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Frontiers of Earth Science

ISSN 2095-0195

ISSN 2095-0209(Online)

CN 11-5982/P

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Front. Earth Sci.    2021, Vol. 15 Issue (4) : 947-955    https://doi.org/10.1007/s11707-020-0835-z
RESEARCH ARTICLE
A high precision model for the terminal settling velocity of drops in fluid medium
Qiu YIN1(), Ci SONG2,3
1. Shanghai Meteorological Service, China Meteorological Administration, Shanghai 200030, China
2. College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China
3. School of Communication and Information Engineering, Shanghai University, Shanghai 200444, China
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Abstract

The terminal settling velocity (TSV) calculation of drops and other spherical objects in fluid medium is a classical problem, which has important application values in many fields such as the study of cloud and precipitation processes, the evaluation of soil erosion, and the determination of fluid viscosity coefficient etc. In this paper, a new explicit approximation model of TSV is established, which combines the theoretical solution of N-S equation about fluid motion around spherical objects and the statistical regression of solution dimensionless coefficients with measurement data. This new model can adapt to different values of drop parameters and medium parameters in a large range of Re. By this model, the relative and absolute calculation errors of TSV are in range of −3.42%–+4.34% and −0.271 m/s–+0.128 m/s respectively for drop radius 0.005–2.9 mm. Their corresponding root mean square values are 1.77% and 0.084 m/s respectively, which are much smaller than that of past theoretical and empirical models.
Keywords terminal settling velocity      drag coefficient      viscous resistance      drop      fluid medium     
Corresponding Author(s): Qiu YIN   
Online First Date: 17 March 2021    Issue Date: 20 January 2022
 Cite this article:   
Qiu YIN,Ci SONG. A high precision model for the terminal settling velocity of drops in fluid medium[J]. Front. Earth Sci., 2021, 15(4): 947-955.
 URL:  
https://academic.hep.com.cn/fesci/EN/10.1007/s11707-020-0835-z
https://academic.hep.com.cn/fesci/EN/Y2021/V15/I4/947
Fig.1  The change of the relative and absolute errors of drop TSV with drop radius when aand bare supposed to be constant as Yin and Xu (1991). (a) Minimizing the RMS relative approximation error; (b) Minimizing the RMS absolute approximation error.
Fig.2  The values of a at different drop radii inversed by the measurement data and the optimized lna–r fitting curve (b=1).
Fig.3  The change of absolute and relative errors of TSV with radius when a(r )and bare given by Eq. (17).
Fig.4  The change of absolute and relative errors of TSV with radius when a(r ) and b are given by Eq. (18).
Fig.5  Comparison of the approximation models of this paper, Stokes approximate solution and Oseen approximate solution with the measurement data in Re<6.
Re <0.5 <6
RMS error Relative error/% Absolute error /(m·s–1) Relative error/% Absolute error /(m·s–1)
Yin-Song 1.63 9.07 × 10−4 1.84 3.50 × 10−3
Stokes 23.98 1.46 × 10−2 38.43 1.45 × 10−1
Oseen 1.68 8.75 × 10−4 8.29 3.69 × 10−2
Tab.1  The RMS errors of the approximation models of this paper, Stokes approximate solution and Oseen approximate solution
Fig.6  Comparison of the approximation model of this paper, the Yin-Xu approcximation model (Eq. (9) and Eq. (15)) and the piecewise empirical model given in Sheng et al. (2003) with the measurement data in r = 0.005–2.5 mm.
RMS error 0.005<r<2.5 0.005<r<1.0 1.0<r<2.5
Relative error/% Absolute error
/(m·s–1)		 Relative error/% Absolute error
/(m·s–1)		 Relative error/% Absolute error
/(m·s–1)
Yin-Song 1.77 7.06 × 10−2 2.11 5.66 × 10−2 0.98 8.58 × 10−2
Yin-Xu 10.42 4.42 × 10−1 11.72 2.75 × 10−1 8.06 6.24 × 10−1
Piecewise empirical model 9.22 3.86 × 10−1 10.28 1.95 × 10−1 7.28 5.69 × 10−1
Tab.2  The RMS errors of the approximation model of this paper, the Yin-Xu approximation model (Eq. (9) and Eq. (15)) and the piecewise empirical model given in Sheng et al. (2003) for different ranges of radius
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