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Pressure and Friction Drag II
Hydromechanics VVR090
Drag and Lift – General Observations I
Inconvenient to separate between pressure and frictional drag.
Total drag force is taken to be the sum of :
• drag in a two-dimensional flow (profile drag)
• drag produced by end effects (induced drag)
Induced drag is related to the lift force.
No lift force Æ no induced drag.
tip vortices
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Drag and Lift – General Observations II
Pressure drag depends on the pressure distribution around the body and the size of the separation zone.
Large zone of separation Æ large drag force
The location of separation points decisive for the magnitude of the pressure drag . Such locations are determined by:
• body shape
• body roughness
• flow conditions
Flow Separation
streamlined body cylindral body
Boundary layer growth starts in the stagnation point.
In the phase of acceleration the boundary layer is stable, whereas during deceleration an unfavorable pressure gradient develops that leads to separation.
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Laminar and Turbulent Boundary Layers
Ideal fluid
Laminar conditions
Turbulent conditions
Drag Coefficients for Different Shapes
Drag coefficient depends on Re (sphere, disk, streamlined body).
Transition to turbulent boundary layer
Laminar flowLittle variation with Re
No separation
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Flow around Sphere
Flow separation behind sphere
Flow separation point
Flow separation point with trip wire
Trip wire
Cricket ball
Drag Coefficient for Laminar Flow
Stokes derived the drag force for laminar conditions (viscous forces dominate):
3= πμ oD V d
General formulation:
212
= = ρD D oD F C A V
Equivalence yields:
2132
πμ = ρo D oV d C A V
George Stokes
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Cross-sectional area:
2
4π
=dA
Solve for drag coefficient:
24 24Re
μ= =
ρDo
CV d
Stokes equation valid for Re < 0.1.
Re ≈ 10 Æ weak separation
Re ≈ 1000 Æ fully developed separation zone
Vortex Shedding
Under certain conditions vortices are generated from the edges of a body in a flow.
Æ Von Karman’s vortex street
Theodore Von Karman
Vortex street behind a cylinder
Vortices at Aleutian Island
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If 6 < Re < 5000, regular vortex sheeding may occur at a frequency n determined by Strouhal’s number:
=o
ndSV
(S = 0.21 over a wide range of Re)
Vincent Strouhal
Periodic vortex shedding may lead to transversal forces on structures (e.g., pipes, chimneys, bridges) resulting in vibration and possible structural damages.
If is close to the natural frequency of the structure, large effects are expected.
Strouhals Number as a Function of Re
Fully developed turbulence, no regular vortex sheddingData for cylinder
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Example I: Vortex Shedding from Antenna Stand
30 m
0.3 m
What is the frequency of the vortices shed?
wind
35 m/s
Standard atmosphere (101 kPa, 20 deg)
Example II: Vortex Shedding from Telegraph Wires
V = 10 m/sWires
diameter = 2 mm
What is the frequency of the vortices shed?
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Ferrybridge Cooling Towers
Three towers collapsed because (November 1965):
• underestmated wind design conditions
• interaction between towers not considered
Tacoma Bridge
Built 1940
Span: 2,800 ft (850 m)
Plate-girder deck: 8 ft (2.4 m)
Wind-induce vibrations caused oscillations of the deck with eventual collapse.
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Example of Drag Force Calculation
• parachute jumping
• sedimentation of particle
• popcorn popper
Basic equation for drag force:
212
= ρD oD C AV
CD obtained from empirical studies
A is the projected area on a plane perpendicular to the flow direction
Empirical Values for the Drag Coefficient CD I
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Empirical Values for the Drag Coefficient CD II
Dolphin drag
Empirical Values for the Drag Coefficient CD III
Lotus
6.400.5919.401.8020.330'90 Esprit Turbo SE
6.400.5919.401.8020.330'89 Esprit Turbo
6.400.5919.401.8020.330'86 Esprit Turbo
6.400.5919.401.8020.330'83 Esprit Turbo
6.400.5919.401.8020.330'94 Esprit S4
6.400.5919.401.8020.330'80 Esprit
6.990.6518.401.7090.380'91 ElanSE
6.990.6518.401.7090.380'95 ElanS2
7.090.6619.691.8300.360'80 Eclat
Cd x ft2Cd x m2Area (ft2 )Area (m2
)Cd
Vehicle Year and Model
Mercedes-Benz Bionic Concept: 0.19
Hummer H2: 0.57
Lotus
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Example I: Parachute Jumping
FG
FD Terminal speed of a person jumping with a parachute?
Assumed data:
M = 100 kg
ρair = 1.2 kg/m3
D = 7 m
Example II: Particle Sedimentation
Sediment particle in water – what is the terminal speed?
Newton-Stokes law of sedimentation
(laminar flow)
FG
FB FD
Example of settling tanks
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Example III: Popcorn Popper
Design the popcorn popper
Unpopped corn:
0.15 g/kernel
6 mm diameter
Popped corn:
18 mm diameter
Allowable air speed produced by the fan?
Fan
Heating coil
Lift Force on Bodies
Important in design of:
• airplane
• pipelines (e.g., on the seafloor)
• pumps and turbines
Flow and pressure distribution around and airfoil
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Principles of Flight
Horizontal and vertical force balance for design
FL = FG
FD = FP
212L L oF C A V= ρ
Lift force: Gliding angle:
tan γ = D
L
CC
Lift Coefficient CL
CL for typical airfoil sections versus angele of attack
Stall speed
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Tip Vortices (Induced Drag) I
Tip Vortices (Induced Drag) II
CD and CL for different wing aspect ratios
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Example: Takeoff Speed of Airplane
What is the necessary angle of attack (a) for a takeoff speed of 140 km/hr?
FG
a
FL
Wingspan: 10 m
Chord length: 1.5 m
Plane weight: 10 kN
Two passengers at 800 N each
Magnus Effect
Heinrich Gustav Magnus
Net force occurs when a sphere or cylinder in a moving fluid is rotating
Top of cylinder: velocities of the moving fluid and the rotating ball enhance each other Æ low pressure
Bottom of cylinder: velocities of the moving fluid and the rotating ball counteract each other Æ high pressure
Pressure difference Æ net force
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Importance of Magnus Effect in Sports I
Golf (hook, slice)Soccer (banana shoot)
Table tennis and tennis (topspin, slice)
Lateral deflection of baseball
Importance of Magnus Effect in Sports II
Spinning baseball (curveball)
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Ship Propulsion
AlcyoneBuckau