Drag Calculation

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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 -




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





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.




dt


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




V VdA


Drag of a 2-D body

Assuming steady flow, above equation becomes

R

n

abhi

pn dA





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





Drag of a 2-D body

Finally, we obtain

D

n u dA





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) )




1 1

2

2


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




b

b


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




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





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





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




V


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





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




viscous

2


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

Drag Calculation

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