turbulent flow first semester 2013-2014[compatibility mode]

CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 1

By

Dr. Ajit Pratap Singh,

Civil Engineering Department,

BITS, Pilani-333031

Turbulent Flow through PipesObjective

� Theoretical discussion on Turbulent flow includingturbulent shear stress and Prandtl’s mixing length theory.

� To explain the development of velocity boundary layer inpipe and to explain how to get length required to establish afully developed flow.

� To study Velocity Distribution in a pipe for turbulent flowand to obtain velocity profile.

� To classify hydrodynamically smooth and rough pipes.

� To measure the pressure drop in the straight section ofsmooth, rough, and packed pipes as a function of flow rate.

� To correlate this in terms of the friction factor and Reynoldsnumber.

� To compare results with available theories and correlations.

� To determine the influence of pipe fittings on pressure drop.

Theoretical Discussion

Fluid flow in pipes is of considerable importance in

process.

•Animals and Plants circulation systems.

•In our homes.

•City water.

•Irrigation system.

•Sewer water system

Turbulent flow�When fluid flow at higher flowrates, the

streamlines are not steady and straight and the

flow is not laminar. Generally, the flow field

will vary in both space and time with

fluctuations that comprise "turbulence”

�In turbulent flow the fluid particles are in

extreme state of disorder, their movement is

haphazard and large scale eddies are developed

which results in complete mixing of the fluid.

�For this case almost all terms in the Navier-

Stokes equations are important and there is no

simple solution

∆∆∆∆P = ∆∆∆∆P (D, µµµµ, ρρρρ, L, U,)

uz

úz

Uz

average

ur

úr

Ur

average

p

P’

p

average

Time

Laminar vs Turbulent Flow

• Laminar • Turbulent

Turbulent flow

Laminar Flow

Turbulent Flow


Definition of turbulent

flow(Hinze)

“Turbulent fluid motion is an irregular

condition of flow in which the various

quantities show a random variation in time

and space, so that statistically distinct

average values can be discerned”

Reynolds Experiment

• Reynolds Number

• Laminar flow: Fluid moves in

smooth streamlines

• Turbulent flow: Violent mixing,

fluid velocity at a point varies

randomly with time

∝>

−

∝<

=2lowfTurbulent4000

flowTransition40002000

flowLaminar2000

Re

Vh

VhVD

f

f

µ

ρ

Laminar Turbulent

Boundary layer buildup in a pipe

Pipe

Entrance

v vv

Because of the shear force near the pipe wall, a boundary layer forms on the

inside surface and occupies a large portion of the flow area as the distance

downstream from the pipe entrance increase. At some value of this distance the

boundary layer fills the flow area. The velocity profile becomes independent of

the axis in the direction of flow, and the flow is said to be fully developed.

Pipe Entrance• Developing flow

– Includes boundary layer andcore,

– viscous effects grow inwardfrom the wall

• Fully developed flow

– In between the entrancesection and section AA, wherethe boundary layer thicknessequal to the radius of the pipe,the velocity of pipe will varyfrom section to section due tovariation in thickness of BL

– Shape of velocity profile issame at all points along pipeafter a section AA. The flow inpipe will be then truly uniformand the flow is said to beestablished.

eL

Entrance length LeFully developed

flow region

Region of linear

pressure drop

Entrance

pressure drop

Pressure

x

≈

flowTurbulent for 50

or 4.4Re

flowLaminar for R 0.07

D

L 1/6

e

e

Problem

• A 15-mm-diameter water pipe is 20 m long

and delivers water at 0.0005 m3/sec at 200C. What fraction of this pipe is taken up by

the entrance region so that after this region

fluid flow becomes fully developed? Take ν= 1.01x10-6 m2/sec.


Fully Developed Turbulent Flow: Overview

One see fluctuation or randomness on the macroscopic scale.

One of the few ways we can describe turbulent flow is by describing it in terms of time-averaged means and fluctuating parts.

mean fluctuating

Turbulent Flow – Shear stressesThere are several theoretical models available for the

prediction of shear stresses in turbulent flow. However,

there is no general, useful model that can accurately

predict the shear stress for turbulent flow.

� We estimate shear stress by using experimental data,

semiempirical formulas and dimensional analysis

Modelling turbulent flow

• Why not solve the Navier-Stokes equations?

– No analytical solution possible

– In a computer, every small whirl would need to

be modelled. Even a 10cm3 volume would

require ~ 100,000,000 nodes

• Need to simplify

– Crossing of streamlines transfers momentum

between parts of the flow


Now, shear stress:

However, for turbulent flow.

Laminar Flow:

Shear relates to random motion as particles glide smoothly past

each other.

Shear comes from eddy motion which have a more random motion

and transfer momentum.

For turbulent flow:

Is the combination of laminar and turbulent shear. If there are no fluctuations, the result goes back to the laminar case. The turbulent shear stresses are

positive, thus turbulent flows have more shear stress.

Turbulent Flow:


The turbulent shear components are known as Reynolds Stresses.

Shear Stress in Turbulent Flows: Turbulent Velocity Profile:

In the outer layer: τtirb > τlaminar 100 to 1000 time greater.

In viscous sublayer: τlaminar > τturb 100 to 1000 times greater.

The viscous sublayer is extremely small.

Apparent shear stress

• Apparent shear stress - Boussinesq(1877)

– Turbulence provides a shear in the flow in

addition to viscous shear

– Even in low viscosity fluids, there will be a

shear

– Propose an apparent viscosity

– In general µT>µ , so ordinary viscosity can be

neglected

dy

udµτ TT =


Reynolds stresses

Object: to include the random fluctuations in the

Navier-Stokes equations for the mean flow.

Method: represent all quantities by the mean plus fluctuation.

uuu ′+= ppp ′+= and so on

(T and ρ must also be considered for compressible flow)

Putting these into the Navier-Stokes equations and separating

out the time averaged and variable terms leads to a

modified set of equations

Reynolds stresses-continuityContinuity - what goes in must come out!

In laminar flow: 0z

w

y

v

x

u=

∂

∂+

∂

∂+

∂

∂

( ) ( ) ( )0

z

ww

y

vv

x

uu=

∂

′+∂+

∂

′+∂+

∂

′+∂In turbulent flow:

Separating: 0z

w

y

v

x

u

z

w

y

v

x

u=

∂

′∂+

∂

′∂+

∂

′∂+

∂

∂+

∂

∂+

∂

∂

Taking a time average: 0z

w

y

v

x

u=

∂

∂+

∂

∂+

∂

∂

0z

w

y

v

x

u=

∂

′∂+

∂

′∂+

∂

′∂Therefore, the fluctuating part

also satisfies the continuity equation

Reynolds stresses - Navier Stokes

Similarly, the N-S equations become (Schlichting, Ch 18)

∂

∂+

∂

∂+

∂

∂=

∂

′′∂+

∂

′′∂+

∂

′∂−

∂

∂+

∂

∂+

∂

∂+

∂

∂−

z

uw

y

uv

x

uuρ

z

wu

y

vu

x

uρ

z

u

y

u

x

uµ

x

pρg

2

2

2

2

2

2

2

x

Shear stresses

Direct stress

Reynolds stresses

• Compared to the laminar Navier-Stokes

equation, one new term has been added.

The other terms have been averaged to

remove the time dependency.

• The terms on the left are the forcing terms,

gravity, pressure, viscosity and turbulence

• The terms on the right are the response

terms)

s

uu

t

u

dt

du:(remember

∂

∂+

∂

∂=

Reynolds stresses in 2D

x

yu

∂

∂+

∂

∂+

∂

∂=

∂

′′∂+

∂

′′∂+

∂

′∂−

∂

∂+

∂

∂+

∂

∂+

∂

∂−

z

uw

y

uv

x

uuρ

z

wu

y

vu

x

uρ

z

u

y

u

x

uµ

x

pρg

2

2

2

2

2

2

2

x

•No z,w terms

•Steady, turbulent flow in x

direction

•Ignore gravity

Reynolds stresses in 2D

y

uρv

y

vuρ

y

uµ

x

p2

2

∂

∂=

∂

′′∂−

∂

∂+

∂

∂−

In 2D, the turbulent N-S equation therefore reduces to:

Note that there are now two shear stress terms.

Re-writing:y

uρvvuρ

y

uµ

yx

p

∂

∂=

′′−

∂

∂

∂

∂+

∂

∂−

vuρy

uµτ ′′−

∂

∂=

In turbulent flow, therefore

the shear stress is given by


Reynolds and Boussinesq

Boussinesq proposed an additive turbulent shear stress:

y

uµ

y

uµτ T

∂

∂+

∂

∂=

So the additive term is equivalent to the Reynolds’ stress.

However, we need to know values for vu ′′

in order to use this

Are the Reynolds’ stresses related to the flow velocity?

Prandtl‘s Mixing Length Theory

• That distance in the transverse direction which

must be covered by a lump of fluid particles

traveling with its original mean velocity in order

to make the difference between its velocity and the

velocity of new layer equal to the mean transverse

fluctuation in turbulent flow

Prandtl‘s Mixing Length Theory

)y(u

lump of

turbulence

x,u

mean

velocity

y,v

mixing length l

defined as that downstream distance, which is

needed for the lump of turbulence to be

completely mixed with the surrounding fluid

turbulent shear

flow along

solid wall(not valid close

to the wall)

lll

y

ulu

y

ulv

==

∂

∂⋅±′⇒

∂

∂⋅+′

21

21 ~~

lump of

turbulence

(mixed)

v~ ′′′′

u~ ′′′′

y

u

∂∂∂∂

∂∂∂∂

Prandtl’s Mixing Length…

•Analogous to the kinetic theory of gases

•Used because ‘it works’

Suppose ‘lumps’ of fluid move

randomly from one shear layer

to another, a distance l apart.

This carries momentum and the

velocity difference must

therefore be related to the

turbulence

y

y1

y2l

(y)u

Prandtl’s Mixing Length

y

uu

∂

∂∝′ l

Turbulence is even in all directions (homogeneous)

y

uuv

∂

∂∝′∝′ l

2

2

y

uvu

∂

∂∝′′ l

So the Reynolds shear stress must be proportional to the

square of the mixing length times the velocity gradient:

CE F312: Hydraulic Engineering by Dr. A. P. Singh

Prandtl’s Mixing Length…

vuρy

uµτ ′′−

∂

∂=

Returning to the equation for the shear stress:

2

2

Ty

uρvu

y

uµ

∂

∂∝′′−=

∂

∂lρ

y

uµ

y

uµτ T

∂

∂+

∂

∂=

2

2

Ty

uρµ

∂

∂= l

This gives a direct relationship between turbulent ‘viscosity’

and velocity gradient in the flow


Total Shear stress at any point

• Total Shear stress at any point is the sum of the viscousshear stress and turbulent shear stress and may beexpressed as

• The significance of Prandtl’s turbulent shear stressequation is that it is possible to make suitable assumptionsregarding the variation of the mixing length

2

2

dy

vdρl

dy

vdµτ

+=


Prantdl’s Mixing Length

• We still need a value for the mixing length, l.

• In free turbulence, l will be constant.

• In wall generated turbulence, l will vary as

the distance from the wall. (l=ky)

• For a smooth wall y=0, l=0

• For a rough wall y=0, l=k (the surface

roughness)


Mixing length measurement in pipes


The Universal Law of The Wall

First define the friction velocity, V*, which is characteristic

of the fluctuating flow:

y

uvuV

*

∂

∂=′′= l

Assuming that the shear stress remains constant throughout,

then V* = const (typically V*~4% u)

y

ukyV*

∂

∂=

Using the relation from above, l=ky, gives the differential

equation


• In general a boundary with irregularities of largeaverage height k, on its surface is considered to berough boundary and the one with smaller k valuesis considered as smooth boundary.

• Hopf found two types of roughness:

– Coarse, dense roughness where f is a function of roughness ratio, k/D, and is independent of the Reynolds number

– Gentle, less dense roughness, where f is a function of both Re and roughness ratio

– The significant factor is the roughness height compared to the laminar sub-layer.

Hydrodynamically Smooth and

Rough Pipe Boundaries




A systematic study is complicated by different types:

1. Shape

2. Height

3. Density



• We should also consider flow and fluid characteristics forproper classification of smooth and rough boundaries.

• If k is the average height of rough projections on the surfaceof the plate and δ is the thickness of the boundary layer,then the relative roughness (k/ δ) is a significant parameterindicating the behavior the boundary surface of a plate.

• If the boundary layer is turbulent from the leading edge ofthe plate, the front portion of the plate will act ashydrodynamically rough followed by transition region andthe downstream portion of the plate will behydrodynamically smooth if the plate is sufficiently long.






δ

Turbulent

Laminar

• As the flow outside the laminar sub-layer is turbulent, eddies of various sizes are present which try to penetrate through the laminar sub-layer. But due to greater thickness of the laminar sub-layer the eddies cannot reach the surface irregularities and thus the boundary act as a smooth boundary.

• In the laminar sub-layer, any vortices generated by the

roughness is damped out, so if k<δ, then the law of friction

for smooth pipes will apply


• With the increase in Reynolds number, the thickness ofthe laminar sub-layer decreases, and it can evenbecome much smaller than the average height k, ofsurface irregularities. The irregularities will thenproject through the laminar sub-layer and laminar sub-layer is completely destroyed. The eddies will thuscome in contact with the surface irregularities and largeamount of energy loss will take place and thus theboundary act as a rough boundary.




From Nikuradse’s experiment

• Hydrodynamically smooth pipe

• Transition region in a pipe

• Hydrodynamically Rough Pipe

50.2δk

'<

0.6δk

25.0'<<

0.6δ

k'

>


• Hydrodynamically smooth Plate

• Plate in Transition region

• Hydrodynamically Rough

5υ

kV s* <

70υkV

5 s* <<

70υkV s* >

Where ks is equivalent

sand grains roughness

defined as that value of

the roughness which

would offer the same

resistance to the flow

past the plate as that of

due to the actual

roughness on the surface

of the plate.


From Nikuradse’s experiment

• Hydrodynamically smooth pipe

• Transition region in a pipe

• Hydrodynamically Rough Pipe

5V

or 50.2δ

k *

'≤

≤

υ

k

70V

3or 0.6δk

25.0 *

'<

<<<

υ

k

70V

or 0.6δk *

'≥

≥

υ

k



The Universal Law of The Wall

+=

+=

=

R

yLog2.5V v v

CyLogK

V v

dy

dvyρkτ

e*max

e*

2

220

=

−

y

RLog2.5V

V

vv e*

*

max

CE F312: Hydraulic Engineering by Dr. A. P. SinghCE F312: Hydraulic Engineering

by Dr. A. P. Singh


Nikuradse’s experimental studies of turbulent

flow in smooth pipes have also shown that

• In smooth pipes of

any size the value of

the parameter

107

δy

V

0.108υyy y for 0.108

υyV

V

11.6υδδ y for 11.6

υyV

''

*

'''

*

*

''*

=⇒

=⇒==

=⇒==


Velocity Distribution for turbulent

flow

• Velocity Distribution in a hydrodynamically smooth pipe

• Velocity Distribution in a hydrodynamically Rough Pipes

5.5υ

yV log 5.75

V

v *10

*

+=

8.5k

y log 5.75

V

v10

*

+

=


Velocity Distribution for turbulent

flow in terms of Mean Velocity (V)

• Velocity Distribution in a hydrodynamically smooth pipe

• Velocity Distribution in a hydrodynamically Rough Pipes

75.1υRV

log 5.75V

V *10

*

+=

4.75k

R log 5.75

V

V10

*

+

=



Law of the Wall

*V

u

υyV

ln*

Turbulent

layer

Laminar

sub-layer

Buffer

zone

5.5


Turbulent Flow – Velocity Profile

For turbulent flow in tubes the time-averaged velocity profile can be

expressed in terms of the power law equation. n =7 is usually a good

approximation. n/1

R

r1

V

u

−=

where V is the velocity

at the centerline


Losses due to Friction/The Friction

FactorFor turbulent flow there is no rigorous theoretical treatment available. In

order to determine an expression for the losses due to friction we must

resort to experimentation.

D

V 2L

hf ∝

By introducing the friction factor, f:

D

V f

2L

hf = where

)2/)(/(2

gVDL

hf

f=

where L=length of the pipe,

D=diameter of the pipe, V=velocity,


Flow in Pipes

Hopf found two types of roughness:

•Coarse, dense roughness where f is a function of

roughness ratio, k/D, and is independent of the Reynolds

number

•Gentle, less dense roughness, where f is a function of

both Re and roughness ratio

The significant factor is the roughness height compared to

the laminar sub-layer.


Nikuradse’s Experiments• In general, friction factor

• Function of Re and

roughness

• Laminar region

– Independent of

roughness

• Turbulent region

– Smooth pipe curve

• All curves

coincide @

~Re=2300

– Rough pipe zone

• All rough pipe

curves flatten out

and become

independent of Re

Re

64=f

( )Blausius

Re4/1

kf =

Rough

Smooth

Laminar Transition Turbulent

Blausius OK for smooth pipe

)(Re,D

eFf =

Re

64=f

2

9.010Re

74.5

7.3log

25.0

+

=

D

e

f


The Friction FactorThe mechanical energy equation can be written:

Knowledge of the friction factor allows us to estimate the

loss term in the energy equation

−=−+−+

ρ−

ρ 24)()

22()(

2

12

21

2212 V

D

Lf

m

Wzzg

VVPP shaft

&

Or in terms of heads:

−=−+−+

ρ−

ρ g

V

D

Lf

gm

Wzz

g

V

g

V

g

P

g

P shaft

24)()

22()(

2

12

21

2212

&



Friction factor: The Moody ChartThe Moody Chart (Figure 14.10 textbook) provides a convenient

representation of the functional dependence f = f(Re, κ/D)

� For laminar flow:

f = 64 / Re

� For turbulent flow:

+−=

f

k

f Re

51.2

3.7

/D log 2

1Colebrook formula

� For turbulent flow, with Re<105 and for hydraulically smooth surfaces:

4/1Re

316.0=f Blasius formula

+

ε+⋅=

3/16

Re

10

D000,201001375.0f



( )

( )

( ) 237.0

10

10

745

Re

0.221 0.0032 f

relation emperical sNikurdse'

8.0fRelog 2.0f

1

found Nikurdse results alexperiment fromBut

91.0fRelog 2.03f

1

pipessmooth for )104 to105 from Refor (Valid 10 ReFor

+=

−=

−=

××>


( )

( ) 74.1Re/klog 2.0f

1

found Nikurdse results alexperiment fromBut

68.1Re/klog 2.03f

1

pipesRough in

flowTurbulent for )104 Refor (Valid 10 ReFor

10

10

35

+=

+=

×>>

Surface Roughness

Additional dimensionless group κκκκ/D need

to be characterize

Thus more than one curve on friction factor-Reynolds number plot

Fanning diagram or Moody diagram

Depending on the laminar region.

If, at the lowest Reynolds numbers, the laminar portion

corresponds to f =16/Re Fanning Chart

or f = 64/Re Moody chart


Variation of friction factor for

Commercial pipes

White Equation

Colebrook formula

+=

−

fRe

R/K18.71log 2.0-1.74

K

Rlog 2.0

f

11010

+−=

fRe

51.2

3.7

k/Dlog 2.0

f

110


• The Colebrook equation is

implicit in f, and thus the

determination of the

friction factor requires

some iteration. An

approximate explicit

relation for was given by

S.E. Haaland in 1983.

• The results obtained from

this relation are within 2

percent of those obtained

from the Colebrook

equation.

+≅

11.1

7.3

/

Re

6.9log 8.1

1 Dk

f



Moody Diagram


Moody Diagram

0.010

0.100

1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08

Re

0.05

0.04

0.03

0.02

0.015

0.010.008

0.006

0.004

0.002

0.0010.0008

0.0004

0.0002

0.0001

0.00005

0

laminar flow

ε/D


Fanning Diagram

f =16/Re

1

f= 4.0 * log

D

ε+ 2.281

f= 4.0 * log

D

ε+ 2.28− 4.0 * log 4.67

D /ε

Re f+1


Following observations from the Moody

chart:

• For laminar flow, the friction factordecreases with increasing Reynolds number,and it is independent of surface roughness.

• The friction factor is a minimum for asmooth pipe (but still not zero because ofthe no-slip condition) and increases withroughness. The Colebrook equation in thiscase (k=0) reduces to the Prandtl equationexpressed as


• The transition region from the laminar to turbulentregime (2300 < Re < 4000) is indicated by theshaded area in the Moody chart. The flow in thisregion may be laminar or turbulent, depending onflow disturbances, or it may alternate betweenlaminar and turbulent, and thus the friction factormay also alternate between the values for laminarand turbulent flow. The data in this range are theleast reliable. At small relative rougnesses, thefriction factor increases in the transition regionand approaches the value for smooth pipes.


Equivalent roughness values for new

commercial pipesMaterial Roughness, k (mm)

Glass, plastic 0 (smooth)

Concrete 0.9 to 9

Wood stave 0.5

Rubber, smoothed 0.01

Copper or brass tubing 0.0015

Cast iron 0.26

Galvanized iron 0.15

Wrought iron 0.046

Stainless steel 0.002

Commercial steel 0.045



Type of Problems

• Determining the head-loss or pressure drop from

the given values of Q, L, D, pipe roughness κ,kinematic viscosity ν.

• Determining the Q from the given values of head-

loss or pressure drop due to friction, L, D, pipe

roughness κ, kinematic viscosity ν.

• Determining the dia of pipe from the given values

of head-loss or pressure drop due to friction, Q, L,

pipe roughness κ, kinematic viscosity ν.


• Type 1

– Calculate Re and k/D from the given data

– Obtain f from the Moody’s chart

• Type 2

– Calculate k/D from the given data and Re√f from

– Using Coolebrook formula and the above equation, Obtain f

– Obtain Re from the Moody’s chart and hence Q

1/2

2

f

LV

D2gh

υVD

fRe

=


• Type 3: Dia is unknown

– Assume a suitable value of f and calculate Dia

from Darcy-Weisbatch equation

– With this trial value of D, calculate k/D and Re

– With this k/D and Re, calculate f from Moody’s

diagram

– Repeat the process till f becomes same


Swamee and Jain in 1986 proposed the following

explicit relations that are accurate to within 2% of

the Moody chart

2000Re hgD

L3.17

3.7D

kln

L

hgD0.965Q

0.5

L

3

20.5

L

5

>

+

−=

υ

8

260.04

5.2

9.4

4.75

L

21.25

103Re5000

10k/D10

gh

LυQ

gh

LQk0.66D

×<<

<<

+

=

−−


CE C371: Hydraulics & Fluid Mechanics by Dr. A. P. Singh

APPARATUS

Pipe Network

Rotameters

Manometers

Problem

• Water at 15 0C is flowing steadily in a 5

cm-diameter horizontal pipe made of

stainless steel at a rate of 0.34 m3/min.

Determine the pressure drop, the head loss,

and the required pumping power input for

flow over a 61m-long section of the pipe.



Problem

• For flow in open channels assume turbulent

shear to the constant т = тo and the mixing

length variation with y is given by l = 0.40 y

for y ≤ 0.20 D, and l = 0.08D for y ≥ 0.20D

where D is the depth of flow. Obtain the

velocity distribution law which will satisfy

the boundary condition, v = V at y = D.


• A 15-mm-diameter water pipe is 20 m long

and delivers water at 0.0005 m3/sec at 20 0C. What fraction of this pipe is taken up by

the entrance region so that after this region

fluid flow becomes fully developed? Take ν= 1.01x10-6 m2/sec.


Problem

• After 15 years of service a steel water main

0.6 m in diameter is found to require 40%

more power to deliver the 300 liters/second

for which it was originally designed.

Determine the corresponding magnitude of

the rate of roughness increase α. Take

kinematic viscosity of water ν = 0.015

Stokes.


Problem• A new reservoir will use gravity to supply drinking water

to a water treatment plant serving several surrounding

towns as shown in Figure. The required flow rate is 0.315

m3/sec. The surface of the reservoir is 61 m above the plain

where the water treatment plant is located, and the supply

pipe is commercial steel, 914.4 mm in diameter. If the

minimum pressure required at the water treatment plant is

347.7 kpa (gage), how far away can the reservoir be

located with this size pipe? Assume that minor losses are

negligible and that the water is at 283.1 K. The average

height of the pipe wall roughness protrusions may be taken

as 0.0458 mm. Take kinematic viscosity of water ν =

0.13x10-5 m2/sec .



• A commercial new galvanized iron service

pipe from a water main is required to

deliver 200 L/s of water during a fire. If the

length of the service pipe is 35 m, the

allowable head loss in the pipe is 50 m and

kinematic viscosity of water at 20 0C is 1.00

x 10-6 m2/sec, what will the pipe diameter to

be used for this purpose?



• Water at 200C is to be pumped from a reservoir (ZA = 2 m)

to another reservoir at a higher elevation (ZB = 9 m)

through two 25-m long plastic pipes connected in parallel.

The diameters of the two pipes are 3 cm and 5 cm. Water

is to be pumped by a 68 percent efficient motor-pump unit

that draws 7 kW of electric power during operation. The

minor losses and the head loss in the smaller single pipes

that connect both the parallel pipes to the two reservoirs

are considered to be negligible. Determine the total flow

rate between the reservoirs and the flow rates through each

of the parallel pipes.

CE F312: Hydraulic Engineering by Dr. A. P. Singh CE F312: Hydraulic Engineering by Dr. A. P. Singh

Pipe Flow Summary

�The statement of conservation of mass, momentum and

energy becomes the Bernoulli equation for steady state

constant density of flows.

� Dimensional analysis gives the relation between flow rate and

pressure drop.

�Turbulent flow losses and velocity distributions require

experimental results.

�Experiments give the relationship between the fraction factor

and the Reynolds number.

� Head loss becomes minor when fluid flows at high flow rate

(fraction factor is constant at high Reynolds numbers).


Images - Laminar/Turbulent Flows

Laser - induced florescence image of an

incompressible turbulent boundary layer

Simulation of turbulent flow coming out of a

tailpipe

Laminar flow (Blood Flow)

Laminar flowTurbulent flow


turbulent flow first semester 2013-2014[compatibility mode]

Engineering

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