CFD-Thermal Interactions Short Course, TFAWS 2003nasa.gov 8/21/03 CFD-Thermal Interactions Short Course, TFAWS 2003 Integrated Fluid-Thermal Analysis from a Thermal-Structures Perspective

8/21/[email protected]

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


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


Thermal-Structural Airframe Concepts for Reusable Launch Vehicles

Cryotank withaeroshell and insulating

thermal protection system

Integrated hot structurewhere thermal protection

also carries load


Airframe Thermal Analysis State-of-the-Art

Through-the-thickness plug

models

Full 3D finite element models

Transient nonlinear problem – Conduction– Radiation exchange– Convection


CFD-Thermal Interactions for Insulated Structures are (Approximately) Decoupled

Heat conducted into the structure is small

radaero

s

qq

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


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


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


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


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


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=


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


“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


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)(


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


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


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


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


||e||

pX=2

1

Convergence of Through-the-Thickness Hierarchical Models of Conduction in Two-Layer

Plate


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


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


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 =


0.880.770.650.530.410.290.180.06

Performance of the Error Estimate on theTwo-Layer Example

px=2

1

||e||

estimate

actual


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||


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


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


Performance of Estimated Error Contributions

T=0

101 102

10-3

10-2

10-1

100

101


||e||

pz=1

pz=2

pz=3

pz=4

pz=5

estimatedmodel error

estimatedFE error

actual total error

3

1


Error Estimates are Sufficiently Accurate to Drive an Adaptive Strategy

T=0

101 102

10-3

10-2

10-1

100

101


||e||

pz=1

pz=2

pz=4

pz=5

estimatedmodel error

estimatedFE error

actual total error

3

1

1

2


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?

Documents

CFD-Thermal Interactions Short Course, TFAWS 2003nasa.gov 8/21/03 CFD-Thermal Interactions Short Course, TFAWS 2003 Integrated Fluid-Thermal Analysis from a Thermal-Structures Perspective