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R. J. Schlamp
,
H. R. Pruppacher
, and
A. E. Hamielec

Abstract

The Navier-Stokes equation of motion for two-dimensional, viscous, steady-state incompressible flow past an infinitely long circular cylinder was solved by numerical techniques for Reynolds numbers between 0.1 and 50. From the streamfunction and vorticity fields the pressure at the cylinder surface, the pressure drag, and the frictional drag were computed, and from the latter two the total drag on the cylinder was derived. The values found for the drag compared well with the best theoretical and experimental values reported in literature, suggesting that our flow fields were sufficiently accurate. These flow fields were used to determine the hydrodynamic interaction between simple columnar ice crystals idealized as circular cylinders of finite length L′, of radius aL ′, and of Reynolds number NRe,L (67.1≤L′≤2440 µm; 23.5≤ aL ′≤146.4 µm;0.2≤NRe,L <20) and spherical water drops of radius aS ′ varying between 2 and 134µm. The flow fields used to describe the flow past drops were numerically computed by a method analogous to that given by LeClair et al. (1970). For atmospheric conditions of −8°C and 800 mb numerical methods were used to determine the trajectory of the drops relative to the cylinder by means of a semi-empirically modified version of the “superposition” model. The model was semi-empirical in that the flow fields used were those determined theoretically by us, while the drag on the columnar crystals was that determined by Jayaweera and Cottis (1969) and by Kajikawa (1971), and the dimensional relationships between the diameter and length of the columnar crystal were those given by the observational relations of Auer and Veal (1970). From the trajectories of the water drops relative to the columnar ice crystals collision efficiencies were computed. Our computations predict that riming on a columnar ice crystal will not commence until the crystal has a diameter which is larger than about 50 µm. This result is in good agreement with field observations reported in literature.

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R. J. Schlamp
,
S. N. Grover
,
H. R. Pruppacher
, and
A. E. Hamielec

Abstract

The aerodynamic interaction between electrically charged cloud drops in the presence of vertical external electric fields was numerically investigated for 800 mb and +10°C. The collector drops had radii between 11.4 and 74.3 µm while the collected drops had radii between 1 and 66 µm. The external electric fields considered ranged between 0 and 3429 V cm−1 (=3.429×105V m−1 and the electric charge on the cloud drops ranged between 0 and 1.1×10−4 (=3.7×10−14 C). The results demonstrate that the presence of electric charges and fields of magnitudes observed during thunderstorm and pre-thunderstorm conditions drastically enhance the collision efficiency of cloud drops. The enhancement was found to he most pronounced for the smallest collector drops studied.

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R. J. Schlamp
,
S. N. Grover
,
H. R. Pruppacher
, and
A. E. Hamielec

Abstract

The numerical model of Schlamp et al. (1976) for determining the collision efficiency of electrically charged or unchanged cloud drops in the presence or absence of a vertical electric field has been extended to study the two following cases, both of which include the presence of a vertical field due to a net positive charge in the upper part of the cloud and a net negative charge in the lower part of the cloud: (i) the larger drop is negatively charged and is initially above the smaller drop, which is positively charged; (ii) the larger drop is negatively charged and it is initially below the smaller drop, which is again positively charged. Also, for the purpose of resolving more accurately the critical electric charge on the drops and the critical electric field necessary to significantly affect the collision efficiency, additional computations have been carried out for charged drops in the absence of an electric field and for unchanged drops in the presence of a vertical electric field.

The sizes of drops considered range from 1–118 μm in radius. The magnitude of the electric charges on the drops range from 0–2.8×10−4 esu, and the electric fields range in strength from 0–3429 V cm−1, which include the charges and fields typically observed in thunderstorms.

It is found that electric fields and charges even of relatively modest values have a profound effect upon the collision efficiency. The results of case (i) show that the electrostatic forces are responsible for determining the shape of the collision efficiency curves with the hydrodynamic forces being of secondary importance. These results are significantly different from either those of case (ii) or those of Schlamp et al. (1976).

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