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
1
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
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.
streamlined body
cylindral body
2
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 flow
Little variation
with Re
No separation
3
Flow around Sphere
Flow separation
point
Flow separation behind
sphere
Trip
wire
Cricket ball
Flow separation point with trip wire
Drag Coefficient for Laminar Flow
Stokes derived the drag force for laminar conditions
(viscous forces dominate):
D = 3πμVo d
General formulation:
D = FD = CD
1
AρVo2
2
George
Stokes
Equivalence yields:
3πμVo d = CD
1
AρVo2
2
4
Cross-sectional area:
A=
πd 2
4
Solve for drag coefficient:
CD =
24μ 24
=
Vo d ρ Re
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
5
If 6 < Re < 5000, regular vortex sheeding may occur at a
frequency n determined by Strouhal’s number:
S=
nd
Vo
(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
Data for cylinder
Fully developed
turbulence, no regular
vortex shedding
6
Example I: Vortex Shedding from Antenna Stand
What is the frequency of the vortices
shed?
30 m
wind
35 m/s
0.3 m
Standard atmosphere
(101 kPa, 20 deg)
Example II: Vortex Shedding from Telegraph Wires
What is the frequency of the
vortices shed?
Wires
diameter = 2 mm
V = 10 m/s
7
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.
8
Example of Drag Force Calculation
• parachute jumping
• sedimentation of particle
• popcorn popper
Basic equation for drag force:
1
D = CD ρAVo2
2
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
9
Empirical Values for the Drag Coefficient CD II
Dolphin drag
Empirical Values for the Drag Coefficient CD III
Cd
Vehicle
Year and
Model
Area (m2
)
Area (ft2 )
Cd x m2
Cd x ft2
Lotus
'80 Eclat
0.360
1.830
19.69
0.66
7.09
'95 Elan
S2
0.380
1.709
18.40
0.65
6.99
'91 Elan
SE
0.380
1.709
18.40
0.65
6.99
'80 Esprit
0.330
1.802
19.40
0.59
6.40
'94 Esprit
S4
0.330
1.802
19.40
0.59
6.40
'83 Esprit
Turbo
0.330
1.802
19.40
0.59
6.40
'86 Esprit
Turbo
0.330
1.802
19.40
0.59
6.40
'89 Esprit
Turbo
0.330
1.802
19.40
0.59
6.40
'90 Esprit
Turbo SE
0.330
1.802
19.40
0.59
6.40
Lotus
Mercedes-Benz Bionic Concept: 0.19
Hummer H2: 0.57
10
Example I: Parachute Jumping
FD
Terminal speed of a person jumping
with a parachute?
Assumed data:
M = 100 kg
ρair = 1.2 kg/m3
D=7m
FG
Example II: Particle Sedimentation
FB
FD
Sediment particle in water – what is
the terminal speed?
Newton-Stokes law of sedimentation
(laminar flow)
FG
Example of
settling tanks
11
Example III: Popcorn Popper
Design the popcorn popper
Heating
coil
Unpopped corn:
0.15 g/kernel
6 mm diameter
Popped corn:
18 mm diameter
Fan
Allowable air speed
produced by the fan?
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
12
Principles of Flight
Horizontal and vertical force
balance for design
FL = FG
FD = FP
Lift force:
FL = CL
Gliding angle:
1
AρVo2
2
tan γ =
CD
CL
Lift Coefficient CL
CL for typical airfoil sections versus
angele of attack
Stall speed
13
Tip Vortices (Induced Drag) I
Tip Vortices (Induced Drag) II
CD and CL for different wing aspect ratios
14
Example: Takeoff Speed of Airplane
FL
What is the necessary angle
of attack (a) for a takeoff
speed of 140 km/hr?
a
FG
Wingspan: 10 m
Chord length: 1.5 m
Plane weight: 10 kN
Two passengers at 800 N each
Magnus Effect
Net force occurs when a sphere or cylinder in a
moving fluid is rotating
Heinrich
Gustav
Magnus
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
15
Importance of Magnus Effect in Sports I
Golf (hook, slice)
Soccer
(banana
shoot)
Table tennis
and tennis
(topspin, slice)
Importance of Magnus Effect in Sports II
Spinning baseball
(curveball)
Lateral deflection
of baseball
16
Ship Propulsion
Buckau
Alcyone
17