PDF Lecture 25 Turbulent Boundary-Layer Flows (3)

Turbulent Boundary-Layer Flows (3)

Contents

25.1 Power Law Formulas for Smooth Walls 25.2 Laws for Rough Walls

Objectives

- Study wall turbulence

- Derive equations of velocity distribution and friction coefficient for both smooth and rough walls

 Logarithmic equations for velocity profile and shear-stress coeff .

~ universal

~ applicable over almost entire range of Reynolds numbers

 Power-law equations

~ applicable over only limited range of Reynolds numbers

~ simpler

~ explicit relations for and

~ explicit relations for in terms of Re and distance 25.1.1 Power-law formulas: Smooth walls

/

u U c_f

 _x

i) Except very near the wall, mean velocity is closely proportional to a root of the distance y from the wall.

ii) Shear stress coeff. c_f is inversely proportional to a root of Re_

where n, A = constants; m = fraction

(A)

(25.1)

1

u ∝ yn

1 , Re

f Rem

c _ U





 

∝

f m

c A

U



  

 

 

The power laws stem from two facts that hold for turbulent boundary layers with negligible pressure gradients when ^Re ^{U 5 105}



   

[Cf] Eq. (25.1) is similar to equation for laminar boundary layer,

 Derivation of power equation Combine Eqs. (24.16) and (25.1)

(24.16):

(25.2)

3.32

f Re c





* 2

cf

u U

2

*

2 2

m

f

U U

u c A

1 2 2

*

2

m m

U

u A

1 2 2

*

*

2

m m

U u u A

2

*

*

m

u m

U B

u

f m

c A

U



  

 

 

Assume depends on y by the same relation, Eq. (25.2), replacing  with y

Divide (25.3) by (25.2)

2

*

*

m

u y m

u B

u (25.3)

(25.4)

2 m

u y m

U u

This result indicates that all profiles are similar and can be represented by a single dimensionless curve like laminar boundary layer profile.

However, turbulent profiles are not truly similar, so Eq. (25.4) will apply for different Reynolds number ranges only if the constant m is varied.

Boundary-layer measurements shows that For 3,000 < Re_ < 70,000;

(25.5)

(25.6)

(25.7)

(25.8)

1 , 0.0466 , 8.74

m 4 A B

1

u y 7

U

1 7

*

*

8.74 u U

u

1 7

*

*

8.74 u y u

u

1 4

0.0466 cf

Re

(25.10)

u y

U F

Re_x1/2

x

[Re] The Blasius solution for laminar boundary layer flows

For steady laminar flow over a flat plate with zero pressure gradient, Prandtl's (1904) 2-D boundary-layer equations become as follows:

2 2 ,

u u u

u v

x y y





    

   u v 0

x y

 

 

 

Blasius (1908) obtained the solution to above PDE by assuming similar profiles along the plate at every x

Blasius obtained the solution in the form of power series after he introduced a stream function for ^y

(25.9)

(25.11)

Blasius, H. (1883-1970): Prandtl’s student

 Relation for 

Adopt integral-momentum eq. for steady motion with

~ integration of Prandtl's 2-D boundary-layer equations

where θ = momentum thickness

Substitute Eq. (25.5) into (A) and integrate

(25.12)

0^h u 1 u

U U dy

p 0 x

2

2 2

0

2 0 2

f f

U c U c U

x

(A)

Substitute Eqs. (25.8) and (25.13) into (25.12) and integrate w.r.t. x

Integrate (25.15) over whole length, l , to get average coefficient (25.13)

7

0 1

7 0.1

72

y y dy

1 1

5 5

0.318 0.318 Re_x

x x

Ux

1 5

0.059 Re

f

x

c

Re_x 107 (25.14)

Re_x 107 (25.15)

,

,

(25.16)

1 5

0.074 Re

f

l

C _{Re 10}⁷

, l

[Re] Derivation of (25.14) and (25.15)

Substitute (25.13) and (25.8) into (B)

(B)

2 2

f 2

U c U

x

2 cf

x

1 4

1 1

4 4

1 4

7 1

0.0466 / (Re )

72 2

7 0.0233 0.0233

72 Re

0.2397 x

x U

x U

Integrate once w.r.t. x

B.C.: at x = 0

→

1 4

0.2397

x C U

0

1 4

0.2397

0 0 C

U

0 C

1 4

0.2397 x U

5 5

4 4

1 4

0.2397 x Ux

→ Eq. (C)(25.14)

4 4

5 5 5

4

1 1

4 5

0.2397 (0.2397)

x x

Ux Ux

1 1

5 5

0.318 0.318 Re_x

x x

Ux

1 1

4 4

0.0466 0.0466 Re

cf

U

(25.17): (C)

Substitute (25.14) into (C)

Integrate (25.15) over

→ (25.15)

1 1 1

4

4 4 1 4

4 5 1

5 1

5

0.0466 0.062

0.318 cf

U U x

Ux U

1 1 1 1

1

5 5 1 5 5

4

5 1

20

0.062 0.062 0.062 0.062

Re_x

U Ux

U x

x U

1 1 1 1 1

0 0 0

5 5 5 5 5

1 0.062 0.062 1 0.062 1 0.076

Re Re

l l l

f

x l

C dx dx dx

l l

Ux U x

l

l

 Effects of roughness

 Rough walls:

- Velocity distribution and resistance = f (Reynolds number, roughness) [Cf] Smooth walls:

- velocity distribution and resistance = f (Reynolds number)

▪ For natural roughness, k is random, and statistical quantity

→ k = k_s = uniform sand grain 25.2.1 Effects of roughness

▪ Measurement of roughness effects

a) experiments with sand grains cemented to smooth surfaces - Nikuradse b) evaluate roughness value ≡ height k_s

c) compare hydrodynamic behavior with other types and magnitude of roughness

▪ Effects of roughness i)

~ roughness has negligible effect on the wall shear

→ hydrodynamically smooth

= laminar sublayer thickness

' 1

ks

'

*

4 u

ii)

~ roughness effects appear

~ roughness disrupts the laminar sublayer

~ smooth-wall relations for velocity and C_f no longer hold

→ hydrodynamically rough

iii) > 15 ~25

~ friction and velocity distribution depend only on roughness rather than Reynolds number

→ fully rough flow condition

' 1

ks

'

ks

▪ Critical roughness, k_crit

If x increases, then c_f decreases, and ' increases.

Therefore, for a surface of uniform roughness, it is possible to be hydrodynamically rough upstream, and hydrodynamically smooth downstream.

Re

'

*

4 4

/ 2

crit

x f

k

u U c x

1

f Re

x

c

k ' ^k

(2) Rough-wall velocity profiles

Assume roughness height k accounts for magnitude, form, and distribution of the roughness. Then

*

u y

u f k (25.18)

Make f in Eq. (25.18) be a logarithmic function to overlap the velocity- defect law, Eq. (24.14), which is applicable for both rough and smooth boundaries.

(24.14):

i) For rough walls, in the wall region

where C₅ = constant = f (size, shape, distribution of the roughness)

*

5.6 log 2.5, 0.15

U u y y

u

* 5

*

5.6 log , 50 ~ 100, 0.15

rough

u k u y y

u y C (25.19)

ii) For smooth walls, in the wall region (24.11):

where C₂= 4.9

Subtract Eq. (25.19) from Eq. (24.11)

→ Roughness reduces the local mean velocity in the wall region

where C₅ and C₆ → Table 9-4 (D&H)

(25.20)

* *

2

*

5.6 log , 30 ~ 70, 0.15

smooth

u u y u y y

u C

*

6

* *

5.6 log

smooth rough

u u u k

u C

u u

u

(3) Surface-resistance formulas: rough walls Combine Eqs. (25.19) and (24.14)

(25.21)

(25.22)

5.6 log 2.5 , 0.15

*

5.6 log

* 5

5.6 log

* 7

U u y y

u

u k

u y C

U C

u k

7

*

2 5.6 log

f

U C

u c k

8

1 3.96 log

f

k C c

[Ex 25.1] Rough wall velocity distribution and local skin friction coefficient - Comparison of the boundary layers on a smooth plate and a plate

roughened by sand grains

▪ Given: t₀  0.485 lb /ft² on both plates U = 10 ft /sec past the rough plate

k_s= 0.001 ft

Water temp. = 58 F on both plates

From Table 1-3 (D&H):

; Eq. (24.16)

1.938 slug ft/ 3 1.25 10 ⁵ft² / sec

0

*

0.485

0.5 / sec 1.938

u ft

2 2

* 0.5

2 2 0.005

f 10 c u

U

*

5

0.5(0.001) 1.25 10 40 u ks

Eq. (25.20): ^*

*

5.6 log u k^s 3.3 u

u

0.5{5.6 log 40 3.3} 2.83 / sec

u ft

ii) For smooth plate, Eq. (24.11):

Check → same result as (a) (b) Velocity on each plate at y =0.007 ft

i) For rough plate Eq. (25.19):

u

*

5.6 log 8.2

u y

u k

0.007

0.5(5.6 log 8.2) 6.47 / sec 0.001

u ft

*

*

5.6 logu y 4.9 u

u

*

5

0.5(0.007) 1.25 10 280 u y

0.5{5.6 log(280) 4.9} 9.3 / sec

u ft

9.3 6.47 2.83 u

(c) Boundary layer thickness  on the rough plate Eq. (25.22):

1 3.96 log 7.55

f ks

c

1 1

log 7.55 1.66

3.96 0.005 ks

46 0.046 0.52 1.4

s

ft in cm

k