## Equations Of Motion For A Rotating Fluid

Equation 6-6 is an accurate representation of Newton's laws applied to a fluid observed from a fixed, inertial, frame of reference. However, we live on a rotating planet and observe winds and currents in its rotating frame. For example the winds shown in Fig. 5.20 are not the winds that would be observed by someone looking back at the Earth, as in Fig. 1. Rather, they are the winds measured by observers on the planet rotating with it. In most applications it is easier and more desirable to work with the governing equations in a frame rotating with the Earth. Moreover it turns out that rotating fluids have rather unusual properties and these properties are often most easily appreciated in the rotating frame. To proceed, then, we must write down our governing equations in a rotating frame. However, before going on to a formal ''frame of reference'' transformation of the governing equations, we describe a laboratory experiment that vividly illustrates the influence of rotation on fluid motion and demonstrates the utility of viewing and thinking about fluid motion in a rotating frame.

6.6.1. GFD Lab III: Radial inflow

We are all familiar with the swirl and gurgling sound of water flowing down a drain. Here we set up a laboratory illustration of this phenomenon and study it in rotating and nonrotating conditions. We rotate a cylinder about its vertical axis; the cylinder has a circular drain hole in the center of its bottom, as shown in Fig. 6.5. Water enters at a constant rate through a diffuser on its outer wall and exits through the drain. In so doing, the angular momentum imparted to the fluid by the rotating cylinder is conserved as it flows inwards, and paper dots floated on the surface acquire the swirling motion seen in Fig. 6.6 as the distance of the dots from the axis of rotation decreases.

The swirling flow exhibits a number of important principles of rotating fluid dynamicsâ€”conservation of angular momentum, geostrophic (and cyclostrophic) balance (see Section 7.1)â€”all of which will be used in our subsequent discussions. The experiment also gives us an opportunity to think about frames of reference because it is viewed by a camera co-rotating with the cylinder.

diffuser

FIGURE 6.5. The radial inflow apparatus. A diffuser of 30-cm inside diameter is placed in a larger tank and used to produce an axially symmetric, inward flow of water toward a drain hole at the center. Below the tank there is a large catch basin, partially filled with water and containing a submersible pump whose purpose is to return water to the diffuser in the upper tank. The whole apparatus is then placed on a turntable and rotated in an anticlockwise direction. The path of fluid parcels is tracked by dropping paper dots on the free surface. See Whitehead and Potter (1977).

FIGURE 6.5. The radial inflow apparatus. A diffuser of 30-cm inside diameter is placed in a larger tank and used to produce an axially symmetric, inward flow of water toward a drain hole at the center. Below the tank there is a large catch basin, partially filled with water and containing a submersible pump whose purpose is to return water to the diffuser in the upper tank. The whole apparatus is then placed on a turntable and rotated in an anticlockwise direction. The path of fluid parcels is tracked by dropping paper dots on the free surface. See Whitehead and Potter (1977).

FIGURE 6.6. Trajectories of particles in the radial inflow experiment viewed in the rotating frame. The positions are plotted every 1/30 s. On the left Q = 5 rpm (revolutions per minute). On the right Q = 10 rpm. Note how the pitch of the particle trajectory increases as Q increases, and how in both cases the speed of the particles increases as the radius decreases.

FIGURE 6.6. Trajectories of particles in the radial inflow experiment viewed in the rotating frame. The positions are plotted every 1/30 s. On the left Q = 5 rpm (revolutions per minute). On the right Q = 10 rpm. Note how the pitch of the particle trajectory increases as Q increases, and how in both cases the speed of the particles increases as the radius decreases.

### Observed flow patterns

When the apparatus is not rotating, water flows radially inward from the diffuser to the drain in the middle. The free surface is observed to be rather flat. When the apparatus is rotated, however, the water acquires a swirling motion: fluid parcels spiral inward, as can be seen in Fig. 6.6. Even at modest rotation rates of Q = 10 rpm (corresponding to a rotation period of around 6 seconds),4 the effect of rotation is marked and parcels complete many circuits before finally exiting through the drain hole. The azimuthal speed of the particle increases as it spirals inwards, as indicated by the increase in the spacing of the particle positions in the figure. In the presence of rotation the free surface becomes markedly curved, high at the periphery and plunging downwards toward the hole in the center, as shown in the photograph, Fig. 6.7.

### Dynamical balances

In the limit in which the tank is rotated rapidly, parcels of fluid circulate around many times before falling out through the drain hole (see the right hand frame of Fig. 6.6); the pressure gradient force directed radially inward (set up by the free surface tilt) is in large part balanced by a centrifugal force directed radially outward.

FIGURE 6.7. The free surface of the radial inflow experiment. The curved surface provides a pressure gradient force directed inward that is balanced by an outward centrifugal force due to the anticlockwise circulation of the spiraling flow.

If Vq is the azimuthal velocity in the absolute frame (the frame of the laboratory) and vq is the azimuthal speed relative to the tank (measured using the camera co-rotating with the apparatus) then (see Fig. 6.8)

where Q is the rate of rotation of the tank in radians per second. Note that Qr is the azimuthal speed of a particle stationary relative to the tank at radius r from the axis of rotation.

4An q of 10 rpm (revolutions per minute) is equivalent to a rotation period T = 60 = 6 s. Various measures of rotation rate are set out in Table A.4 of the appendix.

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