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- Author or Editor: A. E. Hamielec x
- Journal of the Atmospheric Sciences x
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Abstract
Accurate, numerical solutions of the Navier-Strokes equations of motion and the equation of mass transfer have been obtained for the steady-state transfer of a chemically inert substance from the surface of a single rigid sphere moving at its terminal velocity in an unbounded fluid. Local Sherwood numbers have been calculated for spheres with Reynolds numbers in the range 0.05–300 and for a fluid with a Schmidt number of 0.71. The objective of this study was to model the effect of ventilation on the rate of evaporation of cloud drops falling at terminal velocity in air subsaturated with respect to water.
Abstract
Accurate, numerical solutions of the Navier-Strokes equations of motion and the equation of mass transfer have been obtained for the steady-state transfer of a chemically inert substance from the surface of a single rigid sphere moving at its terminal velocity in an unbounded fluid. Local Sherwood numbers have been calculated for spheres with Reynolds numbers in the range 0.05–300 and for a fluid with a Schmidt number of 0.71. The objective of this study was to model the effect of ventilation on the rate of evaporation of cloud drops falling at terminal velocity in air subsaturated with respect to water.
Abstract
Accurate solutions of the steady-state Navier-Stokes equations of motion have been obtained by means of a numerical method to determine the hydrodynamic drag on a rigid sphere falling at its terminal velocity in an unbounded fluid. The calculations were carried out for Reynolds numbers between 0.01 and 400. The numerical solutions were compared with the theoretical results of Stokes, Oseen, Goldstein, Proudman and Pearson, Jenson, Rimon and Cheng, and Carrier, and with the recent experimental data of Maxworthy, and Pruppacher and Steinberger. At the lowest Reynolds numbers the numerical solutions show closest agreement with the theory of Proudman and Pearson and at intermediate Reynolds numbers with the semi-theoretical relationship proposed by Carrier. At higher Reynolds numbers our present results agree well with the calculations of Hamielec et al. for Reynolds numbers of 40 and 100 and with the numerical results of Rimon and Cheng; they depart, however, significantly from the results of Jenson. Over the whole Reynolds number interval 0.01–400 our numerical results are in close agreement with the experimental data of Pruppacher, Pruppacher and Steinberger, and Beard and Pruppacher. It is concluded that our numerical study is unique in that it is able to predict theoretically accurate values for the drag on a sphere over a wide Reynolds number interval. The present study also confirms the findings of Maxworthy, and Pruppacher and Steinberger, that as the Reynolds number approaches zero the drag on a sphere approaches zero via the Oseen drag rather than via the Stokes drag. The significance of the present results to cloud physics is pointed out.
Abstract
Accurate solutions of the steady-state Navier-Stokes equations of motion have been obtained by means of a numerical method to determine the hydrodynamic drag on a rigid sphere falling at its terminal velocity in an unbounded fluid. The calculations were carried out for Reynolds numbers between 0.01 and 400. The numerical solutions were compared with the theoretical results of Stokes, Oseen, Goldstein, Proudman and Pearson, Jenson, Rimon and Cheng, and Carrier, and with the recent experimental data of Maxworthy, and Pruppacher and Steinberger. At the lowest Reynolds numbers the numerical solutions show closest agreement with the theory of Proudman and Pearson and at intermediate Reynolds numbers with the semi-theoretical relationship proposed by Carrier. At higher Reynolds numbers our present results agree well with the calculations of Hamielec et al. for Reynolds numbers of 40 and 100 and with the numerical results of Rimon and Cheng; they depart, however, significantly from the results of Jenson. Over the whole Reynolds number interval 0.01–400 our numerical results are in close agreement with the experimental data of Pruppacher, Pruppacher and Steinberger, and Beard and Pruppacher. It is concluded that our numerical study is unique in that it is able to predict theoretically accurate values for the drag on a sphere over a wide Reynolds number interval. The present study also confirms the findings of Maxworthy, and Pruppacher and Steinberger, that as the Reynolds number approaches zero the drag on a sphere approaches zero via the Oseen drag rather than via the Stokes drag. The significance of the present results to cloud physics is pointed out.
Abstract
Numerical solutions have been found for the vapor density field around a simple ice plate, idealized as an oblate spheroid of axis ratio 0.05, having Reynolds numbers between 0.1 and 20, and failing in a fluid of Schmidt number 0.71. The present solutions are compared with experimental data after Thorpe and Mason for evaporating ice plates, the numerical results of Masliyah and Epstein for oblate spheroids of axis ratio 0.2, and the analytical results of Brenner for thin disks. It is shown that the ventilation coefficient varies linearly with N Sc ⅓ N Rc ½ at higher Reynolds numbers, while as the Reynolds number approaches zero it approaches its stationary value via the analytical solution of Brenner. Over the range of Reynolds numbers investigated, ventilation coefficients for thin oblate spheroids were found to be lower than those for spheres.
Abstract
Numerical solutions have been found for the vapor density field around a simple ice plate, idealized as an oblate spheroid of axis ratio 0.05, having Reynolds numbers between 0.1 and 20, and failing in a fluid of Schmidt number 0.71. The present solutions are compared with experimental data after Thorpe and Mason for evaporating ice plates, the numerical results of Masliyah and Epstein for oblate spheroids of axis ratio 0.2, and the analytical results of Brenner for thin disks. It is shown that the ventilation coefficient varies linearly with N Sc ⅓ N Rc ½ at higher Reynolds numbers, while as the Reynolds number approaches zero it approaches its stationary value via the analytical solution of Brenner. Over the range of Reynolds numbers investigated, ventilation coefficients for thin oblate spheroids were found to be lower than those for spheres.
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.
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.
Abstract
The flow past a thin oblate spheroid falling at terminal velocity in an infinite, viscous fluid was investigated using a numerical solution of the steady-state Navier-Stokes equations of motion. The detailed streamfunction and vorticity yielded the drag, pressure distribution, and the extent of the spheroid's downstream wake. Calculations were performed for spheroids of axis ratios 0.05 and 0.2 and Reynolds numbers between 0.1 and 100. The results were compared with other numerical and analytical solutions to the Navier Stokes equations of motion for viscous flow past oblate spheroids and disks and with experimental results in the literature. Our numerical results for oblate spheroids of axis ratio 0.2 agree well with the numerical results of Masliyah and Epstein and with our own experimental results. Our results for oblate spheroids of axis ratio 0.05 agree well with the numerical computations of Michael and available experimental results on disks, but depart significantly from the numerical results of Rimon. In agreement with our earlier studies on spheres, we find that, as the Reynolds number approaches zero, the drag on an oblate spheroid of any axis ratio approaches its value at zero Reynolds number via the Oseen drag rather than via the Stokes drag. The significance of the present study to cloud physics is pointed out.
Abstract
The flow past a thin oblate spheroid falling at terminal velocity in an infinite, viscous fluid was investigated using a numerical solution of the steady-state Navier-Stokes equations of motion. The detailed streamfunction and vorticity yielded the drag, pressure distribution, and the extent of the spheroid's downstream wake. Calculations were performed for spheroids of axis ratios 0.05 and 0.2 and Reynolds numbers between 0.1 and 100. The results were compared with other numerical and analytical solutions to the Navier Stokes equations of motion for viscous flow past oblate spheroids and disks and with experimental results in the literature. Our numerical results for oblate spheroids of axis ratio 0.2 agree well with the numerical results of Masliyah and Epstein and with our own experimental results. Our results for oblate spheroids of axis ratio 0.05 agree well with the numerical computations of Michael and available experimental results on disks, but depart significantly from the numerical results of Rimon. In agreement with our earlier studies on spheres, we find that, as the Reynolds number approaches zero, the drag on an oblate spheroid of any axis ratio approaches its value at zero Reynolds number via the Oseen drag rather than via the Stokes drag. The significance of the present study to cloud physics is pointed out.
Abstract
A theoretical model to compute the efficiency with which aerosol particles of radius 0.5≤r≤10 μm collide with water drops of radius a=42, 72, 106, 173, 309 and 438 μm falling at terminal velocity in air is presented. Inertial impaction, thermophoresis, diffusiophoresis and electrical effects are considered. The computations were carried out for ambient conditions of 10°C, 900 mb, and 100%, 95% and 75% relative humidity. The drops and particles were assumed to carry electric charges of 0.2 a 2 and 0.2 r 2 [esu], respectively, and charges of 2.0 a 2 and 2.0 r 2 [esu], respectively, where a and r are expressed in centimeters. The external electric field strengths were assumed to range between 0≤E 0≤3×105 V m−1. The results of our computations show 1) that the efficiency E with which aerosol particles collide with the drops considered is significantly raised by phoretic and electric forces over and above the efficiency resulting from inertial impaction, this effect being the more pronounced the smaller the collector drop; 2) hydrodynamic effects as well as phoretic effects tend to promote particle capture in the rear of a drop if particles are sufficiently small, resulting in a minimum of E versus r which lies in the “Greenfield gap” region and thus reinforces the gap; 3) electrical effects tend to eliminate this gap reinforcement; and 4) computations which consider phoretic effects only without simultaneously taking account of the particles’ motion due to the hydrodynamic flow around the collector drop significantly overestimate E for r≲1.5 μm.
Abstract
A theoretical model to compute the efficiency with which aerosol particles of radius 0.5≤r≤10 μm collide with water drops of radius a=42, 72, 106, 173, 309 and 438 μm falling at terminal velocity in air is presented. Inertial impaction, thermophoresis, diffusiophoresis and electrical effects are considered. The computations were carried out for ambient conditions of 10°C, 900 mb, and 100%, 95% and 75% relative humidity. The drops and particles were assumed to carry electric charges of 0.2 a 2 and 0.2 r 2 [esu], respectively, and charges of 2.0 a 2 and 2.0 r 2 [esu], respectively, where a and r are expressed in centimeters. The external electric field strengths were assumed to range between 0≤E 0≤3×105 V m−1. The results of our computations show 1) that the efficiency E with which aerosol particles collide with the drops considered is significantly raised by phoretic and electric forces over and above the efficiency resulting from inertial impaction, this effect being the more pronounced the smaller the collector drop; 2) hydrodynamic effects as well as phoretic effects tend to promote particle capture in the rear of a drop if particles are sufficiently small, resulting in a minimum of E versus r which lies in the “Greenfield gap” region and thus reinforces the gap; 3) electrical effects tend to eliminate this gap reinforcement; and 4) computations which consider phoretic effects only without simultaneously taking account of the particles’ motion due to the hydrodynamic flow around the collector drop significantly overestimate E for r≲1.5 μm.
Abstract
Four theoretical approaches are presented for quantitatively determining the intensity of the internal circulation and the flow patterns inside and outside liquid water spheres falling at terminal velocity in air. The first approach assumes creeping flow outside and inside a water sphere, the second assumes potential flow outside and inviscid motion inside a water sphere, the third makes use of boundary layer theory, and the fourth approach uses a numerical method to solve the full Navier-Stokes equation of motion inside and outside a water sphere. The theoretical predictions are compared with data obtained from new quantitative wind tunnel experiments on spherical and deformed water drops. The results show that the creeping flow analysis greatly underestimates the strength of the internal velocity while the inviscid flow analysis greatly overestimates it. On the other hand, the results of the boundary layer approach and of the numerical approach agree reasonably well with the experimental data for drops with radii <500 μ. For larger drops the results of the boundary layer approach greatly overestimate the strength of the internal circulation and predict a completely wrong trend of the variation of the internal velocity with drop size, while the numerical results, although somewhat overestimating the circulation strength, predict the trend correctly. Reasonably good agreement is also found between the observed flow patterns inside the drop and those numerically predicted. In two appendices the effect of the internal circulation on drop shape and hydrodynamic drag is discussed.
Abstract
Four theoretical approaches are presented for quantitatively determining the intensity of the internal circulation and the flow patterns inside and outside liquid water spheres falling at terminal velocity in air. The first approach assumes creeping flow outside and inside a water sphere, the second assumes potential flow outside and inviscid motion inside a water sphere, the third makes use of boundary layer theory, and the fourth approach uses a numerical method to solve the full Navier-Stokes equation of motion inside and outside a water sphere. The theoretical predictions are compared with data obtained from new quantitative wind tunnel experiments on spherical and deformed water drops. The results show that the creeping flow analysis greatly underestimates the strength of the internal velocity while the inviscid flow analysis greatly overestimates it. On the other hand, the results of the boundary layer approach and of the numerical approach agree reasonably well with the experimental data for drops with radii <500 μ. For larger drops the results of the boundary layer approach greatly overestimate the strength of the internal circulation and predict a completely wrong trend of the variation of the internal velocity with drop size, while the numerical results, although somewhat overestimating the circulation strength, predict the trend correctly. Reasonably good agreement is also found between the observed flow patterns inside the drop and those numerically predicted. In two appendices the effect of the internal circulation on drop shape and hydrodynamic drag is discussed.
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.
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.
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).
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).
