ES 202
Fluid and Thermal Systems
Lecture 29:
Drag and Lift Coefficients
(2/18/2003)
Assignments
• Homework:
– 13-56C, 13-62, 13-63, 13-72C, 13-88
• Reading:
– 13-7 to 13-8
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Announcements
• Guest speaker Dr. John Adams will talk 2 talks today:
– “Hypersonic Systems, Technology, and Testing”, including
relevant remarks on the recent Columbia Space Shuttle tragedy in
O259 at 4:20 pm
– “Flight Mechanics of a Spinning Dimpled Spheroid” in the Khan
Room at 6:00 pm
• Homework assigned this week is just for your learning, no need
to hand it in
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Road Map of Lecture 29
•
Finish up example on drag coefficient of cross-flow cylinder in a wind tunnel
•
Give out answers to in-class drag analyses yesterday
•
Introduce definition of drag coefficients
•
(Combined) Drag coefficients for objects of various geometries
–
•
Categorization of drag components
–
–
–
•
skin frictional drag versus pressure drag
effects of body shape on drag (blunt body versus slender body)
flow separation (an artifact of fluid viscosity)
Exercise on qualitative description of flow acceleration and pressure variation over a
blunt body
–
•
concept of streamlining
notion of stagnation point (high pressure)
Applications:
–
–
truck tipping problem
terminal velocity (balance between weight, drag and buoyancy)
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Answers to Drag Analyses
• Drag analysis on a flat plate:
7
D
U2  w
72
• Drag analysis on a cross-flow cylinder in open air:
41
2
D   U d w
32

• Drag analysis on a cross-flow cylinder in a wind tunnel:
88  1
2
D
 U d w
27  2

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Drag Coefficient
• From the results of drag analysis on a cross-flow cylinder in open air,
4 1

D
d
w
 U2
3 2
frontal area

free -stream seen by the flow
dynamic pressure
a non-dimensional group, the drag coefficient CD , can be defined:
CD 
D
1
U 2 A f
2
• The definition of drag coefficient can also be arrived by means of
dimensional analysis, similar to that on boundary layer thickness.
• Show drag coefficient tables for various geometries
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Categorization of Drag Components
• The total drag force on an object can be broadly classified into two
categories:
Total drag force
Pressure (form) drag
Friction drag
• directly related to
skin friction on surfaces
• dominant on slender bodies
•
•
•
•
indirectly related to fluid viscosity
due to momentum losses through viscosity
mostly involves flow separation
dominant on blunt bodies
• Relative importance between friction drag and pressure drag is
strongly Reynolds number dependent and geometry dependent
(slender versus blunt bodies).
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Fluid Acceleration and Pressure Variation
• Perform a qualitative assessment on the changes in a flow as it
approaches a blunt object.
–
–
–
–
–
–
–
speed decreases, pressure increases from free-stream to stagnation point
highest pressure at stagnation point
flow splits into upper and lower streams
speed increases, pressure decreases from stagnation point to edges
highest speed and lowest pressure at the edges
flow speed decreases and pressure recovers behind the object
too much momentum loss in boundary layer: not enough momentum to
negotiate pressure hill, flow separates
– large pressure difference between front and back sides causes pressure drag
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Example Problem
• Truck tipping problem:
U
–
–
–
–
–
–
–
R2
R1
W
O
recognize blunt body geometry
pressure drag as dominant drag component
moment analysis about Point O to determine minimum wind speed to tip truck
assume drag coefficient is all attributed to pressure drag
assume line of action of pressure drag to be at the geometrical center of truck
at tipping position, R2 = 0
fine points:
• small frictional drag component in tabulated CD value
• asymmetry in problem not accounted for in tabulated CD value
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Terminal Speed of Falling Objects
• Identify the major forces on a falling object
1

D  CD   U 2  A f
2

W
• As the falling object accelerates, the drag force increases rapidly (quadratic
dependence on falling speed).
• At terminal speed, the net force on the falling object is zero, implying a
perfect balance between body weight and drag.
• The force balance sets the condition to determine the terminal speed.
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