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Pressure and Friction Drag I
Hydromechanics VVR090
Fluid Flow About Immersed Objects
Flow about an object may induce:
• drag forces
• lift forces
• vortex motion
Asymmetric flow field generates a net force
Drag forces arise from pressure differences over the body (due to its shape) and frictional forces along the surface (in the boundary layer)
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D’Alemberts Paradox
Jean d’Alembert (1717-1783)
No net force on an object submerged in a flowing fluid.
Ideal fluid Æ no viscosity (implies not friction, no separation)
Drag and Lift on an Immersed Body I
Early studies in naval architecture and aerodynamics.
More recent work in structural engineering (e.g., houses, bridges) and vehicle design (e.g., cars, trains).
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Drag and Lift on an Immersed Body II
Drag and Lift on an Immersed Body
sin cos
cos sin
= θ+ τ θ
= − θ+ τ θ
o
o
dD pdA dA
dL pdA dA
Drag and lift on a small element:
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Total drag and lift force on the body:
sin cos
cos sin
= = θ + τ θ
= = − θ + τ θ
∫ ∫ ∫
∫ ∫ ∫
oA A
oA A
D dD p dA dA
L dL p dA dA
Effects of the shear stress on lift negligible:
cos= − θ∫A
L p dA
sin cos= θ + τ θ∫ ∫ oA A
D p dA dA
Drag force:
Pressure drag (Dp) Frictional drag (Df)
(form drag)
Pressure drag function of the body shape and flow separation
Frictional drag function of the boundary layer properties (surface roughness etc)
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Examples of Flow around Bodies I
sin 0, cos 1
0,
θ = θ =
= = τ∫p f oA
D D dA
Æ No pressure drag
sin 1, cos 0
, 0
θ = θ =
= =∫p fA
D pdA D
Æ No frictional drag
Examples of Flow around Bodies II
Dp >> Df Dp ≈ Df Dp << Df
D1 >> D2 >> D3
1. 2. 3.
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Dimensional Analysis of Drag and Lift
Assumed relationships:
{ }
{ }
1
2
, , , ,
, , , ,
= ρ μ
= ρ μ
o
o
D f A V E
L f A V E
Derived P-terms:
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2 22
2 2
3 2
3 2
Re
M
ρΠ = =
μ
ρΠ = = =
Π =ρ
Π =ρ
o
o o
o
o
AV
V VE a
DA V
LA V
(drag)
(lift)
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Results of Dimensional Analysis
{ }
{ }
2 23
3
1 1Re,M2 2
Re,M
= ρ = ρ
=
o D o
D
D f AV C AV
C f
Total drag force:
Total lift force:
{ }
{ }
2 24
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1 1Re,M2 2
Re,M
= ρ = ρ
=
o L o
L
L f AV C AV
C f
Properties of CD and CL
Bodies of same shape and alignment with the flow have the same CD and CL, if Re and M are the same.
Re describes the ratio between intertia forces and viscous forces
M describes the ratio between inertia forces and elastic forces
Æ M only important at high flow velocities
(e.g. supersonic flows)
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Drag Coefficient for Various Bodies
2D 3D
Drag Coefficient for a Sphere
Mach number effects
Ernst Mach (1838-1916)
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Drag Coefficient at Sound Barrier
CD as a function of M
Example I: Drag Force on an Antenna Stand
30 m
0.3 m
What is the total drag force on the stand and moment at the base?
wind
35 m/s
Standard atmosphere (101 kPa, 20 deg)
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Example II: Sphere Dropped from an Airplane
Plastic sphere falling in air – what is the terminal speed?
FG
FD
50 mm
Relative density of sphere S = 1.3
Standard atmosphere (101 kPa, 20 deg)
Flow around an airfoil
small angle of attack large angle of attack
(drag from frictional losses) (drag from pressure losses)
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Flow Separation
contraction expansion
airfoilcylinder
