Basics of Fluid Mechanics, 2014a

10.3. POTENTIAL FLOW FUNCTIONS INVENTORY 363
the pressure and is defined as
Eu = P 0 − P ∞
1
2 ρU 0 2 (10.171)
In inviscid flow (Euler’s equations) as a sub set of Naiver–Stokes equations the energy
conserved hence (see for discussion on Bernoulli equation),
Dividing equation (10.172) by U 0 2 yields
P 0 = P + 1 2 ρU2 or P 0 − P = 1 2 ρU2 (10.172)
P 0 − P
2
= 1 U 0 2 ρ U 2
2
U =⇒ P 0 − P
1
0 2 ρU 0 2 = U 2
2
(10.173)
U 0
The velocity on the surface of the “solid” body is given by equation (10.168) Hence,
P 0 − P
1
2 ρU 0 2 = 4 sin2 θ (10.174)
It is interesting to point that integration of the pressure results in no lift and no resistance
to the flow.This “surprising” conclusion can by provided by carrying the integration of
around the “solid” body and taking the x or y component depending if lift or drag is
calculated. Additionally, it can noticed that symmetry play major role which one side
cancel the other side.
10.3.1.1 Adding Circulation to a Cylinder
The cylinder discussed in the previous sections was made from a dipole in a uniform
flow field. It was demonstrated that in the potential flow has no resistance, and no lift
due to symmetry of the pressure distribution. Thus, it was suggested that by adding
an additional component that it would change the symmetry but not change the shape
and hence it would provide the representation cylinder with lift. It turned out that this
idea yields a better understanding of the one primary reason of lift. This results was
verified by the experimental evidence.
The linear characteristic (superposition principle) provides by adding the stream
function of the free vortex to the previous the stream function for the case. The stream
function in this case (see equation (10.159)) is
ψ = U 0 r sin θ
(
1 −
( r
a) 2
)
+ Γ
2 π ln a r
(10.175)
It can be noticed that this stream function (10.175) on the body is equal to
ψ(r = a) =0. Hence, the shape of the body remains a circle. The corresponding radial
velocity in cylindrical coordinates (unchanged) and is
U r = 1 r
∂ψ
∂θ = U 0 cos θ
(
1 −
( a
r
) 2
)
(10.176)