Flow About a Cylinder without Circulation (Part - 1) | Additional Documents & Tests for Civil Engineering (CE) PDF Download

Flow About a Cylinder without Circulation

• Inviscid-incompressible flow about a cylinder in uniform flow is equivalent to the superposition of a uniform flow and a doublet.

• The doublet has its axis of development parallel to the direction of the uniform flow (x-axis in this case).

• The potential and stream function for this flow will be the sum of those for uniform flow and doublet.

Potential Function

Stream function

Streamlines

In two dimensional flow, a streamline may be interpreted as

• the edge of a surface, on which the velocity vector is always tangential.

                                                    and

•  there is no flow in the direction normal to the surface (characteristic of a solid impervious boundary ).

 Hence, a streamline may also be considered as the contour of an impervious two-dimensional body .

Figure 22.1 shows a set of streamlines.

1.  The streamline C-D may be considered as the edge of a two-dimensional body .

2.  other streamlines form the flow about the boundary.

In order to form a flow about the body of interest, a streamline has to be determined which encloses an area whose shape is of practical importance in fluid flow. This streamline describes the boundary of a two-dimensional solid body. The remaining streamlines outside this solid region,  constitute the flow about this body.

If we look for the streamline whose value is zero, we will obtain

                                   (22.1)

replacing y by rsinθ, we have

                                   (22.2)

Solution of Eq. 22.2

1. If θ = 0 or θ = π, the equation is satisfied. This indicates that the x-axis is a part of the streamline Ψ = 0.

2. When the quantity in the parentheses is zero, the equation is identically satisfied . Hence it follows that

Interpretation of the solution

There is a circle of radius  which is an intrinsic part of the streamline Ψ = 0.
 

This is shown in Fig.22.2

Fig 22.2    Streamline ψ = 0 in a Superimposed Flow of Doublet and Uniform Stream

Stagnation Points

Let us look at the points of intersection of the circle and x- axis , i.e. the points A and B in the above figure. The polar coordinate of these points are

The velocity at these points are found out by taking partial derivatives of the velocity potential in two orthogonal directions and then substituting the proper values of the coordinates.

                                                                (22.4a)

                                                                                  (22.4b)

The points A and B are  the stagnation points through which the flow divides and subsequently reunites forming a zone of circular bluff body.

The circular region, enclosed by part of the streamline ψ = 0 could be imagined as a solid cylinder in an inviscid flow. At a large distance from the cylinder the flow is moving uniformly in a cross-flow configuration.

Figure 22.3 shows the streamlines of the flow.

1.  The streamlines outside the circle describe the flow pattern of the inviscid irrotational flow across a cylinder.

2.  The streamlines inside the circle may be disregarded since this region is considered as a solid obstacle. 

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 

                                     (22.6)  

From eqs (22.5) and (22.6) we obtain

                                                                           (22.7)  

Lift and Drag

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

:

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

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

                                                                             (22.8)

Similarly, the lift force may be calculated as

                                                                                                                                      (22.9)

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