A New Model for the Equilibrium Shape of Raindrops

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  • 1 Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, IL 61801
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Abstract

The equilibrium shape of raindrops has been determined from Laplace's equation using an internal hydrostatic pressure with an external aerodynamic pressure based on measurements for a sphere but adjusted for the effect of distortion. The drop shape was calculated by integration from the upper pole with the initial curvature determined by iteration on the drop volume. The shape was closed at the lower pole by adjusting either the pressure drag or the drop weight to achieve an overall force balance. Model results provide bounds on the axis ratio of raindrops with an uncertainty of about 1% and very good agreement with extensive wind tunnel measurements for moderate to large water drops.

The model yields the peculiar asymmetric shape of raindrops: a singly curved surface with a flattened base and a maximum curvature just below the major axis. A close match was found between model shapes and profiles obtained from photos of water drops for diameters up to 5 mm. Coefficients are provided for computing raindrop shape as a cosine series distortion on a sphere.

In contrast to earlier models of raindrop shape for the oblate spheroid response to gravity (Green, Beard) or the perturbation response to the aerodynamic pressure for a sphere (Imai, Savic, Pruppacher and Pitter), the present model provides the appropriate large amplitude response to both the hydrostatic and aerodynamic pressures modified for distortion. In addition, the new model can be readily extended to include other pressures such as an electric stress.

Abstract

The equilibrium shape of raindrops has been determined from Laplace's equation using an internal hydrostatic pressure with an external aerodynamic pressure based on measurements for a sphere but adjusted for the effect of distortion. The drop shape was calculated by integration from the upper pole with the initial curvature determined by iteration on the drop volume. The shape was closed at the lower pole by adjusting either the pressure drag or the drop weight to achieve an overall force balance. Model results provide bounds on the axis ratio of raindrops with an uncertainty of about 1% and very good agreement with extensive wind tunnel measurements for moderate to large water drops.

The model yields the peculiar asymmetric shape of raindrops: a singly curved surface with a flattened base and a maximum curvature just below the major axis. A close match was found between model shapes and profiles obtained from photos of water drops for diameters up to 5 mm. Coefficients are provided for computing raindrop shape as a cosine series distortion on a sphere.

In contrast to earlier models of raindrop shape for the oblate spheroid response to gravity (Green, Beard) or the perturbation response to the aerodynamic pressure for a sphere (Imai, Savic, Pruppacher and Pitter), the present model provides the appropriate large amplitude response to both the hydrostatic and aerodynamic pressures modified for distortion. In addition, the new model can be readily extended to include other pressures such as an electric stress.

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