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Terminal Velocity and Shape of Cloud and Precipitation Drops Aloft

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  • 1 Department of Meteorology, University of California, Los Angeles
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Abstract

The terminal velocity of cloud and precipitation size drops has been analyzed for three physically distinct flow regimes: 1) slip flow about a water drop treated as rigid sphere at negligible Reynolds numbers, 2) continuum flow past a water drop treated as a rigid sphere with a steady wake at low and intermediate Reynolds numbers, and 3) continuum flow around a non-circulating water drop of equilibrium shape with an unsteady wake at moderate to large Reynolds numbers. In the lower regime the effect of slip was given by the first-order Knudsen number correction to Stokes drag. In the middle regime a semiempirical drag relation for a rigid sphere was used to obtain a formula for the Reynolds number in terms of the Davies number. In the upper regime a correlation of wind tunnel measurements on falling drops was used in conjunction with sea level terminal velocities for raindrops to obtain a formula for the Reynolds number in terms of the Bond number and physical property number.

The result for the upper regime gave values of the drag coefficient that were consistent with an invariance of shape with altitude in the atmosphere. Simple formulas are given for obtaining the axis ratio and projected diameter as a function of the equivalent spherical diameter. The resulting formulas for the terminal velocity in three diameter ranges (0.5 µm–19 µm, 19 µm–1.07 mm, 1.07 mm–7 mm) may be used to calculate the terminal velocity directly from the equivalent spherical diameter and the physical properties of the drop and atmosphere.

Abstract

The terminal velocity of cloud and precipitation size drops has been analyzed for three physically distinct flow regimes: 1) slip flow about a water drop treated as rigid sphere at negligible Reynolds numbers, 2) continuum flow past a water drop treated as a rigid sphere with a steady wake at low and intermediate Reynolds numbers, and 3) continuum flow around a non-circulating water drop of equilibrium shape with an unsteady wake at moderate to large Reynolds numbers. In the lower regime the effect of slip was given by the first-order Knudsen number correction to Stokes drag. In the middle regime a semiempirical drag relation for a rigid sphere was used to obtain a formula for the Reynolds number in terms of the Davies number. In the upper regime a correlation of wind tunnel measurements on falling drops was used in conjunction with sea level terminal velocities for raindrops to obtain a formula for the Reynolds number in terms of the Bond number and physical property number.

The result for the upper regime gave values of the drag coefficient that were consistent with an invariance of shape with altitude in the atmosphere. Simple formulas are given for obtaining the axis ratio and projected diameter as a function of the equivalent spherical diameter. The resulting formulas for the terminal velocity in three diameter ranges (0.5 µm–19 µm, 19 µm–1.07 mm, 1.07 mm–7 mm) may be used to calculate the terminal velocity directly from the equivalent spherical diameter and the physical properties of the drop and atmosphere.

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