aerodynamic heating distributions. Because of the comprehensive nature of this work, a computationally efficient unsteady aerodynamic theory is essential for model tractability. Therefore, piston theory, which provides a simple point-function relationship between the unsteady pressure and surface deformation, is selected. While piston theory represents a simplistic model for the inviscid aerodynamics, it has been observed in several studies16 to provide reasonably accurate pressure predictions so long as the product of Mach number and surface inclination remains below unity. A third-order expansion of piston theory, given by Eq. (2), is used in this work due to the combined presence of hypersonic flow and moderate panel deformations.21, 47 Note that Eq. (2) uses

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the local flow properties at location 3 as the ambient panel conditions. q3 M3 1 w w + U3 t x +1 1 w w M3 + 4 U3 t x

2

p4 = p3 + 2 II.D.2.

+

+

+1 2 M3 12

1 w w + U3 t x

3

(2)

Aerodynamic Heating

Aerodynamic heating of the panel from the boundary layer flow is modeled using Eckert's reference enthalpy method.48 This semi-empirical method uses boundary layer relations from incompressible flow theory with flow properties evaluated at a reference condition (i.e., the reference enthalpy) to account for the effects of compressibility. Eckert's reference enthalpy method has been used extensively in approximate analyses to efficiently model convective heating of aerospace vehicles,49­54 and has been shown55 to provide reasonable agreement with wind tunnel and flight test data in the range of Mach 1.4 to 15. Note that Eckert's reference enthalpy method differs from Eckert's reference temperature method in that the latter assumes calorically perfect gas, i.e., constant specific heat. Since this study is concerned with hypersonic flow, in which the specific heat varies significantly through the boundary layer due to high temperatures and real gas effects,44 the reference enthalpy method is preferred.48 In this work, temperatureenthalpy tables56 that include the effect of dissociation based on equilibrium air properties are used to implement the reference enthalpy method. The aerodynamic heating distribution is computed by implementing Eckert's reference enthalpy method point-wise along the panel using the equations given below. The point-wise inputs are wall temperature from the thermal model and boundary layer edge properties from the aerodynamic pressure model. Eckert's reference enthalpy is given by Eq. (3).48 In this equation, the adiabatic wall enthalpy is computed from Eq. (4), where the total enthalpy and the recovery factor (for turbulent flow) are given in Eqs. (5) and (6), respectively.48 Local velocity and enthalpy at the edge of the boundary layer are determined from the local edge Mach number and temperature using ideal gas relations44 and the temperature-enthalpy tables,56 respectively. Note that iterations are performed to converge the reference enthalpy and reference Prandtl number. Also, in this investigation, the complete transition from laminar to turbulent flow is assumed to occur upstream of the leading edge of the panel, such that flow over the panel is fully-turbulent. H = He + 0.50(Hw - He ) + 0.22(Haw - He ) Haw = r(H0 - He ) + He H0 = He +

2 Ue 2

(3) (4) (5) (6)

r = (P r )1/3

Aerodynamic heat flux is computed using Eq. (7). In this equation, the Stanton number is determined from the Colburn-Reynolds analogy,48, 55 Eq. (8), in which the local skin friction coefficient for turbulent flow is computed using the Schultz-Grunow formula,48, 55 Eq. (9). The local Reynolds number used in Eq. (9) is defined in Eq.(10) and is computed using the distance from the onset of transition to the point of interest along the panel.55 Note that the location of the onset of transition is assumed to be known, e.g., by the location of boundary layer trips as in experimental studies.39, 40 Qaero = St Ue (Haw - Hw ) St = c = f c 1 f 2 (P r )2/3 (7) (8) (9) (10)

0.370 (log10 Re )2.584 x Ue x

Re = x

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Note in Eqs. (3) and (7), enthalpy at the wall is determined using the wall temperature (from the thermal model) and the temperature-enthalpy tables.56 These tables are also used to determine a temperature from the reference enthalpy in order to evaluate reference density, reference viscosity, and reference Prandtl number, using the ideal gas law,44 Sutherland's law,44 and temperature-dependent Prandtl number tables,38 respectively. Aerodynamic heating is applied in the thermal model using a convection heating boundary condition, Eq. (11), where the convection heat transfer coefficient and adiabatic wall temperature are the input. This type of input is used (instead of heat flux) because aerodynamic heat flux is a strong function of the transient wall temperature. The convection heat transfer coefficient is computed using Eq. (12), which is derived by equating Eqs. (7) and (11). The adiabatic wall temperature is given by Eq. (13), in which the total temperature is computed using Eq. (14). Qaero = hc (Taw - Tw ) hc = St Ue (Haw - Hw ) Taw - Tw (11) (12) (13) (14)

Taw = r(T0 - Te ) + Te T0 = Te 1 + -1 2 Me 2

Note that with this formulation there are two mechanisms for the wall temperature to modify the aerodynamic heat flux. The strongest impact is through the relative magnitude of the wall temperature to the adiabatic wall temperature, as shown by Eq. (11). However, note that the convection heat transfer coefficient also exhibits a dependence, albeit weaker, on wall temperature due to the dependence of reference enthalpy on wall enthalpy in Eq. (3). Additionally, note that both the convection heat transfer coefficient and adiabatic wall temperature are dependent on deformation through the boundary layer edge properties.

III.

Solution Procedures

Three types of fluid-thermal-structural solutions are considered, namely: 1) quasi-static, 2) dynamic response test, and 3) dynamic. Each solution uses a partitioned approach to couple the fluid, thermal, and structural models. A partitioned approach is advantageous for this problem, since it enables the use of relatively independent solution procedures for each discipline. Additionally, this approach is favorable for addressing the disparate time scales that occur naturally in fluid-thermal-structural interaction problems. III.A. Time Scales

The time scales of the fluid, thermal, and structural responses are important considerations in the development of a partitioned solution procedure. For instance, the ratio of the characteristic thermal response time to the structural response time is a key parameter for the occurrence of thermally-induced oscillations.37, 57, 58 This nondimensional parameter (B) is defined in Eq. (15). For values of B >> 1 (i.e., thermal response time much greater than structural response time), thermally-induced vibrations are insignificant and quasi-static thermal-structural analysis is justified.37 Note, in this context thermal-structural analysis implies the absence of fluid flow. In a quasi-static thermal-structural analysis, inertial effects are neglected and structural deformation is a function of the instantaneous thermal loads. This results in a simplified solution procedure that uses a transient thermal analysis and a static structural analysis. B= tT tS (15)

In addition to B >> 1, quasi-static fluid-thermal-structural response requires dynamic aeroelastic stability, i.e., deformation-induced pressures do not result in flutter oscillations or snap-through. Furthermore, dynamic loading (acoustic, gust, etc.) is absent or insufficient to cause structural vibrations. When these conditions are not satisfied, inertial effects cannot be neglected, and a dynamic analysis is required. Note

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that hypersonic flight trajectories subject various regions of the vehicle to combined, dynamic loading and severe heating over extended durations (t >> tT ).8, 9, 11, 12 This indicates the need to investigate the interdependence of structural dynamics and transient thermal loading. Thus, part of the focus of this work is on the development of a long-duration dynamic fluid-thermal-structural solution procedure. The characteristic thermal response time is related to the Fourier number, Eq. (16), which compares a characteristic body dimension (e.g., skin thickness) with an approximate temperature wave penetration depth at a given time.37, 38 Note that the Fourier number involves heat conduction and capacitance terms, thus it is independent of convection and radiation boundary conditions. When the Fourier number has a value of 1, the temperature wave has propagated a distance equal to the characteristic body dimension. Thus, setting F o = 1 yields the characteristic thermal response time, Eq. (17). The characteristic structural response time is simply the period of the lowest natural frequency, Eq. (18).37, 58 The characteristic fluid response time is given by Eq. (19) as the ratio of the characteristic length scale to the speed at which disturbances propagate.59, 60 Note that the characteristic fluid response time is related to a Courant-FriedrichsLewy number of 1 by Eq. (20).59, 60 Fo = kt c h2 c h2 k 1 f1 L U (16) (17) (18) (19)

tT =

tS = tF =

t (20) x The characteristic response times of the representative skin panel (Fig. 3) are listed in Table 6. Thermal and structural time scales are computed using the unheated thermal properties (Table 1) and lowest natural frequency (Fig. 6), respectively. The fluid time scale is based on Mach 12 flow (which is used in subsequent analyses) and the full length of the panel. CF L = U

Table 6. Characteristic response time scales.

Thermal, tT Structural, tS Fluid, tF

7.1 × 10-1 5.0 × 10-3 7.5 × 10-5

s s s

The characteristic structural response time (5.0 × 10-3 s) is two orders of magnitude smaller than the thermal response time of the skin (7.1 × 10-1 s). Thus, thermally-induced vibrations are not expected and quasi-static thermal-structural analysis is justified. However, as noted above, the additional requirements of dynamic stability and insignificant dynamic loading are necessary for the response to be quasi-static. The characteristic fluid response time (7.5 × 10-5 s) is two orders of magnitude smaller than the structural time scale. This indicates that the fluid response is quasi-steady, i.e., governed by the instantaneous structural deformation. Note that the aerodynamics used in this work are quasi-steady models. III.B. Quasi-static Solution Procedure

Based on the above time scale analysis, the response of the representative hypersonic panel is expected to be quasi-static (in the absence of dynamic loading/instability). The quasi-static response is investigated using the solution procedure depicted in Fig. 9. This procedure uses a nonlinear transient thermal solution (MSC.Nastran SOL 159) and a nonlinear static structural solution (MSC.Nastran SOL 106). Nonlinearities in the thermal solution are temperature dependent properties (thermal conductivity and specific heat) and thermal radiation. The structural solution includes nonlinear geometric effects due to large displacements.

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Figure 9. Quasi-static solution procedure.

The staggered solution sequence is: 1) Aerodynamic Pressure, 2) Aerodynamic Heating, 3) Thermal, and 4) Structural. In step 3, the transient thermal analysis is carried out for N time steps, and the resulting temperature distribution (Tstruct ) at time tn+N is the thermal load in the subsequent structural analysis. The pressure (psurf ) used in the structural analysis is initially computed using the deformation (w) at time tn , thus, subiterations are performed to ensure pressure-deformation convergence at time tn+N . Convergence of the pressure load is tested using the relative error defined in Eq. (21). The numerator is the root-meansquare pressure deviation from the previous subiteration (i - 1), and the denominator is the root-meansquare pressure (j is the surface element index).

i EF S = J 1 j=1 J

(pj,i - pj,i-1 ) (pj,i )

2

2

J 1 j=1 J

(21)

The arrows labeled 1 and 2 in Fig. 9 are used to differentiate between a 1-way coupled analysis (arrow 2 not included) and a 2-way coupled analysis (includes both arrows 1 and 2). Thus, the 1-way coupled analysis neglects the dependence of aerodynamic heating on structural deformation (which alters the boundary layer edge properties: pe , Te , Me ). The advantage of 1-way coupling is the aerothermal analysis can be performed for an entire trajectory, followed by structural analyses at specific points. In the 2-way coupled analysis, the deformation at time tn is used to compute the aerodynamic heating input (hc , Taw ), which is held constant for the next N time steps. Thus, 2-way coupling significantly complicates the analysis by requiring a repetitive sequence of fluid, thermal, and structural solutions. In the 2-way coupled solution procedure illustrated in Fig. 9, the impact of deformation on aerodynamic heating is lagged by up to N time steps in the thermal analysis. Thus, convergence studies are performed to test the sensitivity of the solution to the rate at which deformation is updated (every N t). Convergence of the 2-way coupled procedure is expected (and demonstrated in Sec. IV.B), since the characteristic thermal response time is two orders of magnitude greater than the structural time scale (see Table 6), i.e., thermal loads are altered gradually by structural deformation. Conversely, pressure loads are altered nearly instantaneously by structural deformation, thus subiterations are used to ensure pressure-deformation convergence. The 1-way and 2-way coupled quasi-static solutions (ID: S-1 and S-2, respectively) used in this study are summarized in Table 7. The selected time step (0.01 s) results in approximately 70 time steps per characteristic thermal response time (0.71 s), and N = 5 results in approximately 14 deformation updates within this same period of time for 2-way coupling. Note that N = 5 is also used for 1-way coupling. This is selected for direct comparison with the 2-way coupled deformations at any given time. Finally, note that for transient thermal analysis it is reasonable to use a larger time step (e.g., 0.07 s) in conjunction with N = 1 in order to have 10 time steps and deformation updates within the thermal time scale. However, MSC.Nastran transient solutions require a minimum of three time steps (N 3, although a one-step integration scheme, Newmark's method, is employed); thus, N = 5 is chosen for convenience.

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Table 7. Quasi-static solutions.

ID S-1 S-2

Structural Solution Static Static

Thermal Solution Transient Transient

Coupling Type 1-way 2-way

Time Step (ms) 10 10

N (#) 5 5

III.C.

Dynamic Response Test

Dynamic analysis is needed to test the aeroelastic stability of quasi-static response predictions. The dynamic response test procedure developed in this work is illustrated in Fig. 10. This procedure uses a nonlinear transient dynamic structural solution (MSC.Nastran SOL 129), which includes nonlinear geometric effects due to large displacements. The structural solution includes the effect of a steady thermal load (Tstruct ) from a previous quasi-static analysis. The thermal load may be selected to coincide with the initial time, tn , of the dynamic analysis (M = 0) or a later time (e.g., M = N ) in order to test the response to a thermal load perturbation.

Figure 10. Dynamic response test.

The dynamic response test alternates between: 1) Aerodynamic Pressure and 2) Structural solutions in order to converge the pressure-deformation (aeroelastic) response at each time step from tn to tn+N . Subiterations are performed on the entire duration until the solution is converged at each of the N time steps. Convergence of the deformation-induced pressure load is tested using the relative error defined in Eq. (21). Note that Eq. (21) parallels the criteria used within MSC.Nastran, Eq. (22), to test convergence of the nonlinear solution at each time step. In Eq. (22), R is the residual load vector, Fn and Pn are internal and external load vectors, respectively, and indicates the absolute norm. MSC.Nastran's default load convergence criteria is 10-3 . Appropriate values for both criteria, EF S and EP , are investigated in Sec. IV.E.

i EP =

Ri max ( Fn , Pn )

(22)

Two variations of the dynamic response test procedure are considered, as listed in Table 8. The first (RC) uses only one structural analysis for the entire duration of dynamic response from the initial time tn to the final time tn+P , thus N = P (where, P = 1000 time steps). This is referred to as the continuous solution. The second (R-S) uses a series of converged pressure-deformation solutions with multiple restarts to cover the same duration, e.g., N = 5, P = 1000. This is referred to as the staggered solution. The advantage of the continuous solution is simplicity, i.e., no restarts. However, the duration of the response is limited by the need to store the entire solution in random-access memory (RAM). The staggered solution seeks to circumvent this limitation through the use of sequential cold restarts, in which only limited data from the previous time (displacements, velocities, and loads) is used to restart the solution. Note that this cold restart procedure differs from MSC.Nastran's internal restart procedure, which is limited by the same large memory requirement as the continuous solution. Finally, note that the time step (0.1 ms) used for dynamic response tests results in approximately 50 time steps per characteristic structural response time (5.0 ms).

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Table 8. Dynamic response tests.

ID R-C R-S

Structural Solution Dynamic Dynamic

Time Step (ms) 0.1 0.1

N (#) 1000 5

Cold Restarts (#) 0 199

III.D.

Dynamic Solution Procedure

A dynamic fluid-thermal-structural solution procedure is needed to investigate long-duration structural response under transient thermal and pressure loading, and the impact of 2-way coupling on the predicted response. Long-duration implies response times on the order of the thermal time scale (or greater), such that the thermal load is not essentially constant. The dynamic solution procedure investigated in this work is depicted in Fig. 11. This procedure is a combination of the quasi-static solution procedure (Fig. 9) and the staggered dynamic response test (Fig. 10). Note that a continuous dynamic solution is impractical for longduration response due to excessive memory requirements (of MSC.Nastran). Similar to the quasi-static solution, the dynamic solution sequence is: 1) Aerodynamic Pressure, 2) Aerodynamic Heating, 3) Thermal, and 4) Structural. Upon completion of each structural analysis, pressure-deformation (fluid-structure) subiterations are performed, as in the staggered dynamic response test.

Figure 11. Dynamic solution procedure.

The 1-way and 2-way coupled dynamic solutions (ID: D-1 and D-2, respectively) investigated in this study are summarized in Table 9. Time step selection (0.1 ms) is governed by the dynamic structural solution; in this procedure the thermal and structural time step sizes are the same. The 0.1 ms time step results in approximately 50 time steps per characteristic structural response time (5.0 ms), and N = 5 results in 10 deformation updates within this same period of time for 2-way coupling. Note that N = 5 also results in the thermal load being updated approximately 10 times per characteristic structural response time. This is a perceived requirement, in order that application of the transient thermal load does not perturb the structural solution. However, the 0.1 ms time step in conjunction with N = 5 results in > 1400 thermal load updates per characteristic thermal response time (0.71 s), which may be excessive. Thus, further investigation into the required thermal load update rate is warranted. Finally, as noted above, MSC.Nastran transient solutions require a minimum of three time steps (N 3, although a one-step integration scheme, Newmark's method, is employed); thus, N = 5 is chosen for convenience. III.E. Computational Expense

Computational expense is a major consideration for the solution of fluid-thermal-structural interaction problems, since it limits the type and/or duration of analysis that can be performed in a given period of time. The approximate computational cost of each solution procedure is listed in the second column

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Table 9. Dynamic solutions.

ID D-1 D-2

Structural Solution Dynamic Dynamic

Thermal Solution Transient Transient

Coupling Type 1-way 2-way

Time Step (ms) 0.1 0.1

N (#) 5 5

of Table 10 as total computational time per second of response. This total time is broken down into fluid, thermal, and structural contributions in columns three, four, and five, respectively. The data in Table 10 are roughly the mean values from the studies performed, and have a range of approximately ±25% depending on the convergence criteria (10-3 to 10-5 ) used for Eqs. (21) and (22).

Table 10. Computational cost of the quasi-static solutions, dynamic response tests, and dynamic solutions.

Solution Procedure Quasi-static (S-1 and S-2) Dynamic Response Test (R-C and R-S) Dynamic (D-1 and D-2)

a

Computationala Time Per Second of Response (hr) 2 40 50

Fluid (%) 10 30 20

Thermal (%) 10 -- 20

Structural (%) 80 70 60

One 2.6 GHz Opteron processor, 2.0 GB RAM.

The greatest difference in computational expense is between the quasi-static and dynamic solutions. Dynamic solutions increase the cost by at least a factor of 20. This difference is driven by the need to carry out dynamic solutions on the structural time scale, whereas, quasi-static solutions are performed on the thermal time scale. As listed in Table 10, the cost of 1-way coupling and 2-way coupling is the same in this study whether quasi-static or dynamic solutions are employed. This is the result of using the same basic computational procedure for 1-way coupling as for 2-way coupling, but without updating the deformation-altered flow properties in aerodynamic heating computations (i.e., no attempt is made to exploit the simplicity of 1-way coupling). This approach is taken in order to ensure a direct comparison of the responses predicted by 1-way and 2-way coupling. Note in Table 10 that the structural solutions account for the majority of the total computational expense. For quasi-static analysis, this is a direct result of the 2-way coupling procedure, which requires performing relatively expensive nonlinear static structural solutions on the same time scale as the thermal time step. As noted above, the simplicity of 1-way coupling is not exploited in this study. However, for 1-way coupled quasi-static analysis, the majority of the structural solution cost can be eliminated by performing structural analyses only at points of interest. Conversely, for transient dynamic analysis, the cost of the structural solution cannot be reduced by 1-way coupling. However, 1-way coupling does offer savings in the aerodynamic heating and thermal analyses, since these solutions are governed by the thermal time scale (and need not be performed on the smaller structural time scale required by 2-way coupling). This potential savings is small in this study due to the use of relatively simple aerodynamic methods, i.e., the fluid solutions account for only about 20% of the cost of dynamic solutions. However, the overall savings of 1-way coupling may be much more significant for the use of relatively expensive computational fluid dynamics solutions.

IV.

Results and Discussion

The fluid-thermal-structural model and solution procedures are used to investigate the impact of 2way coupling on quasi-static and dynamic response prediction. Initial studies are performed to investigate convergence of the quasi-static solution procedure with respect to: thermal time step, feedback of deformation for 2-way coupling, and deformation-induced pressure load. Subsequently, the dynamic stability

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of quasi-static response predictions is investigated using short-duration dynamic response tests. In addition, the dynamic solution procedure is used to investigate long-duration dynamic response prediction. Unexpected oscillations in the long-duration dynamic response prompt an investigation into convergence criteria and the impact of cold restarts on the dynamic solution. IV.A. Trajectory and Boundary Conditions

The trajectory and structural boundary conditions used in this investigation are a subset of those considered in the authors' previous work.29 Specifically, the Mach 12 trajectory and 10% resistance to in-plane expansion in the flow direction are selected. Under these conditions, 2-way coupling was shown29 to have a significant impact on the quasi-static response within 15 s. Since 15 s transient dynamic responses are computationally feasible (see Table 10), these conditions are also selected to study the impact of 2-way coupling on dynamic response predictions. Note that 10% resistance to in-plane expansion is modeled by placing a linear spring at each node of the trailing edge of the skin (acting in the x-direction), while the leading edge of the skin is fixed in x-translation. The total spring constant is 10% of E11 h, however, the springs are permited to deflect independently. The additional structural boundary conditions are: 1) zero rotation and zero z-translation along the leading (x = 0) and trailing (x = 12 in) edges of the skin, and 2) zero y-translation at the center of the skin (to prevent free-body motion). Note that the panel is free to expand in the y-direction, and 10% resistance to in-plane expansion acts only in the x-direction. The trajectory and panel configuration are summarized in Table 11. The initial conditions and panel orientation are selected to be representative of a hypersonic wind tunnel test,40 since future experimental validation efforts are likely to consider similar conditions and constant Mach numbers. The panel is initially stress-free and at a uniform temperature of 70 F. The undeformed panel is inclined 5 degrees to the freestream flow, and the leading edge of the panel is 60 in downstream of the transition to turbulence. At time zero the representative vehicle (Fig. 7) is exposed to Mach 12 flow at a dynamic pressure of 2000 psf. Note that the freestream pressure and temperature used in the fluid models are determined from the Mach number and dynamic pressure using standard atmosphere tables and ideal gas relations.43 These conditions correspond to an altitude of approximately 104,000 ft.

Table 11. Trajectory and panel configuration.

M q x3 Tinitial

12 2000 psf 5 60 in 70 F

IV.B.

Quasi-static Response

The main objectives of this investigation into quasi-static response are: 1) verify convergence of the quasistatic solution procedure, and 2) demonstrate the effect of 2-way coupling on response prediction. Note that the quasi-static response of the panel studied in this work was investigated extensively in Ref. [29]. However, the quasi-static solution procedure used previously29 did not include fluid-structure subiterations, and time step convergence studies were performed with decoupled models. Thus, the focus of the present study is on convergence of the quasi-static solution procedure with respect to: thermal time step, feedback of deformation for 2-way coupling, and deformation-induced pressure load. IV.B.1. Convergence Studies

Convergence of the quasi-static solution procedure is demonstrated by the responses shown in Figs. 12(a) and 12(b). Figure 12(a) compares the center displacements for 1-way and 2-way coupling predicted in the original work29 to those predicted in the present study. Note in Fig. 12(a) that neither the original nor the present predictions include fluid-structure (F-S) subiterations. The solution procedure in Ref. [29] used values of: t = 0.1 s and N = 10. This results in deformation feedback to aerodynamic heating every 1.0 s.

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The present study tests for convergence by reducing these values to: t = 0.01 s and N = 5, which results in deformation feedback every 0.05 s. The difference in peak displacements between the original and present predictions for 1-way coupling is approximately 1% and for 2-way coupling is approximately 2%. Since 1-way coupling does not include deformation updates in aerodynamic heating, these results indicate that a 1% difference in displacements is due to the order of magnitude change in time step. The additional 1% difference for 2-way coupling is the result of more frequent deformation updates in aerodynamic heating computations.

(a) Study on time step and deformation feedback.

(b) Study on fluid-structure subiterations.

Figure 12. Quasi-static solution procedure convergence studies.

Figure 12(b) compares the center displacements for 1-way and 2-way coupling, with and without the use of F-S subiterations. It is evident in Fig. 12(b) that inclusion of F-S subiterations has a negligible impact (approximately 0.01% difference) on the predicted responses. Note that the inclusion of F-S subiterations is with a convergence criteria of 10-4 for Eq. 21, which results in typically 2­3 subiterations to converge the pressure load. The negligible impact of F-S subiterations on the predicted displacements indicates that the deformations are dominated by thermal loading and/or the covergence criteria is overly conservative. IV.B.2. The Impact of 2-way Coupling

An important finding in the authors' previous work29 is that 2-way coupling can have a significant impact on the predicted quasi-static response. The additional convergence studies performed in the present work confirm this finding, as evident by the approximately 18% difference in peak center displacements between 1-way and 2-way coupled solutions (Fig. 12). Additional results from the present quasi-static solution procedure are presented in Figs. 13(a­f) to further illustrate and quantify the impact of 2-way coupling. Figures 13(a) and 13(b) show the skin deformation and temperature at 15 s from the 1-way (S-1) and 2way (S-2) coupled solutions, respectively. It is evident in Fig. 13(a) that 1-way coupling results in a uniform temperature distribution, except near the stiffened edges (y = 0 and y = 10 in). However, the inclusion of panel deformation in aerodynamic heating predictions (2-way coupling) results in elevated temperatures for surfaces inclined to the flow and lower temperatures for declined surfaces. As illustrated in Fig. 13(c), peak temperature rise is approximately 23% greater for 2-way coupling than 1-way coupling. Note in Figs. 13(a) and 13(b) that both 1-way and 2-way coupling predict panel buckling due to in-plane thermal loading (a result of including resistance to in-plane expansion29 ). The buckling is biased into the flow as a result of the thermal moment imposed by through-thickness (z-direction) temperature gradients in the skin and stiffeners. As illustrated in Fig. 13(d), peak displacements predicted by 1-way and 2-way coupling are approximately the same (0.21 in), however, 2-way coupling results in asymmetric deformation due to nonuniform heating. Another significant difference between the 1-way and 2-way coupled response predictions is the surface ply failure index, as illustrated in Figs. 13(e) and 13(f), respectively. Failure indices are based on the TsaiHill criterion;61 a value of 0 corresponds to zero equivalent stress and values of 1 or greater indicate ply failure. The 1-way coupled solution predicts only localized ply failure due to stress concentrations in the

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(a) 1-way coupled.

(b) 2-way coupled.

(c) Mid-span temperature.

(d) Mid-span deformation.

(e) 1-way coupled.

(f) 2-way coupled.

Figure 13. Skin deformation, temperature, and surface ply failure index from 1-way and 2-way coupled quasi-static solutions.

corners, with moderate stress levels throughout the majority of the skin. However, 2-way coupling predicts ply failure over a relatively large portion of the surface in the leading edge region, in addition to localized ply failure in the corners. This is evidently due to the combination of increased bending, Fig. 13(d), and temperature gradients, Fig. 13(c), in the region near the leading edge. Consequently, stress levels in this

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portion of the skin result in failure indices for 2-way coupling that are approximately 200% greater than those prediced by 1-way coupling. IV.C. Dynamic Response Tests

In order to test the dynamic aeroelastic stability of the quasi-static response predictions, short-duration transient dynamic analyses are performed using the solution procedure depicted in Fig. 10. Note that the continuous dynamic response test procedure (R-C) is used in this investigation, with convergence criteria EF S = EP = 10-4 . Initial conditions for the dynamic analyses are taken directly from the quasi-static results at 15 s. The dynamic analyses include a steady thermal load, which is also taken directly from the quasi-static solutions. The response of the panel is first examined under the thermal load at 15 s. Then, in order to investigate the response to a perturbation, the thermal load at 15.1 s is applied to the structural configuration at 15 s. The dynamic responses are depicted by the center displacement of the skin in Figs. 14(a­d). Figures 14(a) and 14(b) show the dynamic responses from 15 ­ 15.1 s of the 1-way and 2-way coupled quasi-static configurations under their respective 15 s thermal loads. The dynamic response tests indicate that both the 1-way and the 2-way coupled configurations are stable, i.e., the initial (small amplitude) oscillations decay and the dynamic responses match the quasi-static initial conditions. The dynamic responses of the 1-way and 2-way coupled configurations at 15 s to their respective thermal loads from 15.1 s are illustrated in Figs. 14(c) and 14(d). These responses contain initially larger oscillations as a result of the thermal load perturbations; these oscillations also decay, and the dynamic responses are in excellent agreement with the quasi-static deformations at 15.1 s. The results of these dynamic response tests, together with dynamic response tests performed at earlier times in the trajectory (not shown here), indicate that both the 1-way and 2-way coupled solutions predict dynamically stable responses for this trajectory and panel configuration. Furthermore, the dynamic response tests verify the quasi-static response predictions, including the impact of 2-way coupling, presented in the previous section. IV.D. Dynamic Response

The focus of this section is on the development of a procedure for the inclusion of transient thermal loads and 2-way coupling in dynamic analysis, in order to investigate their importance for long-duration response prediction. Long-duration implies response times on the order of the thermal time scale, such that the thermal load is not essentially constant. Therefore, in contrast to the short-duration dynamic response test, the transient nature of thermal loading (and thus, 2-way coupling) may have a significant impact on the predicted response. This study is performed using the solution procedure illustrated in Fig. 11. Three methods of supplying the initial conditions for the dynamic solutions are considered. First, the 1-way (D-1) and 2-way (D-2) coupled dynamic solutions are started at 0 s (undeformed panel at uniform 70 F). These analyses are carried out from 0­10 s. Second, in order to investigate the influence of the convergence criteria (EF S and EP ) on the response from 8­10 s, the dynamic conditions at 8 s from the 0­10 s analyses are used as the initial conditions. Third, the dynamic solutions are started from the quasi-static results at 10 s. This is done to investigate a potential source of error in supplying initial conditions from the 0­10 s dynamic analyses. The 0­10 s dynamic responses (D-1 and D-2) are depicted in Fig. 15 by the center displacement of the skin. Note that the quasi-static responses (S-1 and S-2) are included for reference. It is clear from Fig. 15 that the 1-way and 2-way coupled dynamic responses are in close agreement with the respective quasi-static predictions. However, oscillations are evident in the dynamic solutions beginning at approximately 9.5 s. These oscillations are not expected, since the dynamic response tests performed in the previous section predicted stable responses. Therefore, the focus of the remainder of this section is on the cause (physical or numerical) of the oscillations. The influence of the convergence criteria (EF S and EP ) on the dynamic responses is investigated by restarting the 1-way and 2-way coupled analyses at 8 s with more strict convergence criteria. Note that the convergence criteria used for the 0­10 s responses (Fig. 15) are EF S = 10-4 and EP = 10-3 (MSC.Nastran default). Figure 16(a) shows the predicted center displacements from 9.4­10 s with both convergence criteria set to 10-4 . It is evident that changing EP from 10-3 to 10-4 results in a significant reduction of the oscillations depicted in Fig. 15. However, oscillations are present in the dynamic responses and the 2-way coupled dynamic response is offset from the quasi-static deformation. Figure 15(b) depicts the dynamic

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(a) Thermal load from S-1 at 15 s.

(b) Thermal load from S-2 at 15 s.

(c) Thermal load from S-1 at 15.1 s.

(d) Thermal load from S-2 at 15.1 s.

Figure 14. Dynamic response tests of 1-way and 2-way coupled quasi-static results, initial conditions at 15 s.

responses from 9.4­10 s with both convergence criteria set to 10-5 . A further reduction in oscillations is apparent, however, the 2-way coupled response contains small oscillations near 10 s and is offset from the quasi-static results. Note in Fig. 16(b) that the 1-way coupled dynamic response from 9.4­10 s exhibits no oscillations and close agreement with the quasi-static prediction. However, the 1-way coupled response does contain oscillations, similar to those depicted in Fig. 15(a), at earlier times. These results indicate that numerical errors are present in both the 1-way and 2-way coupled dynamic response predictions. The results of the investigation into convergence criteria indicate that oscillations in the 0­10 s dynamic responses (Fig. 15) are numerical, however, tighter convergence criteria are insufficient for eliminating the oscillations and matching the quasi-static solutions. It is important to note that the initial conditions for these analyses are from the 0­10 s dynamic solutions, therefore, errors introduced prior to 8 s may prevent agreement with the quasi-static results. Thus, an additional study is performed in which the dynamic analyses are started from the quasi-static predictions at 10 s. Figures 17(a) and 17(b) illustrate the 2-way coupled dynamic responses from 10­10.1 s with convergence criteria of 10-4 and 10-5 , respectively. In both Figs. 17(a) and 17(b), an offset between the mean dynamic and quasi-static predictions is evident. The dynamic solution becomes steady at this offset for EF S = EP = 10-4 , as shown in Fig. 17(a). However, with tighter convergence criteria, Fig. 17(b), small amplitude oscillations persist. This trend is counter intuitive, but indicative of numerical error. Note that the 1-way coupled dynamic responses are omitted, because they show substantially the same trends as the 2-way coupled responses. These results indicate that the dynamic solutions contain numerical error that is not sufficiently eliminated through the use of tighter convergence criteria. Additional studies are performed in

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Figure 15. Comparison of 1-way and 2-way coupled dynamic and quasi-static solutions, initial conditions at 0 s.

(a) Dynamic solutions: EF S = EP = 10-4 .

(b) Dynamic solutions: EF S = EP = 10-5 .

Figure 16. The influence of convergence criteria on the dynamic solutions, initial conditions at 8 s from 0 ­ 10 s dynamic solutions.

the following section to further investigate this error by removing the transient component of the thermal load. IV.E. Dynamic Response Tests - Staggered vs. Continuous

The results presented in the previous section indicate that numerical error in the dynamic solution procedure (Fig. 11) can lead to oscillations and/or an offset from the quasi-static response predictions. Note that the quasi-static predictions are used as a baseline, since the continuous dynamic response tests (Fig. 10) performed in Sec. IV.C predict stable responses that are in excellent agreement with the quasi-static solutions. This indicates that the numerical error is the result of using a staggered, rather than a continuous structural dynamic solution. The aim of this study is to test this hypothesis by performing staggered (R-S) and continuous (R-C) dynamic response tests (Table 8) on the quasi-static results. Initial conditions for the dynamic analyses are the quasi-static results at 10 s. The dynamic analyses include a steady thermal load, which is also taken from the quasi-static solutions. Similar to Sec. IV.C, the response of the panel is first examined under the

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(a) Dynamic solutions: EF S = EP = 10-4 .

(b) Dynamic solutions: EF S = EP = 10-5 .

Figure 17. The influence of convergence criteria on the 2-way coupled dynamic solution, initial conditions at 10 s from the quasi-static solution.

thermal load at 10 s. Then, in order to investigate the response to a perturbation, the thermal load at 10.1 s is applied to the structural configuration at 10 s. Additionally, the influence of convergence criteria on the predicted dynamic response is investigated for each thermal load. The staggered and continuous dynamic response predictions under the steady thermal load at 10 s are illustrated by the center displacement of the skin in Figs. 18(a­f). Figures 18(a) and 18(b) show the staggered and continuous response predictions, respectively, for convergence criteria of EF S = EP = 10-3 . Similarly, the responses for EF S = EP = 10-4 and EF S = EP = 10-5 are provided in Figs. 18(c) and 18(d) and Figs. 18(e) and 18(f), respectively. Each of the staggered (R-S) dynamic response predictions, Figs. 18(a,c,e), is offset from the quasi-static displacement at 10 s. The staggered solution for EF S = EP = 10-4 is steady at this offset, whereas, the solutions for EF S = EP = 10-3 and EF S = EP = 10-5 show persistent oscillations with an offset mean. In contrast, the continuous (R-C) dynamic response predictions, Figs. 18(b,d,f), show that the initial oscillations decay and the dynamic responses are in excellent agreement with the quasi-static solution for each convergence criteria. Figures 19(a­f) provide the staggered and continuous dynamic response predictions under the steady thermal load at 10.1 s. The staggered responses (R-S) for convergence criteria EF S = EP of 10-3 , 10-4 , and 10-5 are given in Figs. 19(a,c,e), respectively. As shown in Fig. 19(a), the staggered procedure in conjunction with the thermal load perturbation and EF S = EP = 10-3 results in relatively large amplitude oscillations without decay. The initially large amplitude oscillations do decay for EF S = EP = 10-4 and EF S = EP = 10-5 , as illustrated in Figs. 19(c) and 19(e). However, the staggered response at EF S = EP = 10-4 is offset from the quasi-static solution at 10.1 s, and the the response at EF S = EP = 10-5 contains persistent oscillations with an offset mean value. The continuous dynamic responses (R-C) for convergence criteria EF S = EP of 10-3 , 10-4 , and 10-5 are given in Figs. 19(b,d,f), respectively. Each continuous solution shows that initial oscillations due to the thermal load perturbation decay to a steady result, which is in excellent agreement with the quasi-static solution at 10.1 s. The results of this study confirm that numerical error in the staggered structural dynamic solution is sufficient to alter the dynamic response predictions. Specifically, the chaining (i.e., sequential cold restarts) of nonlinear transient dynamic solutions (MSC.Nastran SOL 129) as implemented in Sec. III.C introduces and/or propagates error that can lead to either spurious oscillations, an offset, or both. Furthermore, this error is not sufficiently reduced through tighter convergence criteria. Note that responses predicted by the continuous dynamic solution procedure are substantially unchanged by convergence criteria EF S = EP of 10-3 to 10-5 , whereas, significant error is evident in the staggered procedure for the highest degree of convergence investigated (10-5 ). Further investigation into the numerical methods employed in MSC.Nastran is required in order to develop a staggered dynamic solution procedure that minimizes the time history of the response that must be stored in RAM. Preliminarily, two potential sources of error in the current implementation are identified based on the description62 of the Newmark method employed in MSC.Nastran SOL 129. First, each cold restart requires an approximation of the initial acceleration, thus the previously computed acceleration is

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(a) Staggered procedure, EF S = EP = 10-3 .

(b) Continuous procedure, EF S = EP = 10-3 .

(c) Staggered procedure, EF S = EP = 10-4 .

(d) Continuous procedure, EF S = EP = 10-4 .

(e) Staggered procedure, EF S = EP = 10-5 .

(f) Continuous procedure, EF S = EP = 10-5 .

Figure 18. Staggered vs. continuous dynamic response tests, initial conditions at 10 s, thermal load at 10 s.

not used. Second, the residual load error from the previous time is not carried over in a cold restart. It is suspected that for any single cold restart, these errors are extremely small. However, it is probable that multiple, sequential cold restarts results in the propagation and summation of these errors.

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(a) Staggered procedure, EF S = EP = 10-3 .

(b) Continuous procedure, EF S = EP = 10-3 .

(c) Staggered procedure, EF S = EP = 10-4 .

(d) Continuous procedure, EF S = EP = 10-4 .

(e) Staggered procedure, EF S = EP = 10-5 .

(f) Continuous procedure, EF S = EP = 10-5 .

Figure 19. Staggered vs. continuous dynamic response tests, initial conditions at 10 s, thermal load at 10.1 s.

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V.

Conclusions

A fluid-thermal-structural model of a carbon-carbon, hypersonic vehicle skin panel is developed to study the effect of 2-way coupling between structural deformation and aerodynamic heating. Partitioned, quasi-static and transient dynamic solution procedures are used to investigate the impact of 2-way coupling on structural response prediction under combined thermal and pressure loading. Studies are performed to evaluate convergence of the quasi-static solution procedure with respect to: thermal time step, feedback of deformation for 2-way coupling, and deformation-induced pressure load. Subsequently, the dynamic stability of quasi-static response predictions is investigated using short-duration dynamic response tests. In addition, a dynamic solution procedure is used to investigate long-duration dynamic response prediction. Unexpected oscillations in the long-duration dynamic response prompt an investigation into convergence criteria and the impact of cold restarts on the dynamic solution. The results of these studies yield several useful conclusions: 1. Convergence of the 2-way coupled quasi-static solution procedure is demonstrated for O(10) thermal time steps and O(10) updates of structural deformation in aerodynamic heating computations within the characteristic thermal response time. In this scenario, the inclusion of fluid-structure subiterations has a negligible impact on the quasi-static response of the panel investigated. 2. Including the dependence of aerodynamic heating on structural deformation results in O(20%) increase in peak skin temperature and O(200%) increase in surface ply failure index for relatively modest peak displacement, O(2%) of panel length. 3. Dynamic aeroelastic stability of the quasi-static response predictions is verified through the use of short-duration transient dynamic response tests that use subiterations to converge the fluid-structural response to a thermal load perturbation. In addition, the steady responses predicted by the continuous dynamic response procedure are in excellent agreement with the quasi-static results. 4. The staggered dynamic solution procedure investigated results in oscillations in the structural response. It is shown that these oscillations are numerical and dependent on the convergence criteria for the nonlinear structural solution. Higher degrees of convergence reduce the amplitude, but do not effectively eliminate spurious oscillations in the response. 5. Comparison of a staggered dynamic response procedure to a continuous procedure indicates that numerical error is introduced and/or propagated by the use of sequential cold restarts in the staggered solution. Preliminary investigation of the nonlinear structural solution (MSC.Nastran) indicates that accelerations and residual load errors from the previous time (not carried forward in the staggered procedure) may need to be carried forward instead of being approximated by each sequential cold restart. Further investigation into the numerical methods employed is required to develop a staggered dynamic solution procedure for long-duration fluid-thermal-structural response prediction.

Acknowledgments

This research was funded in part by: the Short Term Aerospace Research & Development Program (STAR-DP contract number FA8650-10-D-3037), a National Defense Science and Engineering Graduate Fellowship to Adam Culler, and an Air Force Summer Faculty Fellowship to Jack McNamara. The authors gratefully acknowledge the technical insights of Drs. Ravi Chona, Tom Eason and Mike Spottswood, of the Air Force Research Laboratory - Structural Sciences Center, on multi-physics analysis of extreme environment structures. In addition, this work was supported in part by an allocation of computing time from the Ohio Supercomputer Center and by an MSC.Software license from the Biodynamics Laboratory at The Ohio State University.

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