L6_Elements of Unsteady Aerodynamics

6th Elements of UnsteadyAerodynamics

航空科学与工程学院

Xie Changchuan2014 Autumn

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Content1、Basic aerodynamics2、Quasi-steady aerodynamics3、Theodorsen’s unsteady aerodynmics4、Brief introduction of doublet-lattice method

Main AimsUnderstanding the basic concepts, methods and engineering calculationof unsteady aerodynamics concerning the dynamic aseroelasticity of aircraft

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Airflows in natureFlow aroundmountains

Flow on the Earth

Vortexflow oftornado

Unsteady vortexes are major movementpattern

But,where is Steadylaminar flow??

Basic aerodynamics

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Flows around aircraft

Trailing vortex of transporter

F-18 breaks the sound barrier

Detached vortex of delta wing

Basic aerodynamics

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Real flow and the solution of equations

Laminate flow arounda cylinder

Flow around a ball

Laminate flow around a lateral oscillated cylinder

Solutions of N-S and Euler equation

理想流体

小扰动方程的解

Basic aerodynamics

6Laminated flow

Subsonic flow pattern of a wall

Interfering flow offront and hind wing

Basic aerodynamics

Flow patterns around airfoil

Laminatedboundary layer

transitionregion

turbulentboundary layer

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Basic aerodynamics

Concepts: 1、continuity0

limV

mV

ρ→

=

Note: N-S equations has its application range

2、Newton fluid (viscosity assumption)dudy

τ μ=Shear stress:

x

y

uμ Dynamic friction coefficient

3、Ideal fluid Inviscid, no heat conduction

Navier-Stokes equations

Euler equation

4、Irrotational flowFull Potential Equation

Ignoring the rotation effect of fluid particles

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5、compressible and incompressible flow

ρ

6、Lagrangian and Eulerian description

is a function of space and time；

ρ is constant

a∞ → ∞ 0M∞ →

Material (total) derivative：

⎛ ⎞= ⎜ ⎟⎝ ⎠X

D DDt Dt

( , )=x x X t x Eulerian spacecoordination X Coordinaiton

of material

Changing rate of Function along X

Space (local)derivative：

⎛ ⎞= ⎜ ⎟⎝ ⎠x

d ddt dt x

( ) ∂= + = +

∂i i

i

D d dgrad vDt dt dt xF F F Fv F

Local rate Convective derivative

Basic aerodynamicsConcepts:

Changing rate of Function along

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7、Steady and unsteady

Steadyflow

0 ( , , )idv i x y zdt

= =the space derivatives of flow field are aero, i.e. the velocity of any point in field keep constant.

But, it does not means the accelerations of fluid particles are zero.

∂ ∂= + +

∂ ∂x x x x

x yDv dv v vv vDt dt x y

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2

1 3

1 2

=

=

x xy y1 2

1 3

≠

≠x x

x x

v vv v

0

0

∂≠

∂∂

≠∂

x

x

vxvy

0∴ ≠xDvDt

Unsteady flow Steady boundary conditions

Concerned by aeroelasticity

Basic aerodynamics

Concepts:

Unsteady boundary conditions

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9、equations of small velocity potential

Subsonic

8、Assumption of small turbulence

, ,∞∂ ∂ ∂

= = + = = = =∂ ∂ ∂x x y y z zv v v v v v vx y zΦ Φ Φ

Φ Full potential of velocity

Define small disturbingvelocity potential ϕ

, ,∂ ∂ ∂= = =

∂ ∂ ∂x y zv v vx y zϕ ϕ ϕ

∞= +v xΦ ϕ

Supersonic

M not close to 12 2 2

22 2 2(1 ) 0∞

∂ ∂ ∂− + + =

∂ ∂ ∂M

x y zϕ ϕ ϕ

2 2 22

2 2 2( 1) 0∞

∂ ∂ ∂− − − =

∂ ∂ ∂M

x y zϕ ϕ ϕ

Basic aerodynamics

Concepts:

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All equations are Laplace

10、Equations of ideal incompressible flow2 2 2

2 2 2 0∂ ∂ ∂+ + =

∂ ∂ ∂x y zΦ Φ Φ

2 2 2

2 2 2 0∂ ∂ ∂+ + =

∂ ∂ ∂x y zϕ ϕ ϕ

Full velocitypotentialSmall disturbingvelocity potential

2 0∇ =Φ2 0∇ =ϕ

Characters of equations： 1、Linear, superposition principle

2、fundamental (singular) solutions,2D：point source(sink), point vortex, doublet, ……3D：vortex line, vortex panel, doublet panel,……

Solving method of equations:

1、Analytical methods: power series, spectral method……2、Singular solutions method: linear elements,

panel elements……(engineering approach)3、Conformal transformation method: 2D problems4、Numerical methods: numerical differential……(CFD)

Basic aerodynamicsConcepts:

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Quasi steady aerodynamicsGrossman theory

h(t)

α(t)

VAirfoil modeling = attach vortex + free vortex

Semi camber line On trailing line

Airfoil modeling by unsteady vortex

Assumption of quasi steady: ignoring the free vortex effects on attach vortexAccording to the practical experiences, the effects of airfoil thickness and

camber on unsteady aerodynamics could be ignoring too.

Flat airfoil model

a⋅b

αV

b b

E

z

x 0Chord：2b Aero center：1/4 chordElastic center：E, ab after semi chordPitching：α, + nose upPlunging：h, + downwardInflow speed：V

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2 2

0 0( ) ( )

b bL V x dx V x dxρ γ ρ γ= =∫ ∫Aerodynamics

Induced velocity/Down wash

2

0

( )( )2 ( )

=−∫

bw x d

xγ ξ ξ

π ξ

Boundarycondition

( , ) ( , ) 1 ( , )w x t z x t z x tV x V t

∂ ∂= +

∂ ∂

Kutta condition (2 ) 0bγ =

( , ) ( , ) [ (1 ) ] ( , )z x t h x t x a b a x t= + − +Considering the plungingand pitching of plane airfoil

)cos1( θ−= bxLet

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( cos )nn

w V A A nθ∞

=

= − + ∑Write the down washas series form

Already satisfied the Kutta condition

Quasi steady aerodynamics

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Substituting the down wash into boundary condition, it gets

00

1 1( )z zA dx V t

π

θπ− ∂ ∂

= +∂ ∂∫

0

2 1( )cosnz zA n dx V t

π

θ θπ

∂ ∂= +

∂ ∂∫Substitute the motion of plane airfoil into them

)(10 αabh

VaA −−−= 1

bAVα

= − 2 0A =

Lifting 2 12 [ ( ) ]2

= − + + −hL V b a bV V

απρ α

Moment about leading point2

2 21 2

0

1( ) ( )2 2⋅ = = ⋅ + −∫

b

L EbM V x x dx L V b A Aρ γ πρ 31

2 2L EbM L Vbπρ α⋅ = −

Moment about elastic center

2 2 31 1 1 14 ( )[ ( ) ]2 4 2 2+

= − + + − −Ea hM V b a b Vb

V Vαπρ α πρ α


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Lifting coefficient ( ])2

2 1[ + + −= −Lh a bV V

C π αα

Moment coefficientabout leading point

144⋅

= −L Em LC C b

Vπ α

1 1( )4 2 4Em L

a bC CVπ α+

= − −

HomeworkConsidering a plane airfoil in harmonic pitching and plungingmoving respectively, please draw the displacements, lifting coefficient, and moment coefficient about elastic center. Then discuss their phase different and where they come from.Amplitude of pitching angle: 5 degree;Amplitude of plunging: 0.1b; Vibration frequency: k=ωb/V=0.2


Moment coefficientabout elastic center

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Theodorsen theory

h(t)

α(t)

V

2D, incompressible, potential, unsteady

a⋅b

αV

b b

E

z

x 0

Unsteady aerodynamics

Airfoil modeling by unsteady vortex

Airfoil modeling = attach vortex + free vortex

Semi camber line On trailing line

Flat airfoil model

Chord：2b Aero center：1/4 chordElastic center：E, ab after semi chordPitching：α, + nose upPlunging：h, + downwardInflow speed：V

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2 2

2 2 0x zϕ ϕ∂ ∂

+ =∂ ∂

Small disturbing velocity potential(Laplace)

2 0∇ =ϕ

Boundary condition ( , ) ( , ) 1 ( , )w x t z x t z x tV x V t

∂ ∂= +

∂ ∂Far field 0 zϕ → → ∞

Free vortex 0p x bΔ = >

Relationship of circulation, vortex strength and disturbing potential

( , ) ( , ) ( ) ( )x x U Lb b

x t t d d xϕ ϕΓ γ ξ ξ ξ ϕξ ξ− −

∂ ∂= = − = −Δ

∂ ∂∫ ∫

Pressure coefficient is given by unsteady Bernoulli equation

2 2

2 2( ) ( )pc V VV t x V t x

ϕ ϕ Γ Γ∂Δ ∂Δ ∂ ∂Δ = + = +

∂ ∂ ∂ ∂


On airfoil

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Free vortex conditionIs changed to

0Vt x

Γ Γ∂ ∂+ =

∂ ∂( )xt

VΓ Γ= − ( 0, )z x b= >

Induced velocityon airfoil

0

1( , )2

( , )b

z

w x t dz x

tγϕξ

ξ ξπ

∞

−=

∂= = −

∂ −∫For a moving airfoil, induced velocity is equal to down wash velocity,

then the boundary condition on airfoil is satisfied.

Solve the integrate equation to determine the vortex strength or circulation, then the aerodynamics on airfoil can be calculated.

2 1( ) 2 [ () ) ](2

L b V h ab Vb V h a bC kπρ α α πρ α α= − + − − + + −2 2

2

1 1[ ( ) ]2 8

1 12 ( ) [ ( ) ]2 2

( )

EM b ab V h ab Vb b

Vb a V h aC bk

πρ α α α α

πρ α α

= + − − −

+ + + + −

Theodorsen function(2)1

(2) (2)1 0

( ) HH iH

C k =+ H is Hankel function

k is induced frequency

bkVω

=


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.5

0.6

0.7

0.8

0.9

1.0

0.1 0.2 0.3 0.4 0.60.5 0.7 0.8 0.9 1.00

-0.04

-0.08

-0.12

-0.16

-0.20 F(k ) G(k )

When k →∞，F(k) → 0.5, G(k) → 0

( ) ( )( () 1)F k iG kC ik = + = −

Write the Theodorsen function into real part and imaginary part


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Subsonic DLM y

x

j F1

F2 F3

H

j 格

F1F3 1/4 chord lineF1 aero centerH Control point,

at which the boundaryconditions are satisfied

A kind of panel method

Border line at span position should be along the direction of inflow

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1

1 1 cos4 2 j

j

n

i p j j ij jlj

w V c x K dlV

ρ ϕπρ =

= Δ Δ∑ ∫

1

1 cos8 j

j

n

p j j ij jlj

c x K dlϕπ =

= Δ Δ∑ ∫ ( 1, 2, , 1 2 )i n j n= =；，，…，


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Pressure distribution

wDp 12

21 −= VρΔ

D is the matrix of aerodynamic influence coefficient

jj ijjj

ij dlKx

D ∫Δ

= ϕπ

cos8

(i=1,2,…,n;j=1,2,…,n)

qFFw )(bki+′=Down wash of

moving wing

q —— General coordinate vector；F —— Modal shape at all point H；F ′ —— The derivative of matrix F along x direction；b —— Reference length；k —— Reduced frequency

Vbk ω

= V ——Flight speed；ω ——Circular frequency

Δ =p Pq )(21 12 FFDP

bkiV +′= −ρ


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Summary

Documents

L6_Elements of Unsteady Aerodynamics