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CFD-Thermal InteractionsShort Course, TFAWS 2003
Integrated Fluid-Thermal Analysis from a Thermal-Structures Perspective
Kim S. BeyMetals and Thermal Structures
BranchNASA Langley Research Center
Hampton, VA August 21, 2003
GWU M.S. StudentsJames Tomey, Ford Motor Company
Christapher Lang, MTSB LaRCDavid Walker, ATK Thiokol
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Integrated Fluid-Thermal Analysis for High Speed Flight Vehicles
• Coupling at the fluid-thermal interface depends on the type of structure: insulated or non-insulated (hot)
• “Next Generation” Thermal Analysis Methods for Hot Built-up Structures
T
Tw
qw, p p, τw
Structural
Aerothermal
Thermal
u
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Thermal-Structural Airframe Concepts for Reusable Launch Vehicles
Cryotank withaeroshell and insulating
thermal protection system
Integrated hot structurewhere thermal protection
also carries load
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Airframe Thermal Analysis State-of-the-Art
Through-the-thickness plug
models
Full 3D finite element models
Transient nonlinear problem – Conduction– Radiation exchange– Convection
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CFD-Thermal Interactions for Insulated Structures are (Approximately) Decoupled
Heat conducted into the structure is small
radaero
s
0q
≈⇓
≈
Iterate the fluid energy equation at the wall
boundary until
( )44W
W
ffW TT
zT
kq ∞−=∂∂
= εσ
zT
kq ffaero ∂∂
−= ( )44Wrad TTq ∞−= εσ
wsf TTT ==
zTkq s
ss ∂∂
−=1ks <<insulation
substructure1ks >>
Through-the-Thickness Plug Models are Adequate for Insulated Structures since In-plane Temperature
Gradients are Small
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Through-the-Thickness Plug Model of Complex Metallic TPS Concept
Cryogenic fuel
Tank wallPurge cavity
Insulation
Armor TPS panel
Thermallycompliant
sides& support
Box beam& bolt
Sandwich
Insulation
TPS support
Tank wall
BoltPurgecavity
)( aTThq0q
−== or
aqrq
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Thermal Response Predicted with TPS-it
0 2000 4000 6000 8000400
600
800
1000
1200
1400
1600
1800
2000
T (°R)
t (sec)
Substructure,4
TPS bottom surface,3
insulation mid-plane,2
TPS top surface,1
1
23
4
0.01.02.03.04.05.06.07.0
0 1000 2000
Time (sec)
Hea
ting
Rat
e (b
tu/ft
2 -s)
STA 264
STA 827
STA 1238
qw
qw(t)
insulated
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Decoupled CFD-Thermal Interactions Simplify the Design Process
trajectory
α
h
M
q(t), p(t)h(t), M(t), α(t)
CAD geometry
Aero/Aerothermal Environments
AFRSIAFRSI
TABITABI
AETB12AETB12
TPS Sizing and Material Selection
are performed independently
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CFD-Thermal Interactions for Hot (non-insulated) Structures are Coupled
• Heating strongly depends on wall temperature
• Wall temperature strongly depends on thermal energy absorbed by structure
Structure absorbs thermal energy
)( sWW qTT =
)( Waerof
faero Tqz
Tkq =
∂∂
−=
At the fluid-solid interface
Wsf
Ws
sf
f
TTT
qzTk
zT
k
==
=∂∂
=∂∂
−x
zFluid: kf , Tf
Solid: ks, Ts
aeroqradq
qs
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CFD-Thermal Coupling Approaches
Globally Iterative:
Locally iterative:
Fully Coupled:
0Wf TT = CFD soln Wq Thermal solnWfW TT =
0Wf TT = CFD soln
Wq
Thermal solnWfW TT =
• Solid is a fluid with u=v=w=0
• Cast thermal problem in conservation form, use same CFD algorithm, coupled energy equation at interface
u R=
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Integrated Fluid-Thermal-Structural Analysis using Unstructured Meshes
Fully coupled analysis using Taylor-Galerkin finite element formulation
Dechaumphai et. al. LaRC, Circa 1990
Structural analysis of built-up structures rarely use meshes of the “continuum” (3D elasticity equations).
Built-up structures are “modeled” with plates, shells, beams, and 3D elements.
• artificial viscosity for high speed flow
• flux-based formulation for heat transfer
Need thermal analogues
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“Next Generation” Thermal Analysis Methods for Hot Built-up Structures
Hierarchical through-the-thickness modeling
p-version finite elements with
discontinuous Galerkin time marching
Parameters: h, pip, ∆t, pt
Parameter: pz
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Why p-Version Finite Elements?
log ($, N, CPU)
log ||error||
FV, FD, h-FE methods (algebraic convergence)
spectral, p-FE methods without pollution
(exponential convergence)
hp-FE methods on good meshes
Higher accuracy for fixed number of unknownsFewer elements for fixed accuracyElement shapes that reflect actual geometry
pp
33
2210 xaxaxaxaaxT +++++= L)(
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Finite Element Options for Multi-layered Plates
Conventional elements (p=1)
z
x
y
A
B
C
y
C x
T
PiecewiseLinear in (x,y) Piecewise
linear in zA
B
T
z
z
x
y
A
B
C
p-Version elements
Cx
y
T
Polynomial in (x,y) Piecewise
polynomial in z
p1
A
B
T
z
p1
p3
p2
p2
p3
Improve accuracy by adding more elements
Improve accuracy by increasing polynomial degree
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Homogenized Through-the-Thickness Modeling of Conduction in Multi-layered Plates
x
z
y
A
B
C
p-element
Cx
y
T
Polynomial in (x,y)
A
B
T
z
Single polynomial
in z
mathematicallyequivalent
A
Bz
x
y
+ Fewer degrees of freedom thanmultiple layers of p-elements
+ Good for single-layer– Bad for multiple layers
• Lacks convergence withincreasing model order
• Jump in the flux across material interfaces
Hierarchical model• Geometrically collapsed• Thermal “higher-order plate theory”• Structurally compatible
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Optimal Through-the-Thickness Modeling of Conduction in Multi-layered Plates
z
y
xk1
k2
k3
Basis functions are single polynomials defined piecewise by scaling the homogenized basis functions
by the thermal conductivity of each layer
z
ϕ1
Homogenized basis functions
Optimal basis functions
ϕ1
z
ϕ2
z
x 1k1
=
x 1k3
+ c3 =
x 1k2
+ c2 =
A
B
T
zpz
Same number ofDOF’s as homogenizedhierarchical model
Converges with model order and plate thickness
ϕ2
z
Ref: Volgelius & Babuska
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Steady-State Conduction in a Two-Layer Plate
0.880.770.650.530.410.290.180.06
1k1=1
T=0 T=0
2x
k2=10
q=0
z ( ) ( )2
40L
xLxxq −−=
∑∞
=
+
+
=
1 coshsinh
coshsinhsin
nnn
nn
LznD
LznC
LznB
LznA
LxnT
ππ
πππ
Exact Solution
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||e||
pX=2
1
Convergence of Through-the-Thickness Hierarchical Models of Conduction in Two-Layer
Plate
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A Posteriori Error Estimation
• Error in finite element solution• Global error equation
• Can solve this global problem using the same FE approach, but this would be as computationally expensive as obtaining the solution
• Instead, solve a local problem on each element
uue ˆ−=
( )( )( ) ( )uQe
QuQu
ˆ∇⋅∇+=∇⋅∇−=∇⋅∇−=∇⋅∇−
κκ
κ
κ
( )u∇⋅∇+ κ ( )u∇⋅∇+ κ
( ) ( ) ΩΩΩ κκ dvudQvdve KKK ∫ ∇⋅∇+∫=∫ ∇⋅∇− ˆ
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Local Problem for Element Error
• Weak formulation of element error problem
• Approximate boundary flux q~qq +=
( ) ( ) ( ) dsvnudvudQvdve KKKK ∫ ⋅∇+∫ ∇⋅∇−∫=∫ ∇⋅∇ ∂r
κκκ ΩΩΩ ˆ
q is unknown
( ) dsqdQdudsq nK KK nK nnK K θΩθΩθθ κ ∫∫ −−∫ ∇⋅∇=∫ ∂∂ ˆ~=Kq~ polynomial of degree pz such thatFind
Lq Rq
zn p1n ,,, K==θ nodal basis functions
[ ] RqM =~
=q Average flux
=q~ Correction to equilibrate q
Ref: Ainsworth & Oden
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On Each Element, Solve the Local Problem for the Estimated Error
• Approximate element error
• Element error indicator
• Global error estimate
( )∫ ∇⋅∇= KK eee ˆˆ||ˆ|| κ
∑=ΩK
Kee 2||ˆ||||ˆ||
( ) ( ) dsvqdvudQvdve KKKK ∫+∫ ∇⋅∇−∫=∫ ∇⋅∇ ∂ ˆˆˆ ΩΩΩ κκ
ijεψ∑ ∑=+
=
+
=
1p
0i
1p
0jji
x z
(z)(x)φeFind such that
for all admissible v[ ] FeK =
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0.880.770.650.530.410.290.180.06
Performance of the Error Estimate on theTwo-Layer Example
px=2
1
||e||
estimate
actual
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Performance of Error Estimate Steady Conduction with Internal Heat Generation
||e||
1/h (number of elements)
pz=1
pz=2
pz=3
pz=4pz=5
actualestimated
px=2
1
q
insulated
T=0 T=0
Rough Exact Solution
QT=0
101 102
10-2
10-1
100
101
||e||
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Performance of Element Error IndicatorSteady Conduction with Internal Heat Generation
actual error
101 102
10-2
10-1
100
101
32 elementspx=pz=2
||e||K
Max. error
Max. gradient
T=0
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Estimating Contributions of Hierarchical Modeling and Finite Element Error
• Hierarchical modeling error:
• Finite element error:
• Total error:
• Solve local problem twice
HMHM uue −=
uue HMFE ˆ−=
FEHM eeuue +=−= ˆ
222 |||||||||||| FEHM eee +=
( ) ( )
( ) ( ) ijj
p
i
p
jiFE
ijj
p
i
p
jiHM
dzxe
czxe
x z
x z
ψϕ
ψϕ
∑ ∑
∑ ∑+
= =
=
+
=
=
=
1
0 0
0
1
0
ˆ
ˆ
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Performance of Estimated Error Contributions
T=0
101 102
10-3
10-2
10-1
100
101
1/h (number of elements)
||e||
pz=1
pz=2
pz=3
pz=4
pz=5
estimatedmodel error
estimatedFE error
actual total error
3
1
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Error Estimates are Sufficiently Accurate to Drive an Adaptive Strategy
T=0
101 102
10-3
10-2
10-1
100
101
1/h (number of elements)
||e||
pz=1
pz=2
pz=4
pz=5
estimatedmodel error
estimatedFE error
actual total error
3
1
1
2
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Concluding Remarks
• Examples shown here were constructed to have exact solutions to study behavior of the solution method and error estimates.
• Similar approach has been used with same success for transient 3D linear conduction using 2D elements and steady-state 3D nonlinear conduction.
• Can homogenization using p-version finite elements be used to accurately represent the thermal effects of all the internal structure on the surface?