L6_Elements of Unsteady Aerodynamics
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Transcript of L6_Elements of Unsteady Aerodynamics
6th Elements of UnsteadyAerodynamics
航空科学与工程学院
Xie Changchuan2014 Autumn
2
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
4
Flows around aircraft
Trailing vortex of transporter
F-18 breaks the sound barrier
Detached vortex of delta wing
Basic aerodynamics
5
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
01
( 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
Quasi steady aerodynamics
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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
Quasi steady aerodynamics
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 = + = +
∂ ∂ ∂ ∂
Unsteady aerodynamics
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
=
Unsteady aerodynamics
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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
Unsteady aerodynamics
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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= =; ,,…,
Unsteady aerodynamics
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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
Unsteady aerodynamics
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Summary