Superposition of a Uniform Flow
and a Doublet Gives
Flow over a Cylinder 

Flow past a fixed circular cylinder can be obtained by combining
uniform flow with a
doublet. The superimposed stream function and velocity potential are given by
Ψ = Ψ_{uniform
flow} + Ψ_{doublet}
= U r sinθ 
K sinθ/r
and
Φ = Φ_{uniform
flow} + Φ_{doublet} = U r cosθ +
K cosθ/r
respectively. Since the streamline that passes through the stagnation point
has a value of zero, the stream function on the surface of the cylinder
of radius a is then given by
Ψ = U a sinθ 
K sinθ/a = 0
which gives the strength of the doublet as
K = U a^{2}
The stream function and velocity potential for flow past a fixed circular
cylinder become Ψ = Ur [1 
(a/r)^{2}] sinθ
and
Φ = Ur [1
+ (a/r)^{2}] cosθ 


respectively. The plot of the streamlines is shown in the figure. The
velocity components can be determined by:
Along the cylinder (r = a), the velocity components reduce to
v_{r} = 0 and v_{θ} = 2U sinθ
The radial velocity component is always zero along the cylinder while
the tangential velocity component varies
from 0 at the stagnation point (θ = π)
to a maximum velocity of 2U at the top and bottom of the cylinder (θ = π/2
or π/2). The pressure distribution along
the cylinder can be obtained using Bernoulli's equation,

Comparison between Experimental
and Theoretical Potential Flow Results 

where the subscript "o" refers to the upstream condition while
the subscript
"s" refers to the condition along the cylinder. The elevation
changes are assumed negligible. Substituting the expression for v_{θs} into
the above equation and rearranging gives
where C_{p} is the dimensionless pressure coefficient. The plot
of C_{p} as a function of θ is shown
in the figure. The discrepancy between the
experimental and theoretical results, as shown in the figure, is due to
the viscous effects. 
Drag and Lift 

The concepts of drag and lift will be briefly introduced here. The drag developed on the cylinder
can be obtained by integrating the pressure
over the
cylinder
surface
as,
Drag is the resultant force exerted by the fluid on the cylinder, and
its direction is parallel to the upstream uniform flow direction. The
lift is the resultant force acting perpendicular to the uniform flow
direction, and it can be obtained by
When the integrations are carried out for F_{x} and F_{y} (integration details are not given for simplicity), it is found that both drag and lift are zero for potential flow.
The potential flow solutions developed in this
section are based on the assumption of inviscid flow (i.e., zero viscosity),
which implies that drag vanishes. However, as will be discussed in the Lift section, when a real fluid flows past a cylinder, viscous effects
are important near the cylinder. Viscous effects will cause the flow
to separate away from the cylinder, and the drag is nonzero
in actual flow situations. This discrepancy is referred to as d'Alembert's
paradox. 
Superposition of a Uniform Flow, a Doublet and a Vortex
Streamlines of Flow Past a
Rotating Cylinder where
(a) Γ / (4πaU) <1,
(b) Γ / (4πaU)=1 and
(c) Γ / (4πaU) >1 

When the solutions obtained previously for flow past a fixed cylinder
are combined with a vortex, flow past a rotating cylinder can be simulated.
The superimposed stream function and velocity potential now consist of
three components and are given by
and
where Γ is the strength of the vortex circulation.
The radial velocity v_{r} is still zero along
the cylinder surface while the tangential velocity is given by,
From the above equation, it can be found that the location of the stagnation point, θ_{stag}, depends on the strength of the circulation. Setting v_{θ} equal
to zero and yields
The streamlines along with the location of the stagnation point for different dimensionless circulation strengths ( Γ / [2πaU] ) are shown in the figure.
The pressure distribution along the cylinder surface can
be obtained by Bernoulli's equation, giving
This is generally rearranged in terms of the dimensionless pressure coefficient, C_{p},
The drag and lift are obtained by integrating the pressure over the cylinder surface, giving
F_{x} = 0 and F_{y} = ρUΓ
Hence, there is still no drag for a rotating cylinder (inviscid assumption). However, there is lift involved, and it is directly proportional to the density, upstream velocity and strength of the vortex (KuttaJoukowski law). The lifting effect for rotating bodies in a free stream is called the Magnus effect.
