Concept of Circulation in a Free Vortex Flow - Flow of Ideal Fluids - Mechanical Engineering PDF Download

Concept of Circulation in a Free Vortex Flow

Free Vortex Flow

• Fluid particles move in circles about a point.

• The only non-trivial velocity component is tangential.

• This tangential speed varies with radius r so that same circulation is maintained. 

• Thus,all the streamlines are concentric circles about a given point where the velocity along each streamline is inversely proportional to the distance from the centre. This flow is necessarily irrotational.

Velocity components

In a purely circulatory (free vortex flow) motion, the tangential velocity can be written as

                   (21.1)

For purely circulatory motion we can also write

 v_r   = 0               (21.2)

Stream Function

Using the definition of stream function, we can write

Combining Eqs (21.1) and (21.2) with the above said relations for stream function, it is possible to write

                            (21.3)
 

Velocity Potential Function

 Because of irrotationality, it should satisfy

Eqs (21.1) and (21.2) and the above solution of Laplace's equation yields

                   (21.4)

Since, the integration constants C₁ and C₂ have no effect on the structure of velocities or pressures in the flow. We can ignore the integration constants without any loss of generality.

It is clear that the streamlines for vortex flow are circles while the potential lines are radial . These are given by

                   (21.5)

• In Fig. 21.1, point 0 can be imagined as a point vortex that induces the circulatory flow around it.
• The point vortex is a singularity in the flow field (v_θ becomes infinite).
•  Point 0 is simply a point formed by the intersection of the plane of a paper and a line perpendicular to the plane.
• This line is called vortex filament of strength  where  is the circulation around the vortex filament .

Circulation is defined as  

                   (21.6)

This circulation constant denotes the algebraic strength of the vortex filament contained within the closed curve. From Eq. (21.6) we can write

For a two-dimensional flow 

          (according to Fig. 21.2)             (21.7)

After simplification  

                   (21.8)

Physically, circulation per unit area is the vorticity of the flow .

Now, for a free vortex flow, the tangential velocity is given by Eq. (21.1) as

For a circular path (refer Fig.21.2)  

Thus,

Therefore 

= 2πC            (21.9)

It may be noted that although free vortex is basically an irrotational motion, the circulation for a given path containing a singular point (including the origin) is constant (2πC) and independent of the radius of a circular streamline.

• However, circulation  calculated in a free vortex flow along any closed contour excluding the singular point (the origin), should be zero.

Considering Fig 21.3 (a) and taking a closed contour ABCD in order to obtain circulation about the point, P around ABCD it may be shown that

Forced Vortex Flow

• If there exists a solid body rotation at constant ω (induced by some external mechanism), the flow should be called a forced vortex motion (Fig. 21.3 (b).

we can write

    (21.10)

Equation (21.10) predicts that

1. The circulation is zero at the origin

2.  It increases with increasing radius.

3.  The variation is parabolic.

It may be mentioned that the free vortex (irrotational) flow at the origin is impossible because of mathematical singularity. However, physically there should exist a rotational (forced vortex) core which is shown by the dotted line ( in Fig. 21.3a ).

 Below are given two statements which are related to Kelvin's circulation theorem (stated in 1869) and Cauchy's theorem on irrotational motion (stated in 1815) respectively

1. The circulation around any closed contour is invariant with time in an inviscid fluid.--- Kelvin's Theorem
2. A body of inviscid fluid in irrotational motion continues to move irrotationally.------------ Cauchy's Theorem

Combination of Fundamental Flows

1)  Doublet

We can now form different flow patterns by superimposing the velocity potential and stream functions of the elementary flows stated above.     

In order to develop a doublet, imagine a source and a sink of equal strength K at equal distance s from the origin along x-axis as shown in Fig. 21.4.

From any point p(x, y) in the field, r₁ and r₂ are drawn to the source and the sink. The polar coordinates of this point (r, θ) have been shown. 
The potential functions of the two flows may be superimposed to describe the potential for the combined flow at P as

    (21.11)

Similarly

    (21.12)

Expanding θ₁ and θ₂ in terms of coordinates of p and s

Using 

we find

Hence the stream function and the velocity potential function are formed by combining Eqs (21.12) and (21.13), as well as Eqs(21.11) and (21.14) respectively

Doublet is a special case when a source as well as a sink are brought together in such a way that

•   s → 0   and at the same time the

• strength   is increased to an infinite value.

These are assumed to be accomplished in a manner which makes the product of s and (in limiting case) a finite value c

This gives us

Streamlines, Velocity Potential for a Doublet

We have seen in the last lecture that the streamlines associated with the doublet are

If we replace sinθ by y/r, and the minus sign be absorbed in C₁ , we get

                  (21.17a)

Putting    r² = x² + y² we get 

              (21.17b)

Equation (21.17b) represents a family of circles with

• radius :
• centre :

• For x = 0, there are two values of y, one of them=0.

• The centres of the circles fall on the y-axis.

• On the circle, where y = 0, x has to be zero for all the values of the constant.

• family of circles formed(due to different values of C₁ ) is tangent to x-axis at the origin.

These streamlines are illustrated in Fig. 21.5.

Due to the initial positions of the source and the sink in the development of the doublet , it is certain that

• the flow will emerge in the negative x direction from the origin

                                                                 and

•  it will converge via the positive x direction of the origin.

Velocity potential lines

In cartresian coordinate the equation becomes

     (21.18)

Once again we shall obtain a family of circles

• radius:  
• centre:   

• The centres will fall on x-axis.

• For y = 0 there are two values of x, one of which is zero.

• When x = 0, y has to be zero for all values of the constant.

• These circles are tangent to y-axis at the origin.

• In addition to the determination of the stream function and velocity potential, it is observed  that for a doublet

As the centre of the doublet is approached; the radial velocity tends to be infinite.

It shows that the doublet flow has a singularity.

 Since the circulation about a singular point of a source or a sink is zero for any strength, it is obvious that the circulation about the singular point in a doublet flow must be zero i.e. doublet flow  =0

                   (21.19)

Applying Stokes Theorem between the line integral and the area-integral

                 (21.20)

From Eq. 21.20 the obvious conclusion is   i.e., doublet flow is an irrotational flow.