4-Summary, fluid mech

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Summary
SUMMARY
The streamline is a curve everywhere tangent to the local velocity vector. The configuration of streamlines in a
flow field is called the flow pattern. The pathline is the line traced out by a particle. A streakline is the line
produced by a dye introduced at a point in the field. Pathlines, streaklines, and streamlines are coincident in
steady flow if they share a common point but differ in unsteady flows.
In a uniform flow, the velocity does not change along a streamline. In a steady flow, the velocity does not
change with time at any location.
The tangential acceleration of a fluid element along a pathline is
where the first term is the local acceleration and the second term is the convective acceleration. The acceleration
normal to the pathline and toward the center of rotation is
where
r
is the local radius of curvature of the pathline.
Applying Newton's second law to a fluid element in the flow of an incompressible, inviscid fluid results in
Euler's equation
,
where ℓ is an arbitrary direction. Integrating Euler's equation in the radial direction for a rotating flow results in
Integrating Euler's equation along a streamline in steady flow results in the
Bernoulli equation
,
in terms of pressure or
in terms of head where
V
is the speed of the fluid and
C
is a constant along a streamline. The value of
C
may
vary from streamline to streamline.
The rotation of a fluid element is defined as the average rotation of two initially perpendicular lines defining the
sides of the element. If every fluid element in a flow does not rotate, the flow is irrotational and the value for C
in the Bernoulli equation is the same for every streamline.
The
vorticity vector
is defined as
and is equal to twice the fluid rotation vector. In an irrotational flow, the vorticity is zero.
Separation occurs when streamlines move away from the surface of the body and create a local recirculation
zone or wake. Typically the pressure in the recirculation zone assumes the value at the point of separation.
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Summary
Copyright ¨ 2009 John Wiley & Sons, Inc. All rights reserved.
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