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)
1
D’Alemberts Paradox
No net force on an object submerged in a flowing fluid.
Ideal fluid Æ no viscosity (implies not friction, no separation)
Jean d’Alembert
(1717-1783)
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).
2
Drag and Lift on an Immersed Body II
Drag and Lift on an Immersed Body
Drag and lift on a small element:
dD = pdA sin θ + τo dA cos θ
dL = − pdA cos θ + τo dA sin θ
3
Total drag and lift force on the body:
D = ∫ dD = ∫ p sin θdA + ∫ τo cos θdA
A
A
L = ∫ dL = − ∫ p cos θdA + ∫ τo sin θdA
A
A
Effects of the shear stress on lift negligible:
L = − ∫ p cos θdA
A
Drag force:
D = ∫ p sin θdA + ∫ τo cos θdA
A
Pressure drag (Dp)
A
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)
4
Examples of Flow around Bodies I
sin θ = 0, cos θ = 1
D p = 0, D f = ∫ τo dA
A
Æ No pressure drag
sin θ = 1, cos θ = 0
D p = ∫ pdA, D f = 0
A
Æ No frictional drag
Examples of Flow around Bodies II
1.
Dp >> Df
2.
D p ≈ Df
3.
Dp << Df
D1 >> D2 >> D3
5
Dimensional Analysis of Drag and Lift
Assumed relationships:
D = f1 { A, ρ, μ,Vo , E}
L = f 2 { A, ρ, μ,Vo , E}
Derived P-terms:
Π1 =
ρ AVo
= Re
μ
ρVo2 Vo2
Π2 =
= 2 = M2
E
a
Π3 =
D
AρVo2
(drag)
Π3 =
L
AρVo2
(lift)
6
Results of Dimensional Analysis
Total drag force:
1
1
D = f 3 {Re, M} ρAVo2 = CD ρAVo2
2
2
CD = f3 {Re, M}
Total lift force:
1
1
L = f 4 {Re, M} ρAVo2 = CL ρAVo2
2
2
CL = f 4 {Re, M}
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)
7
Drag Coefficient for Various Bodies
2D
3D
Drag Coefficient for a Sphere
Ernst Mach
(1838-1916)
Mach number effects
8
Drag Coefficient at Sound Barrier
CD as a function of M
Example I: Drag Force on an Antenna Stand
What is the total drag force on the stand
and moment at the base?
30 m
wind
35 m/s
0.3 m
Standard atmosphere
(101 kPa, 20 deg)
9
Example II: Sphere Dropped from an Airplane
FD
Plastic sphere falling in air – what is
the terminal speed?
50 mm
Relative density of
sphere S = 1.3
FG
Standard atmosphere
(101 kPa, 20 deg)
Flow around an airfoil
small angle of attack
(drag from frictional losses)
large angle of attack
(drag from pressure losses)
10
Flow Separation
cylinder
contraction
airfoil
expansion
11