457.309.02 Hydraulics and Laboratory .09 Turbulent flow in rough pipe Prepared by Jin Hwan Hwang (2019.04.03)

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Seoul National University

457.309.02 Hydraulics and Laboratory .09 Turbulent flow in rough pipe

Prepared by Jin Hwan Hwang

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Today’s objectives

§ 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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§ Pipe friction in rough pipe will be governed by the size and patterns of the roughness (like as sediment particles)

§ As we learned through the last class, velocity profile follows logarith m no matter what flow is laminar or turbulent.

§ When roughness is high, the viscous layer will be canceled and visc osity may not be the important parameter.

§ Experiment proved that the viscosity can be replaced by the roughn ess in the rough wall condition.

§ In this condition Q (flow rate) can be determined as

Turbulent flow – Rough pipes

u

u_* = 5.75 log y

e + 8.5 (Rough pipe, when e is roughness)

Q = u(2π^{r dr}⁾

0

∫

R ⁼ ²^π^u^*

∫

⁰^R^⎡_⎣⎢^{5.75 log}^⎛_⎝⎜ ^R_e⁻^r^⎞_⎠⎟ ⁺ ^8.5^⎤_⎦⎥^{r dr}

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§ Therefore, the mean velocity is

§ Since

§ With experimental adjustment,

Rough pipes’ friction factor

Turbulent flow – Rough pipes

V

u_* = 5.75 log R

e + 4.75

u_* = V f 8 1

f = 2.03log R

e +1.68

1 = 2.0 log d +1.14

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§ The mean velocity in a 300 mm pipeline is 3m/s. The relative rough ness 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 v elocity 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

– Centerline velocity

– The velocity at 50 mm from the wall – Head loss

Example

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– Friction factor

– Centerline velocity

– The velocity at 50 mm from the wall

– Head loss

Example

1

f = 2.0 log d

e +1.14 u_c

u_* = 5.75 log R

e + 8.5, u_* =V f 8

u₅₀

u_* = 5.75 log y₅₀

e + 8.5 h_L = f l

d V² 2g_n

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§ What parameters do we need to compare to know whether flow is s mooth or rough?

§ To cancel this, we need to see the ratio between this and that.

– This is viscous sublayer thickness and roughness height

§ What is the viscous sublayer thickness in Laminar flow?

– Since viscosity governs overall depth in pipe, R should be the thickness.

– Therefore, in a Laminar flow, no matter what smooth or rough,

§ In turbulent flow,

Classification of smoothness and roughness

f = 64 Re

e δ_ν ⁼

e /d δ_ν ^/^d ⁼

e/d

32.8 / Re f = e d

⎛⎝⎜ ⎞

⎠⎟

Re f 32.8 e Re f = 32.8 e

δ

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§ Now we can plot the results with

Classification of smoothness and roughness

e

d Re f and 1

f − 2.0 log d e

1

f −2.0 logd

e =1.14 Fully rough flow 1

f =2.0 log Re

(

f

)

⁻^0.8

1

f −2.0 logd

e =2.0 log e

dRe f

⎛⎝⎜ ⎞

⎠⎟ −0.8 Smooth flow

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§ Classifications

Classification of smoothness and roughness

For smooth flow: e

d Re f ≤10 For transition flow: 10< e

d Re f < 200 For rough flow: 200 ≤ e

d Re f

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Classification based on velocity profile

§ Another means of classification of the roughness effects is to use th e velocity profiles directly. In all pipes (page 331),

§ In rough pipes, u_c −u

u_* = 5.75 log R y

u

u_* = 5.75 log y

e + 8.5 u_c

u_* = 5.75 log R

e + 8.5 (At center) u_c − u

u_* = 5.75 log R

e − log y e

⎛⎝⎜ ⎞

⎠⎟

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Classification based on velocity profile

§ Now let’s modify the equation – For both (smooth and rough)

§ In smooth flow, A=8.5 from experiment.

§ Check the page 332, equation 9.17, u

u_* = u_c

u_* + 5.75 log y R

= u_c

u_* + 5.75 log e

R + 5.75 log y e

= A+ 5.75 log y e A = u_c

u_* + 5.75 log e R

u

u_* = 5.75 log u_*y

ν ⁺ ^5.5 (Smooth pipe)

= 5.5+ 5.75 log u_*e

ν ⁺ ^{5.75 log} y e

= A+ 5.75 log y e A = 5.5+ 5.75 logu_*e

ν

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Classification based on velocity profile

§ In the previous equation is the Roughness Reynolds Number. u_*e ν

11.6 /δ_ν

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ê ⁼ û^*ê ^/^ν ^≤ ^3.5 Smooth flow