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K. V. Beard

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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K. V. Beard

Abstract

Experimental scavenging efficiencies were measured for freely falling drops of 0.40–0.85 mm diameter and charges of 10−5–10−3 esu supported by the vertical airstream of the UCLA cloud tunnel. The particles in the airstream of 0.4 µm radius and 1.5 gm cm−3 density were produced from indium acetylacetonate using a La Mer generator. Collection efficiencies of 10−4 to 10−3 determined by neutron activation analysis were used to provide a test of an expanded theoretical model of Beard and Grover where collision efficiencies are based on a numerical description of axisymmetric, steady-state flow about rigid spheres up to Reynolds number 400. Scavenging mechanisms were examined in order to determine the important forces on the particle. The coulomb force between a charged drop and charged particle was included in the equation of motion of the particle which moved in a gravitational field with a Stokes-Cunningham resistance in the imperturbed flow of the drop. Numerically evaluated collision efficiencies were found to increase for the smaller particles due to wake capture even without electrostatic effects. A comparison shows that the experimental results for negligibly charged drops scavenging submicron particles are predicted by the theoretical model to within a factor of 2. By use of the numerical results the inconsistencies of previous experiments are in part resolved in accounting for apparent electrostatic and wake effects.

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K. V. Beard

Abstract

The drag force, instantaneous canting angle and response characteristics are discussed for large hydrometeors in horizontally accelerated airflow. Comparisons of calculated canting angles are made with data from an aircraft precipitation spectrometer (Bringi) by assuming an airflow disturbance based on probe geometry. It is concluded that the accelerated airflow is the likely cause of an intrinsic canting angle for raindrops of about 15° but that the force is too weak to appreciably affect the axis ratio. No intrinsic canting was found for graupel.

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K. V. Beard

Abstract

A formula has been developed for computing the terminal velocity V from a reference velocity V 0 by use of an adjustment factor f=V/V 0 for a change in altitude or electric force. The drag coefficients for bodies of regular geometry were analyzed, and found to be sufficiently similar that a single adjustment formula could be used for all hydrometeor shapes. Comparisons with drag data showed that a formula for f, that is only a function of size, air density, viscosity, charge and electric field, predicts V/V 0 to within 2% in most cases. This method also provides a reasonable means of computing V/V 0 for complex geometries such as rimed ice crystals, aggregates and graupel, and for drops undergoing electrical stress.

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K. V. Beard

Abstract

A theory based on the distortion of large water drops is used to calculate the drag and acceleration ofdrops in the laboratory from rest to terminal velocity. The results show that the drag coefficient curve foreach drop size lies between the curves for a sphere and for water drops at terminal velocity. A comparisonwith experimental acceleration data at sea level shows an improvement over the more approximate theoryof Wang and Pruppacher. The calculations of drops accelerating at reduced air density provide a very goodfit to the data of Davies at the reported fall distance of 11m. This new interpretation of Davies' experimentis consistent with the formula of Beard for the terminal velocity of large drops aloft.

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K. V. Beard

Abstract

The velocities of cloud and precipitation drops aloft are obtained from the sea level velocity by multiplication with an adjustment factor. For cloud drops (1–40 µm diameter) the adjustment factor is found from the Stokes-Cunningham equation, and depends upon the Knudsen number and dynamic viscosity. For larger drops (40 µm–6 mm diameter) the adjustment factor is obtained from a semi-empirical fit to the data of Beard (1976) and depends upon the drop diameter, air density and dynamic viscosity. The adjustment factor for each size range is reduced to a simple function of drop size, air temperature and pressure. The velocities aloft using the adjustment method are found to be within 1% of the more precise values of Beard (1976) for reasonable atmospheric conditions. Polynomial formulas are included for calculating the sea level velocities.

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A. R. Jameson and K. V. Beard

Abstract

Radar techniques involving polarization measurements depend critically on the non-spherical shape of the scatterers. In particular, it is often assumed that the equilibrium axial ratio measured under conditions of laminar flow represents the shape of freely falling raindrops. In the atmosphere, however, raindrops apparently deviate significantly from equilibrium toward more spherical shapes. This deviation can be important in quantitative interpretations of polarization measurements by radar.

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K. V. Beard and Harry T. Ochs III

Abstract

The collection efficiency has been measured for 15 size pairs of relatively uncharged drops in over 400 experimental runs. The results indicate that collection efficiencies fall in a narrow range of 0.60 to 0.70 even though the collector drop was varied between 63 and 100 μm radius and the collected drop from 11 to 26 μm radius. The measured values of collection efficiencies were consistently below collision efficiencies based on calculations using rigid sphere hydrodynamics. The coalescence efficiencies computed from the ratio of the theoretical collision efficiencies to the measured collection efficiencies were between 0.6 and 0.8. Our data show fair agreement with one previous coalescence model result, with an existing semi-empirical formula for the coalescence efficiency, and with the predicted trend of the coalescence efficiency to decrease with the size of the collected drop. A plausible interpretation of our experimental findings, consistent with previous coalescence studies, is that contact is prevented during a grazing trajectory by a hydrodynamic deflection which is enhanced by a slight deformation of the collector drop.

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Harry T. Ochs III and K. V. Beard

Abstract

Collection efficiencies for accretion were measured for six pairs of nearly unchanged drops. Cloud droplets of 11 and 17 μm and collector drops between 100 and 400 μm radius were used. The resulting efficiencies were in the 51–70% range and all values were significantly below computed collision efficiencies for rigid spheres. Inferred coalescence efficiencies between 54 and 82% were found to decrease with increasing collector drop and cloud droplet sizes. Drop separation was attributed to the grazing bounce mechanism whereby an air film nullifies the relative closure velocity allowing the tangential velocity of the cloud droplet to carry it past the collector drop.

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S. N. Grover and K. V. Beard

Abstract

Numerical computations were carried out to determine the efficiency with which electrically charged cloud drops and small raindrops of diameters ranging from 84 to 866 µm collide with electrically charged particles of densities between 1 and 4 g cm−3 and diameters between 0.4 and 20 µm. For this purpose, numerical flow fields about rigid spheres of Reynolds numbers 1, 10, 20, 100 and 200 were used to integrate the trajectory of the particle relative to the collector drop. Two approximate methods were used to represent the electrostatic force between the drop and the particle, and the resulting collision efficiencies show that the representation of the drop and particle by two point charges is a sufficient approximation of the electrostatic force for the majority cf atmospheric charges, as long as the drop diameter is larger than 208 µm. For typical charges on drops and particles observed in the atmosphere, and for attractive electrostatic forces, the collision efficiency results for particles smaller than 2 µm in diameter show a significant increase over the collision efficiencies for the case of no electric charges.

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