Lift and Drag for Flow Past a Cylinder without Circulation

Pressure in the Cylinder Surface

Pressure  becomes uniform at large distances from the cylinder ( where the influence of doublet is  small).

Let us imagine the pressure p₀ is known as well as uniform velocity U₀ .
We can apply Bernoulli's equation between infinity and the points on the boundary of the cylinder.

Neglecting the variation of potential energy between the aforesaid point at infinity and any point on the surface of the cylinder, we can write

(22.5)

where the subscript b represents the surface on the cylinder.

Since fluid cannot penetrate the solid boundary, the velocity U_b should be only in the transverse direction , or in other words, only v_θ component of velocity is present on the streamline ψ = 0 .

Thus at                        

From eqs (22.5) and (22.6) we obtain

Lift and Drag

Lift :force acting on the cylinder (per unit length) in the direction normal to uniform flow.

Drag: force acting on the cylinder (per unit length) in the direction parallel to uniform flow.

Fig 22.4 Calculation of Drag in a Cylinder

The drag is calculated by integrating the force components arising out of pressure, in the x direction on the boundary. Referring to Fig.22.4, the drag force can be written as

infinitesimal length on the circumference

Since,   

Similarly, the lift force may be calculated as

The Eqs (22.8) and (22.9) produce D=0 and L=0 after the integration is carried out.

However, in reality, the cylinder will always experience some drag force. This contradiction between the inviscid flow result and the experiment is usually known as D 'Almbert paradox.

Bernoulli's equation can be used to calculate the pressure distribution on the cylinder surface

The pressure coefficient , c_p is therefore

(22.10)

The pressure distribution on a cylinder is shown in Figure below

Fig 22.5 Variation of coefficient of pressure with angle