Post on 21-Jul-2020
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
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
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]
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Motivation
≻ Benchmark case of Isogai, NACA 64A010 aerofoil at zero angle of attack
Time response to initial disturbance
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
(b) unstable (Ma=0.700, VF=1.55)
[Isogai, AIAA Journal 18 (1981) no. 9, pp. 1240–1242]
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Motivation
≻ Benchmark case of Isogai, NACA 64A010 aerofoil at zero angle of attack
Time response to initial disturbance
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
(b) unstable (Ma=0.875, VF=2.84)
./eccomas2008 timme 2088 movies.ppt[Isogai, AIAA Journal 18 (1981) no. 9, pp. 1240–1242]
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Motivation
≻ Benchmark case of Isogai, NACA 64A010 aerofoil at zero angle of attack
Time response to initial disturbance
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
(b) unstable (Ma=0.875, VF=2.84)
[Isogai, AIAA Journal 18 (1981) no. 9, pp. 1240–1242]
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
BIFOR Solver
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
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
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
BIFOR Solver
≻ Coupled full–order fluid/structure 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]
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
BIFOR Solver
≻ Coupled full–order fluid/structure solver
≻ Three main parts
≻ 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
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
BIFOR Solver
≻ Coupled full–order fluid/structure solver
≻ Three main parts
≻ Steady state solver
≻ Consider coupled fluid–structure system ⇒dw
dt= R
`
w(µ), µ´
≻ Equilibrium w0 given by ⇒ R`
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
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
BIFOR Solver
≻ Coupled full–order fluid/structure solver
≻ Three main parts
≻ Steady state solver
≻ Consider coupled fluid–structure system ⇒dw
dt= R
`
w(µ), µ´
≻ Equilibrium w0 given by ⇒ R`
w0(µ), µ´
= 0
≻ Stability determined by eigenvalues λj = γj ± iωj of Jacobian ⇒ A(w0, µ) =∂R
∂w
≻ Extended eigenvalue problem ⇒ REV (λ, p) =
"
(A − λI) p
qTs p − i
#
= 0
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
BIFOR Solver
≻ Coupled full–order fluid/structure solver
≻ Three main parts
≻ Steady state solver
≻ Consider coupled fluid–structure system ⇒dw
dt= R
`
w(µ), µ´
≻ Equilibrium w0 given by ⇒ R`
w0(µ), µ´
= 0
≻ Stability determined by eigenvalues λj = γj ± iωj of Jacobian ⇒ A(w0, µ) =∂R
∂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]
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
BIFOR Solver
≻ Benchmark case of Isogai, NACA 64A010 aerofoil at zero angle of attack
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
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
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 –
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Oscillatory Instability Boundary
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
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]
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Oscillatory Instability Boundary
≻ 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
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Oscillatory Instability Boundary
≻ 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
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
(b) Instability boundary
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Oscillatory Instability Boundary
≻ NACA 00xx at zero angle of attack
3 block, C–type grid
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
(b) Instability boundary
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Oscillatory Instability Boundary
≻ 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
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Oscillatory Instability Boundary
≻ 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)
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Oscillatory Instability Boundary
≻ Resolution of shock wave
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
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Oscillatory Instability Boundary
≻ Resolution of shock wave
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
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Oscillatory Instability Boundary
≻ Resolution of shock wave
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
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
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Oscillatory Instability Boundary
≻ Resolution of shock wave
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
(a) Pressure coefficient cp
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
(b) Pressure change ∂p/∂x
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Oscillatory Instability Boundary
≻ Resolution of shock wave
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
(a) Pressure coefficient cp
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
(b) Pressure change ∂p/∂x
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Oscillatory Instability Boundary
≻ 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)
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Oscillatory Instability Boundary
≻ 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
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Conclusion
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
Conclusion
≻ Bifurcation method using full–order nonlinear aerodynamics
≻ Efficient simulation of aeroelastic instabilities
∗ ∗ ∗
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
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
ECCOMAS 2008 — Timme and Badcock ’Computational Aeroelasticity based on Bifurcation Theory’
ECERTA PROJECT
Computational Aeroelasticitybased onBifurcation Theory
Sebastian Timmesebastian.timme@liverpool.ac.uk
Kenneth J. Badcockk.j.badcock@liverpool.ac.uk