chapterchapter 2 flow kinematicskntu.ac.ir/dorsapax/userfiles/file/mechanical/ostadfile/...pathline...
TRANSCRIPT
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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
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StreamlineStreamline
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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
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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
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PathlinePathline
( , )ii i
dxu x t
dt
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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
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StreaklineStreakline
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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..
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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..
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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(
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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..
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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..
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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.
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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
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streamtubestreamtube
Is a region whose sidewalls are made up of streamlines.Is a region whose sidewalls are made up of streamlines.
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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.
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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
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