< 2.6. An application of the momentum equation > Drag of a 2-D...
Transcript of < 2.6. An application of the momentum equation > Drag of a 2-D...
Aerodynamics 2017 fall - 1 -
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 2017 fall - 2 -
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
• The surface force on def created by the presence of the body
abhi
dAnp ˆ
Aerodynamics 2017 fall - 3 -
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 have
opposite normals n, this R' is precisely equal and opposite
to all the def surface integrals for the control volume.
Aerodynamics 2017 fall - 4 -
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
The right-hand side of this equation is physically the force
on the fluid moving through the control volume
RdAnpdAVnVdvVdt
d
abhi
ˆˆ
Aerodynamics 2017 fall - 5 -
Fundamental Principles & Equations
< 2.6. An application of the momentum equation >
Drag of a 2-D body
Assuming steady flow, above equation becomes
abhi
dAnpdAVnVR ˆˆ
Aerodynamics 2017 fall - 6 -
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'
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
dAinpdAunVD ˆˆˆ
0ˆˆ abhi
dAinp
Aerodynamics 2017 fall - 7 -
Fundamental Principles & Equations
< 2.6. An application of the momentum equation >
Drag of a 2-D body
Finally, we obtain
dAunVD ˆ
Aerodynamics 2017 fall - 8 -
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.
( where dA=dy(1) )
b
h
a
idyudyudAunV 2
22
2
11)ˆ(
Aerodynamics 2017 fall - 9 -
Fundamental Principles & Equations
< 2.6. An application of the momentum equation >
Drag of a 2-D body
Using continuity equation,
b
h
b
h
b
h
b
h
a
i
b
h
a
i
dyuuu
dyudyuudAunV
So
dyuudyu
dyudyu
2122
2
22122
122
2
11
2211
ˆ
...
Aerodynamics 2017 fall - 10 -
Fundamental Principles & Equations
< 2.6. An application of the momentum equation >
Drag of a 2-D body
Therefore,
For incompressible flow, ρ=constant, equation becomes
b
hdyuuuD 2122
b
hdyuuuD 212
Aerodynamics 2017 fall - 11 -
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
• 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
dewq
Aerodynamics 2017 fall - 12 -
Fundamental Principles & Equations
< 2.7. Energy equation >
Energy conservation
The first law of thermodynamics
• 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
321 BBB
Aerodynamics 2017 fall - 13 -
Fundamental Principles & Equations
< 2.7. Energy equation >
Energy conservation
Rate of volumetric heating
Heat addition to the control volume due to viscous effects
Therefore,
v
dvq
viscousQ
viscous
v
QdvqB 1
Aerodynamics 2017 fall - 14 -
Fundamental Principles & Equations
< 2.7. Energy equation >
Energy conservation
Rate of work done by pressure force on S
Rate of work done by body forces
The total rate of work done on the fluid
S
VSdp
v
Vdvf
viscous
vS
WdvVfSdVpB
2
Aerodynamics 2017 fall - 15 -
Fundamental Principles & Equations
< 2.7. Energy equation >
Energy conservation
Net rate of flow of total energy across control surface
Time rate of change of total energy inside v (control
volume)
In turn, B3 is the sum of above equations
S
VeSdV
2
2
v
dvV
et 2
2
Sv
VeSdVdv
Ve
tB
22
22
3
Aerodynamics 2017 fall - 16 -
Fundamental Principles & Equations
< 2.7. Energy equation >
Energy conservation
Energy conservation equation
Sv
viscous
vS
viscous
v
SdVV
edvV
et
WdvVfSdVpQdvq
22
22