eccomas 2008: computational aeroelasticity based on … · 2010-09-30 · yang et al (2004) -...

ECERTA PROJECT

Computational Aeroelasticity

based on

Bifurcation Theory

Sebastian Timme

Kenneth J. Badcock

Overview

≻ Motivation

≻ Bifurcation approach and BIFOR Solver

≻ Oscillatory Instability Boundary

≻ Conclusion

ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’

Motivation


Motivation

≻ Benchmark case of Isogai, NACA 64A010 aerofoil at zero angle of attack

Time response to initial disturbance

Ma = 0.700VF = 1.03

VF = 2.19

(a) stable (Ma=0.700, VF=1.03)

Ma = 0.700VF = 1.55

VF = 2.19

(b) unstable (Ma=0.700, VF=1.55)

./eccomas2008 timme 2088 movies.ppt[Isogai, AIAA Journal 18 (1981) no. 9, pp. 1240–1242]


./eccomas2008_timme_2088_movies.ppt

Motivation



time

α,h

0 200 400 600-0.015

-0.01

-0.005

0

0.005

0.01

0.015α [rad]h

Ma = 0.70VF = 1.03

(a) stable (Ma=0.700, VF=1.03)

time

α,h

0 200 400 600-0.4

-0.2

0

0.2

0.4α [rad]h

Ma = 0.70VF = 1.55


[Isogai, AIAA Journal 18 (1981) no. 9, pp. 1240–1242]


Motivation



time

α,h

0 200 400 600-0.015

-0.01

-0.005

0

0.005

0.01

0.015α [rad]h

Ma = 0.70VF = 1.03

(a) stable (Ma=0.700, VF=1.03)

Ma = 0.875VF = 2.84


./eccomas2008 timme 2088 movies.ppt[Isogai, AIAA Journal 18 (1981) no. 9, pp. 1240–1242]


./eccomas2008_timme_2088_movies.ppt

Motivation



time

α,h

0 200 400 600-0.015

-0.01

-0.005

0

0.005

0.01

0.015α [rad]h

Ma = 0.70VF = 1.03

(a) stable (Ma=0.700, VF=1.03)

time

α,h

0 200 400 600

-0.4

-0.2

0

0.2

0.4α [rad]h

Ma = 0.875VF = 2.84


[Isogai, AIAA Journal 18 (1981) no. 9, pp. 1240–1242]


BIFOR Solver


BIFOR Solver

≻ Coupled full–order fluid/structure solver

≻ Three main parts:

• Steady state solver (Euler, Euler/BL and Full Potential)

• Eigenvalue solver (shifted inverse power method algorithm, Newton eigenvalue solver)

• Unsteady time–accurate solver


BIFOR Solver


≻ Three main parts

≻ Steady state solver (Euler):

• Implicit time marching

• Osher’s approximate Riemann solver

• MUSCL variable extrapolation and van Albada’s limiter

• Preconditioned Krylov subspace method

• Applied within pseudo–time iterations in unsteady calculations

[Badcock et al, Progress in Aerospace Sciences 36 (2000), pp. 351–392]


BIFOR Solver



≻ Steady state solver

≻ Consider coupled fluid–structure system ⇒dw

dt= R

`

w(µ), µ´

≻ Equilibrium w0 given by ⇒ R`

w0(µ), µ´

= 0

• Shock nonlinearity (location and strength) defined in steady flow field


BIFOR Solver





dt= R

`

w(µ), µ´


w0(µ), µ´

= 0

≻ Stability determined by eigenvalues λj = γj ± iωj of Jacobian ⇒ A(w0, µ) =∂R

∂w

• Small number of eigenvalues associated with loss of stability

• Dynamics of system dominated by evolution of these critical modes


BIFOR Solver





dt= R

`

w(µ), µ´


w0(µ), µ´

= 0


∂w

≻ Extended eigenvalue problem ⇒ REV (λ, p) =

"

(A − λI) p

qTs p − i

#

= 0


BIFOR Solver





dt= R

`

w(µ), µ´


w0(µ), µ´

= 0


∂w

≻ Extended eigenvalue problem ⇒ REV (λ, p) =

"

(A − λI) p

qTs p − i

#

= 0

≻ BIFOR solver used for two tasks

• Tracking of aeroelastic modes ⇒ λj = λj(µ) = γj ± iωj

• Detecting of instability point ⇒ λj = ±iωj for µ = µ∗

[Woodgate et al, AIAA Journal 45 (2007) no. 6, pp. 1370–1381]


BIFOR Solver


Highly–resolved transonic instability boundary

xx

xx

x

x

x

x

Ma

VF

0.6 0.7 0.8 0.90.0

1.0

2.0

3.0 BIFOR (2007) - EulerYang et al (2004) - EulerYang et al (2004) - Euler/BLHall et al (2000) - EulerPrananta et al (1998) - TLNSIsogai (1981) - TSD

x

(a) Instability boundary

Real(λ)

Imag

(λ)

-0.1 -0.05 0 0.05 0.10

0.5

1

1.5

2

2.51st mode2nd mode

VF*=0.52

VF*=2.39

VF*=2.39

Ma=0.85

(b) Root loci


BIFOR Solver

Computational costs

Grid dimension Steady state solver Eigenvalue solver Unsteady solver

(CFL=100, conv.=1.e-10) (γj=1.e-07) (∆t=0.05, Nstep=6.e+06)

129 x 33 (4.2k) 6 s 70 s 7 h

257 x 65 (16.7k) 60 s 600 s 30 h

513 x 65 (33.3k) 250 s 1700 s –


Oscillatory Instability Boundary



≻ Oscillatory behaviour observed for transonic flow condition (Ma>Ma∗)

NACA 0012

Ma

VF

0.7 0.75 0.8 0.85 0.9

0.3

0.35

0.4

0.45

(c) Instability boundary

Structural parameter

• aerofoil–to–fluid–mass ratio µs=100

• ratio of natural frequencies ωr=0.343

• radius of gyration rα=0.539

• center of gravity xcg=0.5

• static unbalance xα=−0.2

[Badcock et al, AIAA Journal 42 (2004) no. 5, pp. 883–892]



≻ NACA 0012 at zero angle of attack

3 block, C–type grid

x/c

y/c

-0.5 0 0.5 1 1.5-1

-0.5

0

0.5

13C 129x33

(a) Grid topology

Ma

VF

0.74 0.76 0.78 0.80 0.82

0.3

0.32

0.34

0.36

0.38

0.4 3C 65x333C 129x333C 257x33

(b) Instability boundary



≻ NACA 0012 at zero angle of attack


x/c

y/c

-0.5 0 0.5 1 1.5-1

-0.5

0

0.5

13C 129x33

(a) Grid topology

x

x

x x

x

x

x

x

x

Ma

VF

0.77 0.775 0.78 0.785

0.315

0.32

0.325

∆Ma=2e-3, 3C 129x33∆Ma=2e-4, 3C 129x33

x




≻ NACA 00xx at zero angle of attack


Ma

VF

0.72 0.74 0.76 0.78 0.80 0.82

0.32

0.34

0.36

0.38

0.4

0.42 NACA 0010NACA 0012NACA 0014

(a) Grid topology

x/c

-cp

0.2 0.3 0.4 0.5 0.6

0.2

0.4

0.6

0.8

1

1.2 NACA 0010NACA 0012NACA 0014

Ma=0.78




≻ Time–accurate results crossplotted with bifurcation results

NACA 0012

• Time–response due to initial disturbance

• Plunge rate disturbed by h0 = 0.001U∞

++

+

+

+

++

+

+

+

+−−

−

−

−

−−

−

−

−

−

Ma

VF

0.765 0.77 0.775 0.78 0.785

0.315

0.32

0.325

Bifurcation approachTime-accurate stableTime-accurate unstable

+−h0 = 0.0010U∞

•

(a) Crossplot



≻ Resolution of shock wave

NACA 0012Shock wave physically

• Event of very limited spatial extent

• Thickness of normal shock front of order

δx ≈ 5.E−08c

[Granger, Fluid Mechanics, Holt, Rinehart & Winston]

Shock wave numerically

• Resolved thickness dependent on grid spacing

∆xgrid ≈ 1.E−02c

• Two grids points best possiblex/c

-cp

0.3 0.4 0.5

0.2

0.4

0.6

0.8

1 3C 129x333C 257x333C 513x33

(a) Pressure coefficient (Ma=0.773)




Steady state results at two Mach numbers (3C 129x33)

x/c

-cp

0.2 0.3 0.4 0.5 0.6

0.4

0.6

0.8

1 Trough Ma=0.7706Peak Ma=0.7730

(a) Pressure coefficient cp




Steady state results at two Mach numbers (3C 129x33)

x/c

-cp

0.2 0.3 0.4 0.5 0.6

0.4

0.6

0.8

1 Trough Ma=0.7706Peak Ma=0.7730


x/c

p x

0.3 0.4 0.5

0

1

2

3

4

5

Trough Ma=0.7706Peak Ma=0.7730

(b) Pressure change ∂p/∂x




Steady state results at four Mach numbers (3C 129x33)

x/c

-cp

0.2 0.3 0.4 0.5 0.6

0.4

0.6

0.8

1 Trough Ma=0.7706Peak Ma=0.7730Trough Ma=0.7765Peak Ma=0.7790


x/c

p x

0.3 0.4 0.5

0

1

2

3

4

5

Trough Ma=0.7706Peak Ma=0.7730Trough Ma=0.7765Peak Ma=0.7790





Steady state results at three Mach numbers (3C 129x33)

x/c

-cp

0.3 0.4 0.5

0.4

0.6

0.8

Trough Ma=0.7706Slope Ma=0.7717Peak Ma=0.7730


x/c

p x

0.3 0.4 0.5

0

1

2

3

4

Trough Ma=0.7706Slope Ma=0.7717Peak Ma=0.7730




≻ Resolution of shock wave perceived in flow field

Crossplots of instability boundary and steady state results (3C 129x33)

Ma

VF

ρu

0.765 0.77 0.775 0.78 0.785

0.315

0.32

0.325

0.995

1

1.005

VF

ρux/c ≈ 0.75y/c ≈ 0.04

(a) Aerofoil location (x/c≈0.75, y/c≈0.04)

Ma

VF

ρu

0.765 0.77 0.775 0.78 0.785

0.315

0.32

0.325

0.994

0.995

VF

ρux/c ≈ 1.85y/c ≈ 0.20

(b) Wake location (x/c≈1.85, y/c≈0.20)



≻ Resolution of shock wave perceived in flow field

Crossplots of instability boundary and lift coefficient

Ma

VF c l

0.72 0.74 0.76 0.78 0.80 0.82

0.3

0.32

0.34

0.36

0.38

0.4

0.02

0.025

0.03

Instability boundaryLift coefficient

Incidence = 0.1°

(a) 3C 129x33

Ma

VF c l

0.72 0.74 0.76 0.78 0.80 0.82

0.3

0.32

0.34

0.36

0.38

0.4

0.02

0.025

0.03

Instability boundaryLift coefficient

Incidence = 0.1°

(b) 3C 513x33


Conclusion


Conclusion

≻ Bifurcation method using full–order nonlinear aerodynamics

≻ Efficient simulation of aeroelastic instabilities

∗ ∗ ∗


Conclusion

≻ Bifurcation method using full–order nonlinear aerodynamics

≻ Efficient simulation of aeroelastic instabilities

∗ ∗ ∗

≻ High resolution of instability boundary revealed oscillatory behaviour

≻ Numerical artefact due to shock wave resolution


ECERTA PROJECT

Computational Aeroelasticitybased onBifurcation Theory

Sebastian [email protected]

Kenneth J. [email protected]

eccomas 2008: computational aeroelasticity based on … · 2010-09-30 · yang et al (2004) -...

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