rad Cl

(1)

c V

Cl l

⋅

=

∞ 2

2 1 ρ

c V

Cd d

⋅

=

∞ 2

2 1 ρ

2 2 4 / 1 4

/ 1

2

1 V c

Cm m

⋅

=

ρ ∞

l [N/m] d [N/m] m [Nm/m]

Coefficienti Aerodinamici

(2)

Effetti dello spessore

(3)

-0.1 -0.05 0 0.05 0.1 0.15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

f/c

) (

0 0

0

ol l

Cl Cl

Cl

α α

α

α α

α

−

⋅

=

⇓

⋅

−

=

⋅ +

=

Cl₀

α_0l

Effetti della curvatura

(4)

Profilo Naca 0012

s m

V

m c

rad Cl

/ 30

2 deg 3

2

¹

=

⋅

=

∞

−

α

π

Calcolare l in kgf/m

-0.1 -0.05 0 0.05 0.1

0 0.2 0.4 0.6 0.8 1

l =364 N/m = 37 kgf/m

Esercizio 1

(5)

-0.15 -0.1 -0.05 0 0.05 0.1

-2 -1 0 1 2

Cl Cm A.C.

0012 2412 4412

-0.05 0 0.05 0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-2 -1 0 1 2Cl

Cm 0.5c

0012 2412 4412

-0.05 0 0.05 0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

f/c

f/c Effetti della curvatura

(6)

(7)

• Profilo Naca 2412

05 .

0 deg 4

0064 .

0 71 .

0

4 /

1

= −

=

Cm Cd Cl α

Calcolare il valore del Cm scegliendo come polo dei momenti il bordo di attacco del profilo

-0.05 0 0.05 0.1

0 0.2 0.4 0.6 0.8 1

l

d l.e

Cm_l.e.=-0.23

Esercizio 2

(8)

2

2 1

2

1 ⎟⎟

⎠

⎜⎜ ⎞

⎝

− ⎛

=

⋅

= −

∞ ∞

∞

V V V

p Cp p

ρ

l

d Coefficiente di pressione

(9)

Cm₀=0

(10)

DORSO

VENTRE

Cm₀<0

(11)

(p-p_∞) ds

ds p-p_∞

l

d f_a

∫

⁻ ^⋅ ^⋅

= _∞

s

a p p n ds

f

0

) (

n Integrazione della distribuzione di pressione

(12)

l

d

∫

⋅

−

=

⋅

−

= −

⋅

=

⋅

=

⋅

= −

⋅

=

⋅

−

=

⋅

−

=

⋅

−

=

⋅

−

=

∞

s s

t t

s s

n n

s s

t

s s

n

z d c Cp

dz V

p p c

V C f

x d c Cp

dx V

p p c

V C f

dz p

p ds

p p f

dx p

p ds

p p f

0

0 2

2

0

0 2

2

0 0

2 1

) (

2 1

) (

2 1

) (

sin )

(

) (

cos )

(

ρ ρ

δ δ

fn

ft z

x

(p-p_∞) dz

ds p-p_∞

(p-p_∞) dx

dx dz

(p-p_∞) ds

180°-δ δ

Integrazione della distribuzione di pressione

( )

δ = −cos

(

180−δ

) ( )

;sin δ =sin

(

180−δ

)

cos

Tenendo presente che:

(13)

z

x (p-p_∞) dx

(p-p_∞) dz x1

z1

x-x1 z-z1

∫

⋅

−

− +

⋅

−

=

− =

⋅

− − +

− ⋅

⋅

− −

=

⋅

=

−

⋅

−

− +

⋅

−

⋅

−

=

∞

s s

m

s s

z d z

z Cp x

d x

x Cp

c dz c

z z V

p p

c dx c

x x V

p p

c V

C m

dz z

z p

p dx

x x p

p m

0 0

0 2

2 2

0 0

) 1 (

2 1

) (

) 1 (

2 1

) (

2 1

) 1 (

) (

) 1 (

) (

ρ ρ

ρ

(p-p_∞) ds Integrazione della distribuzione di pressione

(14)

0 0

0

0 0

= x d Cp x

+ z d z Cp +

x d x Cp

= z d z Cp +

x d ) x

x ( Cp

= Cm

s CP s

s

s s

CP CP

∫

⋅

−

⋅

−

⋅

−

⋅

−

⋅

−

C_n Cm_l.e

. l.e C.P.

Cn x Cm

x Cn

Cm_l_._e = − ⋅ ^CP ⇒ ^CP = − ^l^.^e

z

x l

d fn

ft l

d

Centro di pressione

(15)

234 .

0 deg 4

0064 .

0 71 . 0

.

= −

=

e

Cm

l

Cd Cl α

Calcolare la posizione del centro di pressione

X_C.P.=0.33

Esercizio 3

112 .

0 deg 0

0055 .

0 24 .

0

.

= −

=

e

Cm

l

Cd Cl α

Calcolare la posizione del centro di pressione

X_C.P.=0.47

(16)

NACA 2412

x z Cp

1 0.00126 0.43059 0.992863 0.00273 0.23665 0.980187 0.005305 0.15376 0.965251 0.008283 0.08598 0.948591 0.011532 0.03049 0.930885 0.014904 -0.01854 0.912638 0.018292 -0.06238 0.894131 0.021641 -0.10197 0.875499 0.024924 -0.13828 0.856802 0.028131 -0.17316 0.838069 0.031256 -0.20565 0.819311 0.0343 -0.23678 0.800534 0.037259 -0.26729

0.946892 -0.00509 0.22416 0.96423 -0.00386 0.23861 0.979694 -0.00274 0.26151 0.992716 -0.0018 0.30065 1 -0.00126 0.43059

-5 -4 -3 -2 -1 0 1 2

0 0.2 0.4 0.6 0.8 1 1.2

-Cp

-0.05 0 0.05 0.1

0 0.2 0.4 0.6 0.8 1

Integrazione numerica – Esempio Cp numerici o sperimentali

(17)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

-5 0 5 10 15 20 25

alpha(°)

Cl

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-5 0 5 10 15

alpha(°)

Cm

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

-4 -2 0 2 4 6 8 10 12

alpha(°) Cd

α α α

Cm Cd Cl

Integrazione numerica – Esempio di polari numeriche o sperimentali

(18)

alpha CL CD CM

0 0.2394 0.00551 -0.0524 2 0.4531 0.00528 -0.0495 4 0.7104 0.00638 -0.0566

Integrazione numerica – Derivate dei coefficienti aerodinamici

(19)

x z Cp

0.687652 0.053233 -0.43788 0.668821 0.055589 -0.46589 0.64999 0.057854 -0.49401 0.631162 0.060026 -0.52243 0.612339 0.062103 -0.55114 0.593522 0.064082 -0.58029

-5 -4 -3 -2 -1 0 1 2

0 0.2 0.4 0.6 0.8 1

-Cp

0.6 0.8 1

)) ( )

1 (

2 (

) ( )

(

1 1

1 0

n x n

x x Cp x

Cn Cp

x d Cp Cn

n n n

s

− +

+ ⋅

=

⋅

=

∑

∫

− +

Cp(x_n) Cp(x_n+1)

x_n+1x_n Integrazione numerica - Cn

(20)

x z Cp

0.687652 0.053233 -0.43788 0.668821 0.055589 -0.46589 0.64999 0.057854 -0.49401 0.631162 0.060026 -0.52243 0.612339 0.062103 -0.55114 0.593522 0.064082 -0.58029

-5 -4 -3 -2 -1 0 1 2

0 0.2 0.4 0.6 0.8 1

-Cp

0.6 0.8 1

) 2 (

) ( )

(

1 1

1

1 0

n n

n n n

s

z z z

Cp z

Ct Cp

z d Cp Ct

− + ⋅

−

=

⋅

−

=

+

− +

∑

∫

Cp(x_n) Cp(x_n+1)

x_n+1x_n Integrazione numerica - Ct

(21)

-5 -4 -3 -2 -1 0 1 2

0 0.2 0.4 0.6 0.8 1

-Cp

0.6 0.8 1

) 2 (

)) 1 (

) ( )

1 (

) (

(

) 2 (

)) 1 (

) ( )

1 (

) (

(

1 1

1 1 1 1

n n

n

n n

z z z

z z

Cp z

z z

Cp

x x x

x x

Cp x

x x

Cm Cp

−

− ⋅

⋅ +

− + ⋅

−

− ⋅

⋅ +

−

− ⋅

=

+ + +

−

+ +

∑

+

Cp(x_n) Cp(x_n+1)

x_n+1x_n Integrazione numerica - Cm

∫

⁻ ^⋅ ⁺ ⁻ ⁻ ^⋅

= ^s Cp x x d x ^s Cp z z d z Cm

0 0

) 1 (

) 1

x1,z1 (

(22)

z

A.C. x

.

0

.

,

=

Cm

_α _A_C

0

≤ 0 Cm

l

d

Centro Aerodinamico - Definizione

X_A.C. è la X tale che:

Solitamente per profili con curvatura positiva:

(23)

z

x A.C.

l

d

P

C A

C A P

C A

Cm sen

Cd Cl

z z

Cd sen

Cl x

x

Cd sen

Cl z

z

sen Cd

Cl x

x

Cd sen

Cl z

z

sen Cd

Cl x

x Cm Cm

α α

α

α α

α

α α

α

−

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⋅ +

⋅

−

+

⋅

−

⋅

− +

+

⋅ +

⋅

−

⋅

− +

+

⋅ +

⋅

−

⎥ =

⎦

⎢ ⎤

⎣

⎡

⋅ +

⋅

−

⋅

− +

+

⋅ +

⋅

−

∂

= ∂

)) ( )

cos(

( ) (

)) cos(

) ( (

) (

)) cos(

) ( (

) (

)) ( )

cos(

( ) (

)) 0 cos(

) ( (

) (

)) ( )

cos(

( ) (

. .

. . .

.

P(x,z) Centro Aerodinamico

(24)

Z_A.C.

X_A.C. Due incognite Due equazioni Due α (α1, α2)

P

C A

Cm sen

Cd Cl

z z

Cd sen

Cl x

x

Cd sen

Cl z

z

sen Cd

Cl x

x

) 1 ( ))

1 ( )

1 cos(

) 1 ( ( ) (

)) 1 cos(

) 1 ( )

1 ( )

1 ( ( ) (

)) 1 cos(

) 1 ( )

1 ( )

1 ( (

) (

)) 1 ( )

1 ( )

1 cos(

) 1 ( (

) (

. .

α α

α

α α

α

α α

−

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⋅ +

⋅

−

+

⋅

−

⋅

− +

+

⋅ +

⋅

−

⋅

− +

+

⋅ +

⋅

−

P

C A

Cm sen

Cd Cl

z z

Cd sen

Cl x

x

Cd sen

Cl z

z

sen Cd

Cl x

x

) 2 ( ))

2 ( )

2 cos(

) 2 ( ( ) (

)) 2 cos(

) 2 ( )

2 ( )

2 ( ( ) (

)) 2 cos(

) 2 ( )

2 ( )

2 ( (

) (

)) 2 ( )

2 ( )

2 cos(

) 2 ( (

) (

. .

α α

α

α α

α

α α

−

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⋅ +

⋅

−

+

⋅

−

⋅

− +

+

⋅ +

⋅

−

⋅

− +

+

⋅ +

⋅

−

Centro Aerodinamico