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7/29/2019 Drag Calculation
http://slidepdf.com/reader/full/drag-calculation 1/17
Fundamental Principles & Equations < 2.6. An application of the momentum equation >
Drag of a 2-D body
Consider a two-dimensional body in a flow
Aerodynamics 2008 spring - 1 -
7/29/2019 Drag Calculation
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Fundamental Principles & Equations < 2.6. An application of the momentum equation >
Drag of a 2-D body
Surface forces : two contributions
• The pressure distribution over the surface abhi
abhi
pn dA
• The surface force on def created by the presence of the body
Aerodynamics 2008 spring - 2 -
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Fundamental Principles & Equations < 2.6. An application of the momentum equation >
Drag of a 2-D body
Resultant aerodynamic force : R'
Because the body surface and volume surface haveopposite normals n, this R' is precisely equal and opposite
to all the def surface integrals for the control volume.
Aerodynamics 2008 spring - 3 -
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dt
Fundamental Principles & Equations < 2.6. An application of the momentum equation >
Drag of a 2-D body
Considering the integral form of the momentum equation
d
Vdv
V
n VdA
abhi
pn dA R
The right-hand side of this equation is physically the force
on the fluid moving through the control volume
Aerodynamics 2008 spring - 4 -
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V VdA
Fundamental Principles & Equations < 2.6. An application of the momentum equation >
Drag of a 2-D body
Assuming steady flow, above equation becomes
R
n
abhi
pn dA
Aerodynamics 2008 spring - 5 -
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Fundamental Principles & Equations < 2.6. An application of the momentum equation >
Drag of a 2-D body
The x component of R' is the aerodynamic drag D'
D V n u dA
abhi
pn i dA
Because the boundaries of the control volume abhi are
chosen far enough form the body, p is constant along these
boundaries. So, we have
abhi
pn i dA 0
Aerodynamics 2008 spring - 6 -
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Fundamental Principles & Equations < 2.6. An application of the momentum equation >
Drag of a 2-D body
Finally, we obtain
D
n u dA
Aerodynamics 2008 spring - 7 -
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Fundamental Principles & Equations < 2.6. An application of the momentum equation >
Drag of a 2-D body
The momentum-flux integral is zero on the top and bottom
boundaries, since these are defined to be along streamlines,
and hence have zero momentum flux. Only the momentum
flux on the inflow and outflow planes remain. a
2b
2
(V n)u dA i 1u1 dy h
2u
2dy
( where dA=dy(1) )
Aerodynamics 2008 spring - 8 -
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1 1
2
2
Fundamental Principles & Equations < 2.6. An application of the momentum equation >
Drag of a 2-D body
Using continuity equation,
a
i 1u1dy
b
h
2u
2 dy
a
u2 dy i
b
h 2u2u1dy
So...
V n u dA b
h
b
2u2
b
u1dy h u
2 dy
h 2u2 u1 u2 dy
Aerodynamics 2008 spring - 9 -
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b
b
Fundamental Principles & Equations < 2.6. An application of the momentum equation >
Drag of a 2-D body
Therefore,
D 2u
2u
1 u2 dy
h
For incompressible flow, ρ=constant, equation becomes
D
u2 u1
u2 dy
h
Aerodynamics 2008 spring - 10 -
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Fundamental Principles & Equations < 2.7. Energy equation >
Energy conservation
Physical principle :
Energy can be neither created nor destroyed ; it can only
change in form
System and surroundings
q w de
• δq : heat to be added to the system form the surroundings
• δw : the work done on the system by the surroundings
• de : the change of internal energy
Aerodynamics 2008 spring - 11 -
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Fundamental Principles & Equations < 2.7. Energy equation >
Energy conservation
The first law of thermodynamics
B1 B2 B3
• B1 : rate of heat added to fluid inside control volume form
surroundings
• B2 : rate of work done on fluid inside control volume
• B3 : rate of change of energy of fluid as it flows through
control volume
Aerodynamics 2008 spring - 12 -
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Fundamental Principles & Equations < 2.7. Energy equation >
Energy conservation
Rate of volumetric heating
q dv v
Heat addition to the control volume due to viscous effects
Q viscous
Therefore,
B1 qv
dv Q viscous
Aerodynamics 2008 spring - 13 -
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V
Fundamental Principles & Equations < 2.7. Energy equation >
Energy conservation
Rate of work done by pressure force on S
p dS V S
Rate of work done by body forces
f v
dvV
The total rate of work done on the fluid
B2
S
pV dS
v
dv W
viscous
Aerodynamics 2008 spring - 14 -
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Fundamental Principles & Equations < 2.7. Energy equation >
Energy conservation
Net rate of flow of total energy across control surface
V 2
V S
dS e 2
Time rate of change of total energy inside v (controlvolume)
V 2
t e 2
dv
v In turn, B3 is the sum of above equations
V 2
V 2
B3
e dv V dS e t
v 2 S 2
Aerodynamics 2008 spring - 15 -
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viscous
2
Fundamental Principles & Equations < 2.7. Energy equation >
Energy conservation
Energy conservation equation
q dv Qviscous
v
pV S
dS f v
V dv W
V V
2
dS
t e2
dv e V 2
v S
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Aerodynamics 2008 spring - 16 -