Interpretation of the vorticity equation

Since the vorticity is related to the local rotation of fluid elements consider the angular momentum, ${\bf L}$, of a small material element that is instantaneously spherical. Then

$${\bf L}= \frac{1}{2} I \mbox{\boldmath$\omega$}$$ (7.43)

where $I$ is the moment of inertia. The angular momentum will change at a rate determined by the tangential surface stresses alone, which are zero for an invisid fluid, i.e., fluid outside of shocks or boundary layers. The pressure has no influence on ${\bf L}$ at the instant at which the element is spherical since the pressure forces all point radially inward. Therefore, at one particular instant in time, we have

$$\frac{D {\bf L}}{Dt} = 0$$ (7.44)

The convective derivative satisfies the usual rules of differentiation and so we have

$$I \frac{D\mbox{\boldmath$\omega$}}{Dt} = -\mbox{\boldmath$\omega$}\frac{DI}{Dt} .$$ (7.45)

Evidently, the term on the right arises from the change in the moment of inertia of a fluid element. Equation (7.45) is not very useful since it applies only at the initial instant during which the fluid element is spherical. However, it suggests several results. First,there is no reference to pressure in Eq. (7.45), so that we might anticipate that $\mbox{\boldmath$\omega$}$ evolves independently of $p$. Second, if $\mbox{\boldmath$\omega$}$ is initially zero, and the flow is inviscid, then $\mbox{\boldmath$\omega$}$ should remain zero in each fluid particle as it is swept around the flow field. This is the basis of potential flow theory in which we set $\mbox{\boldmath$\omega$}$ = 0 in the upstream fluid, and so we can assume that $\mbox{\boldmath$\omega$}$ is zero at all points. Third, if $I$ decreases in a fluid element (and $\nu = 0$), then the vorticity of that element should increase. For example, consider a blob of vorticity embedded in an otherwise potential flow field consisting of converging streamlines. An initially spherical element will be stretched into an ellipsoid by the converging flow. The moment of inertia of the element about an axis parallel to $\mbox{\boldmath$\omega$}$ decreases, and consequently $\mbox{\boldmath$\omega$}$ must rise to conserve angular momentum. It is possible, therefore, to intensify vorticity by stretching fluid blobs. Intense rotation can result from this process, the familiar bath-tub vortex being just one example. We shall see that something very similar happens to magnetic fields. They, too, can be intensified by stretching.

Figure: Stretching of a material fluid element in converging streamlines concentrated vorticity.

For incompressible flow both ${\bf u}$ and $\mbox{\boldmath$\omega$}$ are both solenoidal. Using the vector relationship

$$\mathbf{ \nabla} \times ( {\bf u}\times \mbox{\boldmath$\omeg... ...bf u}- ( {\bf u}\cdot \mathbf{ \nabla})\mbox{\boldmath$\omega$}$$

Eq. (7.41) may be rewritten as

$$\frac{D\mbox{\boldmath$\omega$}}{Dt} = (\mbox{\boldmath$\omega$}\cdot \mathbf{ \nabla}) {\bf u}$$ (7.46)

Compare this with our angular momentum equation [Eq. (7.45)] for a blob which is instantaneously spherical. We might anticipate that the term on the RHS of Eq. (7.46) represents the change in the moment of inertia of a fluid element due to stretching of that element. In other words, the rate of rotation of a fluid blob may increase or decrease due to changes in its moment of inertia, or change because it is spun up or slowed down by viscous stresses.

Equation (7.46) is the advection equation for vorticity. First consider the two dimensional case where ${\bf u}(x,y) = (u_x, u_y,0)$ and $\mbox{\boldmath$\omega$}(x,y) = (0,0, \omega_z)$ so that Eq. (7.46) becomes

$$\frac{D\omega_z}{Dt} = 0.$$ (7.47)

The vorticity of a fluid element in two dimensional flow is constant.

Figure: Stretching of a vortex tube.

In three-dimensional flows the RHS of Eq. (7.46) is non-zero, and it is this term which distinguishes three-dimensional flows from two-dimensional ones. Comparison of the RHS of Eq.(7.46) with the angular momentum equation suggests that $(\mbox{\boldmath$\omega$}\cdot \mathbf{ \nabla}) {\bf u}$ represents intensification vorticity by the stretching fluid elements. To show this consider, for example, an axisymmetric flow consisting of converging streamlines (in the $r-z$ plane) as well as a swirling component of velocity, $u_\theta$. By writing $\mathbf{ \nabla} \times {\bf u}$ in cylindrical coordinates, we find that, near the axis, the axial component of vorticity is

$$\omega_z = \frac{1}{r} \frac{\partial }{\partial r} (r u_\theta)$$

Now consider the axial component of the vorticity equation Eq. (7.46) applied near $r = 0$. The advection term yields expression

$$(\mbox{\boldmath$\omega$}\cdot \mathbf{ \nabla}) {\bf u}\simeq \omega_z \frac{\partial u_z}{\partial z}$$

This appears on the right of Eq. (7.46) and so acts like a source of axial vorticity. In particular, the vorticity, $\omega_z$, intensifies if $\partial u_z / \partial z > 0$, i.e., the streamlines converge. This is because fluid elements are stretched and elongated on the axis, as shown in Fig 7.1. This leads to a reduction in the axial moment of inertia of the element and so, by conservation of angular momentum, to an increase in $\omega_z$.

More generally, consider a thin tube of vorticity, as shown in Fig 7.2. Let $u_\parallel$ be the component of velocity parallel to the vortex tube and $s$ be a coordinate measured along the tube. Then

$$\vert \mbox{\boldmath$\omega$}\vert \frac{du_\parallel}{ds} = (\mbox{\boldmath$\omega$}\cdot \mathbf{ \nabla}) u_\parallel$$

Now the vortex line is being stretched if the velocity $u_\parallel$ at point $B$ is greater than $u_\parallel$ at $A$. That is, the length of the material element $AB$ increases if $d u_\parallel/ds > 0$. Thus the term $(\mbox{\boldmath$\omega$}\cdot \mathbf{ \nabla}) {\bf u}$ represents stretching of the vortex lines. This leads to an intensification of vorticity through conservation of angular momentum, confirming our interpretation of the RHS of Eq. (7.46).