A tornado consists of a whirling mass of air such that the winds essentially move along horizontal circular streamlines, Fig. 7–29a. Determine the pressure distribution within the tornado as a function of r.

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Fluid Description. We will assume the air is an ideal fluid that has steady flow. Translational motion of the tornado will be neglected.

Free Vortex. If we assume the motion takes the form of a free vortex, then from Eqs. 7–38 and 7–39, the velocity components are

[latex] v_r=\frac{1}{r} \frac{\partial \psi}{\partial heta} ; \quad v_r=0 [/latex] (7-38)

[latex] v_ heta=-\frac{\partial \psi}{\partial r} ; \quad v_ heta=\frac{k}{r} [/latex] (7-39)

[latex] v_r=0 \quad ext { and } \quad v_ heta=\frac{k}{r} [/latex] (1)

Because the flow within a free vortex is steady irrotational flow, the Bernoulli equation can be applied to two points, each lying on different streamlines. If we choose a point within the tornado, and another at the same elevation but remote, where the air velocity V = 0 and the (gage) pressure is p = 0, then using Eq. 7–19, we have

[latex] \frac{p}{\gamma}+\frac{V^2}{2 g}+y= ext { Const. } [/latex] (7-19)

[latex] \begin{array}{r} \frac{p_1}{\gamma}+\frac{V_1^2}{2 g}+z_1=\frac{p_2}{\gamma}+\frac{V_2^2}{2 g}+z_2 \ \frac{p}{ ho g}+\frac{k^2}{2 g r^2}+z=0+0+z \end{array} [/latex]

[latex] p=-\frac{ ho k^2}{2 r^2} [/latex] (2)

Here k is a constant, as yet to be determined. The negative sign in this equation indicates that a suction develops, and both this suction and the velocity will intensify as r becomes smaller.
A free vortex such as this cannot actually exist in a real fluid, since the velocity and pressure would have to approach infinity as r →0. Instead, due to the increasing velocity gradient as r becomes smaller, the viscosity of the air will eventually create enough shear stress to cause the air in the core, or “eye,” to rotate as a solid system having an angular velocity ω. We will assume this transition occurs at a radius of [latex] r=r_0 [/latex], Fig. 7–29a.

Forced Vortex. Because of its “rigid-body” motion, the core becomes a forced vortex, which we analyzed using Euler’s equations of motion in Example 5–1.
In that example, we showed that the pressure distribution is defined by

[latex] p=p_0-\frac{ ho \omega^2}{2}\left(r_0^2-r^2 ight) [/latex] (3)

where [latex] p_0 [/latex] is the pressure at [latex] r_0 [/latex].
In order to fully study the pressure variation, we must match the two solutions at [latex] r=r_0 [/latex] and thereby determine the constant k in Eq. 1.
At [latex] r_0 [/latex] the velocity in the forced vortex, [latex] v_ heta=\omega r_0 [/latex], must equal that in the free vortex, that is,

[latex] v_ heta=\omega r_0=\frac{k}{r_0} \quad ext { so that } k=\omega r_0^2 [/latex]

The pressures in Eqs. 2 and 3 must also be equivalent at [latex] r=r_0 [/latex], so that

[latex] \begin{aligned} -\frac{ ho\left(\omega r_0^2 ight)^2}{2 r_0^2} & =p_0-\frac{ ho \omega^2}{2}\left(r_0^2-r_0^2 ight) \ p_0 & =-\frac{ ho \omega^2 r_0^2}{2} \end{aligned} [/latex]

Therefore, after substituting into Eq. 3 and simplifying, we have for the forced vortex [latex] r \leq r_0 [/latex],

[latex] \begin{aligned} v_ heta & =\omega r \ p & =\frac{ ho \omega^2}{2}\left(r^2-2 r_0^2 ight) \end{aligned}[/latex]

And for the free vortex, [latex] r \geq r_0 [/latex],

[latex] \begin{aligned} v_ heta & =\frac{\omega r_0^2}{r} \ p & =-\frac{ ho \omega^2 r_0^4}{2 r^2} \end{aligned} [/latex]

Using these results, a plot of both the velocity and pressure variations is shown in Fig. 7–29b. Notice that the largest suction (negative pressure) occurs at the center of the forced vortex, r = 0, and the highest velocity occurs at [latex] r=r_0 [/latex]. It is the combination of this low pressure, which can lift houses, cars, and debris off the ground, and the high velocity that makes tornadoes so destructive. In fact tornadoes often reach wind speeds over 322 km/h.
The type of vortex we have considered here, that is, a combination of a forced vortex surrounded by a free vortex, is sometimes called a compound vortex. It not only occurs in tornadoes, but also forms in your bathtub as the water drains from the bottom, or in a river as the water flows off a boat oar or around a bridge pier.

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