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FLUID MECHANICS - THEORY

   

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

     
    Flow Past a Fixed Circular Cylinder


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 = Ψuniform flow + Ψdoublet = U r sinθ - K sinθ/r

and

     Φ = Φuniform flow + Φdoublet = 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

     vr = 0     and     vθ = -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 vθs into the above equation and rearranging gives

     

where Cp is the dimensionless pressure coefficient. The plot of Cp 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 Fx and Fy (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
     
    Flow Past a Rotating Circular Cylinder

Superposition of a Uniform Flow,
a Doublet and a Vortex
Click to view movie (38k)




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 vr 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, θstag, depends on the strength of the circulation. Setting vθ 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, Cp,

  

The drag and lift are obtained by integrating the pressure over the cylinder surface, giving

     Fx = 0      and      Fy = -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.