FULLY DEVELOPED TURBULENT PIPE FLOWCLASS 2 - REVIEW

= 0= 0 = 0

FSx + FBx = /t (cvudVol )+ csuVdAEq. (4.17)

FULLY DEVELOPED, STEADY, NO BODY FORCES, LAMINAR PIPE FLOW

V p2p1

w

w

l

CV

= (r/2)(dp/dx)Eq 8.13a

yx = (du/dy)

u = - (R2/4)(dp/dx)x [1 – (r/R)2]

= UC/L[1-(r/R)2]

Q = 0 uldy;V = Q/A

V/UC/L = 1/2

a

= (r/2)(dp/dx)

laminar

yx = (du/dy)+u’v’

uavg = UC/L(1-r/R)1/n

Q = 0 uldy;V = Q/A

V/UC/L = 2n2/(2n+1)(n+1)

a

turbulent

(empirical)

u(r)/Uc/l = (y/R)1/n = ([R-r]/R)1/n = (1-r/R)1/n

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

r/R

ua

vg

/Uc

l

Laminar Flowu/Uc/l = 1-(r/R)2

n=6-10

u(r)/Uc/l = (y/R)1/n

Eq. 8.30“one of the most important and useful equations

in fluid mechanics” Fox et al.

ENTER ENERGY EQUATION

Allows calculations of capacity of an oil pipe line, what diameter water main to install or pressure drop in

an air duct, ……

V12/ (2) + p1/() + gz1=V2

2/ (2) + p2/() + gz2 + hlT

hlT has units of enery per unit mass [V2]

“one of the most important and useful equations in fluid mechanics” Fox et al.

V12/ (2g) + p1/(g) + z1=V2

2/ (2g) + p2/(g) + z2 + HlT

HlT has units of enery per unit weight [L] from hydraulics during 1800’s

A ½ V(r)2 V(r)dA = (dm/dt) ½ V

2

= [A V(r)3

dA ]/ {(dm/dt) V2}

V12/ (2g) + p1/(g) + z1=V2

2/ (2g) + p2/(g) + z2 + HlT

Turbulent Flow: V(r)/Uc/l = (1-r/R)1/n

Laminar Flow: V(r)/Uc/l = 1 – (r/R)2

= [A V(r)3 dA ]/ {(dm/dt) V2}

= 1 for potential flow = 2 for laminar flow

1 for turbulent flow

V12/ (2g) + p1/(g) + z1=V2

2/ (2g) + p2/(g) + z2 + HlT

= (Uc/l/V)3 2n2 / (3 + n)(3 + 2n)* = 1.08 for n = 6; = 1.03 for n = 10

V(r)/Uc/l = (y/R)1/n

+

Hl

V1avg2/ (2g) + p1/(g) + z1 = V2avg

2/ (2g) + p2/(g) + z2 + Hl

(Eq. 8.30)

(Eq. 8.34)

Hl

V = Q/Area

BREATH

(early 20th Century turbulent pipe flow experiments)

fF = wall /{(1/2) V2}

Similarity of Motion in Relation to the Surface Friction of Fluids Stanton & Pannell –Phil. Trans. Royal Soc., (A) 1914

~1914

fF = wall /{(1/2) V2}

fF = wall /{(1/2) V2}fD = (p/L)D/{(1/2) V2} = (p/L)2R2/2{ ½ V2} = 4wall /{(1/2) V2} = 4 fF

BREATH

(rough pipe turbulent flow experiments)

Original Data of Nikuradze

Stromungsgesetze in Rauhen Rohren, V.D.I. Forsch. H, 1933, Nikuradze

p U?

p uavg2

Newton believed that drag uavg2

arguing that each fluid particle would lose all their momentum normal to the body.

Drag = Mass Flow x Change in Momentum

Drag = dp/dt (UA)U U2A

Drag/Area U2

Sir Isaac Newton (1642 – 1727)

aside

Fully rough zone where have flow separation over roughness elements and p ~ V2

k* = u*/; k* < 4: hydraulically smooth4 < k* < 60 transitional regime; k* > 60 fully rough (no effect)

White 1991 – Viscous Fluid Flow

Curves are from average values good to +/- 10%

BREATH

(Moody Diagram)

Hl = f (L/D)V2/(2g)

f = 64/Re and is proportional to in laminar flow f is not a function of /D in laminar flow f = const. and is not a function of at high

enough Re turbulent flows in a rough pipe f is usually a function of /D in turbulent flows

laminar t u r b u l e n t

fD = (p/L)D/{(1/2) V2} Darcy friction factor

ReD = UD/

For new pipes, corrosionmay cause e/D for old pipesto be 5 to 10 times greater.

Curves are from average values good to +/- 10%

fF = -2.0log([e/D]/3.7 + 2.51/(RefF0.5)]

If first guess is: fo = 0.25[log([e/D]/3.7 + 5.74/Re0.9]-2

should be within 1% after 1 iteration

For turbulent flow in a smooth pipe and ReD < 105,

can use Blasius correlation: f = 0.316/ReD

0.25 which can be rewritten as:

wall = 0.0332 V2 (/[RV])1/4)

For turbulent flow and Re < 105

can use Blasius correlation: fD = 0.316/Re0.25

Which can be rewritten as:

wall =0.0332 V2 (/[RV]) PROOF

fD = 4 fF

0.316 1/4 / (V1/4 D1/4) = 4wall/(1/2 V2)

wall = (0.0395 V2) [1/4 / (V1/4 (2R)1/4)

wall = (0.0332 V2) [ / (VR)]1/4 QED

Question?Looking at graph – imagine that pipe diameter, length,

viscosity and density is fixed.Is there any region where an increase in V

results in an increase in pressure drop?

Question?Looking at graph – imagine that pie diameter and

kinematic viscosity and density is fixed.Is there any region where an increase in V

results in an increase in pressure drop?

Instead of non-dimensionalizing p by ½ V2; use D3 /( 2L)

Laminar flow

Turbulent flow

transition

From Tritton

pD3 /(2L)

Everywhere!!!!!!!

Some history ~

“Moody Diagram”

f = function of V, D, roughness and viscosityf is dimensionless

Antoine Chezy ~ 1770:for channels: V2P = ASextrapolate this for pipe:Hl = (4/C2)(L/D)V2

Gaspard Riche de Prony (1800)Hl = (L/D)(aV + bV2)

C; a and b are not dimensionlessC; a and b are not a function of roughness

Hl

Antoine Chezy

f = function of V, D and roughnessf is dimensionless

Hl

Hl

f is a function of and D

better estimates of f

Could be dropped for rough pipes

Traditional to call f the Darcy friction factor although Darcy never proposed it in that form

Hl

Combined Weisbach’s equation with Darcy and other data,

compiled table for fbut used hydraulic radius.

= w/( ½ Vavg2) prob 8.83

Hl

Eq. 8.34

4000< ReR < 80000

Full range of turbulentReynolds numbers

“ These equations are obviously too complex to be of practical use. On the other hand, if the function which they embody is even approximately valid for commercialsurfaces in general, such extremely important information could be made readily available in diagrams or tables.”

Re

f1/f

Re/f

“The author does not claim to offer anything particularly new or original, his aim merely

being to embody the now accepted conclusionin convenient form for engineering use.”

Hl

f = [p/(g)]D2g/(LV2) f = {[p/L]D}/{1/2V2}

Hl

THE

END