PDF Lecture 24 Turbulent Boundary-Layer Flows (2) - Seoul National University

Turbulent Boundary-Layer Flows (2)

24.1 Velocity Profiles

24.2 Surface Resistance Formulas

Objectives

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

 Relations describing the mean-flow characteristics in boundary layer

- It is useful to predict velocity magnitude and relation between velocity and wall shear or pressure gradient forces.

- It is desirable that these relations should not require knowledge of the turbulence details.

- For laminar boundary layer, it is possible to obtain a solution by integrating the equations of motion (Prandtl’s 2D boundary layers equation, Eq. (17.7)) - However, the turbulent boundary layer is composed of zones of different types of flow, and effective viscosity varies from wall out through the layer.

- Thus, theoretical solution is not practical for the general non-uniform boundary layer → use semiempirical procedure

- No single equation will describe the velocity profile over its entire thickness d.

- Close to the smooth boundary, a law of wall applies.

- For outer reaches of boundary layers for both smooth and rough walls, a velocity-defect law applies.

 Regions of inner law (wall law)

logarithmic

linear

- Inner layer might be divided into three regions

① Viscous sublayer

② Buffer zone

③ Turbulent zone

- Velocity profile is very nearly linear.

- Mean shear stress is controlled by the dynamic molecular viscosity.

- Reynolds stress is negligible, and mean flow is laminar.

- Energy of velocity fluctuation is nearly zero.

2) Buffer zone:

- Viscous and Reynolds stress are of the same order.

- Both laminar flow and turbulent flow exist.

- Sharp peak in the turbulent energy occurs.

du  dy

4 u y* 30 ~ 70

  

3) Turbulent zone - inner region:

- Flow is fully turbulent.

- Intensity of turbulence decreases from its peak value.

- Velocity profile is logarithmic.

* 30 ~ 70 , 0.15

u y y

 ^ and d ^

- Outer layer might be divided into two regions.

① Turbulent zone-outer region:

② Intermittent zone:

- In the intermittent zone, flow is intermittently turbulent and non-turbulent.

0.15d  y  0.4d 0.4d  y  1.2d

d

- All zones merge smoothly into a continuous mean velocity profile.

- On rough wall, laminar sublayer is destroyed by the roughness elements. Overlapping region

- Close to smooth boundaries, molecular viscosity is dominant.

- Law of wall assumes that the relation between wall shear stress and velocity at distance y from the wall depends only on fluid density and viscosity.

- Dimensional analysis yields

( , * , , , ) 0 f u u y   

*

*

u u y

u f 

 

  

  (24.1)

Inner region

- A unique law of wall, Eq. (24.1), shows as a single line up to - At higher values of , a single velocity law no longer holds

because viscosity is no longer a major factor.

* 2,000 u y

 ^

*

u u u y*



- The mean velocity is

- The shear stress may be assumed constant and equal to

- Then Eq. 24.2 becomes u  u

0

0 y

u u

y y

   



   



*

*

u u y u ^ 



(24.2)

(24.3)

 Thickness of laminar sublayer

- Define thickness of laminar sublayer (d' ) as the value of y which makes

(24.4)

where

- c_f decreases slowly with increasing Reynolds number, Thus, increases with distance along the surface.

* * 4

u y ud

 

  

1

* 0 2 2

4 4 4 4

' / 2

/ 2 ^f

f

u U c

c U

   

d  

 

   

 

 

 

2

0 ; .

f 2 f

c u c local shear stress coeff

   

Re] For the laminar sublayer in pipe flows

*

' 11.6 u d  

d '

Re_x Ux

 

- Start with equation of 2D turbulent boundary layer, Eq. (21.15a)

- In the turbulent zone of near wall region, the mean shear stress remains nearly equal to the wall shear;

2

2 2 2

0

du du du

l y

dy dy dy

      ^ ^

  (1)

2 2

2

' ' '

u u u p u u u v

u v

t x y x x y y

^^  ^  ^ ^  ^  ^  ^ ^

  0 2

' 0

' ' 0

u u u p u u

u v u v

t x y x x y y y y

 

^^  ^  ^ ^  ^  ^  ^{ } ^  ^ ^  ^ 

- Hence a solution is obtained if we have a relation for



- We can use Prandtl’s mixing length theory for near wall, l  y

Integrate (1)

Substitute BC [ at ] into (24.5)

Assume

Then (24.6) becomes

*

ln 1

u u y C

  

1

*

1 ln

u y C

u ^  ^ 0

u  y  y'

1

0 1 ln 'y C

  

1

1 ln '

C y

  

2

* *

( / )

' '

( / ) m s

y y C

u m s u

 

  

1 2

* *

1 1 1

ln ' ln ln

C y C C

u u

 

  

 

      

 

(24.5)

(24.6)

(24.7)

0

u* 

 

Substitute (24.7) into (24.5)

Changing the logarithm to base 10 yields

Empirical values of and for inner region of the boundary layer,

→ Prandtl's velocity distribution law; inner law; wall law

*

2 2

* *

1 1 1

ln ln ln

u u y

y C C

u u



   

 

     

*

10 2

*

2.3 log

u u y

u   C

 

  

0.41 ; C2 4.9

  

* *

*

5.6 log 4.9, 30 ~ 70 , 0.15

u u y u y y

u   and d

 

    

(24.8)

(24.9)

 C₂ y 0.15

d ^

x-eq.

y-eq.

Continuity eq.:

2 2

2

' ' '

u u u p u u u v

u v

t x y x x y y

^^  ^  ^ ^   ^ ^  ^  ^

0 (p v' )2

y 

   



u v 0 x y

 

 

 

(24.10a)

(24.10b)

(24.10c)

[Re] Prandtl’s turbulent boundary layer equation

Solution to these equations: u x y t( , , ), ( , , ), ( , , )v x y t p x y t

*

*

5.6 log 4.9

u u y

u 

 

    u  u y( )

- In the outer reaches of the turbulent boundary layer for both smooth and rough walls, Reynolds stresses dominate the viscous stresses to

produce the velocity profile.

- It was observed that the velocity defect (reduction) at y -values was dependent on the magnitude of the wall shear stress

- A logarithmic relation for the function g can be obtained by assuming that Eq. (24.8) will give .

(24.11)

(24.12)

*

U u y

u g d

     

* '

2

*

2.3 log

U u

u C

d

 

 

  

u U at y d

Subtract (24.8) from (24.12)

*

2

*

*

2

*

* *

2 2

*

*

3 3

*

2.3log '

2.3log

2.3 log log '

2.3 2.3

log log

U u

u C

u u y

u C

U u u u y

C C

u

u y

C C

u y d

 

 

d

  

d 

   d

 

  

 

   

        

   

       

   

3

*

2.3 log

U u y

u  d C

        (24.13)

- A single log-equation does not fit the data over the entire boundary layer.

- Instead one equation will fit an inner region overlapping with Eq. (24.8), while second equation will approximate the outer region.

y d

i) Inner region;

→   0.41, C₃  2.5

ii) Outer region;

→  = 0.267, C₃ = 0

→ Eqs. (24.14) & (24.15 ) apply to both smooth and rough surfaces.

→ Eq. (24.14) = Eq. (24.9)

(24.14)

(24.15)

d

*

5.6 log 2.5

U u y

u d

      

 

y 0.15 d ^

*

8.6 log

U u y

u d

     

 

*

*

5.6 log 4.9

u u y

u 

 

  

[Re] Eq. (24.12) - Eq. (24.14) = Eq. (24.9)

Wall law Velocity-defect law Smooth wall Smooth wall Rough wall

Outer region -

Inner region

Laminar sublayer

-

*

*

u u y u ^ 

*

*

5.6 log 4.9

u u y

u 

 

  

 

*

8.6 log

U u y

u d

     

 

*

5.6 log 2.5

U u y

u d

      

  y 0.15

d ^

30 ~ 70 *

0.15 u y y

 d



 Overlapping region

 Local shear-stress coefficient on smooth walls velocity profile ↔ shear-stress equations

where c_f = local shear stress coefficient

[Re] Skin drag force

*

*

/ 2

2

f

f

c U

u U

u c

 

   

2 0 2

0

1

2 2

f f

c U  c U

 

   

0

* 2

cf

u  U

   (24.16)

2 0

1

f 2 f

D  A  A c U D  D_f  D_p

(i) Assume that logarithmic law will give the relation Substituting into Eq. (24.8) yields

Substitute (24.16) into (A)

(24.17)

- is a function of Reynolds number for smooth walls.

- is not given explicitly.

*

4

*

2.3log

U u

u C

d

 

 

  

 

4 4

2 2.3 2.3

log log Re

2 2

f f

f

c c

U C C

c ^d

d

  

   

     

(A)

u U at y d

𝑐_𝑓 𝑐_𝑓

Re_d

(ii) For explicit expression, use displacement thickness and momentum thickness θ instead of d

Clauser:

Squire and Young:

where

*

1 3.96 log Re 3.04

cf  ^d 

1 4.17 log Re 2.54

cf  ^ 

*

*

Re U ; Re U

d 

d 

 

 

Re_d* , Re_  f (Re )_x

(24.18)

(24.19)

(24.20)

iii) Karman's relation

- Assume turbulence boundary layer all the way from the leading edge (i.e., no preceding stretch of laminar boundary layer)

(24.21)

1 4.15log(Re_{x f} ) 1.7

f

c  c 

- Karman's equation is useful for ships and aircraft wings where the laminar boundary layer is insignificant.

iv) Explicit equation by Schultz-Grunow (1940)

Comparison of (24.21) and (24.22)

(24.22)

2.58

0.370 (log Re )

f

x

c 

0.001 ~ 0.01 cf 

 Average shear-stress coefficient on smooth walls

Consider average shear-stress coefficient over a distance l along a flat plate of a width b

1 2

( )

f 2 f

Total frictional drag D    bl C U bl

2 / 2

f

C D

bl U



U

b l

24.2.2 Average shear stress coefficient

i) Schoenherr (1932)

- Assume turbulence boundary layer all the way from the leading edge - Similar to von Karman’s equation

where

ii) Schultz-Grunow

(24.23)

(24.24)

1 4.13log(Re_{l f} )

f

C C 

Re_l Ul

 

6 9

2.64

0.427

, 10 Re 10 (log Re 0.407)

f l

l

C   



Comparison of (24.23) and (24.24)

■ Boundary layer developing on a smooth flat plate

∼ At upstream end (leading edge), laminar boundary layer develops because viscous effects dominate due to inhibiting effect of the smooth wall on the development of turbulence.

∼ Thus, when there is a significant stretch of laminar boundary layer preceding the turbulent layer, total friction is the laminar portion up to x_crit plus the turbulent

portion from x_crit to l.

∼ Therefore, average shear-stress coefficient is lower than the prediction by Eqs.

(24.23) or (24.24).

→ Use transition formula

(24.25)

2.64

0.427

(log Re 0.407) Re

f

l l

C   A



where = correction term =

- Since A is a function of , the curve in Fig. 10 is given for - For flow along smooth walls

- For laminar flow:

/ Re_l

A (Re_crit) , Re_crit Ux^crit

f ^ 

1/2

1.328

f Re

l

C 

Re_crit Re_crit  5 10⁵

300,000  Re_crit  600,000

(17.26)

A 1,060 1,400 1,740 2,080 3,340 Re_{crit 3 10} ⁵ 4 10 ⁵ 5 10 ^{5 6 10 5} 10⁶

[Ex. 24.1] Turbulent boundary-layer velocity and thickness

An aircraft flies at 25,000 ft with a speed of 410 mph (600 ft/s).

Compute the following items for the boundary layer at a distance 10 ft from the leading edge of the wing of the craft.

① Frictional drag = surface resistance = skin drag

② Pressure drag = form drag

~ depends largely on shape or form of the body

For airfoil, hydrofoil, and slim ships: surface drag > form drag

For bluff objects like spheres, bridge piers: surface drag < form drag

(a) Thickness (laminar sublayer; ) at x = 10ft Air at El. 25,000 ft:

Find

Therefore, assume that turbulent boundary layer develops all the way from the leading edge.

4 2

3 10 ft / s

   ^

3 3

1.07 10 slug / ft

   ^

5

4

600( )

Re 5 10

crit 3 10

x

   



0.25 ~ negligible comparaed to 10

xcrit ft l ft

  

7 4

600(10)

Re 2 10

(3 10 )

x

Ux

 ^

   



*

4 u d   d^

 

 

2.58 7 2.58 2.58

0.370 0.370 0.370

0.0022 (log Re ) log 2 10 (7.30)

f

x

c    



2 3 2 2

0

1(1.07 10 )(0.0022)(600) 0.422 /

2 c U^f 2 lb ft

     ^ 

0

* 3

0.422

19.8 / 1.07 10

u  ft s

 ^

  



4

4 4

*

4 4(3 10 )

0.61 10 7.3 10

19.8 ft in

u

d    ^{ }   ^   ^

10 x ft

(b) Velocity at Eq. (24.1):

(c) Velocity at y / d  0.15 Use Eq. (24.14) – outer law

u

*

*

u u y

u ^ 

'

2 2 4

*

4

[ ]

(19.8) (0.61 10 )

79.7 / 13% of (3 10 )

600 /

Cf

y

u u ft s U

U ft s

d d











 

    





u

*

5.6 log 2.5

U u y

u d

      

 

600 5.6 log 0.15 2.5 19.8

u

  

600 91.35 49.5 459.2 / 76% of

u     ft s  U

y d

[Cf]

(d) Distance at and thickness Use Eq. (24.9) – inner law

* *

5.6 log y 2.5

u U u u

d

     

y ^{y /}d  ^0.15 d

*

*

5.6 log 4.9

u u y

u 

 

  

 

0.15 : At y

d ^{ 4}

459 19.8

5.6 log 4.9

19.8 3 10

y



 

   

4 4

19.8 19.8

log 3.26; 1839

3 10 3 10

y y

 

   

   

 

0.028' 0.33 0.8

y   in  cm (B)

Substitute (B) into

0.186' 2.24 5.7 0.15

y in cm

d    

3 4

0.186

At 10': 3049 3 10

' 0.61 10

x d

d ^

    

 ' 0.0003

d d ^

10 ft

[Ex. 24.2] Surface resistance on a smooth boundary given as Ex. 24.1 (a) Find displacement thickness d^*

Neglect laminar sublayer and approximate buffer zone with Eq. (24.14) (A)

(B)

*

0^h 1 u U dy d 



* /

0^h 1 u y , / 1

d h

U

d d d

d d

   





*

( ) / 0.15, U u 5.6 log y 2.5 (24.14)

i y d u

d

  

      

* * *

1 u 5.6u log y 2.5u 2.43ln y 2.5 u

U U d U d U

 

        

Divide (24.14) by U

(17.4)

lnx 2.3logx

Substituting (B) and (C) into (A) yields

(C)

*

( ) / 0.15, U u 8.6 log y (24.15)

ii y d u

d

  

      

* *

1 u 8.6u log y 3.74u ln y

U U d U d

   

        

   

* 0.15 1.0

* *

'/ 0.15

0.15 1

*

0.0003 0.15

*

2.43ln 2.5 3.74 ln

2.43 ln 2.5 3.74 ln

(19.8)

3.74 3.74 0.1184

600

y u y y u y

d d

U U

u y y y y y y y

U u U

d d

d

d d d d d

d d d d d d d

       

            

       

           

  

 

lnx dx xlnxx



(b) Local surface-resistance coeff. c_f Use Eq. (24.18) by Clauser

[Cf] = 0.0022 by Schultz-Grunow Eq.

*

0.1184 11.8%

d

d ^{ }

 

* * 4

3

1 600 0.022

3.96 log Re 3.04 Re 44, 000

3 10 3.96 log 44, 000 3.04

2.18 10 0.00218

f

f

c

c

d d 



     



 

   

cf