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Worked Out Examples

(Kinematics of Fluid Flows)

Example 1. (Use of Continuity euation and !cceleration)"

Consider t#e $elocity field jAyiAxyV

= 221 in t#exyplane% w#ere 1125.0 = smA %

and t#e coordinates are measured in meters. &s t#is a possi'le incompressi'le flow field

Calculate t#e acceleration of a fluid particle at point )1,2(),( =yx .

1. Statement of the Problem

a) Given

Veloit! fiel"# jAyiAxyV

= 22

1, $here 1125.0 = smA .

b) %in"

&hether the veloit! fiel" is a 'ossible inom'ressible flo$ fiel" or not.

eleration of a fli" 'artile at 'oint )1,2(),( =yx .

2. S!stem *iagram

+t is not neessar! for this 'roblem.

3. ssm'tions

Stea"! state on"ition

2 - * flo$ fiel" 'roblem

. Governing ations

ontinit! ation# 0=+

+

+

z

w

y

v

x

u

t

+nom'ressible 2 - * ontinit! eation# 0=

+

yv

xu

eleration#z

Vw

y

Vv

x

Vu

t

V

Dt

VDa

+

+

+

==

2 - * 'roblem#

z

uw

y

uv

x

uu

t

u

Dt

Duax

++

+

==

z

vw

y

vv

x

vu

t

v

Dt

Dvay

++

+

==

Sine ),( yxuu = ),( yxvv = ,

y

u

vx

u

uax

+

=

y

vv

x

vua y

+

=

5. *etaile" Soltion

Veloit! fiel" jAyiAxyV

= 22

1sho$s the om'onents to be#

Axyyxuu == ),(

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2

2

1),( Ayyxvv ==

he ontinit! eation, inom'ressible version of ontinit! eation in this ase, mst be

satisfie" to have a vali" inom'ressible flo$ fiel".

So, he the inom'ressible ontinit! eation#

[ ] ( ) 02

1 2 =+=

+

=

+

AyAyAy

yAxy

xy

v

x

uontinit! eation is satisfie".

herefore, it an be sai" that there eists a 'ossible inom'ressible flo$ fiel".

eleration is#

[ ] [ ] [ ] [ ] [ ] [ ]AxAyAyAxyAxyy

AyAxyx

Axyy

uv

x

uuax

+=

+

=

+

= 222

1

2

1

222222

2

1

2

1xyAxyAxyAax =

+=

[ ] [ ]AyAyAyy

AyAyx

Axyy

vv

x

vuay

+=

+

=

+

= 22222

10

2

1

2

1

2

1

32

2

1yAay =

herefore, at )1,2(),( =yx , the aeleration is#

( ) ( ) ( ) 2221122 /0425.01225.02

1

2

1smmmsmxyAa

x ===

( ) ( ) 2321132 /03125.0125.02

1

2

1smmsmyAay ===

jsmismjaiaa yx +=+= )/03125.0()/0425.0( 22

4. ritial ssessment

+nom'ressible 2 - * ontinit! eation has been satisfie" therefore, the fli" flo$ is

incompressi'lein this flo$ fiel".

Example . (Use of &nte*ral Continuity + !cceleration)

Consider t#e incompressi'le flow of a fluid t#rou*# a no,,le as s#own. -#e area of t#e

no,,le is *i$en 'yA = A0(1 - bx)and t#e inlet $elocity $aries accordin* to U = C(1 + at)%

w#ereA0= 1 ft2%L = 4 ft% b = 0.1 ft-1% a = 2 s-1% and C = 10 ft/s. Find t#e acceleration of a fluid

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particle on t#e centerline atx = L/2for t = 0and 0.5 s. lot axon t#e centerline as a function

ofx/Lfor t = 0and 0.5 s.

1. Statement of the Problem

a) Given

+nom'ressible flo$ of a fli" throgh a no66le (= constant).

rea of the no66le,A = A0(1 bx)

+nlet veloit!, U = C(1 + at)

onstants

A0= 1 ft2

= ! ft

b = 0"1 ft#1

a = 2 s#1

C = 10 ft$s

b) %in"

eleration of a fli" 'artile on the enterline at x = $2for t = 0an" 0"% s.

Plot of axon the enterline as a fntion ofx$for t = 0an" 0"% s.

2. S!stem *iagram

Ao

U

x

L

Ao

U

L

x

# at the inlet

# at a generalx

ontrol Volme

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3. ssm'tions

+nom'ressible bt nstea"! fli" flo$ 'roblem

1 7 * 'roblem alongx"iretion bease the interest is an aeleration, ax, on the

enterline therefore, u = u(x&t)onl!.

. Governing ations

z

Vw

y

Vv

x

Vu

t

V

Dt

VDa

+

+

+

==

1 - * 'roblem z

uw

y

uv

x

uu

t

u

Dt

Duax

++

+

==

8ease u = u(x&t)x

uu

t

uax

+

=

+

=C'CV

A(VV(t

0 9 +ntegral version of mass onservation

+nom'ressible fli" flo$ 'roblem = C' A(V01 inlet () an" 1 otlet () on 1 - * 'roblem 11000 AuAu +=

5. *etaile" Soltion

:btain u = u(x&t)from 11000 AuAu += .AtxuAU += ),(0 0 ;is at generalx, so ),(1 txuu = .