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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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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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Catherine C. Chuang
and
Kenneth V. Beard

Abstract

The model Beard Chuang, using the differential form of Laplace's formula, has been extended to raindrop shapes under the influence of vertical electric fields and drop charges. A finite volume method was used with a boundary-fitted coordinate system to calculate the shape-dependent electric field. The distorted shape was obtained by numerical integration from the upper to lower pole by iteration to achieve the appropriate drop volume and force balance using shape-dependent stresses.

The model prediction of the critical electric field for instability is within a few percent of previous models for a stationary drop, but stability was found to be considerably enhanced for raindrops because of the counteracting aerodynamic distortion. The predicted critical fields for larger raindrops, however, are about 2 kV cm−1 higher than found in the wind tunnel measurements of Richards and Dawson. Model raindrop shapes in a strong, electric field show a pronounced extension of the upper pole, and a flattened base caused by the increased fall speed from vertical stretching. The resultant triangular drop profiles are similar to wind tunnel observations.

The shape of highly charged raindrops have a pronounced oblate distortion caused by the charge enhancement of existing distortion and the fall speed reduction of the aerodynamic asymmetry. The shapes for upward and downward electric forces differ because of the altered fall speed. For the maximum field and charge expected in thunderstorms, a downward force increases the aerodynamic distortion thereby counteracting the electric stretching so that axis ratios are nearly the same as in the absence of electric stresses. In contrast, an upward electric force decreases the aerodynamic distortion, resulting in an enhanced vertical stretching to the extent that large raindrops can become unstable.

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Kenneth V. Beard
and
Rodney J. Kubesh

Abstract

The oscillation frequencies and modes of small raindrops (1.04–1.54-mm diameter) were determined from laboratory experiments using water drops generated at terminal velocity at a fall distance sufficient for initial oscillations to damp out. Frequency information was obtained from fall streaks photographed in backscatter light near the primary and secondary rainbows. Streak data was interpreted with the aid of ray tracing through drops with spherical harmonic perturbations. Axis ratio data was used in conjunction with analyses of spherical harmonic perturbations to help determine the oscillation modes.

Two frequencies were present in all drop sizes. The significant oscillation modes for smaller drops (1.04–1.30 mm) were the transverse modes of the fundamental and first harmonic, whereas the significant oscillation modes for larger drops (1.40–1.54 mm) were the axisymmetric mode of the fundamental and the transverse mode of the first harmonic. Primary resonance appears to be responsible for the transverse modes because of the match in frequencies between the forcing and response and because the spatial pattern of the eddy shedding would tend to force these modes. Secondary resonance would account for the axisymmetric mode in larger drops, since this mode is a subharmonic of the forcing frequency and there is no requirement for the forcing pattern to match the response.

Our study shows that small raindrops oscillate as a resonant response to eddy shedding. The postulated oscillation modes are consistent with scatter and means found in the laboratory data and would produce the trends in axis ratios inferred for small raindrops from field studies (Goddard and Cherry; Chandrasekar et al.). Since the discovered secondary resonance does not require a good frequency match, eddy shedding also may be the cause of raindrop oscillations detected in the field studies for much larger sizes.

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James Q. Feng
and
Kenneth V. Beard

Abstract

An asymptotic analysis with the method of multiple-parameter perturbations has been carried out to examine the basic features of drop oscillations in a uniform flow field. The quiescent drop shape has an oblate deformation resulting from a potential-flow pressure distribution. This equilibrium distortion leads to a frequency splitting that eliminates the so-called mode degeneracies found by Rayleigh for small-amplitude oscillations about an undeformed spherical drop. Our results show that the characteristic frequencies for the zonal (axisymmetric) modes increase whereas those of sectoral modes decrease as the velocity of the external uniform flow increases. This trend agrees with the observational data and the results of the spheroidal model presented by Beard. The fine structure in the frequency spectrum of falling water drops may play a role in mode selection as observed in experiments. In addition, the one-to-one correspondence of characteristic frequencies and oscillation modes provides a theoretical basis for extracting more information from the experimental data. The modification of the drop oscillation mode shapes due to the coupling between the external flow field and oscillatory motion in the drop is also derived from the solutions of higher-order perturbations. Corresponding to each characteristic frequency, the oscillation mode shape contains more than one spherical harmonic.

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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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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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