Circulation

We now discuss one other property of flows, that of Circulation. Consider any closed curve C in a flow as shown. Circulation is defined as the line integral around the curve of the arc length ds times the tangential component of velocity. Shear stress for the element is thus given by

Figure 4.11: Definition of Circulation

$$\Gamma = \oint_C \vec{V}.ds = \oint_C  V  cos\alpha  ds = \oint_C (udx + vdy + wdz)$$ (4.45)

An expression for circulation can be derived by considering a small differential area in the curve, which is shown enlarged in Fig.4.12.

Figure 4.12: Definition of Circulation, continued

If the velocity components at A are (u,v) then we have

$$(u,v)$$ Velocity along AB:

$$(u + {\partial u \over \partial x} dx,v + {\partial v \over \partial x} dx))$$ Velocity along BC:

$$(u + {\partial u \over \partial y} dy,v + {\partial v \over \partial y} dy)$$ Velocity along DC:

$$(u,v)$$ Velocity along AD: (4.46)

Circulation along the boundary of the differential element is given by

$$d\Gamma = (udx+vdy)_{AB} +  (udx+vdy)_{BC} +  (udx+vdy)_{CD} +  (udx+vdy)_{DA}$$ (4.47)

Noting that along AB, dy = 0 etc, the above equation reduces to

$$d\Gamma =  (udx)_{AB} +  (vdy)_{BC} + (udx)_{CD} +  (vdy)_{DA}$$ (4.48)

Upon substituting for velocity components from Eqn.4.46, we have

$$d\Gamma = (udx) +  (v + {\partial v \over \partial x} dx)dy -   (u + {\partial u \over \partial y} dy)dx -  (vdy)$$ (4.49)

which on simplification gives,

$$d\Gamma = \left({\partial v \over \partial x} - {\partial u \over \partial y}\right) dx dy$$ (4.50)

Integrating this for the entire region C gives,

$$\Gamma = \int_C d\Gamma = \int \int \left({\partial v \over \partial x} - {\partial u \over \partial y}\right) dx dy$$ (4.51)

In other words we have,

$$\Gamma = \oint_C (u dx + v dy)= \int \int \left({\partial v \over \partial x} - {\partial u \over \partial y}\right) dx dy$$ (4.52)

Thus we see that a complicated area integral (also a double integral) is reduced to a single integral along the curve.

The other important observation to make is that for an irrotational flow, circulation is zero.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney