Certain flows show a significant amount of rotation while others do not. What cause
these flows to be rotational? In this lesson, we will be learning about the rotationality of
fluid element. It is important the conditions under which the fluid flow is rotational. Some
of the concepts outlined in this lesson were the basis for predicting lift forces on an
aircraft after the first flight by Wright Brothers in 1903. In those days, we did not
have the computing power we have today to solve mathematical models nor measurement techniques
to optimize aircraft wings. The physicists and mathematicians therefore relied on these
techniques to come up with approximate solutions for such engineering problems.
We defined vorticity as the measure of the rotation of a fluid element. In the case of
irrotational flows, each individual fluid particle does not change its orientation
in the flow field. In other words, these particles do not rotate as they travel
through the flow field. This happens only when the fluid particle is under the influence of
either body force or normal pressure force. Rotational component requires the action of
shearing force or shear stress. In fluid dynamics, shear is related to the rate of
angular deformation. Therefore, a fluid cannot rotate without undergoing angular deformation.
The relationship between shear stress and angular deformation is given by viscosity.
Therefore, a flow is rotational only if it is viscous. For irrotational flows,
vorticity is equal to zero. Because of this, the fluid velocity can be expressed as a function
of the gradient of a scalar variable, phi. This variable is defined as the velocity potential.
The circulation over any closed curve or contour is defined as the line integral of
the tangential component of velocity about this closed curve. For irrotational flows, circulation
is also equal to 0. As a consequence, streamlines cannot form closed curves in irrotational flows.
In almost all flows that occur in nature, the individual fluid elements rotate about their
respective axes as they travel through the flow field. Such fluid flows are called rotational.
For these flows, the vorticity vector is non-zero. In fact, from the basics of vector calculus,
the divergence of the vorticity vector for fully rotating flows is 0. This means that
the vortex field is solenoidal. There are no sources or sinks in this vector field.
For fully rotational flows, the vorticity field lines are completely closed loops. By extension,
we can say that the vorticity flux crossing over any close surface is
also 0. This means that the net outflow of vorticity through any close surface is 0.
Similar to the way we introduced streamlines to visualize the flow field, we can introduce
vorticity lines to visualize the rotationality of the flow field. The tangent of a vorticity line
is parallel to the local vorticity vector. These lines can be defined by the following expression.
If we consider a closed loop or curve and draw the vortex lines passing through all points on this
curve, the obtained 3D shape is called a vortex tube. In fact, the flow of vorticity through
any cross-sectional surface of a vortex tube is a constant all along the tube. This is true at every
instant of time. The integral shown here refers to the strength of the vortex or simply vortex
strength. A tornado is an excellent example of a vortex tube occurring in nature. Another example
is smoke ring created in air. Dolphin create vortex tubes in the form of bubble. If we take
a linear vortex tube and join the two ends, it would resemble a smoke ring or bubble ring.
A German physicist Herman von Helmholtz came up with a series of theorems that defined the
motion of a vortex tube. Prandlt, in 1921, made use of these mathematical
theorems developed by Helmholtz to develop the classical lifting theory for line bodies
such as airfoils and wings. He was able to solve for the overall lift force generated on
these line bodies. Helmholtz theorem states four main facts for pure rotational flows.
The first one is Vortex lines move with the fluid, that is fluid elements lying on a vortex line
at some instant continue to lie on that vortex line. This is valid for an ideal rotational flow.
The second fact is the strength of the vortex tube is constant along its
length and in time. This is true unless acted upon by a dissipative force such
as viscous resistance. Smoke rings die out because of this resistive force applied by
the surrounding air. The third one is that fluid elements initially free of vortex
remain free of vortex. The fluid elements which are outside these vortex tubes will
remain outside. This is true unless the vortex tube is acted up on by an external disturbance.
vortex tubes can extend to Infinity, form a closed loop (similar to the case of smoke
rings or dolphin bubble rings) or end at a solid boundary. In fact,
a practical example of this is a tornado. It always attempts to touch the ground somewhere.