< 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




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




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.




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




Drag of a 2-D body

Assuming steady flow, above equation becomes

abhi

dAnpdAVnVR ˆˆ




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




Drag of a 2-D body

Finally, we obtain

dAunVD ˆ




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(




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

...




Drag of a 2-D body

Therefore,

For incompressible flow, ρ=constant, equation becomes

b

hdyuuuD 2122

b

hdyuuuD 212



< 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




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




Energy conservation

Rate of volumetric heating

Heat addition to the control volume due to viscous effects

Therefore,

v

dvq

viscousQ

viscous

v

QdvqB 1




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




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




Energy conservation

Energy conservation equation

Sv

viscous

vS

viscous

v

SdVV

edvV

et

WdvVfSdVpQdvq

22

22

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