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3.1: Introduction to Motion and Velocity Field: Pathlines, Streamlines, and Streaklines

Geometry of Motion

Pathline

Streamline

No flow across a streamline

Local Relative Velocity of Fluid with respect to A Surface : A Preliminary Glimpse at

Flux

Stream Surface and Stream Tube

Streakline

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1. Pathline

2. Streamline

3. Flux through a surface:

Only the normal component of the local relative velocity of fluid with

respect to the surface ( ) can transport fluid across the surface.

dtw

dzdy

u

dxor

wdtdz

dtdy

udtdx

v,v),( txV

dt

xd

dw

dzdy

u

dx

v),(// txVxd

scalar vanishing-non aiswhere

),(

k

txVkxd

nsfV /

Very Brief Summary of Important Points and Equations

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A pathline is the path, or trajectory, traced out by an identified fluid particle.

Note that here is the displacement along the pathline of an identified particle.

dtw

dzdy

u

dxor

wdtdz

dtdy

udtdx

v,v

Pathline

Motion and path of an identified particle (A)

Scene of VARYING TIME / VIDEO

x

y

z

xd

dt

xdV A

A

)(txx A

)( dttxx A )( dttVA

)(tVA

),( txVdt

xd

xd

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A streamline is defined as the line that is everywhere tangent to the

local velocity vector.

xd

Streamline

Scene of FIXED TIME / STILL IMAGE

x

y

z

s (streamline coordinate)

AB

Cxd

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A streamline is defined as the line that is everywhere tangent to the local velocity vector.

Note that here is the displacement along the streamline.

It is customary to denote the coordinate along streamline as s.

xd

Scene of FIXED TIME / STILL IMAGE

x

y

z

x

xd

),( txdxV

),( txV

xdx

s (streamline coordinate)

dw

dzdy

u

dx

v

0),(

txVxd

),(// txVxd

scalar vanishing-non aiswhere

),(

k

txVkxd

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• By its definition, it follows that .

• Hence, there can be no flow across a streamline.

• Note that only the normal component of velocity can transport fluid from one side of the curve to

the other.

0

nV

Some Properties of Streamline: No flow across streamline

No flow across streamline

s (streamline coordinate)

tnt VVVV

Not allowed

)( ˆor ˆ enttangunitst ee

)(ˆ normalunitne

ttt eVVV ˆ

0ˆ

nnn eVV

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Let be the local fluid velocity

be the local surface velocity

Then, the local relative velocity of fluid with respect to the surface is given by

For short, we simply write

• Only the normal component of the local relative velocity of fluid with respect to the surface (

) can transport fluid across the surface.

sfsf VVV

/

fV

sV

Local Relative Velocity of Fluid with respect to A Surface :A Preliminary Glimpse at Flux

nedAAd ˆ ne

ntsf VVVV

/

ttt eVV ˆ

nnn eVV ˆ

te

nsfV /

sfVV /

fV

sV

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Stream Tube

Arbitrary closed curve C

Stream Surface

Arbitrary open curve C

Stream Surface and Stream Tube

Stream Surface

• Starting from an arbitrary open curve C.

• If we trace out streamlines that start from points on this curve, we have a stream surface that contains C.

Stream Tube

• On the other hand, if we choose a closed curve, we have a stream tube.

From the definition of streamline, no flow can cross a stream tube.

Therefore, a stream tube acts like an imaginary pipe/channel.

Due to this property, stream tube is a useful tool for analysis.

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A streakline is the line joining fluid particles that once passed through the same fixed point in space.

(It is helpful to think of a dye streak.)

Note that here is the displacement along the streakline.

One way to think of a streakline that passes through a point P is to think of a still image of a trace of dye from an

injection port at P.

xd

Streakline

Still image of a trace of dye from an injection port at P.

Scene of FIXED TIME / STILL IMAGE

)(txC

)(txB

)(txD

)(txA

xdDye injection

P

)( AA dttx

)( BB dttx

)( CC dttx

)( DD dttx

• Use the current time t as a reference time,

• particle A passed through point P at earlier time of t-dtA

• particle B, at t-dtB

• particle C, at t-dtC

• particle D, at t-dtD

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Coincidence of Pathlines, Streamlines, Streaklines

Unsteady flows:

Pathlines, streamlines, and streaklines are not the same.

Steady flows:

Pathlines, streamlines, and streaklines are identical.

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Flow past an airfoil, visualized by dye in water tunnel.From Van Dyke, M., 1982, An Album of Fluid Motion, Parabolic Press.

Some Images

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Flow past a block showing horseshoe vortex (top-right and bottom),

visualized by smoke-wire.

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Flow past a damper, visualized by smoke-wire.

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Questions

1. What is the dimension of A?

2. Is the velocity field steady?

3. Is the velocity field uniform along any line parallel to the x axis?

4. Is the flow 1-, 2-,or 3-D?

For below, let the velocity be given in m/s and A = 0.3 s-1.

5. Find the pathline of a particle that is located at point at time

6. Find the streamline that passes through the point at time

7. Can we find the streamline in (6) without having to solve for them in (6)? If so, how?

8. Sketch a vector plot.

9. Sketch a few streamlines.

Example 1: Pathlines and Streamlines

Given the velocity field

jAyiAxvuVVtxV yxˆ)(ˆ)(),(),(),(

)8,2(),( oo yx sto 0

)8,2(),( oo yx sto 0

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Questions

1. Find the pathline of a particle that is located at point at time

2. Find the streamline that passes through the point at time

3. Can we find the streamline in (2) without having to solve for them in (2)? If so, how?

4. Find the position and the velocity of the particle that is initially (at time ) located at

at time

5. Sketch a vector plot.

6. Sketch a few streamlines.

Example 2: Pathlines and Streamlines

Given the velocity field

11112 6,2,/ˆ)(ˆ)(),( smbsmasmjbxyiaxtxV

)2/1,2(),( oo yx sto 0

)2/1,2(),( oo yx sto 0

)2/1,2(),( oo yx

sto 0sto 1.0

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Questions

1. Find the pathline of a particle that is located at point at time

2. Find the streamline that passes through the point at time

3. Can we find the streamline in (2) without having to solve for them in (2)? If so, how?

4. Sketch a vector plot.

5. Sketch a few streamlines.

Example 3: Pathlines and Streamlines

Given the velocity field

12 4,1,/ˆ)(ˆ)(),( sbsasmjbxiayttxV

),( oo yx ot

),( oo yx 21 and,, ttto