53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference<BR>20th AI
23 - 26 April 2012, Honolulu, Hawaii
AIAA 2012-1548
Modeling and Analysis of Shock Impingements on
Thermo-Mechanically Compliant Surface Panels
Brent A. Miller∗, Andrew R. Crowell∗, and Jack J. McNamara†
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The Ohio State University, Columbus, OH, 43210, USA
Fluid-Thermal-Structural interactions play an important role in the development of high speed vehicles, impacting various sub-disciplines (i.e., aerodynamic, structural, material, propulsion, and control)
at the micro, component and/or vehicle scales. This study focuses on the development of a partitioned
fluid-thermal-structural procedure aimed at performing a long time record thermo-structural response
prediction of surface panels subject to shock impingements. Specific modeling aspects essential to this
are reduction of the computational aerothermodynamics to a tractable model, and partitioned timemarching of the fluid-thermal-structural problem. Additional factors considered are: 1) the movement
of the shock impingement due to forced motion of a shock generator, 2) panel backpressure, 3) a 140dB
random prescribed pressure load to account for pressure fluctuations associated with turbulent boundary layers, and 4) coupled vs. uncoupled fluid-thermal-structural analysis. Results indicate that quasistatic CFD analysis provides a promising means for generating an aerothermodynamic surrogate model.
Differences between quasi-static and unsteady models were under 6% for both panel temperature rise
and pressure loads for a forced motion analysis. Several studies using the fluid-thermal-structural
model are performed, focusing on the differences between the coupled and uncoupled analyses, as
well as the role of backpressure on the panel response. The effect of the backpressure to the direction
of panel buckling is investigated, and the backpressure required to buckle the panel into the flow is
predicted to be ∼10% of free stream higher for the uncoupled model than the coupled model. However,
generally differences were minor between the coupled and uncoupled analysis. The inclusion of a 140
dB prescribed pressure load, meant to mimic the effect of turbulent boundary layer loadings, results
in negligible temperature differences. However, both the shock motion and this load introduce large
amplitude oscillations at the start of the response, followed by relatively small oscillations once the
buckling amplitude of the panel becomes significant.
Nomenclature
Modal weight
Modal weight normalized by panel thickness
Rayleigh damping matrix
Thermal capacity matrix
Heat flux ODE coefficient
Mass proportional Rayleigh damping coefficient
Plate stiffness
Young’s modulus
Newmark-β force vector
Mechanical load
Thermal load vector
Total distributed forces on panel
Generalized aerodynamic force
Tunnel height
Heat transfer coefficient
Panel thickness
ai
ai
[C]
th C
Ci
c
D
E
F
a
{F
T}
F
f
GAF
H
hht
hp
∗ Ph.D.
Candidate, Department of Mechanical and Aerospace Engineering, Student Member AIAA
Professor, Department of Mechanical and Aerospace Engineering, Senior Member AIAA
† Assistant
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Copyright © 2012 by B. A. Miller, A. R. Crowell, and J. J. McNamara. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
American Institute of Aeronautics and Astronautics
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hw
hw0
Ip
[J]
[K]
KT Kth K
L
Lw
M
[M]
MT
mp
NT
NT
Nm
Nx
n
P
PPL
Pb
Pf
PP P L
Pref
Prms
q
q
q̂
Qth
{R}
Rex
SP L
T
Taw
Tinit
Tref
{T}
t
u0
w
x, z
Shock generator height
Shock generator non-displaced height
Mass moment of inertia of panel
Force Jacobian
Structural nonlinear stiffness matrix
Newmark-β stiffness matrix
Structural stiffness matrix due to thermal effects
Thermal conductivity matrix
Panel length
shock generator total length
Mach Number
Structural mass matrix
Thermal moment on panel
Panel mass per unit length
In-plane thermal force on panel
Averaged in-plane thermal force on panel
Number of mode shapes
Total in-plane force on panel
Unit normal to panel surface
Pressure
Prescribed pressure load
Backpressure on panel
Fluid pressure acting on panel
Random in time prescribed pressure load on panel
Reference pressure, = 20µP a
Root-mean-square pressure
Heat flux
Heat flux vector
Heat flux surrogate correction
Heat load vector
Residual force vector
Reynolds number per unit length
Sound pressure level in dB
Temperature
Flat plat adiabatic surface temperature
Initial panel temperature
Stress free reference temperature of panel
Temperature solution vector
Time
In-plane panel displacement
Transverse displacement of panel into flow
Spatial coordinates
α
∆p
∆tS
∆tT
δw
κ
φ
ρ
ρp
ν
Coefficient of thermal expansion
Distance between shock-generator midpoint and panel leading edge
Structural time step
Thermal time step
shock generator tip displacement
Panel thermal conductivity
Transverse displacement mode shape of panel
Fluid density
Panel density
Poisson’s ratio
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Subscripts
CIM
m
n
P IM
w
∞
Corrected isothermal model
Thermal time step counter
Structural time-step counter
Piecewise isothermal model
At the panel surface
Freestream conditions
Superscripts
avg
i
T
S
Averaged over thermal time step
Subiteration counter
For the thermal model
For the structural model
I.
Introduction
Despite years of focused research, an extensive set of technical challenges have obstructed the development of a responsive, and reusable, hypersonic flight vehicle. This study focuses on one of these challenges,
namely the thermo-mechanical response of skin panels subject to shock-turbulent boundary layer interactions (STBLIs). STBLIs are an important loading case to consider since they amplify turbulent boundary
layer loads, and cause high, localized heating.1, 2 Thus, they present a significant risk to severely damage
surface panels. However, shock impingements, and the associated response of inflicted panels, are challenging to model since: there is a significant amount of uncertainty in modeling of STBLIs, wind tunnel
testing of multi-physics is difficult in the hypersonic flow regime, the impingement location is a function
of the transient thermo-mechanical response, and the thermal response time is on the order of minutes to
hours. This study is focused on the last two issues, and seeks to develop a tractable computational framework for modeling and analysis of STBLIs on thermo-mechanically compliant panels.
There are a number of relevant previous studies to this problem, such as: high fidelity modeling of
STBLIs;3–8 partitioned fluid-thermal-structural analysis;9–16 and shock impingements on compliant surfaces.17–19 Each of these areas are briefly reviewed next.
Computational Fluid Dynamics (CFD) modeling of STBLIs is an active area of research, where considered approaches include Reynolds Averaged Navier-Stokes (RANS),4, 7, 8 large eddy simulation (LES),3, 5
and direct numerical simulation (DNS).3, 4, 6 As expected, these studies have confirmed that DNS provides
the best comparison with experimental results, but at an extreme computational cost; whereas LES and
RANS yield higher levels of error. In particular, RANS cannot account for shock unsteadiness due to interactions with a turbulent boundary layer.1, 3, 4 However, a partial correction has been developed to account
for some of the associated effects.7
Thermo-Structural
Dynamics
Aerothermodynamics
Aerothermal
Aerodynamic
Heating
Tw
Heat Transfer
q
Fluid
Dynamics
Aeroelastic
Aerodynamic
Pressure
Tstruct
w, w
Psurf
Structural
Deformations
q
Tw
Tstruct
Psurf
w
w
- Heat Flux
- Surface Temp.
- Structural Temp
- Surface Pressure
- Structural Disp.
- Structural Velocity
Figure 1. Schematic of the Fluid-Thermal-Structural Interaction problem.
In regards to a fluid-thermal-structural analysis, the complexity involved in coupling the multi-physics
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is highlighted in Fig. 1. Beyond the different coupling mechanisms, another challenge is the disparate time
scales between the fluid, thermal, and structural physics. Generally, characteristic times of the fluid and
the structure are orders of magnitude smaller the thermal response.14 This results in the conflicting requirements of small time steps over large time records. These aspects of the problem are best solved using
a partitioned approach,20 where separate solvers are used to compute the fluid, thermal, and structural
responses and coupling is achieved by passing appropriate boundary conditions at the interfaces of the
domains. Thornton and Dechaumphi,9 and Dechaumphi et al.10 used such an approach to couple quasistatic finite element flow, thermal, and structural models for panels9 and leading edges.10 In a related study,
Loehner et al.11 coupled off-the-shelf CFD, computational thermal dynamics, and computational structural
dynamics codes by interpolating a governing variable to a master surface and projecting that variable to
the other code at each time step. The assembled framework was used to repeat the aerodynamically heated
panel study of Thornton and Dechaumphi.9 Tran and Farhat12 considered the aerothermoelastic stability
of a flat panel and the aerodynamic heating of an F-16 airfoil by coupling a CFD solver with finite element solvers for the thermal and structural domains. Stress and deformations due to temperature were
included in the model; however, the feedback from the deformations was neglected in computing the aerodynamic heating. Culler and McNamara performed dynamic, two-way coupled fluid-thermal-structural
analysis for cylindrical bending of simply-supported panels13 and a stiffened composite panel from the
NASP program.14 In the former, the structural response was computed from von Kármán plate theory21
and the thermal response from a finite difference solution to the 2-D heat equation.22 In the latter, MSC
Nastran R was used for the thermal and structural modeling. For both studies, pressure and heat loads
were modeled using third-order piston theory aerodynamics23 and Eckert’s Reference Enthalpy method,24
respectively. In related work,16 the same fundamental thermo-structural model from Ref. [13] was coupled
with a CFD surrogate model for the pressure and heat flux. Finally, Falkiewicz et al.15 performed a partitioned dynamic aerothermoelastic analysis of a hypersonic control surface using several reduced order
models. Model reduction for the thermal problem was achieved through a proper orthogonal decomposition basis and for the structural problem with a combination of Ritz and free vibration modes to account for
geometric stiffening and material degradation. For the fluid, the pressure was modeled with piston theory,
and the heat load was modeled using a CFD surrogate. An important aspect of the recent literature13–16
is the implementation of different time step sizes for the thermal and structural solvers; with boundary
conditions passed from the structural solver to the thermal solver in a time-averaged sense. Originally proposed by Culler and McNamara,13 this was motivated by the large differences in characteristic time scales
between the two domains. However, it is important to note that none of these latter studies have carried
out a detailed analysis of the numerical accuracy of the time marching procedures.
Recently there has been direct study of shock impingements on compliant surfaces. Visbal18, 19 investigated the fluid-structural response of a panel in Mach 2 flow with an impinging shock for both inviscid18
and laminar flow.19 The analysis was carried out by coupling a CFD solver to a finite difference solver of a
von Kármán plate. These studies found that the dynamic pressure required to incite limit cycle oscillations
was significantly reduced in the presence of shock impingements. The panel response was also strongly
dependent on the shock impingement point. In addition, Visbal19 investigated the effects of a prescribed
time-varying panel backpressure, and found that the associated panel oscillations could reduce the shock
induced flow separation on the outer surface. Separately, the authors of this paper previously developed
a coupled CFD-FEM framework for fluid-thermal analysis of shock impingements.17 The approach was
used to investigate the thermal response of a panel, and also to characterize the different time scales associated with the problem. Shock motion was considered in the study by prescribing sinusoidal oscillations of
both the shock generator and the panel surface. It was found that motion of the shock generator and panel
altered the aeroheating boundary conditions with significant increases in the surface temperature over approximately the distance that the shock moved. Finally, the use of quasi-static coupling for the aerothermal
model resulted in negligible errors (1-2%) in the predicted temperature rise of the panel. However, it is
important to note that these conclusions were made from a relatively short time response record (< 0.1
seconds).
The present work represents an extension to this previous paper,17 and focuses on both the development of an aerothermodynamic surrogate for flow fields with STBLIs and the implementation of a carefully
formulated time marching procedure for partitioned fluid-thermal-structural analysis. Subsequently, the
developed framework is used to investigate, for the first time, the thermo-mechanical response of compliant panels subject to STBLIs. Consideration is given to the importance of coupling in the fluid-thermal-
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structural analysis, panel backpressure, and random prescribed pressure fluctuations associated with turbulent boundary layers.
The remainder of this paper is arranged as follows. Section II details the fluid, thermal, and structural
models for this study, as well as the coupling strategies considered; Section III details the considered panel
configuration and flow environment. In Section IV, the surrogate model development and verification is
discussed. Section V details the results of the different fluid-thermal-structural response studies. Finally,
concluding remarks are given in Section VI.
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II.
Fluid, Thermal, and Structural Modeling
Modeling of the three sub-disciplines is carried out using a RANS CFD surrogate for the flow domain,
a two-dimensional FEM solver for the thermal response, and a moderate deflection Galerkin model for
cylindrical bending. The fluid, thermal, and structural models are alternately integrated forward in time,
using a partitioned approach, where Dirichlet and Neumann boundary conditions are transferred across
the interface.
A.
Fluid Model
As noted above, the fluid modeling in the fluid-thermal-structural analysis is carried out using a CFD surrogate. The process for developing the surrogate is detailed in Section IV. The NASA Langley CFL3D
code25, 26 is used in this study to compute the required CFD solutions to the Navier-Stokes equations. The
CFL3D code uses an implicit, finite-volume algorithm based on upwind-biased spatial differencing to solve
the RANS equations. Multigrid and mesh-sequencing are available for convergence acceleration. The algorithm, which is based on a cell-centered scheme, uses upwind-differencing based on either flux-vector
splitting or flux-difference splitting, and can sharply capture shock waves. The Menter k − ω SST27 turbulence model is used in this study. Furthermore, since it is a RANS code, it is incapable of capturing the local
shock unsteadiness associated with STBLIs.1, 7 Despite this limitation, the use of a RANS code is justified in
the present study due to the exploratory nature of the work and the need for a relatively inexpensive CFD
solver for construction and validation of the surrogate model.
B.
Thermal Modeling
The thermal model is constructed using a two dimensional finite element method to solve the 2-D heat
equation:
∂T (x, z, t)
= 5 κ 5 T (x, z, t)
(1)
ρp cp
∂t
with the following boundary conditions:
−κn · 5T (x, z, t) = q · n
(2)
where n is the unit normal vector of the thermal boundary. The finite element discretization consists of four
node quadrilateral elements using linear shape functions of the temperature. After spatially discretizing
the heat conduction equation, the transient finite element equation is given as:
th n o th C
Ṫ + K {T} = Qth
(3)
where Cth , Kth , and Qth are the thermal capacitance matrix, thermal conductivity matrix, and nodal
heat load vector, respectively. The transient problem is then discretized in time using a second-order accurate Crank-Nicolson scheme discretized about the m + 1/2 step:
th {T}m+1 − {T}m th {T}m+1 + {T}m
C
+ K
= Qth
∆tT
2
m+1/2
+ O ∆tT 2
(4)
Because Eq. (4) is a set of linear equations, it may be solved using matrix inversion without theuse of subiterations. In order to maintain second order accuracy in a coupled analysis, computation of Qth m+1/2
must also be second order accurate.
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C.
Structural Model
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The structure is modeled as a panel in cylindrical bending with von Kármán strains.28 The model includes
thermal strains due to non-uniform temperature through the length and thickness of the panel, as well
as chordwise varying, temperature-dependent elastic modulus and coefficient of thermal expansion. The
equilibrium equations for the continuous panel are:
∂ 2 w(x, t)
∂w(x, t)
∂ 2 w(x, t)
∂2
∂2
m
w(x,
t)
−
I
+
c
D
(x,
t)
+
p
p
∂t2
∂x2
∂t
∂x2
∂x2
∂ 2 MT (x, t)
∂ 2 w(x, t)
= f (x, t) −
− Nx (t)
2
∂x
∂x2
(5)
mp = ρp hp
(6)
h3p
12
(7)
Ip = ρp
D(x, t) = E(x, t)
h3p
12(1 − ν 2 )
(8)
Z
E(x, t)α(x, t) hp /2 T (x, z, t) − Tref dz
1−ν
−hp /2
∂u0 (x, t) 1 ∂w(x, t) 2
hp
+
− NT (x, t)
Nx (t) =
2(1 − ν 2 )
∂x
2
∂x
Z
E(x, t)h hp /2
NT (x, t) =
T (x, z, t) − Tref dz
1−ν
−hp /2
MT (x, t) =
(9)
(10)
(11)
In order to define all variables in Eq. (5) in terms of the transverse displacement and the temperature
distribution, the in-plane displacement term, ∂u0 /∂x, is eliminated from the expression for Nx in Eq. (10).
An implicit assumption in von Kármán plate theory for cylindrical bending is that the in-plane force, Nx ,
must be constant across the panel to satisfy equilibrium conditions. By using this condition, the in-plane
displacement, u0 , is solved in terms of the in-plane forces and transverse displacement:
2 )
Z L(
1 ∂w(ξ, t)2
1 − ν2
[Nx (t) + NT (ξ, t)] −
dξ
(12)
u0 (x, t) =
E(ξ, t)h
2
∂ξ
0
At x = L the in-plane displacement is constrained to zero by the immovable supports, and Nx is solved
from Eq. (12) to yield:
"Z
L
0
1
dx
E(x, tt
"Z
L
Nx (t) =
1
NT (x) =
1−ν
0
#−1 Z
1
dx
E(x, t)
L
hp
2 (1 − ν 2 )
0
#−1 Z
L
Z
∂w(x, t)
∂x
dx − NT (t)
(13)
hp /2
[T (x, z, t) − Tref ] dz dx
α(x, t)
0
2
(14)
−hp /2
where Eq. (14) is dependent on the average temperature change in the panel. Thus, for this formulation,
the in-plane thermal effects are driven by the average chord-wise temperature in the panel, so long as E(T )
and α(T ) do not exhibit strong chord-wise variations.
The four boundary conditions needed to solve the equation of motion are given in Eq. (15) for x = 0 and
L, which corresponds to a clamped-clamped condition.
(15a)
w(0, t) = w(L, t) = 0
∂w(x, t)
∂x
x=0
∂w(x, t)
=
∂x
=0
x=L
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(15b)
The applied force, f (x, t) in Eq. (5), is considered to be a combination of three separate forces:
f (x, t) = Pb − Pf (x, t) − PP P L (t)
(16)
where Pb is the backpressure on the panel, Pf is the pressure from the flow, and PP P L is a prescribed
pressure load (PPL). The backpressure is assumed to be constant in time and uniform across the panel. The
PPL, which is included as an initial approximation for pressure fluctuations due to a turbulent boundary
layer,29 is a randomized in time, spatially uniform load with a zero mean value with respect to time. The
magnitude of the pressure is defined using the sound pressure level in decibels:
Prms = Pref × 10SP L/20
(17)
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29, 30
Following previous work,
the frequency content of the force is specified as a constant power up to the
cutoff frequency of 500Hz.
The damping coefficient, c, in Eq. 5 is modeled as mass proportional Rayleigh damping. The value of
the damping coefficient is set to 2% of the critical damping condition for the first free vibration mode of the
undeformed panel.
The structural model is discretized in space with Galerkin’s Method, using the transverse natural mode
shapes of the undeformed panel as the set of basis functions such that:
w(x, t) ≈
Nm
X
(18)
φi (x)ai (t)
i=0
where φi are the free vibration mode shapes, ai are the corresponding time-dependent weights, and NM
is the total number of assumed modes included in the model. By using the weak form of Eq. (5), the
discretized set of equations are:
[M] {ä} + [C] {ȧ} + [K] − KT {a} = FT + {Fa }
Z
L
Mij =
L
Z
φ0i (x)Ip φ0j (x)dx
φi (x)mp φj (x)dx +
0
(19)
(20)
0
Z
L
Cij =
(21)
φi (x)cφj (x)dx
0
L
Z
φ00i (x)D(x, t)φ00j (x)dx +
Ki,j =
0
K
T
h
2 (1 − ν 2 )
"Z
1
=
1−ν
"Z
ij
L
0
0
L
1
dx
E(x, t)
1
dx
E(x, t
F
a
#−1 "
Z
ak (t)al (t)
#−1 Z
L
φ0k (x)φ0l (x)dx
0
k=1 l=1
L
Z
#Z
(22)
L
φ0i (x)φ0j (x)dx
0
hp /2
Z
(T (x, z, t) − Tref )dzdx
α(x, t)
−hp /2
0
L
φ0i (x)φ0j (x)dx
(23)
0
L
Z
i
M X
M
X
Pb − Pf (x, t) − PP P L (t) φi dx
=
(24)
0
FT i =
Z
0
L
φ00i (x)
E(x)α(x)
1−ν
Z
hp /2
z (T (x, z, t) − Tref )dzdx
(25)
−hp /2
where [M] is the mass matrix, [C] is the damping matrix, [K] is the
nonlinear stiffness matrix, KT is the
thermal stiffness matrix, {Fa } is the mechanical load vector, and FT is the thermalload vector. Note that
[M] and [C] are constant, [K] is dependent on displacement and temperature, and KT is dependent on
temperature.
The equations of motion are discretized in time using the Newmark-β time integration scheme.28, 31 This
method assumes the displacement, velocities, and accelerations as:
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1
an+1 = an + ∆tȧn + ∆t2S än+2β
2
ȧn+1 = ȧn + ∆tS än+γ
(26)
(27)
än+γ = (1 − γ) än + γän+1
(28)
än+2β = (1 − 2β) än + 2βän+1
(29)
where β and γ are parameters of the Newmark-β scheme that are used to define the accuracy and stability
of the solution. For β = 41 and γ = 21 the scheme is second order accurate in time and unconditionally stable
for a linear system.28 Using these values, the discretized equations of motion are then reduced to:
2
4
[M]
+
[C]
{a}n+1 =
[K]n+1 − KT n+1 +
∆t2
∆t
4
4
2
+
[M]
{a}
+
{
ȧ}
+
{ä}
+
[C]
{a}
+
{
ȧ}
n
n
n
n
n
n+1
∆t2
∆t
∆t
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{Fa }n+1 + FT
(30)
Equation (30) is a set of nonlinear equations, which requires computing [K] at the n + 1 step. This is
achieved using Newton-Raphson subiterations.28 Equation (30) may be rewritten as:
K n+1 {a}n+1 = F
(31)
n+1
where K is the summation of the matrices of the left hand side of Eq. (30), and F is the right hand side
of Eq. (30). Note that F n+1 is considered to be known, while K n+1 is not. To solve Eq. (31), the residual
vector {R} is defined as:
i
i
i
{R} ≡ K n+1 {a}n+1 − F n+1
(32)
where the superscript i represents a subiteration counter. The residual vector at the next subiteration may
be approximated using a Taylor series expansion:
i+1
{R}
i
i
≈ {R} + [J]
i+1
{a}n+1
−
i
i
{a}n+1
[J] ≡
+O
∂ {R}
i+1
{a}n+1
−
i
{a}n+1
2
≈0
(33)
i
i
∂ {a}n+1
(34)
i+1
where [J] is the force Jacobian matrix. Truncating the second order and higher terms in Eq. (33), {a}n
i
can be solved for using the inverse of [J] . By iterating the equations in i until {R} is approximately zero,
i+1
0
{a}n+1 becomes a solution for {a}n+1 that satisfies Eq. (31). Note that to initialize the subiterations, {a}n+1
is set equal to {a}n . In this study, convergence is defined as {R} ≤ 107
D.
Fluid-Thermal-Structural Coupling
A loosely coupled partitioned scheme is used to link the models. As opposed to tightly coupled partitioned
or monolithic schemes, loosely coupled schemes are more computationally efficient per time step and are
readily capable of integrating existing computational models together.32, 33 The necessary sequencing of
the models is shown in Fig. 2. The sequencing consists of four models: the thermal model; the structural
model; an aerodynamic heat flux surrogate; and an aerodynamic pressure surrogate. Note the thermal and
structural models are dependent on a time history, while the surrogate models are not. The justification of
a quasi-static representation of the aerothermodynamic loads is detailed in Section IV.
The structural model passes the deformations to both fluid surrogates, and the heat transfer model
passes wall temperatures to both of the surrogate models. The surrogate models pass the pressures and
the heat flux at the boundary. Note it is assumed that strain heat generation and geometrical changes have
a negligible impact on the thermal solution. Thus, the thermal response is dependent on the structural
deformation strictly due to modifications of the heat flux caused by structural deformation.
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As noted previously, an additional advantage of a partitioned time marching scheme is the ability exploit the disparate time scales between the structural and thermal responses13 by taking multiple structural
steps between each thermal step. Thus, at each thermal step the thermal solution is updated using an estimate for temperature and panel displacement to calculate the heating from the heat flux surrogate. After
the thermal model is iterated to the next step, the structure is updated in multiple steps to the same time
as the thermal model, each step using estimates for displacement and surface temperature to calculate the
forces from the pressure surrogate.
Heat Flux Surrogate
ΔtT
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Thermal
T
w, Tw
q
w
ΔtS
Structure
p
w, Tw
Pressure Surrogate
Figure 2. Fluid-thermal-structural coupling of the 4 sub-models.
The estimates for the displacement and surface temperature must be carefully chosen in order to retain
the second order accuracy of the global solution. As shown in Fig. 2, these quantities must be exchanged
in three distinct ways during the sequencing: 1) the temperatures from the thermal model are sent to the
structural model; 2) the displacements and surface temperatures are input into the pressure surrogate; and
3) the displacements and surface temperatures are input into the heat flux surrogate.
Note that in the following discussion, temperatures and displacements used in the structural model are
denoted with a superscript S and the time step counter n, while a superscript T and the time step counter
m are used for the thermal model. This distinction is important because the structural and thermal models
are not updated simultaneously and require calculation of the temperature and displacement at disparate
time steps. To pass the temperature from the thermal model to the structural model at time step n + 1, a
linear interpolation between thermal time steps is used:
S
T
Tn+1
≈ Tm
+
∆tS
T
T
((n + 1) − nT ) Tm+1
− Tm
+ O ∆t2S
∆tT
(35)
where n is the current structural time step, m is the current thermal time step, and nT is the last structural
∆ts
nT . This interpolation gives second order
time step the thermal solution was updated. Note that m = ∆t
T
errors, retaining the second order accuracy of the structural model.
S
S
The pressure surrogate requires an estimate for both Tn+1
and wn+1
. The temperature is already known
at n + 1 from Eq. (35), but the displacement must be estimated using an extrapolation to n + 1:
1
S
wn+1
≈ wnS + ∆tS ẇnS + ∆tS 2 ẅnS + O ∆tS 3
(36)
2
This approximation yields third order errors, satisfying the second order accuracy of the structural model.
T
T
For the heat flux, the surrogate requires an estimate of Tm+1/2
and wm+1/2
to satisfy Eq. (4). The temperature
is estimated using a second-order extrapolation to m + 1/2:
T
Tm+1/2
≈
3 T 1 T
Tm − Tm−1 + O ∆tT 2
2
2
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(37)
Because the displacements are varying within each thermal time step, averaged displacements and their
derivatives are used for extrapolation:
avg
wm−1/2
=
1
∆tT
avg
=
ẇm−1/2
1
∆tT
avg
ẅm−1/2
=
1
∆tT
tm
Z
tm−1
Z tm
tm−1
Z tm
wS dt
(38a)
ẇS dt
(38b)
ẅS dt
(38c)
tm−1
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(38d)
The averaged displacements are second order accurate representations of the instantaneous quantities at
m − 1/2. These averaged terms are then used to extrapolate to the m + 1/2 step:
1
avg
avg
avg
T
+ ∆tT 2 ẅm−1/2
+ ∆tT ẇm−1/2
+ O ∆tT 2
wm+1/2
≈ wm−1/2
2
The errors are of second order, again maintaining the order of accuracy of the thermal model.
III.
(39)
Model Configuration
The configuration examined in this work is shown in Fig. 3. Two dimensional supersonic flow is
bounded by two walls to simulate a wind tunnel. A wedge, with oscillating amplitude in time, generates
an oblique shock wave that impinges on a compliant panel on the opposite wall. Panel deformations create
a shock wave, producing shock-shock interactions (SSI) as shown. In addition, shock-turbulent boundary
layer interactions (STBLI) occur when the shock impinges on the panel. Note, supersonic flow is chosen for
the present study since it is anticipated that this flow condition will be possible for experimental study of
this problem for future validation. Furthermore, it is not anticipated that this assumption will limit the applicability of this work to hypersonic flow, since there is no fundamental difference for STBLIs in supersonic
or hypersonic flow.
Vibrating Panel
Boundary Layer
STBLI
SSI
Supersonic Flow
Oscillating Shock Generator
Figure 3. Schematic of shock-generating wedge and vibrating panel in supersonic flow.
The geometry of the shock generator (wedge) and panel are defined in Fig. 4 and Table 1. The shock
generator height, hw , is prescribed to vary sinusoidally in time about a mean height, hw0 , at a frequency
of 10 Hz, as defined by Eq. (40). Note that this yields a change of the shock generator’s angle into the
flow of about ±3◦ about a 10◦ nominal shape. This frequency was selected based on the consideration that
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L
Δp
w
H
δw
hw
hw0
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Lw
Figure 4. Schematic of the geometry
Table 1. shock generator
and panel geometry
Lw
hw
δwmax
L
∆p
H
8.7146 cm
0.78186 cm
0.2418 cm
25.415 cm
6.412 cm
13.11 cm
shock impingements on surface panels of high speed aircraft may be generated by flexible, low frequency
structures such as a forebody or control surface.
hw (t) = hw0 + δwmax sin 2π(10Hz)t
A.
(40)
Fluid Model Properties
The properties of the fluid domain are shown in Table 2. The freestream properties are based on the standard atmosphere at an altitude of 24 km. Note that the flow is set to transition from laminar to turbulent
flow approximately 2 meters upstream of the panel. Thus, both the shock generator and the panel are
subject to turbulent flow.
Table 2. Flow properties of the
fluid domain.
M∞
P∞
T∞
ρ∞
Rex∞
3.0
2970 Pa
220 K
kg
4.7038 × 10−2 m
3
2.9136 × 106 m−1
The CFD mesh for the domain is shown in Fig. 5. Note that cells are clustered over the wedge, and
panel, and near both surfaces. The portion of the grid upstream of x = 0 is included in order to ensure
a fully-turbulent boundary layer over the wedge and panel. This mesh is an H grid with approximately
500,000 cells. This mesh was chosen based on the convergence study of total drag of the surfaces for three
similar meshes. The difference in drag coefficient between a mesh with 32,000 cells and final mesh is 3.2%,
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and the difference between a 125,000 cell mesh and the final mesh is less than 0.5%. The average y + value
for the first point away from the surface is 0.103. Note that the 500,000 cell grid is used, despite the mesh
independence demonstrated at 125,000 cells, in order maintain grid independence during surface motions.
Figure 5. Two-dimensional computational fluid domain of the panel. Note the scale of the axes are not equal.
B.
Thermal and Structural Model Properties
A schematic of the heat transfer model and structural model are shown in Figs. 6(a) and 6(b), respectively.
The panel is assumed to be thermally insulated on the two ends as well as the side opposite of the flow.
The panel is clamped on both ends so that the transverse displacements and their rotations are zero. The
panel properties, shown in Table 3, are based on steel 4130.34 The numerical properties of the models are
displayed in Table 4. The thermal solution is updated every 10 structural time steps. The number of mode
shapes and size of the time steps were chosen based on a convergence study comparing the solution to a
case using 40 modes and time steps ∆tS = ∆tT = 10µs. Differences in displacement and temperature rise
between the two cases were negligible. The terms in Eqs. (20)-(25) require integration across the panel. This
integration is performed numerically using Simpson’s Method35 with 1001 discrete points across the panel.
L
L
hp
x
z
hp
q
x
f
z
(a) Heat transfer model.
(b) Structural model.
Figure 6. Structural and thermal model configurations.
Table 3. Properties of heat conducting
panel.
ρp
cp
kg
7.85 × 103 m
3
J
4.73 × 102 kg·K
κ
α∗
E∗
L
hp
Tinit
42.2 m2J·K
8.504 µm/mK at 220 K
203.8 GP a at 220 K
25.415 cm
0.07112 cm
220 K
*Function of temperature34
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Table 4. Numerical properties of thermal and structural
models.
Thermal time step
Structural time step
Structural modes
Structural integration points
Thermal elements through length
Thermal elements through thickness
Thermal DOFs
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C.
1000 µs
100 µs
15
1001
1000
4
5005
Model Verification
To the authors’ knowledge, no data exists to validate a comprehensive fluid-thermal-structural model.
Comparative analysis is presented here to verify the individual structural and thermal models of the panel
using the Abaqus R finite element analysis software.
1.
Thermal Verification
The thermal model is verified against an Abaqus R model consisting of 1000 lengthwise and 4 throughthickness quadrilateral elements. A constant in time heat flux profile, shown in Fig. 7(a) is applied to
the upper surface of the panel, with all other surfaces insulated. Both models use a 1 ms time step. The
midplate temperatures up to 10 seconds are shown in Fig. 7(b). Both models predict the same temperature
throughout the time history, and the maximum difference in temperature is 0.0057 K.
(a) Heat flux profile
(b) Mid-plate temperature rise
Figure 7. Comparison of current thermal model to Abaqus R using a constant heat flux profile.
2.
Structural Verification
The structural model is verified using Abaqus R for two separate cases. In the first case a random PPL is
applied to the panel, and the power spectral density (PSD) of the center displacement is compared. In the
second case a rising uniform temperature is applied to the panel, and the center displacement is compared.
The Abaqus R model consists of 1000 cubic-interpolation beam (B23) elements. In order to match the
properties of the Abaqus R beam model to the cylindrical bending panel model, the modulus of elasticity
and coefficient of thermal expansion of the beam are computed using Eqs. (41) and (42). Temperaturedependent E and α and mass-proportional Rayleigh damping are specified in the Abaqus R model.
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Epanel
1 − ν2
≡ αpanel (1 + ν)
Ebeam ≡
αbeam
(41)
(42)
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Figure 8(a) illustrates the random PPL applied to the panel, which has an RMS load of 140 dB. The PSD
of the center displacements are shown in Fig. 8(b) up to 1000 Hz. The two models have excellent agreement,
where both predict peaks near 90 Hz and 340 Hz. Note that the peaks have a wide bandwidth due to the
nonlinear response of the panel. A linear response would show steep peaks at the natural frequencies for
non-symmetric modes of the panel under 500Hz, which are 61Hz and 329Hz for this panel. Because of the
nonlinear terms in the stiffness matrix in Eq. (22), the stiffness increases with displacement, which increases
the instantaneous natural frequencies of the panel and causes the peaks of the PSD to be spread into higher
frequencies. The response is minimal above 500 Hz, as expected due to the 500 Hz cutoff of the PPL.
(b) PSD of the center displacement compared to Abaqus R
(a) 140 dB random prescribed pressure load.
Figure 8. Comparison of current structural model to Abaqus R using a random prescribed pressure load.
A second structural verification case is considered in order to assess the accuracy of the model for moderate deflections of the panel under thermal loads. This is shown in Fig. 9 for a transient, but spatially
uniform, temperature load of the panel. Since the panel is perfectly flat, in addition to this temperature
load a small pressure loading (10−5 Pa) is applied in order to obtain the buckled equilibrium path. Clearly
there is excellent agreement between the two models up to at least 16 panel thicknesses.
(a) Temperature profile as a function of time
(b) Center displacement compared to Abaqus
Figure 9. Comparison of current structural model to Abaqus using a rising temperature profile.
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IV.
Development of a CFD Surrogate Model
A significant modeling challenge associated with fluid-thermal-structural interactions is the need for
a time record that encapsulates the thermal response of the surface. Since the thermal response is generally expected to be on the order of minutes to hours, a full order CFD-based analysis is computationally
impractical for extensive analysis. Thus, a central aspect of this study is on the development of model reduction techniques for computational aerothermodynamics with shock impingements. Typical approaches
for model reduction of unsteady, nonlinear flows include proper orthogonal decomposition (POD),36–38
Volterra series,38, 39 and surrogates.40–45 Each of these seek to identify the primary features of a system from
a limited number of high-fidelity flow solutions. The generation of this dataset requires an initial computational investment. However, this process is efficiently carried out using parallel computing facilities.
POD represents a spectral method, where an orthogonal modal basis is computed from snapshots of the
full-order system response to relevant inputs.38 The Volterra series method uses the assumption that the
response of any nonlinear system is exactly represented by an infinite series expansion of multidimensional
convolution integrals of Volterra kernels. A Volterra series ROM is constructed by computing a truncated
set of kernels from the full-order system response to a set of known inputs.38 Surrogate based approaches
identify a continuous approximate function, i.e., “surrogate function” from a discrete sampling of an unknown, nonlinear function over a bounded set of inputs.46 Methods for constructing the surrogate function
include radial basis functions, neural networks, polynomial response surfaces, and kriging.41, 46
Despite extensive research into aerodynamic ROMs,38 only a limited number of studies are relevant to
supersonic flow. Lucia47 examined POD to model aerodynamic systems with strong shocks and nonlinearity in the parameter space. Tang et al.48 developed a POD based ROM for predicting steady-state pressure
and temperature distributions on the surface of a rigid hypersonic vehicle resembling the X-34. A similar
approach was carried out recently by Crowell and McNamara,43 where unsteady effects in the pressure
were added using a piston theory23 based correction. Also recently, several studies40, 42, 43 have investigated
kriging surrogates for model reduction of the aerothermodynamics. Results from [40] and [42], as well as
similar work done in lower Mach number regimes,41, 44, 45, 49 illustrate that surrogate based approaches are
promising for accurate and efficient modeling of complex fluid dynamic phenomena. In particular, some
of this work suggests that surrogates produce more accurate models than POD.41, 43 However, this previous work did not consider model reduction for flow fields with shock impingement. This is an important
consideration since this introduces significant nonlinearities in the thermo-mechanical loads (i.e., surface
pressure and heat flux), which potentially increases the difficulty in model reduction.
Based on previous work,16, 17, 43 one of the considerations for this study is the assumption of quasi-static
flow conditions for the aerothermodynamic loads. Such an approximation is advantageous since it enables
the generation of a CFD surrogate model using steady-state CFD solutions from a set of instantaneous
positions of the panel and shock generator.16, 43 Thus, the first part of this section further investigates, and
ultimately justifies, the use of a quasi-static surrogate model for the aerodynamic heat and pressure loads.
The second part focuses on the actual construction and verification of the surrogate models.
A.
Assessment of Quasi-Static Flow Approximation
In order to assess the validity of a quasi-static flow assumption, fully unsteady, quasi-static, and steady
flow are considered for a thermally compliant panel undergoing forced vibration and subject to STBLIs.
The time marching scheme used in this assessment is discussed in a prior study;17 note the fluid time step
is set to 0.1ms, the thermal time step is set to 1.0ms, and the method is shown to maintain second order
temporal accuracy.
In order to quantify the accuracy of the different approaches, comparisons are made in terms of the
spatially averaged through-thickness temperature rise in the panel, and the generalized aerodynamic forces
(GAFs). The GAFs are computed as:
Z
GAF =
P φ1 dx
(43)
where φ1 is the first free vibration mode of the panel, nondimensionalized between 0 and 1.
For this analysis, the flow properties listed in Table 2 are used, along with Eq. 40 to define the forced
motion of the shock generator at 10 Hz, as well as Eq. 44 to define the forced motion of the panel in its first
mode shape. Note, the shock generator angle varies ±3o about a nominal 10o wedge, and the panel varies
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± 3 panel thicknesses (hp ) about its undeformed (i.e., flat) position. Comparisons of the spatially averaged
through-thickness temperature rise in the panel at 2 seconds, and from 0 to 2 seconds at the 45% chord are
shown in Fig. 10(a,b). Comparisons of the GAFs from 0 to 2 seconds are shown in Fig. 11(a), and from 0 to
0.1 seconds, in Fig. 11(b). In order to examine the unsteady component of the GAFs, the quasi-static GAFs
are subtracted from the unsteady model GAFs in Fig. 11(c), illustrating the contribution of unsteady effects
on the GAFs; where it is important to note the different magnitude scale of the GAFs in Figs. 11(a) and (c).
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w(t) = 3hp sin 2π(50 Hz)t φ1
(a) Temperature rise at time = 2.0 sec.
(44)
(b) Temperature rise at x = 45% chord.
Figure 10. Spatially averaged through-thickness temperature rise for the unsteady model, quasi-static, and steady cases.
The average error and max error of the quasi-static and steady cases are computed relative to the unsteady case in Table 5. The results in Fig. 10 and Table 5 demonstrate that quasi-static aeroheating loads
are sufficient for fluid-thermal-structural analysis of this 2-D panel configuration. The mean and max errors of the temperature loading are under 1% and 3% respectively. In terms of the pressure loading, from
Fig. 11(a,b) and Table 5 the quasi-static prediction is within average and max errors of 2.7% and 5.9%, respectively. Note that these errors are primarily due to time lag effects, and the error in maximum and
minimum loading between the quasi-static and unsteady models is only 0.16% and 1.18%, respectively.
Comparing Figs. 11(a) and (c), the unsteady contribution to the GAFs is approximately two orders of magnitude smaller than the overall GAFs. Thus, it is assumed that neglecting the full unsteady effects on the
pressure will not result in significant errors, and a quasi-static representation of the pressure loading is acceptable for this configuration. It is interesting that this conclusion differs from that of a previous study by
the authors,43 where unsteady effects were found to be important. This difference is presumably due to the
shock impingement location on the panel, which dominates the pressure load and is apparently readily captured using a quasi-static analysis. Note that, as expected, the steady model provides a poor approximation
in terms of predicting both the temperature rise and GAFs, with errors up to 29% and 53%, respectively.
Table 5. Percent differences in the spatially averaged through-thickness
temperature rise and generalized forces for the quasi-static and steady
cases compared to the unsteady case for 2.0 seconds of response.
Method
Quasi-Static
Steady
Temperature
Avg. (%) Max (%)
1.24
2.71
3.92
28.8
Generalized Forces
Avg. (%) Max (%)
2.73
5.93
24.6
53.4
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(a) Generalized forces, 0 to 2.0 sec.
(b) Generalized forces, 0 to 0.1 sec.
(c) Unsteady component of GAFs, computed from the unsteady
model minus the quasi-static model.
Figure 11. Generalized aerodynamic forces for the unsteady model, quasi-static, and steady cases.
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B.
Surrogate Construction
Kriging is chosen for the construction of the CFD surrogate since it does not require a priori assumptions on the form of the full-order model, and has demonstrated good to excellent accuracy in previous
work.15, 16, 41–43, 50 For this study, the kriging surrogate is computed using the Design and Analysis of Computer Experiments (DACE)51 toolbox in Matlab R . This toolbox solves for the optimum tuning coefficients
of the kriging surrogate through an efficient maximum likelihood estimation, and provides several different options for regression models and correlation functions. In this study, the regression models and
correlation functions were selected through a comparison of surrogate accuracy. The Gaussian correlation
function and a second-order regression model were found to yield the most accurate surrogate.
The process used for generating the surrogate is similar to that used in [15,16,43]. First, input parameters
and bounds are established (e.g., Mach number, altitude, deformation, surface temperature, etc). Latin
hypercube sampling is then used to identify a diverse set of sampling points inside of the defined parameter
input bounds. Next, aeroheating and pressure responses are computed from steady CFD solutions to the
Navier-Stokes equations at each of the sample points. Finally, the model is constructed using the DACE
toolbox.
An important step in this process is parameterization of the model inputs. Scalar inputs, such as Mach
number and altitude, require no special treatment. However, inputs that are formally spatial distributions,
such as surface deformation and temperature, must be parameterized to define them as a set of scalar
inputs. The following discussion details the parameterization process for surface temperature and deformation.
1.
Inclusion of Surface Temperature
In previous work by the authors, surface temperature has been parameterized using either assumed polynomial distributions16, 43 or thermal POD modes.15 Thus, the input for the surrogate model consisted of
participation coefficients for the different assumed distribution functions. A drawback to this approach for
the present study is the presence of a sharp heat flux gradient, due to the shock impingement, on the panel
surface. Thus, a refined method is examined next by characterizing the effect of surface temperature on the
surface heat flux.
The aerodynamic heating is commonly approximated using a constant in time heat flux coefficient,
q(x, t) = hht (x) Tw (x, t) − Taw (x)
(45)
where hht is the heat transfer coefficient, and Taw is the adiabatic wall temperature. The heat flux coefficient
is calculated using the difference in heat flux between the initial surface temperature and the adiabatic
surface temperature of the undeformed panel, shown in Eq. (46).
hht (x) =
q(x, 0) − 0
Tw (x, 0) − Taw (x)
(46)
However, this method neglects the dependence of the heat transfer coefficient on surface temperature.
Thus, a simple interpolation of the heat transfer coefficient is developed as a first model for heat flux. This
model is created by interpolating the heat transfer coefficient for several isothermal cases on flat panels
in the absence of shock impingements. This Piecewise Isothermal Model (PIM) takes as input only the x
location on the panel and the temperature at that location, Tw (x), in order to interpolate the heat flux. Assessment of the method is conducted by generating a non-isothermal temperature profile, Fig. 12(a), again
applied to a flat panel in the absence of a shock impingement. The CFL3D prediction for this temperature
profile is compared to the PIM method and the constant hht method from Eq. 46 in Fig. 12(b).
It is clear that the heat flux is significantly affected by both the magnitude of the temperature and the
presence of spatial thermal gradients. Furthermore, the PIM is most inaccurate during regions with spatial
gradients in temperature, illustrating the importance of these effects on the heat transfer coefficient. In fact
it is this spatial dependence of the heat flux that served as a motivating factor for using assumed spatial distributions of temperature (i.e. polynomials) in previous aerothermodynamic surrogate models.15, 16, 43 Note,
the constant hht method shows similar trends to the PIM method, with an offset due to the assumption of a
constant heat transfer coefficient.
Based on this result, a correction for the PIM method is developed to account for the dependence of
the heat flux on temperature gradients. As shown in Fig. 12(b), the CFL3D solution exponentially decays
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(a) Temperature profile.
(b) Heat flux solutions on a flat panel without shock impingement.
Figure 12. Temperature distribution and heat flux from CFL3D, the piecewise isothermal model (PIM), and the constant hht
method.
back to the isothermal solution when temperature derivatives drop to zero. This portion of the response
mathematically resembles that of a homogeneous, over-damped, harmonic oscillator. Also, over the first half
of the panel, the difference between the PIM solution and the CFL3D solution mathematically resembles the
response of an overdamped harmonic oscillator with forcing. Furthermore, it is deduced that the forcing
function of the associated second-order ordinary differential equation (ODE) is dependent on temperature
gradients, since this represents the only difference between the CFL3D and PIM solutions. Thus, the following general form for a second-order ODE is developed to compute a correction for the PIM method:
d2 q̂
dq̂
dTw
d2 Tw
+
C
+
C
q̂
=
C
+
C
1
2
3
4
dx2
dx
dx
dx2
(47)
where the Corrected Isothermal Model (CIM) heat flux is then:
qCIM = qP IM + q̂
(48)
Solving Eq. 47 for q̂ represents the necessary addition to a piecewise isothermal interpolation in order to
account for spatial temperature derivatives. In order to estimate the coefficients in Eq. 47, 20 cases with
arbitrary temperature profiles are generated. Both the CFL3D and PIM heat flux solutions are obtained,
and Eq. 48 is re-arranged for q̂, where:
q̂ = qCF L3D − qP IM
(49)
2
d Tw
w
Next, derivatives in q̂, dT
dx , and dx2 are computed for the 20 sample cases. The coefficients, Ci , are then
determined from the a least squares fit to Eq. 47 over these 20 cases. The values obtained from this analysis
are shown in Table 6.
Table 6. Coefficients for the correction equation, Eq. 47.
83.03
243.66
-2822.62
-105.46
C1
C2
C3
C4
Lastly, boundary conditions are required to solve Eq. 47. The first boundary condition is:
q̂(0) = 0
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(50)
which is equivalent to the assumption that there are no upstream temperature gradients at the leading edge
of the panel. The second boundary condition is:
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dTw (0)
dq̂(0)
=−
.
(51)
dx
dx
This boundary condition was tested over a wide range of values and found to have a negligible effect
w (0)
on the overall solution. However, through experimentation, − dTdx
is found to be in general a good
dq̂(0)
approximation for dx . Once both q̂ and qP IM are computed, the CIM is computed from Eq. 48, and is
valid for any temperature profile. In order to partially validate this, the previous case is repeated in Fig. 13.
The mean error for the corrected method has decreased from 10.7% to 1.2%, and the maximum error has
decreased from 34% to 4.0%, which equates to a maximum error of 0.03W/cm2 . In order to further validate
this approach, 25 subsequent test cases were generated. The CIM produced an overall mean error of 0.86%
and max error of 4.0%.
(a) Temperature profile.
(b) Heat flux.
Figure 13. Temperature distribution and heat flux on a flat plate for CFL3D, piecewise isothermal model (PIM), and corrected
isothermal model (CIM).
Note that pressure is also a function of surface temperature. However, the dependence on surface temperature is relatively weak, demonstrated by the fact that the PIM produces mean and maximum errors
of only 0.26% and 0.86%, respectively. Therefore no additional correction is necessary to the pressure to
account for temperature gradients. This offers a significant computational savings, since the pressure is
updated more frequently than the heat flux, and the correction (i.e., solution to Eq. 47) must be computed
every time the surface temperature changes.
Based on these results, surface temperature is parameterized in terms of a isothermal surface temperature. Thus, a single parameter is used for the temperature, where interpolation is done between isothermal
solutions at a corresponding x location. Subsequently, heat flux output from the surrogate is corrected for
spatial temperature derivatives using the CIM.
2.
Inclusion of Deformation and Shock Generator Angle
Parameterization of the panel surface deformation is done similar to previous studies,15, 16, 43 where the
inputs to the surrogate are the time dependent amplitudes of the free vibration modes, ai in Eq. (18). For
this study, the parameterization is done about the nondimensional amplitudes, a¯i = hapi . Note, from a
preliminary uncoupled analysis of the structure, the amplitudes of modes 7 through 15 are found to be
approximately three or more orders of magnitude smaller than the first mode, and therefore only the first
6 free vibration modes are included as inputs to the surrogate. The last parameter for inclusion as an input
to the surrogate is the shock generator wedge angle. For this study the angle is limited to vary between 7o
and 13o .
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The bounds of the parameter space for the surrogate are given in Table 7. The surface temperature is
set to vary from slightly below the freestream temperature (220K) to just under the Mach 3.0 total temperature (616K). The bounds of the deformation amplitudes are based off of a preliminary uncoupled fluidthermal-structural analysis, where a single CFD flow solution is combined with Eqs. 43 and 45 to compute
the pressure and heat flux loads, respectively. Latin hypercube sampling is used to generate 1000 sample
points throughout the parameter space, and the DACE toolbox computes the kriging surrogate, relating the
input parameters to the CFL3D output from the sample points. The surrogate is complete when the CIM
correction is applied to the heat flux output from the surrogate for temperature distributions computed
using the thermal model.
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Table 7. Parameter bounds of the surrogate.
200 K
-20.0
-5.0
-2.0
-1.0
-1.0
-1.0
7.0o
C.
≤ Tw ≤
≤ ā1 ≤
≤ ā2 ≤
≤ ā3 ≤
≤ ā4 ≤
≤ ā5 ≤
≤ ā6 ≤
≤ Shock Generator Angle ≤
600 K
20.0
5.0
2.0
1.0
1.0
1.0
13.0o
Verification of the Surrogate
For verification of the accuracy of the surrogate models, 25 test cases with random temperature profiles, deformations, and shock generator angles (inside of the bounds listed in Table 7) are computed using CFL3D.
One of these cases is shown in Fig. 14. The mean and maximum errors for all 25 cases are 3.60% and 28.9%,
respectively. Note, without the CIM correction the mean and maximum errors are 11.4% and 57.4%, respectively. The mean and maximum errors from all 25 cases for the aerodynamic pressure portion of the
surrogate are 1.71% and 20.1%, respectively.
As indicated in Figs. 14(c,d), including the effects of arbitrary temperature, deformation, and shock
location can result in large maximum errors (>20%). However, the average error of surrogate is relatively
low (<4%), particularly considering the nonlinearity of the response with shock impingement.
As a final verification of the surrogate, the previous aerothermal study comparing the unsteady, quasistatic, and steady methods is repeated with results from the surrogate model included. Results are in
Fig. 15 and Table 8. Clearly the surrogate provides an excellent model for the aerothermodynamic loads.
The mean and max errors of the surrogate are very similar to the quasi-static model. In terms of the heat
flux the mean and max errors are 1.0% and 4.8% respectively, for the GAFs they are 2.9% and 5.1% relative
to the unsteady model. The error at the peak loading for the surrogate model is 1.67%, and the error at the
minimum loading is 1.09%.
Table 8. Percent differences in the spatially averaged through-thickness
temperature and generalized forces for the surrogate, quasi-static, and
steady cases.
Method
Quasi-Static
Steady
Surrogate
Temperature
Avg. (%) Max (%)
1.24
2.71
3.92
28.8
1.04
4.79
Generalized Forces
Avg. (%) Max (%)
2.73
5.93
24.6
53.4
2.85
5.15
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(a) Temperature profile.
(b) Deformation profile.
(c) Heat flux.
(d) Pressure.
Figure 14. Example case with shock impingement, for the heat flux, and pressure from CFL3D and the surrogate. Shock generator
angle = 12.71o .
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(a) Temperature rise at time = 2.0 sec.
(b) Temperature rise at x = 45% chord.
(c) Generalized forces, 0 to 2.0 sec.
(d) Generalized forces, 0 to 0.1 sec.
Figure 15. Spatially averaged through-thickness temperature and generalized aerodynamic forces for the unsteady, quasi-static,
steady, and surrogate models.
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V.
Fluid-Thermal-Structural Analysis
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The fluid-thermal-structural model is used to investigate two main cases: 1) a coupled fluid-thermalstructure model using the surrogate to compute the pressure and heat loads (’Coupled’), and 2) an uncoupled case where the initial conditions are used for the loads (’Uncoupled’). For the uncoupled model,
the heat flux is linearized between the initial temperature of the panel and the adiabatic wall temperature
for a flat plate. Both models may include a 140dB prescribed pressure load (PPL) to simulate the panel’s
response to fluctuating pressures due to a turbulent boundary layer. An important consideration for both
of these models is the backpressure acting on the panel. A very low backpressure will bias the panel to
always buckle out of the flow, and a very high backpressure will buckle the panel into the flow. Thus, as
a first assessment of the models, a range of backpressures are considered in order to identify the critical
backpressure that causes the panel to switch from buckling out of the flow to buckling into the flow. Next,
a backpressure is chosen to force all of the models to buckle out of the flow, and observations are made
over 90 seconds of aerothermoelastic response. Finally, a backpressure which will force all of the models to
buckle into the flow is implemented, and again 90 seconds of aerothermoelastic response are computed.
A.
Effect of Backpressure on Buckling Direction
Panel buckling direction can have significant effects on the transient behavior of the panel and the aerodynamics, including changes in aerodynamic loading, panel temperature, and the stability of the panel
against aeroelastic flutter phenomenon.29 Thus, the critical backpressure that determines the buckling direction for the coupled and uncoupled models is investigated. The nondimensional displacements for the
first four seconds of response are shown in Fig. 16. For the uncoupled model without PPL, in Fig. 16(a),
the critical backpressure occurs between 2045Pa and 2050Pa above the freestream pressure (P∞ = 2970Pa).
When the PPL is included, Fig. 16(b), the panel always buckles out of the flow for backpressures below
2025Pa above freestream and into the flow for backpressures above 2075Pa above freestream. Between
2025Pa and 2075Pa, the panel may buckle in either direction due to the random nature of the PPL. For the
coupled model without PPL, Fig. 16(c), the critical backpressure is significantly less than the uncoupled
cases; between 1675Pa and 1775Pa above freestream. Note that the panel may buckle in either direction
between this range due to snap-through behavior caused by the shock motion. The effect of the PPL on the
coupled model, Fig. 16(d), causes the range of critical backpressures to move slightly higher to 1750Pa and
1825Pa above freestream. Note also, all of the results other than the uncoupled without PPL model predict
the panel to snap into and out of the flow for up to the first three seconds of the response. For the coupled
models, the frequency of the snap through occurs at 10 Hz, and is driven by the 10 Hz forced motion of the
shock generator, which is not included in the uncoupled analysis. However, the uncoupled case with PPL
does show some chaotic snap through behavior, due solely to the random 140dB PPL.
The difference in critical backpressure between the coupled and uncoupled models is approximately
300Pa, or 10% of the freestream pressure. The reason for this discrepancy is not immediately clear, but it
may be due to either the motion of the shock or the coupling of temperature and deformation to the aerodynamic surrogate, or a combination of the two. In order to clarify the importance of each aspect of the
coupled model, two additional cases are considered. First, the coupled model is considered with a 10o stationary shock (Coupled-SS), thus showing the importance of coupling alone. Second, the uncoupled model
is updated to included the moving shock generator (Uncoupled-MS); however structural deformations and
temperature coupling are still neglected, thus showing the importance of the shock motion alone. Results
for the first four seconds of response for these two cases are presented in Fig. 17.
For the Uncoupled-MS model, Fig. 17(a), the critical backpressure is between 2035Pa and 2075Pa above
freestream. This range is similar to the original uncoupled results, Figs. 16(a-b). Thus, the inclusion of the
shock generator motion on the pressure and heat loads only increases the range where either buckling into
or out of the flow may occur, it does not lower the range by 300Pa. The results for Coupled-SS model,
which has a stationary shock, Fig. 17(b), predicts the critical backpressure to be between 1975Pa and 2000Pa
above freestream. Thus, including deformations and temperature coupling but holding the shock generator stationary has increased the range of critical backpressure by approximately 200Pa above the original
fully coupled result, Fig. 16(c). Therefore neglecting either the shock generator motion or the structural
deformation and temperature coupling will result in critical backpressure uncertainty on the order of 10%
of the freestream pressure.
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(a) Displacement for the uncoupled results without the PPL
(b) Displacement for the uncoupled results with the PPL
(c) Displacement for the coupled results without the PPL
(d) Displacement for the coupled results with the PPL
Figure 16. Varying backpressures for the coupled and uncoupled models with and without 140dB random prescribed pressure
loads (PPL). Legend denotes backpressure above freestream, 2970Pa.
(a) Displacement for the Uncoupled-MS model, which includes
the moving shock generator.
(b) Displacement for the Coupled-SS model, with a stationary
shock generator.
Figure 17. Panel buckling direction due to various backpressures (PPL not included). Legend denotes backpressure above
freestream, 2970Pa.
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B.
Freestream Backpressure
The next case considered has a backpressure equal to the freestream pressure of 2970Pa, which results in a
panel buckled out of the flow for both coupled and uncouple analyses. Results for 90 seconds of response
are shown in Fig. 18. Note, Fig. 18(d-f) depict the response at the 40% chord since this generally corresponds
to the point of maximum differences between the coupled and uncoupled analysis.
It is clear from Figs. 18 that the inclusion of the 140dB PPL has no noticeable effect on the chord-wise
temperature distribution of the panel. However, some difference is apparent between the uncoupled and
coupled analysis, with a 7% higher average temperature for the uncoupled model at 90 seconds. This
results in approximately a 4% larger maximum displacement at 90 seconds for the uncoupled analysis.
Furthermore, note that the coupled analysis produces an asymmetric panel displacement, while the uncoupled displacement response is symmetric. This is due to the coupling of the pressure load to the panel
displacements. In terms of the transient displacements, Figs. 18(d-f), both cases with the 140dB PPL have
relatively minor oscillations compared to the models without the PPL. Also, it is clear from Fig. 18(f) that
the coupled analysis exhibits a harmonic oscillation due to the 10 Hz motion of the shock generator. Finally,
note both the displacements and average temperatures are nearing a steady-state condition by 90 seconds
of response.
C.
Backpressure 2500Pa Above Freestream
The last case considered consists of a backpressure of 2500Pa above freestream. This backpressure was chosen to bias the panel to buckle into the flow for both coupled and uncoupled analyses. First, a comparison
of the temperature rise without the PPL is shown in Fig. 19, for the coupled model with freestream and
2500Pa above freestream backpressures, as well as the uncoupled model. Note, the temperature rise for all
of the uncoupled models is exactly the same whether the panel buckles into or out of the flow, since the
panel deformation is not coupled to the heat load.
In comparing the temperature profiles, Fig. 19(a) for the two coupled models, some significant differences are observed, owing to panel buckling direction. For the freestream backpressure, panel buckles out
of the flow, the maximum temperature occurs at the 75% chord, where as for the 2500Pa above freestream
backpressure case, panel buckles into the flow, the maximum temperature occurs at the 45% chord. Also,
the temperature differences between the two coupled models are as high as 55K at 90 seconds. However,
the average temperatures are still similar for all three models, Fig. 19(b). The average temperatures are only
4% different between the uncoupled and 2500Pa coupled model at 90 seconds. This is significant since, as
noted previously, the cylindrical bending displacements are driven by the average chord wise temperature.
Finally, note the uncoupled model predicts the highest maximum temperature rise, approximately 355K
above the inital temperature (220K).
The next set of results are only for the cases with a backpressure of 2500Pa above freestream. The
displacements are shown in Fig. 20. Considering first the displacement profiles, Fig. 20(a), the coupled
models again predict asymmetric deformation due to structural-pressure coupling, while the uncoupled
models do not. The coupled models also have approximately 3% lower maximum displacements relative
to the uncoupled models. For the transient displacement at the 40% chord, Figs. 20(b-d), there are several
noteworthy differences between the coupled and uncoupled models. Again, the cases with the 140dB PPL
predict minor oscillations relative to the models without the PPL, but the primary oscillations of the coupled
models are driven by the shock generator, as seen in Fig. 20(d). Comparing Figs. 20(d) and 18(f), it can
be noted that the oscillations caused by the shock generator are much larger for the case with the panel
buckling into the flow 20(d); where the motion of the shock changes hwp by approximately 1.5 compared to
0.2 in 18(f). In Figs. 20(b,c) the coupled models predict the panel snapping into and out of the flow for the
first second of the response, before eventually remaining buckled into the flow.
VI.
Conclusions
An aerothermoelastic model is developed for analysis of panel structures subject to shock-turbulent
boundary layer interactions (STBLIs). Specifically, the development and verification of fluid, thermal, and
structural models as well as a partitioned second-order accurate solver are presented in this work. The
model configuration includes a 10 Hz oscillating shock generator which creates an impinging shock on a
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(a) Avg. through thickness temperature rise at 10 sec. and 90sec.
(b) Spatially averaged temperature rise.
(c) Nondimensional displacement at 10 sec. and 90 sec.
(d) Nondimensional displacement at 40% chord
(e) Nondimensional displacement over first 4 seconds, at 40%
chord.
(f) Nondimensional displacement over last second, at 40%
chord
Figure 18. Spatially averaged through-thickness temperature rise and nondimensional displacement for coupled and uncoupled
results with and without 140dB random prescribed pressure loads (PPL). Backpressure = P∞ .
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(a) Temperature rise at 90sec.
(b) Avg. temperature rise.
Figure 19. Spatially averaged through-thickness temperature rise for two coupled models and the uncoupled model.
(a) Nondimensional displacement at 10 sec. and 90 sec.
(b) Nondimensional displacement at 40% chord
(c) Nondimensional displacement over first 4 seconds, at 40%
chord.
(d) Nondimensional displacement over last second, at 40%
chord
Figure 20. Spatially averaged through-thickness temperature rise and nondimensional displacement for coupled and uncoupled
results with and without 140dB random prescribed pressure loads (PPL). Backpressure = P∞ +2500Pa.
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thermo-mechanically compliant panel. Several studies are conducted including: 1) unsteady, quasi-static,
and steady fluid-thermal analysis of the panel undergoing force oscillations, 2) construction of a quasi-static
surrogate for the pressure and heating loads, 3) identification of the critical backpressure which results in a
change of buckling direction of the panel for the fluid-thermal-structural analysis, 4) coupled vs. uncoupled
fluid-thermal-structural analysis over 90 second time records, and 5) the effect of a random in time 140dB
prescribed pressure load (PPL) on the aerothermoelastic models. Results from these studies allow one to
reach several useful conclusions:
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1. A quasi-static representation of the fluid is found to be acceptable in an fluid-thermal analysis for a
low frequency (≈ 10 Hz) sinusoidally moving shock interacting with a 50 Hz oscillating panel. Specifically, after 2.0 seconds of response, the temperature rise predicted from a quasi-static approximation
has average and max errors of 1.2% and 2.7% relative to the unsteady model. In terms of pressure
loading, the quasi-static approximation averaged 2.7% error, with a max error of 5.9%.
2. A steady flow approximation in the fluid-thermal analysis over predicts the temperature rise by up
to 28% at 2 seconds relative to the unsteady model. The pressure loads are inaccurate by as much as
53%.
3. A novel surrogate is developed for the fluid, based on quasi-static CFD solutions. Temperature dependence is incorporated through an isothermal interpolation and an overdamped second order ordinary
differential equation correction. Errors for the surrogate average under 4%, and are as high as 29%,
relative to 25 test cases with random combinations of inputs. The relatively high maximum errors are
due to the nonlinearity of the shock-impingement. The surrogate is also tested in the fluid-thermal
analysis, and is found to have maximum errors under 3% in predicting the temperature rise and under
6% for the pressure loading relative to the unsteady model.
4. Analysis of the critical backpressure required to change the direction of the panel buckling between
the coupled and uncoupled models results in differences on the order of 10% of the freestream pressure (P∞ = 2970Pa).
5. Neglecting shock generator motion on the coupled model results in critical backpressure near the uncoupled model prediction. Neglecting structural deformation and thermal coupling from the model
also results in critical backpressures near the uncoupled model. Thus, inclusion of all forms of coupling are necessary to predict the critical backpressure within approximately 300Pa of the coupled
result.
6. 90 seconds of aerothermoelastic response are generated for the coupled and uncoupled models with
a backpressure set to freestream. With this configuration all models predict the panel to buckle out of
the flow. The inclusion of the PPL has no noticeable effect on the temperature rise. At 90 seconds, the
average temperatures of the uncoupled results are 7% higher than the coupled results, and maximum
displacements are 4% larger. Small oscillations are observed for the PPL cases. The coupled cases
also exhibit harmonic motion, on the order of 0.2 panel thicknesses, due to the motion of the shock
generator.
7. Backpressures of 2500Pa above freestream are also considered for the models. This pressure results in
the panel buckling into the flow. Chordwise temperature distributions for the 2500Pa coupled model
are up to 55K different from the freestream case. However, average temperatures are very similar
to each other and to the uncoupled case (<4%). The coupled models also exhibit larger oscillations
due to shock motion, on the order of 1.5 panel thicknesses. Maximum displacements between the
coupled and uncoupled models are within 3%. The addition of the PPL has no noticable effect on the
temperature, and only provides minor oscillations in the displacement.
Acknowledgments
The authors gratefully acknowledge government support awarded by AFOSR Grant FA9550-11-1-0036,
with Dr. John Schmissuer as Program Manager and the DoD Science Mathematics and Research for Transformation (SMART) Scholarship. Additionally, this work is supported in part by an allocation of computing
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time from the Ohio Super Computer Center. The authors also appreciate the technical insights of Drs. Ravi
Chona, S. Michael Spottswood, and Tom Eason of the AFRL/RBSM Structural Sciences Center.
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