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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