Page 104 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 87

%Reference axes
Fig. 2.26

2.7.6 Rotational and irrotational flow

It will be made clear in Section 2.8 that the generation of shear strain in a fluid element,
as it travels through the flow field, is closely linked with the effects of viscosity. It is also
plain from its definition (Eqn (2.76)) that vorticity is related to rate of shear strain.
Thus, in aerodynamics, the existence of vorticity is associated with the effects of
viscosity.* Accordingly, when the effects of viscosity can be neglected, the vorticity is
usually equivalently zero. This means that the individual fluid elements do not rotate,
or distort, as they move through the flow field. For incompressible flow, then, this
corresponds to the state of pure translation that is illustrated in Fig. 2.26. Such a flow is
termed irrotational flow. Mathematically, it is characterized by the existence of a velocity
potential and is, therefore, also called potential flow. It is the subject of Chapter 3.
The converse of irrotational flow is rotational flow.

2.7.7 Circulation
The total amount of vorticity passing through any plane region within a flow field is
called the circulation, r. This is illustrated in Fig. 2.27 which shows a bundle of vortex
tubes passing through a plane region of area A located in the flow field. The
perimeter of the region is denoted by C. At a typical point P on the perimeter, the
velocity vector is designated q or, equivalently, t. At P, the infinitesimal portion of C
has length 6s and points in the tangential direction defined by the unit vector t (or i‘).
It is important to understand that the region of area A and its perimeter C have no
physical existence. Like the control volumes used for the application of conservation
of mass and momentum, they are purely theoretical constructs.
Mathematically, the total strength of the vortex tubes can be expressed as an
integral over the area A; thus

(2.80)

where n is the unit normal to the area A. In two-dimensional flow the vorticity is in the
z direction perpendicular to the two-dimensional flow field in the (x, y) plane. Thus
n = k (i.e. the unit vector in the z direction) and = Ck, so that Eqn (2.80) simplifies to

(2.81)

* Vorticity can also be created by other agencies, such as the presence of spatially varying body forces in the
flow field. This could correspond to the presence of particles in the flow field, for example.