fluid mechanics flowing fluids engineering fluid mechanics 8/e by crowe, elger, and roberson...
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Fluid Mechanics
FLOWING FLUIDS
Engineering Fluid Mechanics 8/E by Crowe, Elger, and RobersonCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
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Streamlines & Flow Patterns
Flow Pattern: Construction of streamlines showing the flow direction
Streamlines (light blue): Local velocity vector is tangent to the streamline at every point along the line at a single instant.
Flow through an opening in a tank & over an airfoil section.
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Streamline & Pathline
Streamline: line drawn through flow field such that local velocity vector is tangent at every point at that instant
– Tells direction of velocity vector
– Does not directly indicate magnitude of velocity
• Pathline: shows the movement of a particle over time
► In unsteady flow, all can be distinct lines.
► The latter two tells us the history of flow as the former indicates the current flow pattern.
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Examples...
Predicted streamline pattern over the Volvo ECC prototype.
Pathlines of floating particles.
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TYPES OF FLOW
Uniform: Velocity is constant along a streamline(Streamlines are straight and parallel)
0
s
V
0
s
V
Non-uniform: Velocity changes along a streamline (Streamlines are curved and/or not parallel)
Express velocity V = V(s,t)
Vortex flow
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Steady: streamline patterns are not changing over time
Unsteady: velocity at a point on a streamline changes over time
0
t
V
0
t
V
Flow patterns can tell you whether flow is uniform or non-uniform, but not steady vs. unsteady… Why?
Because streamlines are only instantaneous representation of theflow velocity.
TYPES OF FLOW
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LAMINAR & TURBULENT FLOW
(a) Experiment to illustrate the type of the flow (b) Typical dye streaks for different cases
(a) (b)
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Engineering Fluid Mechanics 8/E by Crowe, Elger, and RobersonCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
LAMINAR & TURBULENT FLOW
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DIMENSIONALITY OF FLOW FLIED
→ Characterized by the number of spatial dimensions needed to describe velocity field.
1-D flow:
Axisymmetric uniform flow in a
circular duct
2-D flow:
Uniform flow in a square duct
3-D flow:
Uniform flow in an expanding
square duct
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FLOW ACCELERATION (rate of change of velocity with time)
• Consider a fluid particle moving along a pathline...
• There are two components of acceleration:
Tangential to pathline
at : the time-dependent acceleration related to change in speed.
Normal to pathline
an : the centripetal acceleration related to motion along a curved pathline.
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Flow Acceleration
Local acceleration – occurs when flow is unsteady (direction or magnitude is changing with respect to time)
Convective acceleration – occurs when flow is nonuniform (acceleration can depend on position in a flow field)
Local acceleration – occurs when flow is unsteady
Centripetal acceleration – occurs when the pathline is curved(normal to the pathline & directed toward the center of rotation)
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Example: Convective Acceleration
The nozzle shown below is 0.5 meters long. Find the convective acceleration at x = 0.25 m. The equation describing velocity variation is provided below.
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Problem 4.17:
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Problem 4.17: (Solution)
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Example: