Aeroelasticity & Experimental Aerodynamics (AERO0032-1)

Lecture 3 Unsteady Aerodynamics – Theodorsen

T. Andrianne

2015-2016

From Lectures 1 & 2

•  Quasi-steady aerodynamics ignores the effect of the wake on the flow around the airfoil

•  Unsteady aerodynamics à Reduction of the magnitude of

the aerodynamic forces à Wagner function (time domain) à Concept of aerodynamics

states à Strong effect on the flutter speed

2D oscillations

3

Consider a 2D airfoil oscillating sinusoidally in an airflow.

à Changes in the circulation around the airfoil

Kelvin’s theorem: “conservation of circulation“ à Change in circulation over the entire flowfield is zero

€

∂Γ∂t

= 0

Γ ≡ −V ⋅ds

C∫

V

C

ds

Kelvin’s Theorem

à Any increase in the circulation around the airfoil must result in a decrease in the circulation of the wake.

“The circulation in the wake balances the changes in circulation over the airfoil.”

à For the oscillating airfoil problem, this means that: where Γ0 is the total circulation at time t=0. à The wake cannot be ignored in the calculation of

the forces acting on the airfoil.

4

€

Γairfoil t( ) + Γwake t( ) = Γ0

Pitching and Heaving

5

Wake shape of a sinusoidally pitching and heaving airfoil.

Positive vorticity is denoted by red and negative by blue

Experimental results

6

Numerical simulation results (top) VS flow visualization in a water tunnel (bottom) by Jones and Platzer.

How to model this?

Simulation results are useful but – Not always accurate (e.g. problems concerning

starting vortices) – Not practical. If the motion (or any of the parameters)

is changed, a new simulation must be performed. à Analytical mathematical models of the

problem exist (developed in the 1920’s and 1930’s)

Most popular models:

– Wagner – Theodorsen

7

Theodorsen (1935)

Only three major assumptions: •  The flow is always attached

à The motion’s amplitude is small •  The airfoil is a flat plate Initially Theodorsen worked on a flat plate with a control surface (3 d.o.f.s), so asymmetric airfoils can also be handled. •  The wake is flat If the motion is small (first assumption) then the flat wake assumption has little influence on the results. à The wake vorticity travels at the free stream airspeed 8

Basis of the model

•  The model is based on elementary solutions of the Laplace equation:

•  Such solutions are: –  The free stream: –  The source and the sink: –  The vortex: –  The doublet:

9

€

∇2φ = 0

€

φ =U cosαx +U sinαy

€

φ =σ2πln r =

σ2πln x − x0( )2 + y − y0( )2

€

φ =µ2πcosθr

=µ2π

xx 2 + y 2

€

φ = −Γ2π

θ = −Γ2πtan−1 y − y0

x − x0

'

( )

*

+ ,

Mapping

Airfoil = circle that can be mapped onto a flat plate through a conformal transformation:

10

Joukowski’s conformal transformation

•  Complex variable z as z = x+iy. in the working plane •  Complex variable za, in the physical plane :

11

R

x

y

xa

ya

-2R 2R

€

za = xa + iya = z +R2

z

Singularities Three types of singularities are used:

– A free stream of speed U and zero angle of attack – A pattern of sources of strength +2σ on the top

and surface of the flat plate, balanced by sources of strength -2σ on the bottom surface

– A pattern of vortices +ΔΓ on the flat plate balanced by identical but opposite -ΔΓ vortices in the wake

12

x

y 2σ

2σ 2σ

-2σ -2σ -2σ

-ΔΓ +ΔΓ U

Working plane à Physical plane

13

b

x

y

2σ 2σ

2σ x1,y1

x1,-y1 -2σ -2σ -2σ

X0,0 b2/X0,0 -ΔΓ +ΔΓ

b xa

ya

-b

Airfoil Wake

Airfoil and wake

•  The airfoil is a flat plate with a source distribution that changes in time

à The +2σ and -2σ source contributions do not

cancel each other out. •  The wake of the airfoil is a flat line with vorticity

that changes both in space and in time. à The +ΔΓ and -ΔΓ vorticity contributions do not cancel each other out.

14

Airfoil and wake

Airfoil and wake = slits (~cracks) à Different parts of the circle map to different parts of the airfoil

b x

y

-b

Outside circle

Inside circle

Circle upper surface

Circle lower surface

Boundary conditions (1)

Attached flow aerodynamic problems à 2 boundary conditions:

–  Impermeability: the flow cannot cross the solid boundary

– Kutta condition: the flow must separate at the trailing edge

+ Kelvin’s theorem

16

Boundary conditions (2)

•  Impermeability condition fulfilled by the source and sink distribution

•  Kutta condition fulfilled by the vortex distribution

•  Kelvins’ theorem is automatically fulfilled because for every vortex +ΔΓ there is a countervortex –ΔΓ

à Total change in vorticity is zero

17

Impermeability (1)

Impermeability à zero normal flow on a solid surface

For a moving airfoil, the velocity induced by the source distribution normal to the airfoil’s surface must be equal to the velocity due to its motion and the free stream:

where n is a unit vector normal to the surface w is the external upwash.

18 €

∂φ∂n

= −w

Impermeability (2)

Across the solid boundary of a closed object the source strength is given by

(assuming that the potential of the internal flow is constant) à  σ = w à The strength of the source distribution is defined by the airfoil’s motion.

19

€

σ = −∂φ∂n

Airfoil’s motion

Total upwash due to the pitch-plunge motion: where xf is the position of the flexural axis goes from -1 to +1.

20

€

w = − Uα + ˙ h + b x 1 +1( ) − x f( ) ˙ α ( )

x1 =xb

x is measured from the half-chord

α h

x

x=-b=-c/2

x=b=c/2

Potential induced by sources

Potential induced by a source located at (x,y) at point (x1,y1) Using sources of strength 2σ, the potential induced by a source at x1 , y1 and a sink at x1 , -y1 is given by

Using non-dimensional coordinates

21

€

dφ x1,y1( ) =σ2πln x − x1( )2 + y − y1( )2 =

σ4πln x − x1( )2 + y − y1( )2[ ]

€

dφ x1,±y1( ) =σ2πln

x − x1( )2 + y − y1( )2

x − x1( )2 + y + y1( )2&

' ( (

)

* + +

€

dφ x 1,±y 1( ) =σ2πln

x − x 1( )2 + y − y 1( )2

x − x 1( )2 + y + y 1( )2&

' ( (

)

* + +

€

where x =xb

, y = 1− x 2

Total source potential

Total potential induced by the sources and sinks

Substituting for σ from the upwash equation we get

22

€

φ x ,y ( ) =b2π

σ lnx − x 1( )2 + y − y 1( )2

x − x 1( )2 + y + y 1( )2&

' ( (

)

* + + −1

1

∫ dx 1

φ x, y( ) = b2π

Uα + h+ b x1 +1( )− x f( ) α( ) ln x − x1( )2 + y − y1( )2

x − x1( )2 + y + y1( )2"

#$$

%

&''−1

1

∫ dx1

σ = w = − Uα + !h+ b x1 +1( )− x f( ) !α( )

After integration … •  On the upper surface

•  On the lower surface

23

φupper x, y( ) = b Uα + !h− x f !α( ) 1− x 2 + b2 !α2

x + 2( ) 1− x 2

φ x, y( )lower = −φupper x, y( )

(1)

Total source potential

φ x, y( ) = b2π

Uα + h+ b x1 +1( )− x f( ) α( ) ln x − x1( )2 + y − y1( )2

x − x1( )2 + y + y1( )2"

#$$

%

&''−1

1

∫ dx1

Pressure on the surface

Aerodynamic forces on the airfoil: à Pressure on its surface. àUnsteady Bernoulli equation:

where p = static pressure ρ = air density q = local air velocity

The local velocity on the surface is tangential to the surface. As the flat plate lies on the x-axis:

24

€

p = −ρq2

2+∂φ∂t

&

' (

)

* + + Constant

€

q =U cosα + u =U cosα +∂φ∂x

≈U +∂φ∂x

Pressure difference

•  On the upper surface:

•  On the lower surface:

•  Pressure difference:

25

pu = −ρ12U +

∂φ∂x

"

#$

%

&'2

+∂φ∂t

"

#$$

%

&''+Constant

pl = −ρ12U −

∂φ∂x

"

#$

%

&'2

−∂φ∂t

"

#$$

%

&''+Constant

Δp = pu − pl = −2ρ U ∂φ∂x

+∂φ∂t

#

$%

&

'(= −2ρ

Ub∂φ∂x

+∂φ∂t

#

$%

&

'(

Non-circulatory lift

•  The non-circulatory lift is given by

•  Substituting for

with à φ(1)=φ(-1)=0

26

lnc = Δpdx0

c∫ = b Δpdx

−1

1∫

lnc = −2ρbUb∂φ∂x

+∂φ∂t

"

#$

%

&'dx

−1

1∫ = −2ρbφ

−1

1− 2ρb ∂φ

∂tdx

−1

1∫ = −2ρb ∂φ

∂tdx

−1

1∫

lnc = ρπb2 !!h − x f −

c2

"

#$

%

&' !!α +U !α

"

#$

%

&'

Δp = −2ρ Ub∂φ∂x

+∂φ∂t

#

$%

&

'(

φ x, y( ) = b Uα + !h− x f !α( ) 1− x 2 + b2 !α2

x + 2( ) 1− x 2

Non-circulatory moment

•  The non-circulatory moment around the flexural axis is given by:

•  Substituting for Δp we get

where the first integral was evaluated by parts. Carrying out the integrations we get:

27

€

mnc = Δp x − x f( )dx0

c

∫ = b Δp b x +1( ) − x f( )dx −1

1

∫

€

mnc = −2ρbU x ∂φ∂x dx

−1

1

∫ − 2ρb∂φ∂t

x b + b − x f( )dx −1

1

∫

= 2ρbU φdx −1

1

∫ − 2ρb∂φ∂t

x b + b − x f( )dx −1

1

∫

€

mnc = ρπb2 x f −c2

% &

' (

˙ ̇ h − x f −c2

% &

' (

˙ ̇ α % & *

' ( + −

ρπb4

8˙ ̇ α + ρπb2U ˙ h + ρπb2U 2α

Circulatory forces

Up to now we’ve only satisfied the impermeability condition. à Now we need to satisfy the Kutta condition using the vortex distribution.

28

x

y

X0,0 b2/X0,0

-ΔΓ +ΔΓ

Potential induced by vortices

•  The potential induced by the vortex pair at (X0,0) and (b2/X0,0) is

•  In non-dimensional coordinates:

•  Define •  And remember that on the circle:

à 29

€

φΔΓ =ΔΓ2π

tan−1 yx − X0

− tan−1 yx − b2 /X0

'

( )

*

+ ,

€

X 0 +1/ X 0 = 2x 0 or X 0 = x 0 + x 02 −1

€

y = 1− x 2

€

φΔΓ = −ΔΓ2πtan−1

1− x 2 x 02 −1

1− x x 0

€

φΔΓ =ΔΓ2π

tan−1 y x − X 0

− tan−1 y x −1/X 0

'

( )

*

+ ,

(2)

= Contribution to the pressure difference at one point x on the flat plate by one vortex located at x0

Pressure difference

•  Unsteady Bernoulli equation:

•  Theodorsen assumes that the vortices

propagate downstream at the free stream velocity:

à

€

Δp = pu − pl = −2ρ U∂φ∂x

+∂φ∂t

' (

) *

€

∂φ∂t

=∂φ∂x0

U

€

Δp = −2ρU∂φ∂x

+∂φ∂x0

'

( )

*

+ , = −2ρ

Ub

∂φ∂x

+∂φ∂x 0

'

( )

*

+ , €

where x0 = bx 0

.... = Δp x, x0( ) = −ρU ΔΓbπ

x0 + x1− x 2 x0

2 −1

$

%&&

'

())

Lift - Integrate over wing

•  Integrating over the wing i.e. lift force induced by of one vortex

•  Substituting:

… à

31

lc x0( ) = Δp x, x0( )0

c∫ dx = b Δp x, x0( )

−1

1∫ dx

lc x0( ) = − ρUΔΓπ x0

2 −1x0 + x1− x 2

$

%&

'

()

−1

1∫ dx = −ρUΔΓ x0

x02 −1

Δp x, x0( ) = −ρU ΔΓbπ

x0 + x1− x 2 x0

2 −1

$

%&&

'

())

Lift - Integrate over the wake

•  Integration over the wake: from the trailing edge up to infinity

i.e. lift force contribution from all the vortices of the wake

•  We can define

à The circulatory lift becomes

32

lc = −ρUx0x02 −1

ΔΓ1

∞

∫

€

ΔΓ = bVdx 0

lc = −ρUbx0x02 −1

V dx01

∞

∫

(3)

Circulatory Moment

•  The circulatory moment around the flexural axis becomes

•  Substituting for Δp

… à

33

€

mc x 0( ) = Δp x ,x 0( ) x − x f( )0

c

∫ dx = b Δp x ,x 0( ) b x +1( ) − x f( )−1

1

∫ dx

€

mc = −ρUbb2

x 0 +1x 0 −1

− ecx 0

x 02 −1

$

% & &

'

( ) ) Vdx 01

∞

∫

The nature of V

•  V is a non-dimensional measure of vortex strength at a point x0 in ‘flat plate space’

•  Vortices don’t change strength as they travel downstream

à V is a function of space •  If we use a reference system that travels with

the fluid à V is stationary in value

•  If we use a fixed system, à V is a function of both time and space

34 V = f Ut − x0( )

Kutta condition (1) Which value to give to V ?

•  It can be obtained from the Kutta condition •  One of the forms of the Kutta condition is: ‘The local velocity at the trailing edge must be finite’

The airfoil lies on the x-axis à Restriction to the horizontal velocity component

where φtot = total potential caused by both the sources and vortices.

35 €

∂φtot

∂x x =1

= finite

Kutta condition (2)

•  From equations (1), (2) and (3), the total potential is

•  The horizontal airspeed is

36

φtot x( ) = b Uα + !h− x f !α( ) 1− x 2 + b2 !α2

x + 2( ) 1− x 2

−b2π

tan−1 1− x 2 x02 −1

1− xx0V (x0, t)dx01

∞

∫

∂φtot∂x

= −b Uα + !h− x f !α( ) x1− x 2

−b2 !α22x 2 + 2x −1

1− x 2

+b2π

x02 −1

1− x 2 x − x0( )V (x0, t)dx01

∞

∫

Kutta condition (3)

•  The total potential can be written as

•  At the trailing edge à denominator becomes 0

à For the horizontal velocity to be finite, the numerator must also become zero at the trailing edge:

37

∂φtot∂x

=11− x 2

−b Uα + !h− x f !α( ) x − b2 !α2

2x 2 + 2x −1( )"

#$

+b2π

x02 −1

x − x0( )V (x0, t)dx01

∞

∫'

())

Uα + !h− x f !α( )+ 3b !α2 =12π

x02 −1

1− x0V (x0, t)dx01

∞

∫

€

x = 1

Kutta condition (4)

… Cleaning up we get

= Necessary vortex strength for the Kutta condition to be satisfied. = Most important result of Theodorsen’s approach.

38

−12π

x0 +1x0 −1

V (x0, t)dx01

∞

∫ =Uα + !h+ 3c4− x f

$

%&

'

() !α (4)

Circulatory lift

•  Remember that the circulatory lift is given by

•  Divide this lift by equation (4) to obtain:

where

39

lc = −ρUbx0x02 −1

V (x0, t)dx01

∞

∫

→ lc = πρUcC Uα + !h+ 3c4− x f

#

$%

&

'( !α

#

$%

&

'(

C =

x0x02 −1

V (x0, t)dx01

∞

∫

x0 +1x0 −1

V (x0, t)dx01

∞

∫

lc

Uα + !h+ 3c4− x f

"

#$

%

&' !α

=

−ρUb x0x02 −1

V (x0, t)dx01

∞

∫

−12π

x0 +1x0 −1

V (x0, t)dx01

∞

∫

Circulatory moment

•  Similarly:

•  Divide by equation (4) to obtain

40

mc = −ρUbb2

x0 +1x0 −1

− ec x0x02 −1

"

#$$

%

&''V (x0, t)dx01

∞

∫

mc = −πρUcb2− ecC

"

#$

%

&' Uα + !h+

3c4− x f

"

#$

%

&' !α

"

#$

%

&'

l = lnc + lc = ρπb2 !!h − x f −

c2

"

#$

%

&' !!α +U !α

"

#$

%

&'

+πρUcC Uα + !h+ 3c4− x f

"

#$

%

&' !α

"

#$

%

&'

Total lift

Total lift = circulatory + non-circulatory

Notice that the non-circulatory terms are the added mass terms.

41

(5)

Total moment

Total moment = circulatory + non-circulatory

Some terms drop out:

42

m =mnc +mc = ρπb2 x f −

c2

"

#$

%

&' h − x f −

c2

"

#$

%

&' α

"

#$

%

&'−

ρπb4

8α

+ρπb2U h+ ρπb2U 2α −πρUc b2− ecC

"

#$

%

&' Uα + h+

3c4− x f

"

#$

%

&' α

"

#$

%

&'

(6)

m = ρπb2 x f −c2

"

#$

%

&' !!h − x f −

c2

"

#$

%

&' !!α

"

#$

%

&'−

ρπb4

8!!α − ρπb2 3c

4− x f

"

#$

%

&'U !α

+πρUec2C Uα + !h+ 3c4− x f

"

#$

%

&' !α

"

#$

%

&'

Discussion

•  Theodorsen’s approach has led to equations (5) and (6) for the full lift and moment acting on the airfoil.

•  The main assumptions of the approach are: – Airfoil = flat plate – Attached flow everywhere – The wake is flat – The wake vorticity travels at the free stream

airspeed •  In all other aspects it’s an exact solution •  However, it’s not complete yet. à What is the value of C ? 43

Prescribed motion

à We need to know •  The only way to know V is the prescribe it. •  However, prescribing V directly is not

useful. •  It’s better to prescribe the airfoil’s motion

and then determine what the resulting value of V will be.

44

C =

x0x02 −1

V (x0, t)dx01

∞

∫

x0 +1x0 −1

V (x0, t)dx01

∞

∫

V (x0, t)

Sinusoidal motion (1)

Sinusoidal motion = most logical choice for prescribed motion

45

Slowly pitching and plunging airfoil

Vorticity variation with x/c in the wake: à Sinusoidal near the airfoil

•  For small amplitude and frequency oscillations à V(x0,t) is sinusoidal near the airfoil

•  V(x0,t) is periodic in both time and space:

–  V(x0,t)=V(x0,t+2π/ω) –  V(x0,t)=V(x0+U 2π/ω,t)

à Phase angle of V(x0,t) is given by ωt+ωx0/U

46

Sinusoidal motion (2)

Vortex strength

47

x0

V(x0,t)

α(t)

h(t)

For sinusoidal motion:

To inject in

α =α0ejωt

h = h0ejωt

V =V0ej ωt+ω

Ux0

!

"#

$

%&=V0e

j ωt+bωUx0

!

"#

$

%&

C =

x0x02 −1

V (x0, t)dx01

∞

∫

x0 +1x0 −1

V (x0, t)dx01

∞

∫

Theodorsen Function

48

For sinusoidal motion Theodorsen’s function can be evaluated in terms of Bessel functions of the first and second kind:

A much more practical, approximate, estimation is:

with k=ωb/U = REDUCED FREQUENCY

Theodorsen Function

49 k=ωb/U = REDUCED FREQUENCY

Theodorsen Function

50

Theodorsen’s function = frequency domain equivalent of the Wagner function (time domain)

k=ωb/U

€

Φ t( ) = 1−Ψ1e−ε 1Ut / b −Ψ2e

−ε 2Ut / b

Usage of Theodorsen

51

•  Theodorsen’s lift force is now given by

•  Theodorsen’s function can be seen as an analog filter : attenuates the lift force by an amount that depends on the frequency of oscillation

•  Theoretically, Theodorsen’s function can only be applied in the case where the response of the system is exactly sinusoidal

lc = πρUcC k( ) Uα + !h+ 3c4− x f

"

#$

%

&' !α

"

#$

%

&'

QS vs US

52

lc = πρUcC k( )h0 jω exp jωt

lc = πρUch0 jω exp jωtQuasi-steady lift:

Theodorsen lift:

Circulatory lift of a purely plunging flat plate: h=h0exp jωt.

k2=10 k1

k3=100 k1

k1=ωb/U k2=10 k1 k3=100 k1

k1=ωb/U

Lift and moment

53

Full equations for the lift and moment around the flexural axis using Theodorsen :

Aeroelastic equations

•  The full aeroelastic equations are:

•  For sinusoidal motion they become:

•  Substituting for l(t) and m(t) yields …

54

m SS Iα

!

"##

$

%&&!!h!!α

'()

*)

+,)

-)+

Kh 00 Kα

!

"

##

$

%

&&

hα

'(*

+,-=

−l t( )m t( )

'()

*)

+,)

-)

€

−ω 2m + Kh −ω 2S−ω 2S −ω 2Iα + Kα

%

& '

(

) * h0α0

+ , -

. / 0 e jωt =

−l t( )m t( )+ , -

. / 0

Equations of motion ?

55

As the system is assumed to respond sinusoidally à No sense in writing out complete eqs of motion Combining the lift and moment with the structural forces gives

Validity of this equation

•  This algebraic system of equations is only valid when the airfoil is performing sinusoidal oscillations.

•  For an aeroelastic system such oscillations are only possible when: –  The airspeed is zero and there is no structural damping

à Free sinusoidal oscillations –  There is an external sinusoidal excitation force

à Forced sinusoidal oscillations –  The airfoil is flying at the critical flutter condition

à Self-excited sinusoidal oscillations à Very useful for calculating the critical flutter condition

56

Flutter Determinant

57

•  Non-trivial system of equations: à 2x2 determinant must zero, i.e. D = 0, where

•  D is called the flutter determinant and must be solved for the flutter frequency ωF and airspeed, UF

•  D is complex à Re(D)=0 AND Im(D)=0. à 2 equations with 2 unknowns

Solution

58

•  The flutter determinant is nonlinear in ω and U. •  It can be solved using a Newton-Raphson scheme •  Given an initial value ωi, Ui, a better value can be

obtained from

where F=[Re(D) Im(D)]T. •  The initial value of ωi is usually one of the wind-off

natural frequencies

Theodorsen assumes a sinusoidal motion à Equations only valid at the flutter point à Variation of the eigenvalues with airspeed not available Flutter speeds from Theodorsen are less conservative than the quasi-steady results.

Effect of Flexural Axis

59

Theodorsen assumes: •  Attached flow à small motion

•  Airfoil = flat plate •  Flat wake + wake vorticity travels at the free stream airspeed à Unsteady circulatory lift :

Summary

60

lc = πρUcC k( ) Uα + !h+ 3c4− x f

"

#$

%

&' !α

"

#$

%

&'

C(k) = function depending on the reduced frequency k = ωb/U

QS, k à 0 and C(k) à 1 (real)