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Kenneth V. Beard

Abstract

Potential flow oscillations about an equilibrium raindrop distortion were modeled for ellipsoidal variations driven by changes in surface and gravitational potential energy with linear dissipation of kinetic energy. The model was found to be quantitatively similar to the Navier-Stokes results of G. B. Foote for axisymmetric oscillations without gravity. Computed frequencies and average axis ratios for vertical and horizontal oscillations with gravity were compared to wind tunnel observations of oscillating water drops and raindrop camera data. Simple formulas with good accuracy were developed for the time variation and time average axis ratios as a function of oscillation amplitude.

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Kenneth V. Beard

Abstract

The altitude factor for adjusting raindrop velocities from sea level depends primarily on air density and drop size. An adjustment factor is formulated as the air density ratio raised to a power m that varies linearly with drop diameter from 0.4 at 1 mm to 0.5 at 5 mm: m(D) = 0.375 + 0.025D (mm). The improved velocity adjustment can be directly incorporated into the standard equation for calculating the size distribution from the Doppler spectrum. An integral mean value () is provided to adjust sea level values of rainfall rate and mean (and mean square) Doppler velocities. The recommended value of varies from 0.41 to 0.46 depending on application. The simplest adjustment is obtained using 0.42 for rainfall rate and 0.45 for mean Doppler velocity.

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Ali Tokay and Kenneth V. Beard

Abstract

The size spectra of oscillating raindrops were determined from photographic measurements in Illinois showers at night. The oscillations were detected from modulations in the fall streaks produced by backscattered light near the primary rainbow. Drop sizes were determined from the fall speed using strobe lights. A continuous record of raindrop size distributions was obtained from a disdrometer located beneath the camera sample volume. Results show that oscillations begin near 1-mm diameter, at the onset size for vortex shedding. This finding is consistent with the authors' recent laboratory and field studies for small raindrops. The size spectra show that all raindrops above 1 mm are oscillating, out to the largest size measured of 4.2-mm diameter.

Extrinsic sources of drop oscillations were evaluated using a collision model with viscous decay and using the pressure forcing from turbulence and wind shear in the surface layer. Based on the disdrometer size distributions, the number of oscillating drops produced by collisions was negligible at all rainfall rates compared to the observed number of oscillating drops. Forcing by turbulence and wind shear at the observed wind speeds was found to be well below the threshold of detection. The most plausible explanation for the observed oscillations must involve intrinsic mechanisms capable of maintaining oscillations against viscous decay. Likely sources are the positive feedback to particular oscillation modes caused by vortex shedding and caused by the aerodynamic pressure and drag fluctuations associated with the oscillations themselves.

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Kenneth V. Beard and Catherine Chuang

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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David B. Johnson and Kenneth V. Beard

Abstract

When raindrops collide, some of the kinetic energy involved in the collision will be available to initiate or sustain oscillations in the surviving drops. This paper presents results of a simple model of drop collisions that generates an estimate of the expected distribution of energies in an ensemble of colliding raindrops as a function of drop size and rain intensity. The results indicate that drop collisions can be an effective source of raindrop oscillations and that within any one rain shaft, it tends to produce a range of oscillation energies from intense to imperceptible. In every case, however, the fraction of drops oscillating and the severity of the oscillations increase with increasing drop size and rainfall intensity.

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Brian A. Tinsley and Kenneth V. Beard
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Kenneth V. Beard and Harry T. Ochs III

Abstract

No abstract available

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Rodney J. Kubesh and Kenneth V. Beard

Abstract

The natural oscillations of moderate-size raindrops were studied in a seven-story fall column using a computer-controlled generator to produce isolated water drops at terminal speed. Instantaneous shapes were photographed to obtain oscillation sequences of single drops by a multiple-strobe technique. The oscillation frequencies were determined from fall-streak modulations that were photographed in backscattered light of the primary rainbow. Measurements were made at three levels for 2.0- and 2.5-mm diameter drops to assess the role of aerodynamic feedback as the source of drop oscillations.

Variations as large as 15% in axis ratio were observed at the bottom of the fall column, even though the initial oscillations were predicted to die out by viscous decay theory. Practically all oscillations were at the fundamental and first harmonic frequencies. The oscillation modes deduced from the axis ratio scatter indicated that the axisymmetric modes died away slowly and that transverse modes persisted. The slow decay of the axisymmetric modes is postulated to be caused by positive feedback of shape-induced changes in pressure and drag from the initial oscillations. The transverse mode is believed to persist because of transverse pressure perturbations associated with eddy shedding. Various types of feedback are considered that could explain the broad coupling between eddy shedding and oscillations.

The mean experimental axis ratios were higher than equilibrium values—an apparent consequence of shape changes from transverse modes. The deviation from equilibrium shape was generally consistent with previous field measurements of raindrop axis ratios. Use of empirical mean axis ratios in differential reflectivity calculations would change equilibrium values of ZDR by 20%–30%.

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Harry T. Ochs III and Kenneth V. Beard

Abstract

A closed parcel model which simulates condensation, collection and breakup was used to evaluate the effects of recently measured collection efficiencies on precipitation development. Computations were made using theoretical collision efficiencies assuming coalescence efficiencies of 100% and, for comparison, with semiempirical coalescence efficiencies of 50–100%. The model indicated that the development of a detectable radar echo was (retarded in time) by 250 to 500 meters from the effects of limited coalescence and that the concentration of precipitation drops was generally reduced by several orders of magnitude.

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Kenneth V. Beard and Andrew J. Heymsfield

Abstract

Velocity adjustments are evaluated for altitude changes using Reynolds number-Davies number correlations of the form Re = aXb which have been obtained from empirical fall velocities of ice particles. In general, the altitude adjustment was found to vary with both pressure and temperature, except for a temperature-independent range near b ≈ 0.7. A quantitative evaluation of b, using the drag on a sphere, shows that altitude adjustments for precipitation particles are less sensitive to changes in temperature than pressure, and that the net adjustment is reduced by compensation between the two effects. A comparison between the X-Re method of Heymsfield and Kajikawa (1987) and the Reynolds number method of Beard (1980), developed from drag data using models of hydrometeor shapes, yields similar velocity adjustments for altitude changes. The agreement suggests that X-Re formulas, based on X for ice particles of one type, but different masses, can also be used for altitude adjustments because the shape is relatively invariant for the small changes in X typical of altitude adjustments. For larger changes in X the constant shape method of Beard is suited to calculating velocity adjustments for charged particles whereas the empirical X-Re formulas of Heymsfield and Kajikawa are appropriate for computing velocities changes from riming.

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