ME2342_Sec3_Kinematics
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Transcript of 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
(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
2.2
1) Streamlines. Streamlines are lines that are everywhere
tangentto the velocity field.
Stagnation Point Flow:
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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]
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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)]
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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.