Proposal of a correlation for boundary layer flow over real roughened surfaces

(1)

P

ROPOSAL OF A

C

ORRELATION FOR

B

OUNDARY

L

AYER

F

LOW OVER

R

EAL

R

OUGHENED

S

URFACES

G

IORGIO

M

ELIS

T

UTOR

:

P

ROF

.

N

ATALINO

M

ANDAS

D

OTTORATO DI

R

ICERCA IN

P

ROGETTAZIONE

M

ECCANICA

(2)

(3)

ABSTRACT

!

"#

$%&%

'

%

(

(4)

)

*

(5)

Acknowledgments

#

,

-.

.

/

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!0% 1

.

0

2 %

3#

%

!

#

# %

+

(6)

!

,

0 /

$ $

4 4 4 2

4 4 "

5 4 6

1

7 $$ 2

0

8 " 4 $

8 " " 2

,

8 " " 4 #

44 " " "

44 " " 6 %

44 " " 9 %

,

4"

" " : 0

;

4"

" " 5 ,

<

=

.

49 " 6 .

>

&

49 " 9

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4:

" :

45 " 5 /

!

47 $$$ %

%

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6 4

.

4?

6 " 2

"@

6 " 4 %

A#

.

"4

$' &

"6

9 4

.

"6

9 4 4 .

"6

(7)

9 4 " %

"6

9 " 2

0 "9

9 " 4 .

%

>B

"9

9 " "

C

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' #

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"8

: 4 %

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"8

: 4 4 %

A#

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"8

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%

6@

: 6 2

0 #

64 : 6 4 20$

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64 : 6 " 0

;

6"

: 6 6

.

66 '$

,

6:

5 4 $

6:

5 " %

.

6:

5 " 4 %

A#

.

67 5 6 2

0

68 5 6 4 0

;

9"

5 6 "

.

9"

5 6 6

.

96 5 9 ,

99 '$$ ,

55 7 4 ,

55 7 4 4 >

%

2

57 7 "

5?

7@

!

! D %

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75 ! 4 !

.

75 ! " +

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7?

(8)

List of Figures and Tables

Figure 1.1 Efficiency losses of steam turbine stages due to surface

roughness.

2 Figure 1.2 Power output losses of steam turbine stages due to surface

roughness.

2 Figure 1.3 Non-dimensional velocity profiles for smooth and rough

surfaces.

k

is an arbitrary roughness height

4 Figure 2.1 Boundary layer on a flat plate (vertical thickness

exaggerated for clarity) [56].

10 Figure 3.1 Texture of a manufactured surface.

19 Fig. 4.1 Plates' shape and roughness measurement points.

24 Fig. 4.2 Wind tunnel test section.

25 Fig. 4.3 Plate geometry and axial locations for BL measurements.

26 Fig. 5.1 MISES output: skin friction coefficient vs. axial direction

x

.

31 Figure 6.1 Required values of peak feed

PF

and mean roughness

R

a

.

36 Figure 6.2 Shape/density parameter for all plates,

β

= 0°.

38 Figure 6.3 Non-dimensional velocity profiles at different Re, smooth

plate, non-tilted case. Comparison with empirical law.

46 Figure 6.4 Velocity profiles for different plates (smooth and rough) at

Re

1

.

47 Figure 6.5 Velocity profiles for different plates (smooth and rough) at

Re

2

.

47 Figure 6.6 Velocity profiles for different plates (smooth and rough) at

Re

3

.

(9)

Figure 6.7 Non-dimensional velocity profiles at different Re, smooth

plate, incidence angle 7°. Comparison with the non-tilted

case.

48 Figure 6.8 Velocity profiles for different plates (smooth and rough) at

Re

3

, incidence angle 7°.

49 Figure 6.9 Velocity profiles for plate 02 at Re

1

. Effect of the

staggering

angle.

49 Figure 6.10 Velocity profiles for plate 02 at Re

2

. Effect of the

staggering

angle.

50 Figure 6.11 Velocity profiles for plate 02 at Re

3

. Effect of the

staggering

angle.

50 Figure 6.12 Velocity profiles for plate 02 and plate 03 at Re

3

,

β

= 15°.

51 Figure 6.13 Velocity profiles for plate 02, plate 03 and plate 04 at Re

3

,

β

= 30°.

51 Figure 6.14 Displacement thickness: Reynolds number influence.

52 Figure 6.15 Momentum thickness for all plates at Re

1

: comparison with

empirical law and roughness influence.

52 Figure 6.16 Momentum thickness for all plates at Re

2

: comparison with

empirical law and roughness influence.

53 Figure 6.17 Momentum thickness for all plates at Re

3

: comparison with

empirical law and roughness influence.

53 Figure 6.18 Shape factor for all plates at Re

3

: comparison with

numerical results (MISES).

54 Figure 6.19 Skin friction coefficient for all plates, Re1

.

54 Figure 6.20 Skin friction coefficient for all plates, Re2

.

55

(10)

Figure 6.22 Displacement thickness: Reynolds number influence.

Incidence angle 7°.

56 Figure 6.23 Momentum thickness for all plates at Re

3

: comparison with

empirical law and roughness influence. Incidence angle 7°.

56 Figure 6.24 Shape factor for all plates at Re

3

: comparison with

numerical results (MISES). Incidence angle 7°.

57 Figure 6.25 Skin friction coefficient for all plates, Re

3

. Incidence angle

7°.

57 Figure 6.26 Displacement thickness for plate 02 and plate 03 at Re

3

,

β

= 15°.

58 Figure 6.27 Momentum thickness for plate 02, plate 03 and plate 04 at

Re

3

,

β

= 30°.

58 Figure 6.28 Skin friction coefficient for plate 02 at Re

3

. Effect of the

staggering

angle.

59 Figure 6.29 Law of the wall for all plates at Re1

.

59 Figure 6.30 Law of the wall for all plates at Re2

.

60 Figure 6.31 Law of the wall for all plates at Re3

.

60 Figure 6.32 Law of the wall for all plates at Re1

. Incidence angle 7°.

61 Figure 6.33 Law of the wall for all plates at Re2

. Incidence angle 7°.

61 Figure 6.34 Law of the wall for all plates at Re3

. Incidence angle 7°.

62 Figure 6.35 Law of the wall for plate 02 at Re3

.

Staggering

angle effect. 62

Figure 6.36 Law of the wall for plate 02 at Re3

.

Staggering

angle effect. 63

Figure 6.37 Arbitrary roughness height for all plates. Non-

staggered

(11)

Figure 6.38 Correlation between velocity and roughness height for all

plates at Re

3

.

64 Figure 6.39 Correlation between velocity and roughness height for all

plates at Re

3

. Incidence angle 7°.

64 Figure 6.40 Correlation between velocity and roughness height for plate

04 at Re

3

.

Staggering

angle influence.

65

(12)

Nomenclature

2

1

2

w f

C

U

τ

ρ

∞

=

_%

2 1

δ

=

H

_%

s

k

%

ν

δ

k

+

=

/

=

PF

.

a

R

!

tm

R

!

= =

Re

m

S

ρ

τ

τ w

u

=

v

y

δ

+

₌

/

=

U

u

_τ +

₌

/

=

inf

U

)

*

Greek Symbols

α

$

β

δ

2 _*

)

1

δ

#

(13)

2

δ

v

u

τ

ν

δ

=

'

41 .

0 =

κ

E

µ

#

µ

ν

ρ

=

E

ρ

#

0

d

w y

U

y

τ

µ

=

;

s

Λ

%

A

(14)

(15)

I. Introduction

1.1 Background

C

F" 6G

%

)

*

)

* F9G $

=

4 4

4 "

%

4 "

)+.

$.*

)0.

*

)

* F46G

(16)

Figure 1.1 Efficiency losses of steam turbine stages due to surface roughness.

(courtesy of Encotech Inc.)

Figure 1.2 Power output losses of steam turbine stages due to surface

roughness. (courtesy of Encotech Inc.)

(17)

(

F5 7G

F:9G

%

=

$

)

H :*

;

)

I 5@*

$

): H

H 5@*

F::G

%

)

*

F:5G

0 $

$

=

F:9G

F:5G

=

)

4 6* $

(18)

!

F8G ;

F:5G

x/c

C

f 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.001 0.002 0.003 0.004 0.005 0.006 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re1

x/c

C

x/c

C

y

+

U

+ 101 ₁₀2 ₁₀3 ₁₀4 ₁₀5 0 5 10 15 20 25 30 35 40 Empirical [54] Smooth Plate01 Plate02 Plate03 Plate04 Re1

y

+

U

y

+

U

y

+

U

+ 101 ₁₀2 ₁₀3 ₁₀4 ₁₀5 0 5 10 15 20 25 30 35 40 Smooth Rough

Figure 1.3 Non-dimensional velocity profiles for smooth and rough surfaces.

k

is an arbitrary roughness height

$

!

0 F9?G

(19)

-(

'

F49G

F6"G

F9" 98G

)

*

$

F9:G

!

-2

(

F6G

=

(20)

1 (

1.2 Motivation for this Survey

(

$

;

)

*

!

(

(21)

$

1.3 Thesis Overview

(

,

$$

!

,

$$$

(

!

,

$'

(

(22)

$

,

'

A

(

,

'$

,

(

,

=

)

*

,

'$$

(23)

II. Boundary Layer Theory

2.1 Introduction

$

)

F:9 :5G*

2.2 Basic Concepts

δ

=

'

δ

)

*

(

88J

4 4@

5

!

" 4

δ

(24)

Figure 2.1 Boundary layer on a flat plate (vertical thickness exaggerated for

clarity) [56].

!

(

=

$

= (

/

= (

A

=

)

48:5* $

C

F:9G

(

E

(

K

(25)

dx

dU

U

H

dx

d

C

_f _∞ ∞

+

=

2

₍

₂

₎

2

2 δ

δ

(

)

*

(

2.2.1 Displacement Thickness

%

−

=

δ

0 inf 1

1 _U

dy

U

!

2.2.2 Momentum Thickness

F:9G

−

=

δ

0 inf inf 2

1 _U

dy

U

!

2.2.3 Shape factor

2 1

δ

=

H

(26)

L6M9

2.2.4 Skin Friction Coefficient

%

=

K

2 '

2

1

_∞

=

U

C

w f

ρ

τ

;

K

0 =

=

y w

_dy

dU

µ

τ

!

(

2.2.5 Law of the Wall

$

=

K

u y

y

τ

ν

+

₌

U

u

_τ +

₌

K

(27)

ρ

τ

τ w

u

=

δ

ν

K

τ ν

ν

δ

u

=

v

u y

y

τ

δ

ν

+

₌

K

B

y

U

+

=

1 _ln

+

κ

=

)

.

E

*

!

=

F:"G

2 )

*

=

!

=

!

F"8G

κ

E

=

K

41 .

0 =

κ

(28)

N :

$

I 6@

=

)

*

)

N

*

2.2.6 Coles’ wall-wake Profile

,

=

K

Π

+

=

+ +

δ

κ

y

w

x

B

y

U

1 ln

(

)

) A

δ

*

K

δ

π

δ

2 sin

2

y

w

=

⋅

Π

=

,

K

1 )

(

₌

1

₋

Π

∞

δ

κ

τ

u

U

x

2.3 Pressure Gradient Effects

(29)

!

1 E

0 (

);

*

!

" #

(

δ

$

%

1 F:5G

(

2.4 Roughness Effects

!

$

=

-)

4874 F9?G*

-)4875*

$

)

(30)

.

δ

ν

;

) A

δ

ν

*

:

5@

=

I 5@

2.5 Riblets

F44G

,

O

F68G $

,

#

), #*

$

=

,

=

(31)

2.6 Numerical Analysis

)#/%*

/

=%

(

C

-(

(32)

III. Surface Roughness Statistics

3.1 Texture Parameters

%

1 (

K

N)P

PQP

P*A

(33)

Figure 3.1 Texture of a manufactured surface.

!

C

%

=

1 =

= (

&

F5G

(34)

(

!

F49G

E

!

3.2 Blade Roughness

!

F9 :G

=

(

F4 6 45G

%

#

F:7G

(

)

2 3 %

E

3 %

*

+

(35)

,

F49G

F

G

=

(

3.2.1 Shape/Density Parameter

!

(

F: 4@ 94 96G

A

Λ

%

#

F:7G

=

$

K

6 . 1 −

=

Λ

s f f s

_A

A

S

(36)

'

%

F: 4@G

'

(37)

IV. Experiments

4.1 Roughness Measurement Protocol

4.1.1 Plates Manufacturing

%

K

=

)

9 4*

>

)()*

!

)

*

(

1 9 4

(

4.1.2 Surface Roughness Measurement

.

)

%'

(38)

!

.

()

Figure 4.1 Plates' shape and roughness measurement points.

4.2 Boundary Layer Measurements

4.2.1 Plate Test Set of Göttingen

R

=

!

#0 >B

F47 "6G

=

)

4@

?

_{* %}

)

9 "*

P3 P4 P1 P2

(39)

$

!

)

=

*

7S

1 +

#

=

!

9 4

(40)

4 N 4 :9 4@

5 =4

(

U

inlet

)

1

=

25 .

05 m/s

"

N 7 95 4@

5 =4

(

U

inlet

)

2

=

23 .

34 m/s

6 N 8 :: 4@

7 =4

(

U

inlet

)

3

=

30 .

14 m/s

)

=

*

).

*

)

*

)

*

9 6

Figure 4.3 Plate geometry and axial locations for BL measurements.

=

.

6J

=

@"

)

+ $ TEST SECTION PLATE LENGTH X1 X2 X3 X4 X5 FLOW DIRECTION

(41)

*

=

#

=

%

.

!

$

.

)

*

PF

[mm]

R

a

[

µ

m]

Staggering

Angle

β

[°]

Plate 01

7.8

4.2 0, 20, 40

Plate 02

3.1

4.2 0, 15, 30

Plate 03

3.1

1.6 0, 15, 30

Plate 04

10.4

4.2 0, 30, 60

Table 4.1 Summary of the plates measuring conditions.

4.2.2 Measurement Uncertainty

! .

(42)

;

)

*

(43)

V. Data Analysis

5.1 Surface Roughness Data

,

$' !

A

5.1.1 Shape/Density Parameter

$

=

O

)

,

$'*

A

)

,

$$$*

=

)

(

*

(

'

(

K

!

F4@G

'

(44)

5.2 Numerical Simulations

/

"#

$%&%)T

#

4884* F9@G

$

=

&

(

!

: 4

$%&%

,

'$ $

)

*

(45)

Figure 5.1 MISES output: skin friction coefficient vs. axial direction

x

.

5.3 Boundary Layer Data

.

$

)

,

$'*

(46)

)

(

,

$$*

F7 "7G !

F48G

K

F7G

.

%

!

(

)

τ

*

,

O

F6GK

(

=

>B

.

E

(

)

,

$$*

(

$

1 (

5.3.2 Laws of the wall

2

(47)

(

,

$$

,

<

!

=

)

!*

E

κ

>

F96GK

)

3 .

0

1 ln(

1

+ + + +

₌

₋

₌

₊

∆

U

_smooth

U

_rough

k

κ

v

k

δ

+

₌

)

*

,

-,

-5.3.3 Tilted plates

$

O

(48)

$

)

*

;

=

;

E

(

!

=

$

=

!

$

=

(

E

O

$

F6@G

(49)

VI. Results and Correlations

6.1 Introduction

$

,

!

A

$

!

6.2 Surface Roughness Parameters

!

=

.

()

'

)()*

) *

(50)

!

A

!

)

*

)

*

k

+

∆

U

10-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 0 1 2 3 4 5 6 7 Grigson Re1 Re2 Re3

k

+

∆

U

+ 10-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 0 1 2 3 4 5 6 7 Grigson Plate01 Plate02 Plate03 Plate04

Re

1

, 0°

k

+

∆

U

Re

2

, 0°

k

+

∆

U

Re

3

, 0°

k

0 2 4 6 8 10

PLATE01

PLATE02

PLATE03

PLATE04

R

a

[

µ

m

],

P

F

[m

m

]

0 2 4 6 8 10 12 Ra PF

PLATE01

PLATE02

PLATE03

PLATE04

(51)

6.2.1 Shape/Density Parameter

!

,

'

A

6 . 1 −

=

Λ

s f f s

A

S

( )

m _{m q}

S S S

=

2 cos

m pm q f

S

_R

S

β

=

f f

S

A

=

( )

1 2 ₂ 2

2 cos

2

m s s m q pm

S

A

=

S

= ⋅

S

+

R

β

&

(

β

$

A

K

( )

2.6

cos

m s pm q

S

R

β

Λ ∝

(52)

m pm

S

R

( )

S

m q

( )

R

pm q

$

Λ

)

.

*

&

()

A

)

.

*

&

A

Λ

A

5 "

k

0 2 4 6 8 10

PLATE01

PLATE02

k

PLATE03

PLATE04

+

∆

U

10-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 0 1 2 3 4 5 6 7 Grigson Re1 Re2 Re3

k

+

∆

U

10-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 0 1 2 3 4 5 6 7 Grigson Plate01 Plate02 Plate03 Plate04

Re

1

, 0°

k

+

∆

U

Re

₂

, 0°

k

+

∆

U

Re

3

, 0°

R

a

[

µ

m

],

P

F

[m

m

]

0 2 4 6 8 10 12 Ra PF

PLATE01

PLATE02

PLATE03

PLATE04

Λ

s 200 400 600 800 1000

PLATE01

PLATE02

PLATE03

PLATE04

(53)

6.3 Boundary Layer Features

!

=

5 6

/

-$

F:5G

=

&

5 9=5

5 5

@"

$

K

• $

$

• '

$

(

• $

δ

(54)

=

$

)

*

$

δ

.

!

A

$

5 49

@9K

5 47

=

(55)

!

)

5 4:=45*

%

5 4?

$%&%

$

%

,

'

5 48="4

(

=

%

)/

2

/

1

*

!

)

5 48*K

!

5 48

!

/

+

5 48

5 "4

@4

/

+

K

%

=

(56)

.

@"

6.3.1 Law of the Wall

%

4866

/

F:6G

,

$$

=

$

(

)

* $

,

O

F:9G $

5 "8=64

=

)!

*

6.3.2 Tilted Plates

=

)

5 7*

=

)

5 ?*

=

(57)

)

5 ""=":*

!

=

5 "9 !

!

5 6"=69 %

E

=

!

=

)

*

A

5 " 4

6.3.3 Staggered

Plates

@"

5 8=44

=

)

5 4"=46*

)

5 "5="?*

$

(58)

*

F44G

@9

-=

%

5 "5

) *

A

/

!

)9@SU9:S*

(

&

)

5 6:*

)

5 65*

6.4 Correlations

$

=

5 "8=65

,

'

(

E

(59)

/

'

>

F96G

$

A

K

2 1 C S tm

k C

= ⋅Λ ⋅

R

;

K

1

0.0041

C

=

1

0.68 C

=

.

!

.

= =#

+

$

'

=

5 67

!

.

K

%

(60)

/

F:6 :9G

()

>

5 6?=

9@

!

@4

6

)

α

N @S

β

N @S*

!

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Re1 Re2 Re3 Plate 04

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0003 0.0006 0.0009 0.0012 0.0015 0.0018 Smooth Plate01 Plate02 Plate03 Plate04 Re3

x/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 MISES Smooth Plate01 Plate02 Plate03 Plate04 _Re₃

x/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 0.0024 Re1 Re₂ Re3 Plate 04

x/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re1

x/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re2 24 Jan 2007

x/c

δ

2

/c

U/U

_inf

y

0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 Smooth Plate01 Plate02 Plate03 Plate04 Re3

U/U

_inf

y

0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 Smooth Plate01 Plate02 Plate03 Plate04 Re2

U/U

_inf

y

0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 Smooth Plate01 Plate02 Plate03 Plate04 Re1 Re1y_vs_Uinf24 Jan 2007

U/U

_inf

y/

δ

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 Empirical [56] Re1 Re2 Re3

Figure 6.3 Non-dimensional velocity profiles at different Re, smooth plate,

(61)

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Re1 Re2 Re3 Plate 04

x/c

δ

1

/c

U/U

_inf

y/

δ

x/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 MISES Smooth Plate01 Plate02 Plate03 Plate04 _Re₃

x/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 0.0024 Re1 Re₂ Re3 Plate 04

x/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 .0004 .0008 .0012 .0016 0.002 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re2

x/c

δ

2

/c

U/U

_inf

y

U/U

inf

y

U/U

_inf

y

0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 Smooth Plate01 Plate02 Plate03 Plate04 Re1

Figure 6.4 Velocity profiles for different plates (smooth and rough) at Re

1

.

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Re1 Re2 Re3 Plate 04

x/c

δ

1

/c

U/U

_inf

y/

δ

U/U

inf

y

x/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 MISES Smooth Plate01 Plate02 Plate03 Plate04 _Re 3

x/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 0.0024 Re1 Re2 Re3 Plate 04

x/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re2 24 Jan 2007

x/c

δ

2

/c

U/U

_inf

y

0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 Smooth Plate01 Plate02 Plate03 Plate04 Re3 e3y_vs_Uinf24 Jan 2007

U/U

_inf

y

(62)

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Re1 Re2 Re3 Plate 04

x/c

δ

1

/c

U/U

inf

y/

δ

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 Empirical [56] Re1 Re₂ Re3

U/U

_inf

y

x/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 4 8 2 6 2 MISES Smooth Plate01 Plate02 Plate03 Plate04 _Re₃

x/c

1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 0.0024 Re1 Re2 Re3 Plate 04

x/c

δ

2

/c

x/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re2 e2 24 Jan 2007

x/c

δ

2

/c

U/U

_inf

y

U/U

_inf

y

Figure 6.6 Velocity profiles for different plates (smooth and rough) at Re

3

.

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0 0.0008 0.0016 0.0024 0.0032 0.004 Re1 Re₂ Re3 Plate 04,α=7 Red224 Jan 2007

x/c

H

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.4 0.8 1.2 1.6 2 2.4 2.8 MISES Smooth Plate01 Plate02 Plate03 Plate04 Re3,α=7° H_Re324 Jan 2007

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.0006 0.0012 0.0018 0.0024 0.003 ∆p=0 [56] Smooth Plate01 Plate02 Plate03 Plate04 Re3,α=7° d224 Jan 2007

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0000 0.0003 0.0006 0.0009 0.0012 0.0015 0.0018 0.0021 0.0024 0.0027 0.0030 0.0033 Smooth Plate01 Plate02 Plate03 Plate04 Re3,α=7° d124 Jan 2007

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.001 0.002 0.003 0.004 0.005 0.006 Re1 Re2 Re3 Plate 04,α=7° Red124 Jan 2007

U/U

inf

y

0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 Smooth Plate01 Plate02 Plate03 Plate04 Re3,α=7° Re3y_vs_Uinf24 Jan 2007

U/U

_inf

y/

δ

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 No tilting Re1 Re2 Re3

Figure 6.7 Non-dimensional velocity profiles at different Re, smooth plate,

(63)

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.0008 0.0016 0.0024 0.0032 0.004 Re1 Re2 Re3 Plate 04,α=7° Red224 Jan 2007

U/U

_inf

y/

δ

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 No tilting Re1 Re2 Re3

x/c

H

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.4 0.8 1.2 1.6 2 2.4 2.8 MISES Smooth Plate01 Plate02 Plate03 Plate04 _Re₃_,α=7°

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.0006 0.0012 0.0018 0.0024 0.003 ∆p=0 [56] Smooth Plate01 Plate02 Plate03 Plate04 Re3,α=7° d224 Jan 2007

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0000 0.0003 0.0006 0.0009 0.0012 0.0015 0.0018 0.0021 0.0024 0.0027 0.0030 0.0033 Smooth Plate01 Plate02 Plate03 Plate04 Re3,α=7° d124 Jan 2007

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.001 0.002 0.003 0.004 0.005 0.006 Re₁ Re2 Re3 Plate 04,α=7° Red124 Jan 2007

U/U

_inf

y

0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 Smooth Plate01 Plate02 Plate03 Plate04 Re3,α=7°

Figure 6.8 Velocity profiles for different plates (smooth and rough) at Re

3

,

incidence angle 7°.

y

0 0.2 0.4 0.6 0.8 1 0 2 4 6 Smooth Plate02 Plate03 Plate04 β=30°, Re3 02_03_04_Re3_2beta_dim24 Jan 2007

y

0 1 2 3 4 5 6 Smooth Plate02 Plate03 β=15°, Re3 02_03_Re3_beta1_dim24 Jan 2007

y

0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 Smooth β=0° β=15° β=30° Plate02, Re3 Plate02_Re3_dim24 Jan 2007

U/U

y

0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 Smooth β=0° β=15° β=30° Plate02, Re2 Plate02_Re2_dim24 Jan 2007

U/U

y

0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 Smooth β=0° β=15° β=30° Plate02, Re1

(64)

U/U

_inf

y

U/U

_inf

y

0 0.2 0.4 0.6 0.8 1 0 2 4 6 Smooth Plate02 Plate03 Plate04 β=30°, Re3 02_03_04_Re3_2beta_dim24 Jan 2007

U/U

y

0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 Smooth Plate02 Plate03 β=15°, Re3 02_03_Re3_beta1_dim24 Jan 2007

U/U

_inf

y

0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 Smooth β=0° β=15° β=30° Plate02, Re3 Plate02_Re3_dim24 Jan 2007

U/U

_inf

y

0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 Smooth β=0° β=15° β=30° Plate02, Re2

Figure 6.10 Velocity profiles for plate 02 at Re

2

. Effect of the

staggering

angle.

U/U

_inf

y

U/U

_inf

y

U/U

_inf

y

0 0.2 0.4 0.6 0.8 1 0 2 4 6 Smooth Plate02 Plate03 Plate04 β=30°, Re3

U/U

y

U/U

_inf

y

(65)

U/U

_inf 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 Smooth β=0° β=15° β=30° Plate02, Re1

U/U

_inf

y

U/U

_inf

y

U/U

_inf

y

U/U

_inf

y

0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 Smooth Plate02 Plate03 β=15°, Re3

Figure 6.12 Velocity profiles for plate 02 and plate 03 at Re

3

,

β

= 15°.

U/U

inf

y

U/U

_inf

y

U/U

_inf

y

U/U

y

U/U

_inf

y

(66)

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Re1 Re2 Re3 Plate 04

x/c

δ

1

/c

U/U

inf

y/

δ

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 Empirical [56] Re₁ Re2 Re₃

U/U

inf

y

U/U

_inf

y

U/U

inf

y

0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 Smooth Plate01 Plate02 Plate03 Plate04 Re3 Re3y_vs_Uinf24 Jan 2007

x/c

δ

2

/c

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re2 2AllRe224 Jan 2007

x/c

δ

2

/c

x/c

H

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.4 0.8 1.2 1.6 2 MISES Smooth Plate01 Plate02 Plate03 Plate04 _Re 3

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 0.0024 Re₁ Re₂ Re₃ Plate 04

Figure 6.14 Displacement thickness: Reynolds number influence.

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Re1 Re2 Re3 Plate 04

x/c

δ

1

/c

U/U

_inf

y/

δ

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 Empirical [56] Re₁ Re₂ Re₃ SmoothProfile24 Jan 2007

U/U

y

U/U

inf

y

U/U

inf

y

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re2 AllRe224 Jan 2007

x/c

δ

2

/c

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 0.0024 Re₁ Re₂ Re₃ Plate 04

x/c

δ

2

/c

Figure 6.15 Momentum thickness for all plates at Re

1

: comparison with

(67)

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Re1 Re2 Re3 Plate 04

x/c

δ

1

/c

U/U

inf

y/

δ

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 Empirical [56] Re1 Re₂ Re3

U/U

_inf

y

0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 Smooth Plate01 Plate02 Plate03 Plate04 Re1

U/U

inf

y

U/U

inf

y

x/c

H

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 0.0024 Re₁ Re₂ Re₃ Plate 04

x/c

δ

2

/c

x/c

δ

2

/c

x/c

δ

2

/c

Figure 6.16 Momentum thickness for all plates at Re

2

: comparison with

empirical law and roughness influence.

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Re1 Re2 Re3 Plate 04

x/c

δ

1

/c

U/U

inf

y/

δ

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 Empirical [56] Re₁ Re₂ Re₃ SmoothProfile24 Jan 2007

U/U

y

U/U

_inf

y

U/U

_inf

y

x/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.4 0.8 1.2 1.6 2 MISES Smooth Plate01 Plate02 Plate03 Plate04 _Re₃

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 0.0024 Re₁ Re₂ Re₃ Plate 04

x/c

δ

2

/c

x/c

2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re2 Re224 Jan 2007

x/c

δ

2

/c

(68)

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.0004 0.0008 0.0012 0.0016 0.002 Re1 Re2 Re3 Plate 04

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.0003 0.0006 0.0009 0.0012 0.0015 0.0018 Smooth Plate01 Plate02 Plate03 Plate04 Re3 d124 Jan 2007

U/U

inf

y/

δ

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 Empirical [56] Re1 Re₂ Re3 SmoothProfile24 Jan 2007

U/U

y

U/U

inf

y

0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 Smooth Plate01 Plate02 Plate03 Plate04 Re2 Re2y_vs_Uinf24 Jan 2007

U/U

inf

y

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 0.0024 Re₁ Re₂ Re3 Plate 04 Red124 Jan 2007

x/c

δ

2

/c

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0004 0.0008 0.0012 0.0016 0.002 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re2 d2AllRe224 Jan 2007

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.0004 0.0008 0.0012 0.0016 0.002 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re3

x/c

H

Figure 6.18 Shape factor for all plates at Re

3

: comparison with numerical

results (MISES).

U

+ 101 ₁₀2 ₁₀3 ₁₀4 ₁₀5 0 5 10 15 20 25 30 35 40 Empirical [54] Smooth Plate01 Plate02 Plate03 Plate04 Re₃ U+Re324 Jan 2007

U

+ 101 ₁₀2 ₁₀3 ₁₀4 ₁₀5 0 5 10 15 20 25 30 35 40 Empirical [54] Smooth Plate01 Plate02 Plate03 Plate04 Re2 U+Re224 Jan 2007

y

+

U

+ 101 102 103 104 105 0 5 10 15 20 25 30 35 40 Empirical [54] Smooth Plate01 Plate02 Plate03 Plate04 Re1 +Re124 Jan 2007

x/c

C

f 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.001 0.002 0.003 0.004 0.005 0.006 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re₃ CfRe324 Jan 2007

x/c

C

f 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.001 0.002 0.003 0.004 0.005 0.006 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re2 CfRe224 Jan 2007

x/c

C

(69)

+

U

+ 101 ₁₀2 ₁₀3 ₁₀4 ₁₀5 0 5 10 15 20 25 30 35 40 Empirical [54] Smooth Plate01 Plate02 Plate03 Plate04 Re₃ U+Re324 Jan 2007

y

+

U

y

+

U

x/c

C

x/c

C

x/c

C

Figure 6.20 Skin friction coefficient for all plates, Re2

.

U

y

+

U

+ 101 102 103 104 105 0 5 10 15 20 25 30 35 40 Empirical [54] Smooth Plate01 Plate02 Plate03 Plate04 Re1

x/c

C

x/c

C

f 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.001 0.002 0.003 0.004 0.005 0.006 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re2 CfRe224 Jan 2007

x/c

C

f 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.001 0.002 0.003 0.004 0.005 0.006 Empirical [56] Smooth Plate01 Plate02 Plate03 Plate04 Re₃

(70)

x/c

H

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0006 0.0012 0.0018 0.0024 0.003 ∆p=0 [56] Smooth Plate01 Plate02 Plate03 Plate04 Re₃,α=7°

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0000 0.0003 0.0006 0.0009 0.0012 0.0015 0.0018 0.0021 0.0024 0.0027 0.0030 0.0033 Smooth Plate01 Plate02 Plate03 Plate04 Re3,α=7° d124 Jan 2007

U/U

inf

y

U/U

inf 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 No tilting Re1 Re₂ Re3

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0008 0.0016 0.0024 0.0032 0.004 Re₁ Re₂ Re₃ Plate 04,α=7°

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.001 0.002 0.003 0.004 0.005 0.006 Re₁ Re₂ Re₃ Plate 04,α=7°

Figure 6.22 Displacement thickness: Reynolds number influence. Incidence

angle 7°.

x/c

H

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0000 0.0003 0.0006 0.0009 0.0012 0.0015 0.0018 0.0021 0.0024 0.0027 0.0030 0.0033 Smooth Plate01 Plate02 Plate03 Plate04 Re3,α=7° 24 Jan 2007

U/U

inf

y

U/U

inf 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 No tilting Re₁ Re₂ Re₃

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0008 0.0016 0.0024 0.0032 0.004 Re₁ Re₂ Re₃ Plate 04,α=7°

x/c

δ

1

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.001 0.002 0.003 0.004 0.005 0.006 Re₁ Re₂ Re₃ Plate 04,α=7°

x/c

δ

2

/c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0006 0.0012 0.0018 0.0024 0.003 ∆p=0 [56] Smooth Plate01 Plate02 Plate03 Plate04 Re3,α=7°

Figure 6.23 Momentum thickness for all plates at Re

3

: comparison with

Proposal of a correlation for boundary layer flow over real roughened surfaces

P

ROPOSAL OF A

C

ORRELATION FOR

B

OUNDARY

L

AYER

F

LOW OVER

R

EAL

R

OUGHENED

S

URFACES

G

IORGIO

M

ELIS

T

UTOR

:

P

ROF

.

N

ATALINO

M

ANDAS

D

OTTORATO DI

R

ICERCA IN

P

ROGETTAZIONE

M

ECCANICA

ABSTRACT

!

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Acknowledgments

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Table of Contents

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