Vortex

Corresponding Wikipedia article: Vortex

The vortex is a circular motion within a continuum. The known theory of vortices is applicable only for the 3-dimensional space, because it uses the differentiation formulas for products of the vector fields.

The vorticity measure is the curl of a velocity field. According to one of the curl definitions by the circulation, choosing a circle of radius $$r$$ as the contour, the curl becomes: $curl\,\mathbf{\overrightarrow{v}}=\lim_{S\to 0}\frac{1}{S}\oint{\mathbf{\overrightarrow{r}}\times\mathbf{\overrightarrow{v}}\mathrm{d}\alpha}=\frac{1}{\pi}\lim_{r\to 0}\oint{\frac{\mathbf{\overrightarrow{r}}\times\mathbf{\overrightarrow{v}}}{r^2}\mathrm{d}\alpha}\tag{1}$

Then, using the kinematic formula $$\mathbf{\overrightarrow{\omega}}=(\mathbf{\overrightarrow{r}}\times\mathbf{\overrightarrow{v}})/r^2$$, the curl of a velocity field is the twice angular velocity of infinitesimally small continuum element: $curl\,\mathbf{\overrightarrow{v}}=2\mathbf{\overrightarrow{\omega}}\tag{2}$

The Euler’s equation ("Continuum", 4), using the known identity $$(\mathbf{\overrightarrow{v}}\cdot\nabla)\mathbf{\overrightarrow{v}}=\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\,\mathbf{\overrightarrow{v}}$$, is converted to the Gromeka-Lamb equation: $\rho\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{\rho}{2}grad\;v^2-\rho\mathbf{\overrightarrow{v}}\times curl\,\mathbf{\overrightarrow{v}}+grad(P+U)=\rho\mathbf{\overrightarrow{f}}\tag{3}$ $\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\frac{1}{2}grad\;v^2-\mathbf{\overrightarrow{v}}\times curl\,\mathbf{\overrightarrow{v}}+\frac{1}{\rho}grad(P+U)=\mathbf{\overrightarrow{f}}\tag{4}$

Applying the curl operator to (4) and using the identity $$curl\;grad\;X=0$$ produces: $\frac{\partial}{\partial t}curl\,\mathbf{\overrightarrow{v}}-curl(\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}})+grad\frac{1}{\rho}\times grad(P+U)=curl\,\mathbf{\overrightarrow{f}}\tag{5}$

The strict theory of vortices is applicable either to the incompressible flows ($$\rho=const$$) or to the compressible barotropic flows. In both cases, the equation (5) is converted into an equation, which is independent of density and pressure: $\frac{\partial}{\partial t}curl\,\mathbf{\overrightarrow{v}}-curl(\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}})=curl\,\mathbf{\overrightarrow{f}}\tag{6}$

The vortical force field ($$curl\,\mathbf{\overrightarrow{f}}\neq 0$$) obviously causes an accelerated circular motion, so $$curl\,\mathbf{\overrightarrow{f}}=0$$ in the steady-state equation (6): $curl(\mathbf{\overrightarrow{v}}\times curl\,\mathbf{\overrightarrow{v}})=0\tag{7}$

The steady-state continuity equation ("Continuum", 5) is: $div\;\rho\mathbf{\overrightarrow{v}}=\mathbf{\overrightarrow{v}}grad\;\rho+\rho\;div\mathbf{\overrightarrow{v}}=0\tag{8}$

For an incompressible flow ($$grad\;\rho=0$$), the velocity is a vortical field ($$div\mathbf{\overrightarrow{v}}=0$$). For a compressible flow, the velocity is a vortex field too, if its density is constant along the flow: $\mathbf{\overrightarrow{v}}grad\;\rho=0\tag{9}$

As a result, a stable (steady-state) vortex is generally described by the system of equations:

Within the vortices, the centrifugal force is compensated by the force of the static pressure gradient, which is decreasing towards the center of a vortex due to increase in velocity and in dynamic pressure.

The conservative and pressure forces are shaping a vortex, and they move it over a continuum. This follows from the Kelvin’s circulation theorem, whereby any conservative force does not change a vortex circulation, and consequently the energy of this force can change velocity of the entire vortex: $\frac{\mathrm{d}}{\mathrm{d}t}\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=\oint{\frac{\mathrm{d}\mathbf{\overrightarrow{v}}}{\mathrm{d}t}\mathrm{d}\mathbf{\overrightarrow{l}}}=-\oint{\frac{1}{\rho}grad(P+U)\mathrm{d}\mathbf{\overrightarrow{l}}}=0\tag{10}$

For a barotropic flow: $\oint{\frac{1}{\rho}grad(P+U)\mathrm{d}\mathbf{\overrightarrow{l}}}=\oint{\frac{1}{\rho}grad\;f(\rho)\mathrm{d}\mathbf{\overrightarrow{l}}}=\oint{\frac{\mathrm{d}f(\rho)}{\rho\mathrm{d}\rho}grad\;\rho\;\mathrm{d}\mathbf{\overrightarrow{l}}}=\oint{\frac{\mathrm{d}f^*(\rho)}{\mathrm{d}\rho}grad\;\rho\;\mathrm{d}\mathbf{\overrightarrow{l}}}=\oint{grad\;f^*(\rho)\mathrm{d}\mathbf{\overrightarrow{l}}}=0\tag{11}$

The proof of (10) uses the kinematic Kelvin's theorem: $\frac{\mathrm{d}}{\mathrm{d}t}\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=\oint{\frac{\mathrm{d}\mathbf{\overrightarrow{v}}}{\mathrm{d}t}\mathrm{d}\mathbf{\overrightarrow{l}}}\tag{12}$

The streamline is a line, the tangent of which is parallel to the velocity vector at a given point. The Bernoulli's law follows from (3), and it is valid for any streamline, because any parameter along it is a function of single coordinate, and the force vector $$\mathbf{\overrightarrow{v}}\times curl\,\mathbf{\overrightarrow{v}}$$ is always perpendicular to the flow, so it does not produce a tangent acceleration.

The vortex line is a line, the tangent of which is parallel to the velocity curl vector at a given point. The Bernoulli's law is also valid for any vortex line.

The vortex tube is a closed surface, which is produced by the vortex lines. An example of a vortex tube is the inner surface of a vortex in a liquid or a gas with the maximum speed within it and the resting continuum inside it.

The velocity curl flux through a tube surface is: $\oint{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\int_{S_1}{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}-\int_{S_2}{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}\tag{13}$ where $$S_1$$ and $$S_2$$ are the cross sections of the tube. On the other hand, according to the divergence theorem and the identity $$div\;curl\,\mathbf{\overrightarrow{v}}=0$$: $\oint{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\oint{div\;curl\,\mathbf{\overrightarrow{v}}\mathrm{d}V}=0\tag{14}$ Thus, the velocity curl flux through any tube section is constant over its length, and according to the Stokes theorem, it’s equal to a circulation along the section contour: $\int{curl\,\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{S}}}=\oint{\mathbf{\overrightarrow{v}}\mathrm{d}\mathbf{\overrightarrow{l}}}=const\tag{15}$ The equation (15) is known as the 3rd Helmholtz's theorem, from which follow:

• The flow velocity within a tube is inversely proportional to the length (radius) of the tube section contour.
• The helical flow is possible along the tube of any shape without the flow discontinuity.
• The tube is continuous between the continuum interfaces. A stable vortex is either located between the interfaces or closed into the ring.

A funnel is a typical vortex, which occurs, for example, when a fluid is poured from a reservoir through a small hole at its bottom. The flow velocity increases as it get closer to the hole, and a weak fluid rotation around it can raise a powerful whirl.

The vortex ring is a vortex, which is closed into the annular solid of revolution. Generally, the generatrix and the guide of this solid have an arbitrary shape. The vortex ring is the only possible form of a stable vortex, which may not consume energy from the environment. Within the real fluids and gases, the viscous friction dissipates the energy of a vortex, and it decays. The streamlines of the vortex ring can be:

• Circular, along a generatrix of the solid. Such vortices arise within the fluids and gases by the effect of a jet. For example, the stable smoke rings arise when the jet of smoke passes through a narrow opening.
• Helical, which consists of a motion both along a generatrix and along a guide of the solid.

Exmple. «A storm in a teacup».

The rotation of a liquid in a cup causes the centrifugal force and boosts pressure at the walls of the cup. An annular cylindrical vortex appears at certain speed and lifts from the bottom a stuff, which is heavier than the liquid (tea leaves, for example).

The flow direction along the generatrix of a vortex ring is determined by the pressure gradient. The greatest pressure is always at the edge of the cup bottom, and it’s a sum of the liquid column pressure and the cup walls pressure. If the walls pressure is high enough, the stream may run from the bottom edge to the bottom center, ascending along the rotation axis and descending along the cup walls.

The decaying vortex causes the accumulation of heavy objects (tea leaves) at the bottom center, because the flow force is sufficient to move the objects over the bottom, but not enough to lift them off the bottom.