ES 202 Fluid and Thermal Systems Lecture 28: Drag Analysis on Flat Plates and Cross-Flow Cylinders (2/17/2003)

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ES 202Fluid and Thermal SystemsLecture
28Drag Analysis on Flat Plates and Cross-Flow
Cylinders(2/17/2003)
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Road Map of Lecture 28

• Finish up drag analysis over a flat plate
• Visualization from MMFM
• flow separation over a sphere at high Reynolds
  numbers (serves as motivation to drag analysis)
• flow pattern over a cylinder from low to high
  Reynolds numbers
• Drag analysis on a cross-flow cylinder in open
  air
• Drag analysis on a cross-flow cylinder in a wind
  tunnel
• emphasize the difference between the open air
  problem and the wind tunnel problem

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Drag Analysis over a Flat Plate

• Consider a uniform flow over a flat plate and
  assume the velocity distribution within the
  boundary layer follows a 1/7-power law
• Determine the total drag on a flat plate of
  length L and width w.

U
y
d
x
L
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Main Points

• Boundary layer slows down the flow next to the
  plate causing a mass and momentum deficit.
• The mass deficit in the boundary layer causes
  mass to flow out of the top boundary, governed by
  mass conservation.
• As the mass flows out of the top boundary, it
  takes streamwise momentum along with it.
• In open air, the pressure is atmospheric on all
  control surfaces, leaving the drag force on the
  bottom surface the only force in the streamwise
  direction.
• In the momentum balance equation, two
  integrations need to be performed to find out the
  mass and momentum outflow on the exit surface.
  The integration procedure is not necessary in a
  simplified one-dimensional analysis .

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Visual Learning

• Motivational visualization flow separation
  behind a tennis ball at high Reynolds numbers
  from MMFM
• high-light the wake region
• Can you use Bernoullis equation in the wake
  region? Why?
• Flow pattern over a cross-flow cylinder at
  various Reynolds numbers from MMFM
• at low Reynolds numbers, no separation behind the
  cylinder
• at increasingly high Reynolds numbers, flow
  separation becomes more severe.
• emphasize the Reynolds number dependency of the
  flow
• reinforce the meaning and physical significance
  of Reynolds number

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Drag Analysis on a Cross-Flow Cylinder in Open Air

• In an experiment to determine the drag on a
  cylinder due to a uniform cross-flow U, a
  circular cylinder of diameter d was immersed in a
  steady, two-dimensional incompressible flow in
  open air. Measurements of velocity and pressure
  were made at the boundaries of a fixed control
  volume shown below. The pressure was uniform
  over the entire control surface. The streamwise
  velocity component is indicated in the following
  figure. Based on the measured data, determine
  the drag coefficient on the cylinder (based on
  the projected frontal area.)

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Main Points (Similar to Flat Plate)

• Similar to the boundary layer over a flat plate,
  the wake behind a cylinder causes a mass and
  momentum deficit.
• The mass deficit in the boundary layer causes
  mass to flow out of the top and bottom
  boundaries, governed by mass conservation.
• As the mass flows out of the top and bottom
  boundaries, it takes streamwise momentum along
  with it.
• In open air, the pressure is atmospheric on all
  control surfaces leaving the drag force on the
  cylinder surface the only force in the streamwise
  direction.
• In the momentum balance equation, two
  integrations need to be performed to find out the
  mass and momentum outflow on the exit surface.
  The assumed linear velocity profile is easier to
  be integrated, as compared with the 1/7-power law
  in a boundary layer.

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Drag Analysis on a Cross-Flow Cylinder in a Wind
Tunnel

• In an experiment to determine the drag on a
  cylinder due to a uniform cross-flow U, a
  circular cylinder of diameter d was immersed in a
  steady, two-dimensional incompressible flow in a
  wind tunnel. Measurements of the velocity
  profile downstream of the cylinder were made as
  shown below. Assume the tunnel walls to be
  relatively frictionless, compared to the flow
  around the cylinder. Determine the drag
  coefficient on the cylinder (based on the
  projected frontal area.)

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Main Points

• The major difference between the open air
  problem and the wind tunnel problem is due to
  the flow confinement effects from the top and
  bottom walls.
• The cylinder blockage causes the outer inviscid
  flow to speed up. The exit flow speed in the
  inviscid region can be obtained by mass
  conservation.
• The flow acceleration causes pressure to drop.
  The corresponding pressure drop can be obtained
  quantitatively by the Bernoullis equation.
• In the wind tunnel problem, the drag force on the
  cylinder surface is no longer the only force in
  the streamwise direction. There is an additional
  force due to pressure variation.
• Again, two integrations need to be performed to
  find out the mass and momentum outflow on the
  exit surface.