ME2342_Sec3_Kinematics

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P. S. Krueger ME/CEE 2342 3 - 1

ME/CEE 2342:

Paul S. Krueger

Associate Professor

Department of Mechanical Engineering

Southern Methodist UniversityDallas, TX 75275

[email protected]

(214) 768-1296Office: 301G Embrey

Fluid Mechanics

Section 3 Fluid Kinematics (Motion)

[Chapter 4 in the text book]
mailto:[email protected]:[email protected]


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Example: Flow Out of a Pipe

Two Options:1) Track the velocity of each individual fluid particle.

2) Observe the velocity of the fluid particles moving through

each point in space

Eulerian and Lagrangian Flow Descriptions


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Option 1 is called the Lagrangian description of the flow.

This method provides the velocity of each particle i in the

flow for all time:

where

( )

Option 2 is called the Eulerian description of the flow. In

this case the velocity is given at each point in the flow for all

time and is represented as a velocity field:( ) ( ) ( )zyxu ,,,,,,,,,,,, tzyxwtzyxtzyxutzyx ++= v

222wuV ++= vu = flow speed

Both descriptions provide the same information in different

ways. For fluid mechanics, the Eulerian description isusually more convenient


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Streamlines, Streaklines, and Pathlines

To assist visualization or interpretation of complex flows, we

often construct/insert lines in the flow that represent fluid

motion. Three common types are

x

y

Streamlines and Vector Field

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

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1) Streamlines. Streamlines are lines that are everywhere

tangentto the velocity field.

Stagnation Point Flow:



2) Pathlines. Pathlines follow the path of individual fluid

particles. Conceptually these can be generated by

placing a dot of dye in a liquid flow and tracing its path asit moves.

Example: Bubbles follow pathlines

[Source: Cengel and Cimbala (2010), Fig. 4-21]



3) Streaklines. Streaklines are the path traced out by

particles injected into the flow at a fixed point.

Conceptually, we can create streaklines by using adropper to put drops of dye into the flow at a fixed point

and then connecting the dots to form a streakline.

Example: Smoke visualization of flow over a wing

[Source: Bertin, Aerodynamics for Engineers, Prentice Hall (2002)]



For a general flow, all three flow lines may give different but

complimentary pictures of the flow (see Example 4-5 in the

textbook).

Forsteady flows (flows that do not change in time), each

fluid particle follows the one preceding it, so streamlines,

streaklines, and pathlines are identical for steady flows.

We will deal primarily with steady flows and focus on

streamlines. For a 2D flow:

By definition of streamlines, the coordinates (xs, ys) of all

streamlines must obey


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Examples:

1) Linear Flow


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Separate variables and integrate:


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2) Stagnation Point Flow

( )yxu 0 yxLU =


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Sketch of a few streamlines:


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The Acceleration Field and Material

Derivative

Given the velocity field, we can find the acceleration of the

fluid particles at each point in space by taking the total

derivative:


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In component form

Short hand:

( )uuuu

a +

==

tDt

D

where

( )( ) ( ) ( )

zyx

+

+

= zyx

D()/Dt is called the material derivative and refers to the rate

of change of a quantity () for a fluid particle at (x, y, z) at

instant t. We applied it to u to obtain a, but it can be appliedto any fluid property (temperature, density,).


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Note: The fluid velocity can be changed by either of two

effects:

1) Unsteady effects: rate of change at a fixed location

2) Convective effects: Change caused by moving to a

location where flow properties are different


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Example: Converging Nozzle

Find the acceleration field.

x-component:


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y and z-components:

Note: ax 0 even though flow through the nozzle is steady.

This occurs because flow speed increases in x (u/x 0).As you move from A to B to C, u increases and so the

acceleration is non-zero.

ME2342_Sec3_Kinematics

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