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Pressure and Friction Drag I
Hydromechanics VVR090
ppt by Magnus Larson; revised by Rolf L Jan 2014
SYNOPSIS
1. General description
2. Applications
3. Forces on immersed bodies
4. Dimensional analysis
5. Drag coefficients and lift coefficients
6. Examples/Problems
7. Flow separation. Introduction.
1. General Description
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)
D’Alemberts Paradox
Jean d’Alembert
(1717-1783)
Ideal fluid no viscosity (implies no friction, no separation)
No net force on an object submerged in a flowing fluid.
2 Applications
• Naval architecture and aerodynamics.
• Structural engineering (e.g., houses, bridges)
• Vehicle design (e.g., cars, trains).
Applications
3. Forces on an Immersed Body
sin cos
cos sin
o
o
dD pdA dA
dL pdA dA
Drag and lift on a small element:
Total drag and lift force on the body:
sin cos
cos sin
o
A A
o
A A
D dD p dA dA
L dL p dA dA
Effects of the shear stress on lift negligible:
cos A
L p dA
sin cos o
A A
D p dA dA
Drag force:
Pressure drag (Dp) Frictional drag (Df)
(form drag)
Dp = function of the body shape and flow separation
Df = function of the boundary layer properties (surface
roughness etc)
Examples of Flow around Bodies I
sin 0, cos 1
0,
p f o
A
D D dA
No pressure drag
sin 1, cos 0
, 0
p f
A
D pdA D
No frictional drag
Examples of Flow around Bodies II
Dp >> Df Dp Df Dp << Df
D1 >> D2 >> D3
1. 2. 3.
4. Dimensional Analysis of Drag and Lift
Assumed relationships:
1
2
, , , ,
, , , ,
o
o
D f A V E
L f A V E
Derived -terms:
1
2 22
2 2
3 2
3 2
Re
M
o
o o
o
o
AV
V V
E a
D
A V
L
A V
(drag)
(lift)
Results of Dimensional Analysis
2 2
3
3
1 1Re,M
2 2
Re,M
o D o
D
D f AV C AV
C f
Total drag force:
Total lift force:
2 2
4
4
1 1Re,M
2 2
Re,M
o L o
L
L f AV C AV
C f
5. Drag coefficients CD and lift coefficients 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
M only important at high flow velocities
(e.g. supersonic flows)
Drag Coefficient for Various Bodies
2D3D
Re Re
CDCD
Drag Coefficient for a Sphere
Mach number effects
Ernst Mach
(1838-1916)
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)
6. Examples / problems
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)
7. Flow Separation
contraction expansion
airfoilcylinder
Flow around an airfoil
small angle of attack large angle of attack
(drag from frictional losses) (drag from pressure losses)