fm lecture notes 5

8/16/2019 FM Lecture Notes 5

1/20

CHAPTER 4

EULER’S EQUATION

Engineering Fluid Mechanics 8/E by Crowe, Elger, and Roberson

Copyright © 2005 by John Wiley & Sons, Inc !ll rights reser"ed

Dr . Ercan Kahya

#l$id %echanics


2/20

Review of Definitions

• Steady flow: velocity is constant with respect to time• Unsteady flow: velocity changes with respect to time

• Uniform flow: velocity is constant with respect to position

• Non-uniform flow: velocity changes with respect to position

Local acceleration:

– change of flow velocity with respect to time

– occurs when flow is unsteady

• Convective acceleration:

– change of flow velocity with respect to position

– occurs when flow is non uniform‐


3/20

EULER’S EQUATION

'o predict pressure variation in moving fluid

E$ler(s E)$ation is an e*tension o+ the hydrostatic e)$ation +or

accelerations other than gra"itational

RES-'E. #R/% !-1I3 EW'/ SEC/. -!W '/ ! #-I.

E-E%E' I THE FLOW OF INOM!"E##I$LE% IN&I#I' FL(I'

!ss$4e that the "isco$s +orces are ero


4/20

EULER’S EQUATION

lρaγsinα

Δl

ΔP=−−

l l ma F =Σ

lρaγz)( =+

∂

∂− p

l

n the ! direction"

for e!ample# !ρaγz)( =+∆

∆

− p x

!$% ρaγz)(γz)( x p p ∆−=+−+

&%' and &$' refer to the location with respect to the direction l (hen l ! direction" then &%' is theright*most point+ hen l z direction" &%' is the highest point+)

,a-ing the limit of the two terms at left side at a given time as Δl . /

When “a = ! " #uler e$uation reduces to hydrostatic e$uation%

ACCELERATION IS IN THE DIRECTION OF

DECREASING PIEZOMETRIC PRESSURE!!!


5/20

EULER’S EQUATION

/pen tan6 is accelerated to the right at a rate a*

#or this to occ$r 7 a net +orce 4$st act on the li)$id in the *8direction

'o acco4plish this7 the li)$id redistrib$tes itsel+ in the tan6 9!(:(C.;

– ,he rise in fluid causes a greater hydrostatic force on the left than the right side

. this is consistent with the re0uirement of &1 ma'

– 2long the 3ottom of tan-" pressure variation is hydrostatic in the vertical direction

!n e*a4ple o+ E$ler E)$ation is to the $ni+or4 acceleration o+ in a tan6


6/20

EULER’S EQUATION

• &he component of acceleration in the l direction: a x

cosα

2pply the a3ove e0uation along 2454

2pply the a3ove e0uation along 67

lρaγz)( =+∂

∂− p

l

α Cos ρadl

d x=− γz)(

α α

sin* == g

Cosa

dl

dz x

g

a x*tan =α


7/20

Ea"#$ 4.%& E'#$r’( $)'a*+,n

'he tr$c6 carrying gasoline 9) * +,+- .N/m0 and is slo1ing do1n at a rate o+

=05 4>s2

?; What is the press$re at point !@

2; Where is the greatest press$re & at what "al$e in that point@

lρaγz)( =+

∂

∂− p

l

Α


8/20

Solution:

!pply E$ler(s e)$ation along the top o+ the tan67 so is constant

!ss$4e that deceleration is constant

ress$re does not change with ti4elρaγz)( =+− p

dl

d

lρa=−dl dp

C l p +−= lρa

)γz()γz( toptopbottombottom p p +=+

ress$re "ariation is hydrostatic in the "ertical direction

!long the top the tan6

zρaγz)( =+− pdz

d E$ler(s e)$ation in "ertical direction<

9ote that a A0;


9/20

Centripetal (Raial! A""eleration

ar A centripetal 9radial; acceleration, 4>s2

Bt A tangential "elocity, 4>s

r A radi$s o+ rotation, 4 A ang$lar "elocity, rad>s

r r

V a t r

%%

ϖ == #or a li)$id rotatingas a rigid body<

- / r


10/20

#ressure Distri$ution in Rotatin% &low

When +low is rotating, +l$id le"el will rise away +ro4 the direction o+ net acceleration

C g r z p =−+

%

%%

ω

γ

Pressure variation in

rotating flow

! co44on type o+ rotating +low is the +low in which the +l$id rotates as a

rigid body

!pplying E$ler E)$ation in the direction nor4al to strea4lines and o$tward+ro4 the center o+ rotation 9O" I'E3R!'I3 E-ER ED!'I/ I 'E

R!.I!- .IREC'I/ #/R ! R/'!'I3 #-/W; res$lts in

ote that this is not the :erno$lli e)$ation


11/20

E'aple )*):

g

r

z

p

g

r

z

p

%%

%

%

%

%%

%

$

%

$$ ω

γ

ω

γ −+=−+

#ind the ele"ation di++erencebetween point ? and 2

p? A p2 A 0 and r ? A 0 , r 2A 0254 then F g

r

z z %

%

%

%

%$

ω

−=

2 G ?A 005?4 & ote that the s$r+ace pro+ile is parabolic


12/20

#ressure Distri$ution in Rotatin% &low

g

r z

p

g

r z

p

%%

%

%

%

%%

%

$

%

$$

ω

γ

ω

γ −+=−+

p = pressure, Pa

γ = specific weight, !m" z = elevation, m

# = rotational rate, radians!second

r = distance from the axis of rotation

$nother independent e%uation&

The sum of water heights in left and

right arms should remain unchanged


13/20

+ernoulli E,uation

Integrating E$ler(s e)$ation along a strea4line in a steady +low o+ an

incompressi2le% inviscid +l$id yields t he :erno$lli e)$ation<

C z

P

g

V

=++ γ %

%

z# Position

p8H# Pressure head9%8%g# 9elocity head

7# ntegral constant


14/20

Appli"ation of +ernoulli E,uation

'ernoulli #$uation:

( )ie*ometric pressure : p + γz ( +inetic pressure : ρV 2 /2

,or the steady flow of incompressile fluid inviscid fluid

the sum of these is constant along a streamline


15/20

A""#+ca*+,n ,0 1$rn,'##+ E)'a*+,n& S*a2na*+,n T'3$

γ

+=

γ

+ %%%$

%$ p

g%

9 p

g%

9

)(%

$%

%

$ P P V −= ρ

)(P %$ d l d P +== γ γ

))((%%

$ d d l V γ γ

ρ

−+=

gl V %$ =


16/20

Sta%nation Tu$e

p$ 8 γ

p%8γ

∆h)9% 8%g

$ %

γ +=

γ + %

%%$

%$ p

g%

9 p

g%

9

hg% p p

g%9 $%$ ∆=

γ −

=

9%/ : z$ z%


17/20

Appli"ation of +ernoulli E,uation: #itot Tu$e

%

$

h$

;


18/20

-ENTURI .ETER

'he Bent$ri 4eter de"ice 4eas$res the +low rate or "elocity o+ a +l$id thro$gh a pipe 'he

e)$ation is based on the :erno$lli e)$ation, conser"ation o+ energy, and the contin$ity

e)$ation

S,#$ 0,r 0#,5 ra*$

S,#$ 0,r "r$(('r$ 6+00$r$n*+a#

http://www.ajdesigner.com/venturi/venturiflow.phphttp://www.ajdesigner.com/venturi/venturi_pressure_differential.php


19/20

Class E'er"ises: 9roble4 2;


20/20

Class E'er"ises: 9roble4 5;

fm lecture notes 5

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