Aerodynamic Studies of Falling Snowflakes

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1 June 1965 Choji Magono and Tsutomu Nakamura 139 Aerodynamic Studies of Falling Snowflakes By Choji Magono Faculty of Science, Hokkaido University, Sapporo and Tsutomu Nakamura Institute of Low Temperature Science, Hokkaido University, Sapporo (Manuscript received 18 November 1964) Abstract Simultaneous observations were made of the fall velocity, size and mass of snowflakes in the steady falling state. It was found that the fall velocity depended upon both the size and the density of the flakes, but could not be represented as a function of their mass. An aerodynamic equation which expresses the fall velocity as a function of the density of a snowflake a, was obtained ; u=330( - )1/4 which agrees very well with the empirical data, where p is the density of air. 1. Introduction The fall velocity of snowflakes is considered to be important in the interpretation of radar echoes, particularly those of the RHI type. Magono (1953, 1954) attempted aerodynamic investigation of the fall velocity using the shape, size and density of the falling snowflakes, but since he did not obtain simultaneous density measurements, his work was insufficient. Langleben (1954) and Imai et al (1955) observed falling of snowflakes and determined experimental formulae for the fall velocity as a function of the diameter of the melted flakes. They only observed small snowflakes, and while their method of representing the fall velocity may be convenient in radar meteorology, it is not aerodynamically correct. The problem of measuring the density of falling snowflakes has since been overcome and the results of a detailed study of falling snowflakes made at Sapporo during the winter seasons of 1958 to 1962 are given here. 2. Methods 2.1 Fall velocity Naturally falling snowflakes were introduced Fig. 1 a. Observation hut. through a hole, cm, in the ceiling of an observation hut, as is shown in Fig. 1 a. When observations were made at night, the falling snowflakes were illuminated by a narrow beam of light as they passed through the hole (A in Fig. 1 a). The time required for a snowflake to fall from A to the floor (4 m) was measured with a stop watch. During the daytime, the time required for a

2 140 Journal of the Meteorological Society of Japan Vol. 43, No. 3 snowflake to fall from B to the floor (3m) major and minor diameters. However, when was measured instead. These distances were they fell on the paper, they were slightly sufficient to permit measurement of the fall depressed which made it impossible to determine their vertical dimensions by measuring velocity. 2.2 Density of falling snowflakes their heights on the paper. Therefore the The snowflakes fell on dyed filter paper mean of their major and minor diameters, placed just above the floor of the but in ab was assumed to give their vertical dimensions. When the snowflakes were very which the air temperature was approximately 0C. Before each snowflake melted, large, this assumption was not valid and a its outline was traced on the filter paper. correction was made, as will be explained When this outline was very irregular, an later, but with the exception of very large ellipsoidal area equal to the area enclosed by flakes, the density of falling snowflakes may the outline was used for the measurement of be expressed as size, as is shown in Fig. 1 b. Thus the hori-fig. 1 b. Method of measuring size and mass of falling snowflakes.zont where M is the mass of the snowflakes as determined by the method described above. 3. Results 3.1 Vertical and horizontal dimensions of falling snowflakes As previously stated, it was assumed that the vertical dimension of a falling snowflake is equal to the mean of its horizontal diameters. To confirm this assumption, photographs were taken of snowflakes in falling state and the vertical dimensions of the flakes were measured from the photographs. In Fig. 2 these vertical dimensions are plotted Fig. 2. Relation between horizontal and vertical dimensions of falling snowflakes.

3 June 1965 Choji Magono and Tsutomu Nakamura 141 against their horizontal dimensions. As may be seen, for snowflakes smaller than 10 mm, the vertical dimension is roughly the same as the horizontal dimension, but for flakes larger than 10 mm, the horizontal dimension is considerably larger than the vertical one. This difference is reasonable from aerodynamic considerations. Therefore, in these investigations, the vertical and horizontal dimensions were assumed to be equal in snowflakes smaller than 10 mm, and in larger ones, the vertical dimension was assumed to be 90% of the horizontal dimension. As will be discussed, snowflakes smaller than 10 mm are heavier and more dense (>0.02gr cm-3). 3.2 Density and size of falling snowflakes From the aerodynamical point of view, the fall velocity of snowflakes is presumed to depend upon their size, shape and density, but investigations made by Litovinov (1956) and Magono (1953) indicate that the fall velocity of very large snowflakes is not dependent upon the size of the flake. This is presumed to be explained by the tendency of snowflakes to decrease in density as they increase in size. To confirm this, the relation between the density and the size of falling snowflakes was examined and the results are shown in Fig. 3. These measurements contain all kinds of snowflakes. As was expected, the results show that the larger the snowflake, the smaller its density. With the possible exception of dry crystals, this relation may be roughly represented by where and are the densities of the snowflakes and air, respectively, and d is the diameter of the flakes in cm. 3.3 Fall velocity of snowflakes The fall velocities of snowflakes in relation to their diameter and their density are given in Figs. 4-7 for densities ranging from to 0.6 gr cm-3. As these figures indicate, if the values for the fall velocity in relation to the size of the flakes are considered as a whole, they are fairly scattered, but if the values in each density range are considered, it becomes apparent that there is a real relation and the greater the size, the greater the Fig. 3. Relation between density and diameter of falling snowflakes.

4 142 Journal of the Meteorological Society of Japan Vol. 43, No. 3 Fig. 4. Fall velocity of snowflakes with densities of 0.05~0.06 and 0.1~0.3 gr cm-3. Fig. 5. Fall velocity of snowflakes with densities of 0.07~0. 09 and 0.4~0. 6 gr cm-3.

5 June 1965 Choji Magono and Tsutomu Nakamura 143 Fig. 6. Fall velocity of snowflakes with densities of 0.004~0.006, ~0.014 and gr cm ~0.04 Fig. 7. Fall velocity of snoflakes with densities of 0.007~0.009 and 0.015~0.024 gr cm-3.

6 144 Journal of the Meteorological Society of Japan Vol. 43, No. 3 velocity, at least for densities greater than 0.05 gr cm-3. It seems therefore that the fall velocity of snowflakes is more dependent upon the density of the flakes than upon their size. 4. Aerodynamic consideration of the fall velocity of snowflakes 4.1 Aerodynamic consideration Considering the Reynolds number of falling snowflakes at terminal velocity, we have the following relation ; where d is the mean diameter of the falling snowflakes, g the gravitational acceleration, C the drag coefficient and u the terminal velocity. With the exception of C the values are obtained from empirical data. In the case of a smooth solid sphere, the value of C is supposed to be about 0. 5, but for the snowflakes of irregular shape it is not yet known up to the present. For the calculations, the value of C used was 1.3 which was determined by the method of least squares, using the data in the density ranges from 0.1 to 0.3 gr cm-3. Therefore the most probable fall velocity for snowflakes is given by The curve A in Fig. 4 was obtained from (3) when =0. 2 gr cm-3. This expression is valid when the density is greater than 0.05 gr cm-3, as is shown by curve B in Fig. 4 ( =0.05), and by curve B in Fig. 5 ( =0.08), but when the density was higher than this, there was less agreement, as is shown by curve A, Fig. 5 ( =0.5). In this range, the empirical values were much lower than those calculated from (3). This disagreement may have resulted from errors in measurement of the size of the flakes, since flakes with such high densities are very small and the method used is inaccurate for snowflakes as small as 3 mm in diameter. In addition, it is difficult to determine whether such small flakes are really snowflakes or single snow crystals. For comparison, Gunn and Kinzer's values for the fall velocity of raindrops (Gunn and Kinzer, 1949) are shown in curve, R, Fig. 5. In any case, equation (3) is valid for a wide range of densities greater than 0.05 gr cm-3, but is inadequate for snowflakes with smaller densities, whose observed fall velocity is nearly constant, as is shown in Figs. 6 and 7. As previously reported, Magono (1953, 1954) considered that for snowflakes with densities smaller 0.05 gr cm-3, the drag force encountered is related not only to the horizontal dimension of the falling flake, but also to its volume, since some amount of air penetrates the flake. With this consideration, the following relation expresses the fall velocity snowflakes with densities smaller than 0.05 gr cm-3, where coefficient A and B are : It was assumed here that the amount of air which penetrated the snowflake was proportional to and that the drag coefficient resulting from its horizontal dimension also decreased by an amount proportional to The previously determined value of 1.3 was used for C, and, in a similar manner, the value for the coefficient x in (5) was determined as 8.1 from the empirical data obtained in the density range 0.03 to 0.04 gr cm w 3 and for y in (6), as 20, from the 0.05 to 0.06 gr cm-3 range. Thus, for snowflakes with densities smaller than 0.05 gr cm-3, As is shown in curves A ( =0.03), B ( = 0.01), and C ( =0.005) in Fig. 6 and A ( = 0.02) and B ( =0.008) in Fig. 7, the agree-

7 Tune 1965 Choji Magono and Tsutomu Nakamura 145 ment is fairly good in these density ranges. Equation (7) is slightly different from that given by Magono (1954), but it is theoretically better, since empirical values were used for the density of the falling flakes, and because it agrees more closely with the empirical data, as is illustrated by curves M and A in Fig Fall velocity and mass of falling snowflakes In radar meteorology, it is more convenient to represent the fall velocity of snowflakes as a function of their mass or their diameter when melted as Langleben (1954) and Imai et al (1955) have done, rather than as a function of their size and density, although the former method is not theoretical. The present authors plotted the fall velocity of snowflakes against their mass, and the results are shown in Fig. 8. Curves obtained from Langleben's equation and Imai's curve concerned with dendritic crystals are also included for comparison. It may be seen that the values are widely scattered, although their equations appear to indicate the means. It is obviously impossible to represent fall velocity solely as a function of mass. Langleben's and Imai's equations appear to represent their empirical data fairly well, but their observations covered only short periods of time and did not include extremely large or partially melted snowflakes. 4.3 Density and fall velocity of snowflakes As previously stated, the fall velocity of snowflakes depends upon the density of the flakes rather than upon their size. The relation between these two factors for a wide range of densities is shown in Fig. 9. The following empirical formula (curve A in Fig. 9) was obtained from the values in Fig. 9. where is the density of air. This equation has only one variable,, the density of snowflakes. The equation may be derived theoretically from (2), where the drag coefficient, C, is 1.3 and from (1) obtaining which agrees well with the experimental expression, (8). 5. Discussion According to wind tunnel experiments, the drag coefficient of phenomena with Reynolds numbers of approximately 1000, is 0. 5, and this value increases as the Reynolds number Fig. 8. Relation between mass and fall velocity.

8 146 Journal of the Meteorological Society of Japan Vol. 43, No. 3 Fig. 9. Relation between density difference ( - ) and fall velocity. decreases. Falling snowflakes have Reynolds numbers of approximately 1000, but the value of 1.3 for the drag coefficient as determined in this investigation is considerably larger than would be expected from the wind tunnel experiments. In his experiment with hail, List (1959) also obtained values for the drag coefficient which ranged from 0.4 to 1.2. Spherical bodies with smooth hard surfaces are usually used in the wind tunnel experiments, but falling snowflakes have irregular shapes and very complex surfaces. It is considered, therefore, that there is much more air turbulence around falling snowflakes than around a smooth sphere in a wind tunnel, and this turbulence increases the cross-section for the air resistance of the snowflakes. If we imagine a smooth spherical shape enveloping the snowflake and if we assume the diameter of the sphere to be (1.3/0.5)1/2 times the diameter of the snowflake, a value of 0.5 may be used for the drag coefficient in place of 1.3. From this, it is assumed that the observed discrepancy is not significant and results solely from the difference in the determination of the size of the snowflakes. With reference to the relation between the density and the size of a snowflake, the aerodynamic explanation of the fall velocity is very simple if (1) is correct, in other words if the assumption that the larger the flake, the smaller the density, is valid. If the properties of snowflakes were constant irrespective of their size, we would not expect such a relationship, but there are many kinds of snowflakes, and the dense, partially melted ones and those which make up graupel are usually small in size, while almost all of the large flakes are light in specific gravity because of dendritic crystals. Therefore, if we know the distribution of density against the size of the snowflake, the experimental equation (1) is statistically correct. Acknowledgments The authors wish to express their thanks to N. Hino for the use of his observation hut. This research was supported by the Special Fund for Scientific Research of the Educational Ministry of Japan. References Gunn, R. and G. D. Kinzer, 1949: The terminal velocity of fall for water droplets in stagnant air. J. Meteor., 6, Imai, I., M. Fujiwara, I. Ichimura and Y. Toyama, 1955: Radar Reflectivity of Falling Snow.

9 June 1965 Choji Magono and Tsutomu Nakamura 147 Papers in Meteor. and Geophys., 6, Langleben, M. P., 1954: The terminal velocity of snowflakes. Quart. J. Roy. Meteor. Soc., 80, List, R., 1959: Zur Aerodynamik von Hagelkorner. Z. angew. Math. Phys., 10, Litvinov, I. V., 1956: Determination of the steady state velocity of falling snow particles. Izv. Akad. Nauk, SSSR, Geophysical Ser. 7, Magono, C., 1953: On the growth of snow flake and graupel. Sci. Rep., Yokohama National Univ., Sec. I, No. 2, Magono, C., 1954: On the fall velocity of solid precipitation elements. ibid. No. 3,

Observation of Falling Motion of Columnar Snow Crystals

276 Journal of the Meteorological Society of Japan Vol. 54, No. 5 Observation of Falling Motion of Columnar Snow Crystals By Masahiro Kajikawa Department of Earth Science, Akita University, Akita, Japan