Flag Flutter:Potential Flow Around a Rectangular Plate

Singularities in Mechanics: Description and Formation (Paris, 21-25 January 2008)

CollaboratorsLionel Schouveiler IRPHELaurent Duchemin IRPHE

Olivier Doaré ENSTAEmmanuel de Langre LadHyX

Romain Lagrange IRPHE

IRPHE, CNRS & Aix-Marseille Université, Marseille, France

ANR Contract “DRAPEAU”

Christophe Eloy

FILM EXP.

• Pressure: destabilising

• Elasticity: stabilising

Instability Mechanism

P0

P > P0

P < P0

The flutter wind speeds obtained in the N–S simulation are also plotted and compared inFigure 8(b) (m ¼ 0 "39). The flutter speeds calculated by these two methods are in goodagreement. The computation time for potential flow analysis is very short, about 10 s persheet using a personal computer (200MHz Intel Pentium MMX Processor). Theparametric study that follows was conducted using potential flow analysis.

Figure 9 shows the relationship between U#S and m for the parametric and experimental

analyses. Aerodynamic drag along the sheet surface was added to the potential flowanalysis. The results for CD ¼ 0 "0, 0 " 1 and 0 " 2 are shown. According to Fairthorne(1930), the drag coefficient of flag paper is approximately 0 " 1 at m ¼ 0 "1. Therefore, theCD ¼ 0 "1 results are considered to correspond to actual conditions. The range of primaryflutter with CD ¼ 0 "0 and 0 " 1 is shown in the lower part of Figure 9. Qualitatively, U#

S

tends to decrease with increasing m, which is in agreement with the experiment results.However, there is a considerable quantitative discrepancy between the results of thepotential flow analysis and the experiment. The effect of CD is not large enough to accountfor the magnitude of the discrepancy. Four mechanisms may be considered as contributingto this discrepancy: the effect of the aspect ratio of the sheet, span-wise deformation,stabilization due to deformation of the sheet surface by wind pressure, and the increase inthe rigidity during paper flutter compared to the static value obtained in the tensile testand subsequently used in the formula for dimensionless flutter speed. The last mechanismwas determined from observations in natural frequency tests using small sheets of paper ina vacuum vessel. However, the cause was not confirmed in the present study, and remainsto be investigated.

The calculated solutions from the four potential theory approaches (methods 2, 5, 6,and 7) are compared in Figure 9. The solutions by methods 6 and 7 agree well in the range0 "3 $ m $ 1, and the solutions by methods 2, 5, and 7 agree well for m > 1. These resultsconfirm the reliability of the method proposed in this study (method 7) over a wide rangeof mass ratio.

1

10

100

1000

0.01 0.1 1 10

US*

4th mode 3rd mode 2nd mode

!

Figure 9. Relationship between dimensionless flutter speed and mass ratio. Present experiment: *, flag-typepaper;*, long-type paper; 4, elastic sheet; other experiment:}, Huang (1995);&, Kornecki et al. (1976). Presenttheory: very thick black line, CD ¼ 0; medium black line, CD ¼ 0 "1; ordinary line, CD ¼ 0 "2; other theories: +,

Kornecki et al. (1976), }, Huang (1995); thick gray line, Guo & Pa.ııdoussis (2000).

Y. WATANABE, K. ISOGAI, S. SUZUKI AND M. SUGIHARA554

• [Ref] Watanabe, Isogai, Suzuki & Sugihara, J. Fluids Struct. (2002)

Literature

1/M∗

V∗

c

Equation of Motion

• Linearised Euler-Bernoulli beam equation

x

y

y(x,t)U

: flexural rigidity : mass per unit areaEI

ρ

ρ∂2

ty + EI∂4

xy = ∆P (x)

Galerkin Method

• Galerkin–Fourier expansion

• PDE eigenvalue problem

0 0.2 0.4 0.6 0.8 1!2

!1

0

1

2

0 0.2 0.4 0.6 0.8 1!2

!1

0

1

2

0 0.2 0.4 0.6 0.8 1!2

!1

0

1

2

x/L

y1

y2

y3

(−ρω2I + EIK − P (ω)

)A = 0

y(x, t) =∑

n

Anyn(x)eiωt

ρ∂2

ty + EI∂4

xy = ∆P (x)

Flow Around the Plate

Potential Flow

• Perturbation potential:

• Perturbation pressure (Bernoulli equation)

-1 1

Uvy(x)eiωt

x

y

φ(x, y)eiωt

∆φ = 0{

x ∈ [−1 1]for

P = (∂t + U∂x) φ eiωt

∂yφ|y=0

= vy = (∂t + U∂x) yn

vy(x) =

∫ 1+

−1−

δφ(ξ)dξ

2π(x − ξ)2

Inverse Problem

• Green’s representation theorem

• Inverse problem (Fredholm equation of 1st kind)

∆φ = 0{

∂yφ|y=0

= vy

-1 1

U

x

y

G(r) =1

2πln r

M

P

+

+

(in 2D)

FP

φ(M) = −

∫P∈S

δφ ∂ypG(|MP |)dSp

Vorticity Distribution

• Symmetry of perturbation potential

• Vorticity distribution

• Inverse problem for vorticity

vy(x) =

∫ 1+

−1−

δφ(ξ)dξ

2π(x − ξ)2FP

-1 1 x

yφ(−y) = −φ(y)

vx = ∂xφ

vx(−y) = −vx(y)

-1 1 x

y

γ(x) = −∂xδφ

Cvy(x) =

∫ 1

−1

γ(ξ)dξ

2π(x − ξ)

Oscillating Plate

• Inversion formula (Söhngen, 1939)

• Particular case:

• is the circulation around the plate:

C

γ(x) = −

2

π

√1 − x

1 + x

∫ 1

−1

√1 + ξ

1 − ξ

vy(ξ)

x − ξdξ +

α

√1 − x

2

vy = 1

C

γ(x) = −2x

√1 − x2

+Γo

π√

1 − x2

Γo =

∫ 1

−1

γ(ξ)dξΓo

∆P (x) = −γ(x) − iω

∫x

−1

γ(ξ)dξ

vy(x) =

∫ 1

−1

γ(ξ)dξ

2π(x − ξ)

x

y

• Flow with no circulation

• Flow with circulation

• Wake behind the plate advected at velocity

• Using Kutta hypothesis

Γ = Γoeiωt

γ|x=1+ = −

1

U

dΓ

dt= −

iω

UΓo e

iωt

γ(x > 1) = −

iω

UΓo eiωteiω(1−x)/U

U

∆P (x) = 2C(ω)

√1 − x

1 + x− 2 iω

√1 − x2

∆P (x) =2x

√1 − x2

− 2 iω√

1 − x2

(Kelvin’s circulation theorem)

C(ω) =H

(2)1 (ω)

H(2)1 (ω) + iH(2)

0 (ω)(Theodorsen function)

3D Flow?

• Inverse problem (Fredholm equation of 1st kind)

• Expansion in powers of or (lifting-line theory)

vz(x, y) =A

4π

∫∫S+Σ

γy(ξ, η) dξdη

(y − η)2

(1 +

A(x − ξ)

[A2(x − ξ)2 + (y − η)2]1/2

)

3D Problem

-1 1

U

x

1

-1

S Σ

L

H

A = L/H

FP

A A−1

y

!1!0.5

00.5

1

!1

!0.5

0

0.5

10

1

2

3

4

5

x

y

Singularities in the Pressure Field

• Leading-edge singularity

• Trailing-edge and side edges

P

P ∼ 1√n

P ∼√

n

Vorticity Distribution

• Vorticity lines

• Vortex lattice method

x

zy

x

Γj

xj

yj

vi =

∑

j

KijΓj

[Ref] Tang, Yamamoto & Dowell, J. Fluids Struct. (2003)

y

• Assuming

• The pressure along can be determined

〈p(y)〉 = p∞

(1 −

1

kxH+ O

(e−kxH

))

Fourier Space

!1 ! 0.5 0 0.5 1

! 0.2

0

0.2

0.4

0.6

0.8

1

0.1

1

10

100

y

p(y)

p∞

kxH

y

XY

0Y(X,T)

Z

L

H—2

H—2-

U H

kx

x

y

vz ∼ eikxx

Results of the Stability Analysis

• Parameters:

• Dimensionless numbers

Flutter Modes

2 4 6 8 10 12 14

0

2

4

6

8

10

12

14

16

18

20

2 4 6 8 10 12 14

!2

!1.5

!1

!0.5

0

0.5

1

U∗

ω∗

σ∗

U∗

U∗=

√ρ

EILU ω

∗=

Lω

UH

∗=

H

LM∗

=

ρairL

ρ

M∗

= 2.5H∗

= ∞

1

2

3

4

1

2

3

4

Shape of Flutter Modes

2 4 6 8 10 12 14

0

2

4

6

8

10

12

14

16

18

20

U∗

ω∗

1

2

3

4

mode 1 mode 2 mode 3

mode 4

Unstable modes (50x slower)

• mode 2

• mode 3

M∗

= 0.74

M∗

= 1.94

L = 8 cm

L = 21 cm

Mode 2

ω∗

= 1.82 ω∗

= 2.01

Mode 3

ω∗

= 2.5

ω∗

= 3.5

• Mass ratio:

10!1 100 1010

5

10

15

20

25

Stability curve

U∗

c

H∗

M∗

= 0.6

Stability curve

• Experiments for and

10!1 100 1010

5

10

15

20

25

mode 3

mode 2

H∗

= 1

H∗

= ∞

H∗

= 0.2

M∗

U∗

c

H∗

= 1 H∗

= 0.25

The flutter wind speeds obtained in the N–S simulation are also plotted and compared inFigure 8(b) (m ¼ 0 "39). The flutter speeds calculated by these two methods are in goodagreement. The computation time for potential flow analysis is very short, about 10 s persheet using a personal computer (200MHz Intel Pentium MMX Processor). Theparametric study that follows was conducted using potential flow analysis.

Figure 9 shows the relationship between U#S and m for the parametric and experimental

analyses. Aerodynamic drag along the sheet surface was added to the potential flowanalysis. The results for CD ¼ 0 "0, 0 " 1 and 0 " 2 are shown. According to Fairthorne(1930), the drag coefficient of flag paper is approximately 0 " 1 at m ¼ 0 "1. Therefore, theCD ¼ 0 "1 results are considered to correspond to actual conditions. The range of primaryflutter with CD ¼ 0 "0 and 0 " 1 is shown in the lower part of Figure 9. Qualitatively, U#

S

tends to decrease with increasing m, which is in agreement with the experiment results.However, there is a considerable quantitative discrepancy between the results of thepotential flow analysis and the experiment. The effect of CD is not large enough to accountfor the magnitude of the discrepancy. Four mechanisms may be considered as contributingto this discrepancy: the effect of the aspect ratio of the sheet, span-wise deformation,stabilization due to deformation of the sheet surface by wind pressure, and the increase inthe rigidity during paper flutter compared to the static value obtained in the tensile testand subsequently used in the formula for dimensionless flutter speed. The last mechanismwas determined from observations in natural frequency tests using small sheets of paper ina vacuum vessel. However, the cause was not confirmed in the present study, and remainsto be investigated.

The calculated solutions from the four potential theory approaches (methods 2, 5, 6,and 7) are compared in Figure 9. The solutions by methods 6 and 7 agree well in the range0 "3 $ m $ 1, and the solutions by methods 2, 5, and 7 agree well for m > 1. These resultsconfirm the reliability of the method proposed in this study (method 7) over a wide rangeof mass ratio.

1

10

100

1000

0.01 0.1 1 10

US*

4th mode 3rd mode 2nd mode

!

Figure 9. Relationship between dimensionless flutter speed and mass ratio. Present experiment: *, flag-typepaper;*, long-type paper; 4, elastic sheet; other experiment:}, Huang (1995);&, Kornecki et al. (1976). Presenttheory: very thick black line, CD ¼ 0; medium black line, CD ¼ 0 "1; ordinary line, CD ¼ 0 "2; other theories: +,

Kornecki et al. (1976), }, Huang (1995); thick gray line, Guo & Pa.ııdoussis (2000).

Y. WATANABE, K. ISOGAI, S. SUZUKI AND M. SUGIHARA554

Conclusion

• Flow inherently singular

• Good agreement for 1D mode + 3D flow

• Importance of 3D effects

1/M∗

U∗

c

√M∗