## 1. Introduction

All cloud drops and raindrops^{1} experience an acceleration toward the ground by the pull of the gravitational force. Due to the existence of the viscosity, they attain a constant falling speed when the gravitational force is balanced by the viscous frictional force. This constant speed at which drops fall is called the terminal velocity. The terminal velocity of raindrops is one of the major factors that determine the rate of removal of liquid water from the atmosphere, which can strongly affect weather predictions (Parodi and Emanuel 2009; Hagos et al. 2018). An accurate prediction of the terminal velocity is therefore one of the necessary conditions for improving the accuracy of weather and climate simulations. Because the general equation governing the free-falling motion of water drops of even moderate size (on the order of 10^{−1} mm) is nonlinear, an analytic solution of the terminal velocity exists only for drops that are small enough (on the order of 1 *μ*m) for the Stokes law to be applicable. In atmospheric modeling, the terminal velocity is commonly retrieved from parameterizations via some empirical formulas derived from experimental data.

There have been numerous experimental reports (Gunn and Kinzer 1949; Beard and Pruppacher 1969; Yu et al. 2016; Chowdhury et al. 2016) on the measurement of the terminal velocity of water drops. The laboratory experiment conducted by Gunn and Kinzer (1949) produced detailed data on the terminal velocity of drops with diameters ranging from 0.0783 to 5.8 mm at a temperature of 20°C and pressure of 1013 hPa. In their experiment, droplets were electrically charged and allowed to fall freely in a rain tower. Two electrode detectors that could detect the passing of charged droplets at separate heights were used to calculate the time span, and thus the terminal velocity could be obtained. Beard and Pruppacher (1969) pointed out that the environmental condition of 50% relative humidity in Gunn and Kinzer’s experiment might have generated considerable errors in the measurement of the terminal velocity, especially for smaller drops, because some water might have evaporated from the drop before its mass was measured. Instead, they used a wind tunnel with 100% relative humidity to measure the terminal velocity, and their results indeed suggested a slight overestimation of the terminal velocity of the smallest droplets measured by Gunn and Kinzer (1949). Although wind tunnel method might be prone to measurement error due to non-uniform velocity profile in the tunnel, the measured values for larger droplets agreed well with Gunn and Kinzer’s experimental data (Beard 1976). These values were used in the Beard (1976) parameterization of the terminal velocity.

More recent experimental studies (Yu et al. 2016; Chowdhury et al. 2016) used a high-speed camera to measure the terminal velocity. However, it is prone to errors due to pixel resolution and determination of centroid, and which may lead to larger measurement errors for cloud drops and small raindrops (Yu et al. 2016). The experimental data of Gunn and Kinzer (1949) and Beard and Pruppacher (1969) have remained as the reference solution of the terminal velocity under standard conditions (20°C and 1013 hPa) up to the present day. Although these data are rather complete, there is a severe limitation to the range of applicability since the experiments were performed only at one particular set of environmental conditions. As cloud drops or raindrops usually fall from high altitudes at temperatures and pressures that are far different from the standard conditions of 20°C and 1013 hPa, an extrapolation is necessary to deduce the terminal velocity at arbitrary temperatures and pressures.

Finding a fixed functional relationship among dimensionless numbers is a common way to predict the terminal velocity at high altitudes. As is shown in section 2, four dimensionless numbers are required to completely describe the solution to the free-falling droplet problem, and the Reynolds number is the only dimensionless number that contains the terminal velocity. However, it is a daunting task to find an equation that relate the Reynolds number to the other dimensionless numbers; therefore, assumptions are usually made to reduce their number. For example, Beard and Pruppacher (1969) attacked this problem by assuming that there exists a fixed relationship between the Reynolds number and the drag coefficient. This relationship does not depend on environmental conditions if the drop is a solid object. In other words, experiments of a free-falling solid object conducted at arbitrary temperatures and pressures will lead us to the same relationship. Nonetheless, drops have finite viscosity and surface tension. Two dimensionless numbers are insufficient to fully describe the solution, thus the relationship actually depends on environmental conditions. This dependency has not been studied in a quantitative way.

Most microphysical schemes adopt a much simpler method to formulate the effect of the reduced air density on the terminal velocity at high altitudes. A simple function of the ratio of the corresponding air densities or pressures is directly related to the ratio of the terminal velocity aloft to that at the surface. However, the accuracy of this method has not been rigorously examined. It is worth mentioning that Yu et al. (2016) compared their measurements with Gunn and Kinzer’s data modified by this method at nonstandard environmental conditions of 30.2°C and 956.4 hPa, and they found a mean measurement error of approximately 6% (1.86%) for drops of diameters smaller (larger) than 1 mm. Due to measurement uncertainties of a similar magnitude as the mean measurement error, it is unclear whether this method is the primary cause of the error.

Although the terminal velocity of pure water drops was thoroughly studied in previous experiments, uncertainties about the influence of varying environmental conditions on the terminal velocity have not been well investigated. The purpose of the present work is to revisit this problem and tackle it with a numerical approach. We adopted the recently developed numerical solver of the two-phase flow Navier–Stokes equation (Ong and Miura 2018, 2019), which can explicitly resolve capillary waves, to investigate the motion of an axisymmetric free-falling drop by direct numerical simulations (DNS), with the aim of shedding light on the relationship between the terminal velocity and environmental conditions. We also compared existing empirical formulas with our simulation results.

Besides temperature and pressure, there are other factors that can also affect the falling speed and terminal velocity. Examples are reduction of the surface tension coefficient by aerosols and shape oscillation by capillary force. It is known that chemical compounds that are commonly observed in the atmosphere can act as surfactants to reduce the surface tension coefficient *σ* (Gill et al. 1983; Taraniuk et al. 2008). Although no experimental evidence had been given to indicate that there exists a significant fluctuation in falling speed due to shape oscillation (Beard 1976) and terminal velocity due to reduced surface tension (Müller et al. 2013), we can examine the correctness of the above statement by simulations with a two-phase flow model.

Since our two-phase flow model is currently limited to two-dimensional simulations, only free-falling drops of diameters smaller than 0.5 mm without unstable higher modes of asymmetric oscillations are considered in this study. Attempts to apply the axisymmetric model to larger raindrops lead to nonphysical diverging solutions. This size limitation is certainly a major drawback of the present study, it may serve as the first example of successful applications of the two-phase flow model in the atmospheric science. Once it is verified that this approach can be successfully applied on small raindrops, we will step forward to 3D calculations of raindrops with diameters larger than 0.5 mm, the difficulty of which is associated with their chaotic motion. For this purpose, a new surface reconstruction scheme for 3D simulations has recently been developed (Ong and Miura 2018), but a development of a full 3D two-phase flow model warrants further investigations. Simulations of large raindrops with unsteady wakes are inevitably left for future work.

In summary, we prepared five sets of numerical experiments to achieve our objectives of 1) verification and validation of our numerical model for free-falling water drops by comparing numerical solutions with existing experimental data and numerical data, e.g., Gunn and Kinzer (1949), Wang and Pruppacher (1977), and LeClair et al. (1972); 2) investigating the influence of environmental conditions on the terminal velocity at high altitudes and evaluating the accuracy of the existing empirical formulas for the terminal velocity aloft; 3) examining the dependency of the relationship between the Reynolds number and the drag coefficient (Re–*C*_{D}) on environmental conditions by comparing Re and *C*_{D} of free-falling small raindrops under standard conditions and at high altitudes; 4) determining the influence of oscillatory motion on the falling speed; and 5) determining the shape deformation effect on the motion of a small raindrop with a reduced surface tension coefficient. The detailed setup of these five sets of numerical experiments is given in section 3. Based on the results of the second and third set of experiments, we will propose a new empirical formula of the terminal velocity aloft to account for reduced air density.

A simple description of the mathematical model of a free-falling drop and existing parameterizations of terminal velocity at any temperature and pressure are given in section 2. The numerical setup is described in section 3. Simulation results are presented in section 4. The results and findings are summarized in section 5.

## 2. Terminal velocity

To facilitate the elaboration of the experimental methods and computation of the terminal velocity in later sections, we briefly describe the fundamental equation of the incompressible two-phase flow and the four dimensionless numbers that determine its solution in the first half of this section. The empirical formulas of the terminal velocity aloft that are commonly used in atmospheric models are briefly explained in the second half of this section.

### a. Governing equation

**u**is the flow velocity,

*p*is the pressure,

*μ*is the viscosity, superscript T denotes transpose of a tensor,

*σ*is the surface tension coefficient,

*κ*is the interfacial curvature,

*ρ*is the difference between the local density and the density of air,

*g*is the gravitational constant, and

*ρ*and viscosity

*μ*are continuous Heaviside functions:

*H*is the Heaviside function,

*ρ*

_{w}is the density of water,

*ρ*

_{a}is the density of air,

*μ*

_{w}is the viscosity of water, and

*μ*

_{a}is the viscosity of air. We can nondimensionalize the above equations by using the following scale factors:

*x*and

*z*represent the spatial coordinates,

*D*is the equivalent diameter,

**u**is the velocity,

*U*is the characteristic fall speed of the droplet,

*t*is time, and

*p*is the pressure. The superscript asterisk (*) denotes that the variables are dimensionless. Substituting Eqs. (4) to (7) into the fundamental equation, Eq. (1) then becomes

*gD*

^{2}

*ρ*

_{a}(

*R*

_{ρ}− 1)/

*σ*, and the Morton number

*R*

_{ρ}, and

*R*

_{μ}. It is notable that only the Reynolds number includes the characteristic speed

*U*. Since the free-falling motion is considered here, it is possible to find a correlation between the Reynolds number and the remaining four dimensionless numbers. For instance, we can take the characteristic speed

*U*to be the Hadamard–Rybczynski (Hadamard 1911; Rybczynski 1911) terminal velocity:

*U*, surface tension does not appear because the surfactants are not included and the droplet is assumed to be spherical in the Hadamard–Rybczynski solution. Substituting Eq. (16) into Eq. (9), the Reynolds number can be written as

### b. Formulations in previous studies

*C*

_{D}, the Best number Be =

*C*

_{D}Re

^{2}, and the Reynolds number can be chosen. This approach was adopted in the work of Beard and Pruppacher (1969). They assumed a unique functional relationship between the Reynolds number and the drag coefficient. This functional relationship is commonly approximated by a piecewise power-law function to simplify the calculation of the bulk terminal velocity:

*a*and

*b*are constants that can be determined from experimental data. We can rewrite Eq. (18) to express the velocity in terms of physical properties:

*A*and

*B*in Eq. (20) are treated as constants, and the equation is multiplied by an extra correction term

*α*to take into account the change in environmental conditions:

*α*is set to (

*ρ*

_{o}/

*ρ*)

^{0.5}. Here, the subscript

*o*means that the variable is measured on the ground. This parameterization is based on the assumption that the drag coefficient and Δ

*ρ*in Eq. (20) are constant for water drops of all diameters. In Rutledge and Hobbs (1984), the correction term is an approximated form of the empirical formula of Foote and du Toit (1969), (

*p*

_{o}/

*p*)

^{0.4}. In Heymsfield et al. (2007), the form is slightly modified to (

*ρ*

_{o}/

*ρ*)

^{0.54}, which is derived from the ensemble fall speed of snow crystals in ice clouds and has been applied to drops in, for instance, CAM3 model (Morrison and Gettelman 2008). However, the drag coefficient is known to vary with the Reynolds number even under standard conditions (Gunn and Kinzer 1949), the accuracy of the empirical formula of Foote and du Toit (1969) has not been investigated (see below), and the validity of the correction term in Heymsfield et al. (2007) has not been explicitly demonstrated for drops. Without any sufficient theoretical or experimental justifications, these forms of the correction term are widely used in atmospheric models.

*U*

_{0}(

*D*),

*ρ*

_{0}(

*D*), and

*T*

_{0}are the terminal velocity, density, and temperature, respectively, at surface level. As the experimental procedure and data are unpublished, the accuracy of this formula is unclear and need to be examined rigorously.

In the next section, our numerical model and procedure to simulate free-falling water drops for the purpose of this study are concisely explained.

## 3. Numerical method

An improved immersed boundary method (IBM) (Ong and Miura 2018, 2019) was adopted for DNS of a free-falling water drop. This two-phase flow model was chosen because it has been shown to be robust even when the pressure difference due to surface tension and distortion of the interface are large (Ong and Miura 2019). The drop diameters considered in this study ranged from 0.025 to 0.5 mm. In particular, free-falling water drops of diameters 0.025, 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5 mm with an initial velocity of zero were simulated. The reason for the lower limit of 0.025 mm is that creeping motion (Stokes flow) is a good assumption for droplet flow with diameters as small as 0.025 mm (Gunn and Kinzer 1949). The terminal velocity can be deduced analytically and compared with the numerical solution. Capillary oscillations appear to magnify and numerical solutions become unstable when the raindrop diameter is larger than approximately 0.5 mm.

When a water drop falls through the air in non-Stokes flow, it takes a relatively large distance, compared to its diameter, before the falling speed approaches its asymptotic value, i.e., the terminal velocity. To reduce the required size of the computational domain, the noninertial domain was used (Komrakova et al. 2013; Ong and Miura 2019). In the noninertial domain, the drop is kept approximately at a fixed position by adjusting the inflow speed at the lower boundary such that the downward acceleration of the drop is zero, with the gravitational force acting vertically toward the lower boundary. Ideally, the domain size should be infinitely large so that the flow field around the drop is not affected by the existence of the wall. In our model, we set the computational domain size to be at least twice the thickness of the boundary layer (see appendix A for the validity of this choice). Based on the boundary layer theory of Tomotika (Tomotika 1935; Abraham 1970), the boundary layer thicknesses are approximately 25*D* and 10*D* for free-falling spheres with diameters of 0.025 and 0.05 mm, respectively. The thickness decreases as the diameter increases. Therefore, three different domain sizes were used for simulations of free-falling water drops with diameters of 0.025, 0.05, and above 0.1 mm, respectively, as shown in Fig. 1. Grid cells with the highest resolution were placed in the vicinity of the drop (yellow region in the figure). We set their grid size to 0.0125*D* in all simulations. All physical quantities were nondimensionalized. Further details regarding the numerical schemes can be found in Ong and Miura (2019).

The physical properties of the drop water and the surrounding air in the numerical experiments are given in Table 1. The viscosity of dry air at different temperatures was deduced from Eq. (3) in Kadoya et al. (1985). The viscosity of pure water was obtained based on the work of Sengers and Watson (1986). The surface tension coefficient was interpolated from tables provided in the handbook of Lide (2005). The density of air was derived from the gas law, and the density of water was calculated from the interpolation equation of Tanaka et al. (2003).

Physical properties of air and water at altitudes of 1000, 2000, and 3000 m assuming a fixed lapse rate of −6.5°C km^{−1}, surface temperature of 20°C, and pressure of 1013 hPa. The density, viscosity, and surface tension coefficient are rounded to four significant figures.

As explained in the introduction, five sets of experiments were prepared to achieve the objectives of this study. These experiments and their corresponding dimensionless numbers are listed in Table 2. The diameter was nondimensionalized to 1 in all simulations. In the first set of experiments A–G, the environmental temperature and pressure were set to the standard conditions. We compared the simulated terminal velocity with the experimental data of Gunn and Kinzer (1949) to validate the numerical model. We also examined the convergence of the numerical solution to the Hadamard–Rybczynski solution at small Reynolds numbers, which is an important indication of whether the numerical solution can converge properly to the reference solution.

Summary of numerical experiments conducted in this work and the corresponding dimensionless numbers. All values are rounded to four significant figures.

In the second set of experiments A1–G3, a fixed lapse rate of −6.5°C km^{−1} was assumed in order to observe the change in the terminal velocity when water drops fall at different altitudes. Free-falling water drops at three specific altitudes of 1000, 2000, and 3000 m were simulated, with corresponding environmental temperatures of 13.5°, 7°, and 0.5°C, respectively. The ratio of the simulated terminal velocity aloft to terminal velocity under standard conditions for each drop was then compared with existing empirical formulas [Eqs. (21) and (22)].

In the third set of experiments H–S, two separate Re–*C*_{D} power-law curves [Eq. (18)] were obtained within the diameter ranges of 0.28–0.32 mm and 0.48–0.5 mm, respectively. Two more similar curves were obtained from experiments E1–E3 and G1–G3. Then, the variation in the exponent *b* of deduced from these four curves were compared to examine the error in the computation of terminal velocity aloft when a fixed Re–*C*_{D} curve is assumed.

Experiment G was repeated with the raindrop initially perturbed to an oblate spheroidal shape of eccentricity 0.8062 to study the influence of the oscillatory motion. The diameters along the long *a* and short *c* semiaxes were 1.3*D* and *D*/1.3, respectively. The grid and four dimensionless numbers remained unchanged. Finally, experiments E–G were repeated with *σ* artificially reduced by factors of 2, 10, and 100 (*σ* = 3.638 × 10^{−2}, 7.275 × 10^{−3}, and 7.275 × 10^{−4} N m^{−1}) to examine the impact of the shape deformation on the terminal velocity and tangential velocity in the fifth set of experiments. *σ* in experiment E of a free-falling 0.3 mm raindrop was further reduced to 7.275 × 10^{−5} N m^{−1} by a factor of 1000 to gain insight into the influence of large deformation on the terminal velocity. Note that the viscosity and density ratios are unaffected by simply changing *σ*, while the Eötvos and Morton numbers are modified according to their definition.

## 4. Results

### a. Terminal velocity at 20°C and 1013 hPa

The falling speed profiles of water drops computed from experiments A–G, with diameters ranging from 0.025 to 0.5 mm, are shown in Fig. 2. All numerical solutions asymptotically reached steady state solutions. Note that the falling distance traveled until the terminal velocity was reached was less than about 1.5 m for a 0.5 mm raindrop and, as expected, was smaller for smaller drops. The time required for these drops to reach their terminal velocity was less than 1 s. The conversion formula from dimensionless time to physical time can be found in the appendix of Ong and Miura (2018).

For comparison, the falling speed profiles predicted by the empirical formula of Wang and Pruppacher (1977) are drawn as solid curves on the same figure. Clearly, the growth pattern of simulated falling speeds agreed with the empirical formula. Let *Z*_{99%} and *T*_{99%} respectively be the distance and time traveled by a water drop to reach its 99% terminal velocity from rest. Table 3 compares *Z*_{99%} and *T*_{99%} obtained from current simulation results and the empirical formula. The simulation results were in good agreement with the empirical formula when *D* ≥ 0.1 mm, with the maximum percentage difference of 5.1%. However, the percentage difference was larger than 9% when *D* = 0.025 and 0.05 mm. Considering that the maximum error between the simulated terminal velocity and Beard’s (1976) terminal velocity was only 3.2%, this large percentage difference was possibly due to the slow acceleration of small drops. In other words, a small change in the terminal velocity leads to large changes in *Z*_{99%} and *T*_{99%}.

A comparison of the simulated falling distance (*Z*_{99%}) and falling time (*T*_{99%}) obtained from numerical simulations and the empirical formula of Wang and Pruppacher (1977). *Z*_{99%} (*T*_{99%}) is defined as the distance (time) traveled until the drop reaches its 99% terminal velocity. The value in parentheses beside each numerical result denotes its percentage difference from the corresponding value predicted by the empirical formula. For comparison, numerical time steps are also shown in the second column of the table. We decreased the time step for smaller drops to avoid unstable solutions because smaller drops have larger water surface tension and higher capillary wave speeds.

Figure 3 shows a plot of the terminal velocity against the drop diameter. A comparison between the numerical and interpolated experimental data of the terminal velocity with various parameterizations (Gunn and Kinzer 1949; Beard 1976; Atlas et al. 1973) is summarized in Table 4. The discrepancy from Gunn and Kinzer’s experimental data was below 4.5% for diameters larger than 0.3 mm, increasing rapidly to more than 10% for a 0.1 mm water drop. There are two main reasons for this overestimation and the decreasing trend of the discrepancy with the diameter. The first reason is discretization error, which is the main source of error in numerical methods. It is shown in appendix B that the magnitude of the error is 1.9%. The second reason is measurement error in Gunn and Kinzer’s experiment. According to the speculation of Beard and Pruppacher (1969), the drops might have evaporated slightly before their diameters were determined at the end of fall because the relative humidity of the air in their experiment was only 50%. Therefore, the terminal velocity was possibly overestimated, especially for small drops.

Terminal velocities of axisymmetric water drops of different diameters obtained from the numerical simulations and theoretical solution, and their discrepancies. *U*_{H} and *U*_{S} are the Hadamard–Rybczynski and Stokes terminal velocities, respectively. The discrepancies from Gunn and Kinzer (1949) and the empirical formula of Beard (1976) are shown in the third and fourth columns. The terminal velocities of Gunn and Kinzer (1949) at these particular diameters were estimated by direct interpolation of their data points. The smallest drop they measured had a diameter of 0.0783 mm, and thus the simulated terminal velocities of drops with diameters of 0.025 and 0.05 mm were not compared with their data. All values are rounded to four significant figures.

A much better agreement was achieved when comparing the simulated terminal velocities with Beard (1976) parameterization. The overall discrepancy was below 3.2%. Because it is known that Atlas et al. (1973) formula has a better accuracy when the raindrop diameter is larger than ~0.5 mm, a large discrepancy from the simulated terminal velocities was observed.

The red dashed line in Fig. 3 represents the terminal velocity predicted by the Hadamard–Rybczynski solution of the creeping flow of a spherical droplet in a uniform flow. The discrepancies between the numerical results and these theoretical values are also summarized in Table 4. We see that the discrepancy of the simulated terminal velocity from the Hadamard–Rybczynski solution Δ_{HR} was below 1.9% when the diameter was as small as 0.025 mm. The creeping flow assumption for a spherical droplet is approximately valid for diameters smaller than 0.1 mm. Note that the number of grid cells is almost the same in the experiments A and B, whereas the domain is larger in the experiment A. This might have led to a larger discretization error in the numerical solution of the experiment A. The Stokes solution for a free-falling spherical solid object is also shown in the last two columns of Table 4. The discrepancy between the Stokes velocity and simulated terminal velocity is larger than Δ_{HR} because of the nonzero viscous stress exerted from the internal circulation.

The wake and internal circulation of water drops with diameters 0.025, 0.4, and 0.5 mm at terminal velocity are shown in Fig. 4. At low Reynolds numbers, there was no visible wake or separation of the boundary layer behind the droplet (Fig. 4a), and thus the Hadamard–Rybczynski solution may be valid with good accuracy. A stationary wake appeared behind 0.4 and 0.5 mm raindrops with moderate Reynolds numbers. Moreover, the axis ratio of the 0.5 mm raindrop, which is defined as the ratio of the semimajor axis over the semiminor axis, increased merely to 1.001.

We further examined quantitatively the maximum and averaged tangential velocity on drop surface, eddy’s location inside the drop, and wake’s size behind the drop. These data are compiled in Table 5. All angles are measured from the front stagnation point, and a velocity vector is positive if it points rearward. At low Reynolds numbers, the center of the internal circulation and maximum tangential velocity were located approximately on the plane perpendicular to the falling direction. This is in accordance with the Hadamard–Rybczynski theory. The internal circulation was pushed forward (into falling direction) slightly as the Reynolds number increased, and its distance from the drop center was nearly constant. These results are in agreement with the simulation results of a spherical water sphere by LeClair et al. (1972). A similar trend was observed for the maximum tangential velocity, with its angular position decreasing from 67.15° when Re = 8.88 to 57.21° when Re = 65.4. Nevertheless, LeClair et al. (1972) did not observe a notable change in the location of the maximum tangential velocity in their simulations when the Reynolds number is between 10 and 100. Note that the wake started to appear when the diameter was larger than 0.3 mm.

Numerical data of the averaged tangential velocity on drop surface (*U*), location of the center of internal circulation, and wake’s size. All angles are measured from the front stagnation point. *R*_{eddy} is the distance from the center of internal circulation to the drop center and *θ*_{eddy} is its angular position. *L*_{wake} is the vertical length of the wake and *θ*_{wake} is its minimum angular position on the interface.

Figure 5a shows the ratios of tangential velocities *D* < 1 mm.

Interestingly, the wake that appeared behind the 0.4 and 0.5 mm raindrops was not strong enough to cause a flow separation on the surface, because *D* ≥ 0.3 mm due to the presence of the wake.

### b. Terminal velocity aloft

With the assumption of a fixed lapse rate of −6.5°C km^{−1}, the ratio of the terminal velocity aloft to the terminal velocity under standard conditions, *R*_{tv}, at the three altitudes of 1000, 2000, and 3000 m obtained from simulations A1–G3 are summarized in Table 6. The terminal velocity increased with the altitude, mainly because the air density was lower at higher altitudes while the density of water was nearly unchanged. *R*_{tv} increased with diameters when *D* < 0.3 mm; however, it remained almost constant when *D* ≥ 0.3 mm. The increment of the terminal velocity was 4.6% for a 0.025 mm cloud drop, and it grew to 10.1% for a 0.5 mm raindrop at 3000 m. Comparison with existing parameterizations of the correction term in Eq. (21) is also shown in the same table.

Ratios *R*_{tv} of the terminal velocity at altitudes of 1000, 2000, and 3000 m with a fixed lapse rate of −6.5°C km^{−1} over the terminal velocity at the surface (standard conditions). Δ_{1} = [(*ρ*_{0}/*ρ*)^{0.5} − *R*_{tv}]/*R*_{tv} × 100%, Δ_{2} = [(*P*_{0}/*P*)^{0.4} − *R*_{tv}]/*R*_{tv} × 100%, Δ_{3} = [(*ρ*_{0}/*ρ*)^{0.54} − *R*_{tv}]/*R*_{tv} × 100%, and Δ_{4} = [(*U*_{f}/*U*_{0}) − *R*_{tv}]/*R*_{tv} × 100%. Δ_{5} is the percentage error of the proposed parameterization in which the velocity is calculated from Eq. (23), and the exponent *b* is taken from Table 7. All values are rounded to four significant figures.

It is evident that a simple correction term consisting of the density or pressure ratio with a fixed exponent, or even the more complicated empirical formula of Foote and du Toit (1969), is not an ideal solution for the accurate prediction of the terminal velocity at arbitrary temperatures and pressures. The discrepancy between simulated *R*_{tv} and existing parameterizations generally increased with altitude. It is also clear that this discrepancy became larger as the Reynolds number decreased toward the range in which the Hadamard–Rybczynski solution is valid. The largest discrepancy was about 10% at an altitude of 3000 m for cloud drops with diameters smaller than 0.05 mm. It can be shown that *R*_{tv} is linearly proportional to the ratio of the density difference in the creeping flow, whereas the exponent of the density or pressure ratio in the parameterizations considered in this study is between 0.4 and 0.54. The underestimation of this exponent thus leads to a large discrepancy for low Reynolds numbers.

The empirical formula of Foote and du Toit (1969) [Eq. (22)] yields values of the terminal velocity aloft with the smallest discrepancies. However, the discrepancy is still larger than 4% at the high altitude of 3000 m. Even though the discretization error in the simulated terminal velocity is 1.9%, the ratio of the terminal velocities is relatively insensitive to the grid refinement operation (see appendix B). Thus, a discrepancy as small as 4% cannot be solely due to numerical errors. It is likely that this discrepancy is a result of imperfections in the empirical formula.

### c. New empirical formula

*R*

_{tv}can then be expressed by the following equation:

*C*

_{D}curves generated from the numerical experiments H–O, in which only the diameter was slightly perturbed to values between 0.28 and 0.32 mm, and experiments E1–E3, in which the 0.3 mm raindrop fell at different altitudes. The exponent

*b*in Eq. (23), or Eq. (18), is the gradient of the Re–

*C*

_{D}curve on a logarithmic scale. A plot of Re–

*C*

_{D}curves obtained from these experiments is shown in Fig. 6a. These two curves diverge, and thus

*b*is not of the same value. By performing a least squares fitting of the power law [Eq. (18)] to each curve,

*b*is equal to −6.265 × 10

^{−1}and −7.211 × 10

^{−1}for the solid (experiments E1–E3) and dashed (experiments H–O) curves, respectively. This amounts merely to about 1% error in the estimation of the terminal velocity at the altitude of 3000 m. Recall that the discretization error in the simulated terminal velocity is around 1.9%, the error above is considered small. Nevertheless, The error increased gradually to about 1.5% for the 0.5 mm raindrop at 3000 m (Fig. 6b).

*b*= −5.767 × 10

^{−1}as deduced from the numerical results of experiments P–S and

*b*= −7.252 × 10

^{−1}from experiments G1–G3.

In fact, the error in the estimation of the terminal velocity by treating a water drop as a solid object increases with Reynolds number when *D* ≤ 0.5 mm. This can be observed in Figs. 7a and 7b. Figure 7a is a plot of the Re–*C*_{D} linear spline curve by joining the Reynolds numbers and drag coefficients at terminal velocity obtained from numerical experiments A–G under standard conditions. Since *b* is a gradient of the Re–*C*_{D} curve in logarithmic scale, we can estimate its value by calculating the gradient of line segments at *D* = 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5 mm. Let us denote this approximation of *b* from the curve in Fig. 7a by *b*′. On the other hand, the true value of *b* for each diameter can be derived from the gradient of the least squares fitting curves obtained in the experiments A1–G3 (e.g., see the dashed lines in Fig. 6). Note that *b*′ is a rather crude approximation because the data points are scarce. However, the motivation here is to illustrate the trend and compare it with the true values of *b*. This comparison is shown in Fig. 7b.

The linear spline curve of *b* in Fig. 7b can be used to create a new parameterization of the correction term *α* in Eq. (21) with improved accuracy by replacing *α* with the right-hand side of Eq. (23). The downside of using this complicated correction term is that it cannot be integrated analytically over the entire mass spectrum to obtain the total mass flux in closed form. An alternative approach is to approximate the exponent *b* by a piecewise constant function of the diameter by taking averaged values between all adjacent data values in Fig. 7b. The new piecewise *b* function is presented in Table 7. The maximum error in the estimation of the terminal velocity aloft at each diameter is shown in Table 6. We see that the maximum error is smaller than 1% for all diameters. The new piecewise function can thus be a good replacement for a smooth *b* function with the advantage of straightforward integration over the mass spectrum. Nevertheless, this alternative approach is not without clear disadvantages. The terminal velocity predicted from the piecewise function becomes discontinuous and may produce numerical errors (Khvorostyanov and Curry 2002).

Piecewise function of the exponent *b*.

Up to this point, only discrepancies at specific diameters were considered. Whereas in the dynamics calculation of atmospheric models, a statistical distribution of cloud drops and raindrops is defined and bulk quantities are used to predict prognostic variables. Instead of a discrepancy at a particular diameter, discrepancies in the bulk mass flux [*m*(*D*) and *N*(*D*) are the individual drop mass and number concentration density distribution, respectively] are a more useful guide to measure the impact of inaccurate values of the terminal velocity. To investigate discrepancies in the bulk mass flux, the normalized gamma distribution defined by Testud et al. (2001) was adopted. Typical values of the median drop diameter (*D*_{m} = 1 mm), shape parameter, and liquid water content (LWC = 0.5 g m^{−3}) for stratiform precipitation (Thompson et al. 2015) were used. Since the maximum diameter examined by the axisymmetric model is 0.5 mm, we set the upper limit of the integral equal to 0.5 mm. Bulk mass fluxes were computed at an altitude of 3000 m with three parameterizations of the correction term: 1) (*ρ*_{0}/*ρ*)^{0.5}, 2) Eq. (22), and 3) Eq. (23) with the proposed parameters in Table 7. The results are compiled in Table 8. We see that the discrepancy between the existing and proposed parameterizations varies between 5.1% and 4.1%. The magnitude of the discrepancy is literally determined by the discrepancy in the estimation of the terminal velocity aloft of larger drops (≥0.3 mm), because the total mass and terminal velocity of small drops (<0.3 mm) is much less than that of larger drops in the gamma distribution.

Bulk mass fluxes of drops smaller than 0.5 mm in normalized gamma distribution calculated using the terminal velocity aloft with the correction terms (1) (*ρ*_{0}/*ρ*)^{0.5}, (2) Eq. (22), and (3) Eq. (23) denoted by ^{−2} s^{−1}. Since the upper limit is 0.5 mm, these are only partial mass fluxes. The mass flux of drops smaller than 0.5 mm is approximately 44% (2%) of total mass flux when *μ* = 1 (4) if the terminal velocity of Beard (1976) is used for large raindrops. *D*_{m} is the mean drop diameter, LWC is the liquid water content, and *μ* is the shape parameter of the gamma distribution. *D*_{peak} is the diameter at which the peak of the distribution is found.

### d. The effect of shape on the terminal velocity

The free-falling water drops in all scenarios provided in Table 2 maintained an approximate spherical shape at terminal velocity (see Fig. 4). It was mentioned in the introduction that shape oscillation and deformation due to reduction in the surface tension coefficient are another factors that might affect the falling speed profile and internal circulation. The free-falling 0.5 mm raindrop simulation with its initial shape deformed to an spheroidal shape was performed to quantitatively study the influence of the oscillatory motion to the falling motion.

Figure 8 shows the time series of the axis ratio *a*/*c*, where *a* (*c*) is the length of the semimajor (semiminor) axis, and the velocity fluctuation, which is defined as the difference between the falling speeds in experiment G and this experiment. The periodic oscillation was due to capillary waves caused by the surface tension force. This high-frequency wave was quickly dampened within 0.1 s. The maximum magnitude of the velocity fluctuation was only on the order of 1 × 10^{−4} m s^{−1}. Therefore, the influence of shape oscillation on the falling speed was insignificant. Note that a negative bias was observed after the velocity fluctuation flattened out. This occurred because the boundary of the raindrop was approximated by linear splines in the IBM, which led to a small difference in the raindrop mass due to different initial shapes. Nevertheless, the bias was much less than the maximum velocity fluctuation caused by the oscillation.

In the sensitivity test of the surface tension coefficient, 0.3, 0.4, and 0.5 mm raindrops with reduced surface tension were simulated to examine the influence of shape deformation on the terminal velocity and tangential velocity. The relationship between the ratio of the terminal velocity of raindrops with reduced surface tension to the original terminal velocity and the axis ratio is illustrated in Fig. 9a. There are two notable physical properties we can deduce from this figure. First, the terminal velocity is weakly dependent on the shape. The terminal velocity decreased by less than 1% when the axis ratio increased by more than 40% in the 0.3 mm raindrop simulation where the surface tension coefficient was reduced by a factor of 1000. For the 0.5 mm raindrop, a 12% increase in the axis ratio resulted in only about a 1.3% decrease in the terminal velocity. Therefore, shape is a relatively minor factor in the determination of the terminal velocity for small raindrops. This agrees with the conclusion drawn from the previous result of the free-falling oscillatory raindrop simulation that large deformation does not significantly change the falling speed profile. Second, the change in the terminal velocity increases with the diameter. In other words, the terminal velocity of larger raindrops is more sensitive to the shape deformation.

In the real atmosphere, *σ* rarely ever decreases by more than a factor of 2 for precipitation water (Seidl and Hänel 1983). Whereas the reduction in the terminal velocity is only about 0.5% when *σ* is reduced by as much as a factor of 10 based on the current results. Therefore, the effect of reduced surface tension can be safely neglected for small raindrops of diameter up to 0.5 mm.

From Fig. 9b, we see that *υ*_{i}/*U*, the ratio of averaged tangential velocity on the raindrop surface to the terminal velocity, is linearly proportional to the axis ratio. Interestingly, this proportionality remained linear even when the axis ratio was as large as 1.4. The tangential velocity increased as raindrop shape became more elliptic because the pressure difference between the front and rear stagnation points increased while the flow remained laminar (Fig. 10). The center of the internal circulation was stretched outward and rearward. The angular position of the maximum tangential velocity was also pushed rearward. Since the raindrop shape became elongated in the direction perpendicular to the falling direction, the wake subsequently increased to a larger size. This can be seen by comparing the wake’s size in Figs. 4b,c and 10b,c. Furthermore, the front part of small raindrops tended to be distorted into a U shape when the surface tension was weak (Fig. 10a).

## 5. Conclusions

The terminal velocity of water drops under standard conditions has been investigated in previous experimental studies. In the present study, the IBM was employed to study the terminal velocity and shape of free-falling water drops numerically at temperatures and pressures that were not restricted to standard conditions. The validity of the IBM as a robust numerical model for free-falling water drop simulations was demonstrated by comparing the simulated terminal velocity of drops smaller than 0.5 mm with the experimental data of Gunn and Kinzer (1949) and the parameterization of Beard (1976). The simulated terminal velocity agreed with the experimental data to within 4.5% when the diameter was larger than 0.3 mm. The discrepancy grew to 14.64% when the diameter was 0.1 mm, and it was possibly due to a combination of errors in simulations (discretization and wall effect) and those in laboratory measurements (evaporation of drops). Nevertheless, the numerical solution agreed with the Hadamard–Rybczynski solution to within 2% for diameters smaller than 0.05 mm. Moreover, the discrepancy with Beard (1976) parameterization was below 3.2% for all diameters examined in this study.

Falling speed profiles and flow geometry were also compared with Wang and Pruppacher (1977) and LeClair et al. (1972). In spite of larger discrepancies in *Z*_{99%} and *T*_{99%} for cloud drops, which are possible due to the sensitive dependence of the falling distance and time on the terminal velocity, good agreement was achieved for drops larger than 0.5 mm. Together with the result of the oscillating 0.5 mm raindrop simulation, the improved IBM was demonstrated to be a reliable model to simulate a free-falling, deforming water drop under general conditions.

Free-falling water drop simulations at altitudes of 1000, 2000, and 3000 m with a fixed lapse rate of −6.5°C km^{−1} were conducted to quantitatively examine the accuracy of common parameterizations of terminal velocity aloft [Eqs. (21) and (22)]. It was found that there were significant discrepancies between the terminal velocities predicted by these empirical formulas and the simulated terminal velocity. The discrepancy was between 9.38% and 11.8% when the diameter was 0.025 mm at 3000 m. It generally decreased with increasing diameter, and increased with increasing altitude. The empirical formula of Foote and du Toit (1969) had the smallest discrepancies of 9.3% and 4.2% for diameters of 0.025 and 0.5 mm, respectively, at 3000 m.

By analyzing the *C*_{D}–Re power-law curves computed from the simulations conducted at standard conditions and high altitudes, it was found that the exponent *b* indeed depends on the environmental conditions because two dimensionless numbers are not sufficient to characterize the free-falling motion of a drop. The terminal velocity deduced from the assumption of a fixed *C*_{D}–Re curve derived under standard conditions could contain a percentage error as large as 1.5% for a 0.5 mm raindrop at an altitude of 3000 m. When the drop diameter was smaller than 0.3 mm, this assumption produced a good approximation of the terminal velocity aloft with an error smaller than 1%.

A new parameterization of the terminal velocity aloft with an overall discrepancy from the simulated terminal velocity aloft better than 1% was proposed. This new parameterization was based on the piecewise *C*_{D}–Re power-law relationship (Fig. 7b and Table 7). From Table 6, it can be seen that the accuracy of this new formula is better than that of existing empirical formulas [Eqs. (21) and (22)].

The simulations of free-falling 0.3, 0.4, and 0.5 mm raindrops with reduced surface tension showed that the impact of shape deformation on the terminal velocity was relatively small, with less than 1.3% of reduction in the terminal velocity when the axis ratio increased by 12%. The averaged dimensionless tangential velocity was linearly proportional to the axis ratio. Changes in the shape and terminal velocity were negligibly small when the surface tension coefficient was reduced by a factor of 10, suggesting that the influence of reduced surface tension caused by the presence of soluble aerosols on the shape and terminal velocity of small raindrops is negligible. It should be noted that the IBM does not directly simulate surfactants. Instead, it only simulates a liquid drop with lowered surface tension that has the same density and viscosity as water. It was pointed out by Müller et al. (2013) that surfactants have a considerable impact on the internal circulation and shape of large raindrops. The IBM can be improved by including the surfactant effect to study the influence of surfactants on small raindrops in the future.

In conclusion, the improved IBM was verified by comparing the falling speed, terminal velocity, and internal circulation of axisymmetric cloud drops and raindrops under standard conditions. It was successfully applied to free-falling axisymmetric drops at high altitudes, and with reduced surface tension to quantitatively study the influence of water viscosity and surface tension on the terminal velocity and internal circulation. Since the error in the estimation of terminal velocity aloft generated by using a fixed *C*_{D}–Re curve was found to increase with the diameter (Figs. 6 and 7), the terminal velocity aloft of large raindrops (*D* > 0.5 mm) should be revised in the future. To resolve this problem, a full 3D model is necessary because the motion of large raindrops is asymmetric and chaotic sometimes. This will be our next goal with the foundation laid by this work.

This work was partly supported by a Grant-in-Aid for Scientific Research on Innovative Areas 6102 (KAKENHI Grant JP19H05699) from the JSPS of Japan, by the ArCS (JPMXD1300000000) and ArCS II (JPMXD1420318865) projects of the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) in Japan, by the Environment Research and Technology Development Fund (2-1703, 2-2003) of the Environmental Restoration and Conservation Agency in Japan, and by JSPS KAKENHI Grant-in-Aid for Scientific Research (B) 25287119, Grant-in-Aid for Scientific Research (B) 16H04048, and Grant-in-Aid for Transformative Research Areas (B) 20H05731.

## Data availability statement

For readers who are interested in the model or the data used in this work, please email the corresponding author to gain access.

# APPENDIX A

## Domain Size

For all the simulations that were performed in this work, we set the computational domain size to be at least twice larger than the boundary layer thickness. To show that the domain size is sufficient enough to keep the error in the simulated terminal velocity due to the wall effect below 1%, the free-falling 0.1 mm water drop simulation was repeated with a larger domain. The original size of the domain can be found in Fig. 1 in the main text. Extra grid cells are adjoined to the original domain such that its width and height become 41.4 and 69, which are twice as large as the width and height of the original domain. The simulated terminal velocity of the 0.1 mm drop in this extra-large domain was 0.2483 m s^{−1}. The difference from the terminal velocity obtained in experiment C was merely 0.0005 m s^{−1}, or 0.2%. This shows that the original domain size is sufficient.

# APPENDIX B

## Grid Refinement Test

A grid refinement test was performed to demonstrate the reliability of the simulated terminal velocity, and provide a possible range of discretization errors for the results presented in this study. Because substantial computational resources and time are needed to perform a free-falling raindrop simulation with doubly refined grid resolution, only simulations of free-falling raindrops with diameters from 0.48 to 0.5 mm were chosen for this test. This is because the thickness of the boundary layer of larger raindrops is smaller and consequently the gradient of the velocity field is higher, thus the discretization error is expected to be larger in contrast to smaller raindrops. In these simulations, the grid size in the vicinity of the raindrop (see Fig. 1 in the main text) was reduced by a factor of 2 to 0.006 25. In other words, the number of grid cells in the entire domain was quadrupled.

A comparison of the Reynolds number and the drag coefficient of raindrops at terminal velocity obtained from this refinement test and those from simulations with standard grid size is shown in Fig. A1. Although the discretization error in the simulated terminal velocity between low- and high-resolution simulations of free-falling raindrops was as large as 0.94%, the gradient of the Re curve increased by only about 0.18% when the doubly refined grid resolution was used. Since it has been shown that the order of convergence of the improved immersed boundary method is roughly equal to one (Ong and Miura 2019), i.e., the discrepancy between two numerical solutions is reduced by half if the grid spacing is halved, we expect that the discrepancy of the terminal velocity from the true value is at best 1.9%.^{B1} Furthermore, the error in the estimation of gradient of the Re–*C*_{D} in logarithmic scale, *b*, is expected to be 0.36% at best in this case.

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^{1}

We follow the AMS *Glossary of Meteorology* to define drops of diameters smaller than 0.2 mm as cloud drops in this study. While drops with diameters between 0.2 and 0.5 mm are defined as drizzle drops and drops larger than 0.5 mm are called raindrops in the AMS *Glossary of Meteorology*, we call all precipitation drops larger than 0.2 mm raindrops for simplicity. The word “drops” refers to water drops of all sizes.

^{B1}

Since *U*_{Δx=0.006 25D} = [(1 − 0.94)/100]^{−1}*U*_{Δx=0.0125D}, *U*_{truth} is the true value of the terminal velocity. Thus, (*U*_{truth} − *U*_{Δx=0.0125D})/*U*_{truth} = 1.9% in the best-case scenario.