In this section, the solutions for potential flow past a fixed and rotating circular cylinder will be obtained by the method of superposition.

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 a²

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,

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.

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.

Combination of Uniform Flow and Doublet

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 (Kutta-Joukowski law). The lifting effect for rotating bodies in a free stream is called the Magnus effect.