1

CHAPTERCHAPTER 22

FLOW KINEMATICSFLOW KINEMATICS

2

CONTENTSCONTENTS

•• Flow LinesFlow Lines

•• Circulation and VorticityCirculation and Vorticity

•• Stream tubes and Vortex tubesStream tubes and Vortex tubes

•• Kinematics of streamlines and vortex linesKinematics of streamlines and vortex lines

IntroductionIntroduction

ThereThere areare threethree typestypes ofof flowflow lineslines whichwhich areare usedused

frequentlyfrequently forfor flowflow visualizationvisualization purposespurposes::

StreamlineStreamline

PathlinePathline

StreaklineStreakline

3

StreamlineStreamline

4

Is a line whose tangent is everywhere parallel to the velocity vector.Is a line whose tangent is everywhere parallel to the velocity vector.

In a In a 22--D flow field:D flow field:

Similarly in other planes:Similarly in other planes:

In a In a 33--D flow field:D flow field:

In tensor notation:In tensor notation:

dy v

dx u

dz w

dx u

dz w

dy v

dx dy dzds

u v w

( , ),ii i

dxu x t t fixed

ds

5

If the streamline which passes through the point is required, If the streamline which passes through the point is required, EqsEqs

are integrated and the initial conditions that when are integrated and the initial conditions that when

are applied. This will result in a set of equations of the form are applied. This will result in a set of equations of the form

0 0 0( , , )x y z

0 0 00, , ,s x x y y z z

0 0 0( , , , , )i ix x x y z t s

Example

(1 2 )

0

u x t

v y

w

. ' : 1, 1, 0I C s x y s

(1 2 )dx

x tdsdy

yds

(1 2 )1

2

. 't s

s

x C efrom I C s

y C e

(1 2 )t s

s

x e

y e

@ 0t

s

s

x e

y e

x y

6

PathlinePathline

( , )ii i

dxu x t

dt

7

Is a line which traced out in time by a given fluid particle as it flows.Is a line which traced out in time by a given fluid particle as it flows.

To find pathline which passes through the point last To find pathline which passes through the point last EqsEqs. Are . Are

integrated with initial conditions to obtain a set integrated with initial conditions to obtain a set

of equations of the form of equations of the form

0 0 0( , , )x y z

0 0 00, , ,t x x y y z z

0 0 0( , , , )i ix x x y z t

Example

(1 2 )

0

u x t

v y

w

. ' : 1, 1,@ 0I C s x y t

(1 2 )dx

x tdtdy

ydt

(1 )1

2

. 't t

t

x C efrom I C s

y C e

(1 )t t

t

x e

y e

1 ln yx y

8

StreaklineStreakline

9

IsIs aa lineline whichwhich tracedtraced outout byby aa neutrallyneutrally buoyantbuoyant markermarker fluidfluid whichwhich isis

continuouslycontinuously injectedinjected intointo aa flowflow fieldfield atat aa fixedfixed pointpoint inin spacespace..

AnAn exampleexample ofof aa streaklinestreakline isis thethe continuouscontinuous lineline ofof smokesmoke emittedemitted byby aa

chimneychimney atat pointpoint P,P, whichwhich willwill havehave somesome curvedcurved shapeshape ifif thethe windwind hashas aa

timetime--varyingvarying directiondirection..

10

SmokeSmoke isis beingbeing continuouslycontinuously emittedemitted byby aa chimneychimney atat pointpoint PP ,, inin

thethe presencepresence ofof aa shiftingshifting windwind.. OneOne particularparticular smokesmoke puffpuff AA isis alsoalso

identifiedidentified..

TheThe figurefigure correspondscorresponds toto aa snapshotsnapshot inin timetime whenwhen thethe windwind

everywhereeverywhere isis alongalong oneone particularparticular directiondirection..

11

12

To find To find streaklinestreakline which passes through the point which passes through the point EqsEqs. of pathline . of pathline

Are integrated with initial conditions to obtain a Are integrated with initial conditions to obtain a

set of equations of the form set of equations of the form

0 0 0( , , )x y z

0 0 0, , ,t x x y y z z

0 0 0( , , , , ),i ix x x y z t t

Example

(1 2 )

0

u x t

v y

w

. ' : 1, 1,@I C s x y t

(1 2 )dx

x tdtdy

ydt

(1 )1

2

. 't t

t

x C efrom I C s

y C e

(1 ) (1 )t t

t

x e

y e

@ 0t 1 ln yx y

ey

ex

)1(

13

Gragh of solved examplesGragh of solved examples

14

Steady & unsteady flowsSteady & unsteady flows

ConsiderConsider fluidfluid flowflow characterizedcharacterized byby velocityvelocity vectorvector u(u(x,y,zx,y,z))..

IfIf uu atat allall positionspositions ((x,y,zx,y,z)) doesdoes notnot varyvary withwith timetime thenthen thethe flowflow isis

steadysteady..

IfIf uu variesvaries withwith timetime thethe flowflow isis unsteadyunsteady..

Note :

If the flow is steady then pathline, streamlines, and streaklines If the flow is steady then pathline, streamlines, and streaklines coincidecoincide..

15

CirculationCirculation

.d u lu

dl

ContainedContained withinwithin aa closedclosed contourcontour inin aa bodybody ofof fluidfluid isis definedasdefinedas thethe integralintegral

aroundaround thethe contourcontour ofof thethe componentcomponent ofof thevelocitythevelocity vectorvector whichwhich isis locallylocally

tangenttangent toto thethe contourcontour..

16

VorticityVorticity

In tensor notation:In tensor notation:

u

j jki ijk i

k j k

u uue

x x x

rotational

alIrrotation

0

0

Vorticity of an element of the fluid is curl of its velocity vector.Vorticity of an element of the fluid is curl of its velocity vector.

17

Relationship between circulation and vorticity (applying stokes theorem):Relationship between circulation and vorticity (applying stokes theorem):

. ( )A A

d dA dA u l u n n

Free vortexFree vortexForced vortexForced vortex0,0 0,0

18

streamtubestreamtube

Is a region whose sidewalls are made up of streamlines.Is a region whose sidewalls are made up of streamlines.

19

VortexlineVortexline

Is a line whose tangents are everywhere parallel to the vorticity vector.Is a line whose tangents are everywhere parallel to the vorticity vector.

Vortextube:Vortextube: Is a region whose sidewalls are made up of vortexlines.Is a region whose sidewalls are made up of vortexlines.

20

Kinematics of streamlinesKinematics of streamlines

Continuity Eq. for incompressible fluid:Continuity Eq. for incompressible fluid:

Integrating over some volume Integrating over some volume V:V:

Applying Gauss theorem:Applying Gauss theorem:

but,but,

Note that on the walls of the stream tube:Note that on the walls of the stream tube:

u

0V

dV u

0s

ds u n1 2

0A A

ds ds u n u n

u n 1

1

A

ds Q u n2

2,A

ds Q u n

1 2Q Q

21

Kinematics of vortex linesKinematics of vortex lines

We know:We know:

Similar to previous section: Similar to previous section:

But,But,

0V

dV 0s

ds n1 2

0A A

ds ds n n

1

1

A

ds n2

2,A

ds n

1 2 1 2A A

ds ds n n

1 1 2 2A A

u

22