4-2 Turbulent Flow in Pipes

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Seoul National University

Ch. 4 Steady Flow in Pipes

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Contents

4.3 Turbulent flow in smooth pipe 4.4 Turbulent flow in rough pipes

4.5 Classification of smoothness and roughness

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Today’s objectives

 Review the shear stress for laminar and turbulent flows

 Apply Prandtl’s theory to the smooth pipe flows

 Figure out the wall law for flow in smooth pipe

 Similarly to the smooth pipe case, turbulent flow in rough pipe will be understood.

 Classifying the pipes whether they are smooth or rough

 Evaluating pipe friction factors in the given condition

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 Shear stress between adjacent layers in a simple parallel flow is determined by the viscosity and the velocity gradient

 If laminar flow is disturbed by an obstacle or roughness, then disturbances must be damped by the molecular viscosity (stable)

 However, in turbulent flow, inertia overcomes the viscous force and disturbance grows and becomes random motion.

1. Shear stress

2 2 I

e

V

F V l Vl Vl Vl

R F Vl

ρ ρ

µ µ µ ρ ν

= = = = =

4.3 Turbulent flow in smooth pipe

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 turbulent shear stress (Reynolds stress)

 Boussinesq modeled the Reynolds stress with gradient transport theorem (simply mimicking the molecular viscosity),and Prandtl

suggested the mixing length model

 Finally, total shear stress is the sum of the viscous shear stress and the turbulent shear stress

u v' '

τ = −ρ

2

' ' dv 2 dv 2 dv

dy dy d

u v l

l y

τ = −ρ = = ρ ^  ρ  

  =  

   

  

' '

m t

dv dv dv

dy d d v

y u

τ τ= + =τ µ + = µ y − ρ

(6)

 Shear stress in turbulent pipe flow

- t is maximum at the pipe wall due to viscous shear stress in the viscous sublayer.

- t is decreasing linearly with y from the wall.

0

0 1 y r

R R

τ τ= ^ − ^ = τ

 

y r

(7)

 This figure shows linear variation of total shear stress in a turbulent pipe flow.

 For smooth pipes, discussion (Section 7.3) strongly suggests the existence of a viscous sublayer near the pipe walls.

 Employing the Prandtl relationship by assuming the viscous stress is negligible over most of the flow

(9.12) where y is measured from the pipe wall

2. Prandtl’s mixing length theory

2 2

0 1 0 1

dv dv y dv y

dy dy R dy R

l l

τ ρ= ^ ^ − µ =τ ^ − ^ ⇒ ρ ^ ^ =τ ^ − ^

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3. Velocity profile of turbulent flow in pipe

 From Nikuradse’s measurement in experiments: smooth or rough but uniform sand grain, all velocity profiles could be represented by

(9.13) Differentiate Eq. 9.13 wrt y, then , and from (9.12)

Near wall, then in a pipe flow, length scale follows;

2 2

2 2 0 *

0

1 1 1 1

dv y

l y y

dy y R

y y

l l v

R R

ρ τ τ

ρ ρ

  ∝   ∝  − → ∝ − ∝ −

     

   

(9.14)

3.1 Nikuradse’s empirical equation

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Seoul National University Eq. (9.12) now becomes

Integrating Eq. 9.15 produces

(9.13)

2

0 2

0

2

*

dv

dy y

v dv

dy y y

τ ρκ τ ρ

κ κ

 

  =

 

= = _(9.15)

*

ln ln

c

v R C

C v

v R

v κ

κ

= +

= −

1 ln ( 0.4)

vc y

v R

v κ

κ

− = − =

3.2 Theoretical equation from Prandtl’s mixing length model

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Seoul National University [Remark]

 Near wall ,

Then (9.12) can be written

2

0 2 2

0 *

*

/

ln dv

dy y

v dv

dy y y

v v y C

τ ρκ τ ρ

κ κ

κ

 

 

 

=

+

 v=v_c at y=R (center)

 So,

*

1 ln ( 0.4)

vc y

v R

v κ

κ

− = − =

In all pipes (no matter whether rough or smooth)

2

2 2

0 1

dv y

y dy R

ρκ ^ ^ =τ ^ − ^

 

(

^y  ^R

)

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 For smooth pipe, at very near wall (viscous sublayer), velocity profile is linear and to match with Nikuradse’s experiments, then

(9.17)

 In terms of common logarithms,

 This is the general equation of the velocity profile for turbulent flow in smooth pipes.

4. Wall law

*

=2.5ln +5.5 ν

v y v

v

*

v =5.75log +5.5 ν

v y v

c

* *

(9.13) : v =2.5ln y + v

v R v

(9.17)

ν µ

= ρ

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 For smooth pipe, a viscous sublayer must exist near the smooth wall, and the velocity profile is given by

(9.7)

 The nominal extent of the viscous sublayer, y’, is obtained by finding the intersection of the viscous profile of Eq. 9.7 with the turbulent profile given by Eq. 9.17. The sublayer thickness 𝛿𝛿_𝜈𝜈 is given as

(9.18)

*

v

v = v y ν

* *

*

v v

v

v = =5.75log +5.5 ( )

ν ν

v v

at y y

δ δ

= ′

* ν

ν

δ ν

=11.6 δ =11.6 v

ν v

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Velocity distribution near a smooth wall

*

v

v = v y ν

*

v =5.75log +5.5 ν

v y v

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 Mean velocity, V

(9.19)

 Maximum velocity (velocity at the center)

 Subtract the mean from the maximum, then adjust to the experiment (9.20)

5. Mean velocity

R *

0 *

* 2

*

R 0

*

Q= v(2πrdr)=2πv 5.75log v (R-r) +5.5 rdr V Q v R

ν

= =5.75log +1.75

πR

v v ν

 

 

 

∫

c *

*

=5.75log +5.5 ν

v v R

v

c =1+4 v

V .07 f/8

v =V f/8*

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 Friction factor f can be obtained by introducing Eq. 9.4 into Eq. 9.19, and adjusting the result based on the experiments

(9.21)

 Introducing Eq. 9.4 into Eq. 9.18 yields the expression for the laminar sublayer thickness

(9.22)

 It means that laminar sublayer is decreasing with increase of Reynolds number

→ rough wall

6. Friction factor in turbulent flow of smooth pipe

( )

2.0 log 0 8

1 Re f .

f = −

v

*

11.6 11.6 32.8 d v d V f/

δ ν ν

= =

= d

8 Re f

Re Vd

= ν

(16)

( )

2.0 log 0 8

1 Re f .

f = −

(17)

 Rewrite Eq. (9.22)

 Substituting this into Eq. 9.21 yields the equation below

(9.23)

 For turbulent flow over smooth walls, the friction factor is a function of the ratio of the sublayer thickness to the pipe diameter.

Re 32.8

v

f

d

= δ

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Example problem #1 (pp. 334-335)

 Water at 20°C flows in a 75 mm diameter smooth pipeline. According to a wall shear meter, τ₀= 3.68 N/m². Calculate the thickness of the viscous sublayer, the friction factor, the mean velocity and flowrate, the centerline velocity, the shear stress and velocity 20 mm from the pipe centerline, and the head lost in 1,000 m of this pipeline.

– Thickness of the viscous sublayer – Friction factor

– Mean velocity – Flowrate

– Centerline velocity – Shear stress

– Head loss

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Example problem 1

– Thickness of the viscous sublayer

– Friction factor

– Mean velocity

– Flowrate

– Centerline velocity

– Shear stress

– Head loss

0

* ν

*

τ ν

v = =0.061m/s; δ =1 =0.19 mm

ρ 1.6

v

1 32.8

2.0 log / 0.8 f 0.018

f δ_ν d

 

=  − → =

 

*

=5.75logv +1.75; V=1.2 ν

R V

v 9 m/s

0.0057 3 /

Q =VA= m s

1.54 / 1 4.07 8

c

v f

v m s

V = + → =

2

25 2.45 /

y mm N m

τ ₌ =

2

2 20.4

L

h f l V m of w

g er

d at

= =

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 Before the generalization by Prandtl, von Karman, and Nikuradse, Blasius (1913) developed simpler equations.

 Blasius work has limited scope and empiricism.

• Valid only for 3,000<Re<100,000

• Often called seventh-root law because the turbulent velocity profile is given by

7. Blasius’ equation

/7

c

v y

1

v =   R

   

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 For 3,000<Re<100,000, Blasius showed that the friction factor could be closely approximated by the equation

(9.24)

 Derivation of Blasius velocity profile and shear stress.

 Blasius then assumed the velocity profile as a power relationship

( )( )

21 0

c

m

m 2

v y

v R

Q=V = v y 2

=

πR π R-y -dy

  

 

  

∫

(22)

Seoul National University Therefore,

But, wall shear stress could not be affected by pipe radius (R), so m = 1/7.

22

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Finally,

→ seventh-root law

At this moment,

/7

c

v y 1

v = R

  

1/4 2

c 0

c

ρ v τ μ

=0.0464 2

ρ v R

 

 

 

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 For the conditions of previous examples, calculate the friction factor, wall shear stress, centerline velocity and the velocity 25 mm from the pipe centerline using the seventh root law.

Check whether we can use Blasius eq. The Reynolds number is Then

The centerline velocity.

Example problem #2 (pp. 337-338)

6 2

0.075

Re 96, 750 100, 000

1 1.29 /

1 0 /

m

Vd m s

m s

ν ⁻

= = × = <

×

1.54 m/s in

the previous example

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 The shear stress

Example

3.68 Pa given in the previous example

0.25 2 c 0

c

ρv τ =0.0464 ν

v R 3.70Pa

2 =

 

 

 

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 Pipe friction in rough pipe will be governed primarily by the size and patterns of the roughness

 Velocity profile follows logarithm when the flow is turbulent.

 When roughness is high, the viscous layer will be canceled and viscosity may not be the important parameter.

 Experiment by Nikuradse proved that the viscosity can be replaced by the roughness in the rough wall condition.

4.4 Turbulent flow in rough pipes

*

=5.75l

v y

v og e +8.5

*

Smooth pipe: v =5.75log +5.5 ν

v y v

 

 

 

e = sand grain diameter

1. Velocity profile

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The mean velocity is

Since

With adjustment by experimental data by Nikuradse,

Rough pipe’s friction factor

*

=5.75log +

V R

v e 4.75

*

=2.03log v =V f

8

1 R

f e +1.68

=2.0log +

1 d

1.14

*

c =5.75l

v R

v og e +8.5

0^R (2 ) 2 * 0^R 5.75log R r 8.5

Q v rd rdr

r v e

π π ^ ^ ⁻ ^ ^

=

∫

=

∫

  + 

Q (flow rate) for pipe flow can be determined as

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( )

2.0 log 0 8

1 Re f .

f = −

2.0 log 1 1

1 d . 4

f = e +

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 The mean velocity in a 300 mm pipeline is 3m/s. The relative roughness of the pipe is 0.002, and the kinematic viscosity of the water is 9 x 10^-7m²/s. Determine the friction factor, the centerline velocity, the velocity 50 mm from the pipe wall, and the head lost in 300 m of this pipe under the assumption that the pipe is rough.

– Friction factor

– Velocity at y=50 mm from the wall – Head loss

IP 9.6; pp. 339-340

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– Friction factor

– The velocity at 50 mm from the wall

– Head loss

Example

2.0 log 1.14 0.023

1 4

f = de + → =f

* 0.16

8f 2 /

V

v = = m s

5

50 0

*

50 3.1

5.75log 8.5 7 /

v y

v m s

v = e + → =

2

2 10.7

L L

n

f l V m o

h f water

d h

g →

= =

*

5.75log 8.5 3.61 /

c

v R

v = e + → v = m s

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 For a turbulent flow in a smooth pipe, the characteristic length is the viscous sublayer thickness δ_v, and for a rough pipe it is the

roughness height e.

 In case of transition, must be a significant parameter.

 In laminar flow, viscous sublayer thickness is the radius of pipe since viscosity governs the whole flow in pipe.

 In turbulent flow,

4.5 Classification of smoothness and roughness

Eq. 9.22 e δ_ν

ν R δ =

32.8

ν d δ ₌

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 Now we can plot the friction factor versus

 For fully rough flow,

(9.31) Thus, it is convenient to plot versus

 For smooth flow,

(9.33)

2.1 Classification by friction factor

Re or

v

e f e

d δ

2.0 log 1 1

1 d . 4

f − e =

( )

2.0 log Re 0.8

2.0 log 2.0 log R

1 e 0.8

1 f

f

d f

e

e d f

= −

 

− =  −

 

2.0 og

1 l

f e

− d e Re

d f

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2.0 log 2.0 log Re 0.8 Smoot

1

h flow

d e f

e d

f

 

− =  −

1 9.28

Re( / )

: 2.0 log 1.14 2 log 1 (For commertial roughness) Colebrook

f f

d

e e d

 

− = −  + 

 

 

(34)

 Classifications

For smooth flow: Re 10; 0.3

For transition flow: 10< Re 200 For rough flow: 200 Re ; 6

v

e e

d e d

e e

d f f

f

δ

≤ ≤

<

≤ ≤

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 Colebrook Equation

• Nikuradse experimental results cannot be applied directly to

commercial pipes since the roughness patterns are entirely different, much more variable than the artificial roughness used by Nikuradse.

• Colebrook suggested a single equation for a highly turbulent flow which can be applied to both smooth and rough commercial pipes.

• Distinctions between smooth, transition, and rough flow are not present.

Commercial pipe

1 9.28

2.0 log 1.14 2 log 1 (For commertial Re

roughness) d

e e

d

f f

 

 

− = −  + 

 

 

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2.2 Classification based on velocity profile

 In all pipes, use Eq. 9.13 by Nikuradse

 In rough pipes, begin with Eq. 9.29

*

c =5.75l v -v

v og R

y

*

=5.75l

v y

v og e +8.5

c

*

=5.75log +8.5 (At cen

v R

v e ter)

c

*

=5.75 log -log

v -v R y

v e e

=5.75log R y

 

 

 

(2) (3) (1)

Subtract (2) from (3)

(4)

This means that Eq. 9.13 can be used for both smooth and rough pipes. ³⁶

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Classification based on velocity profile

 Now let’s modify the equation

For both smooth and rough, Eq. 9.13 For smooth flow, Eq. 9.17

→ Thus, in rough flow, A=8.5 from experiment.

c

* *

= v +5.75l v

v v og y

R

c

*

= v +5.75log e +5.75l

v R og y

e

c

*

y e

v e

v

=A+5.75log A= +5.75log

R

*

=5.75log v y +5.5 (Smooth p v

v ipe)

ν

= v e*

5.5+5.75log +5.75log ν

y e

*

=A+5.75log A=5.5+5.75

e log

y v e

(9.36) ν (9.37)

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Classification based on velocity profile

 Plot A versus (Roughness Reynolds Number) for Nikuradse’s data^{v e}^* ν

( )

ν *

δ <3.5 Smooth flow ν

3.5< δ <70 Transition flow ν

70

11.6/ e= v e

< δ v Wholly rough flow ν

11.6/ e= e

1.845 0.544 (70)

(3.5)

*

11. ν 6v δv =

38

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 Check Example #1 whether or not the flow is truly rough as assumed.

 The mean velocity in a 300 mm pipeline is 3m/s. The relative roughness of the pipe is 0.002 and the kinematic viscosity of the water is 9 x 10^-7m²/s. Determine the friction factor, the centerline velocity, the velocity 50 mm from the pipe wall, and the head lost in 300 m of this pipe under the assumption that the pipe is rough.

IP 9.7; p. 344

v* f 0.16

=V 8 = 2m s/

*

-7 2

0.162 m/s 0.0006 m

= =108 > 70 rough pipe

ν 9 10

e

m /s

v ×

× →

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Velocity profile equations

Laminar flow

Turbulent flow

Smooth pipe Rough pipe

Whole pipe

Near wall (Viscous sublayer)

-

2

*

v v y

= y-

v ν 2R

 

 

 

*

v v y

=5.75log +5.5

v ν

1 7

*

v y

: =

v R

Blasius  

  

*

v y

=5.75log +8.5

v e

*

v v y

v = ν _* ^*

v v y v = ν

c

*

-v =-2.5ln (All pi

v y

v R pes)

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𝑅𝑅 𝑚𝑚 : 0.01 𝜐𝜐_∗ 𝑚𝑚/𝑠𝑠 : 0.0057

0.0 0.1 0.1 0.2 0.2 0.3

0.0000 0.0500 0.1000 0.1500

y (m)

u (m/s)

smooth pipe (5.5) rough pipe (8.5)

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Friction factor equations

Laminar flow

Turbulent flow

Smooth pipe Rough pipe

64 f = Re

v

1 32.8

=2.0log δ -0.8 f

d

 

 

 

v

*

δ =11.6 v ν

0.25

0.316

: Re

Blasius f =

1 2.0logd 1.14

f = e +