A new model for settling velocity of non-spherical particles

The settlement of non-spherical particles, such as propagules of plants and natural sediments, is commonly observed in riverine ecosystems. The settling process is influenced by both particle properties (size, density, and shape) and fluid properties (density and viscosity). Therefore, the drag law of non-spherical particles is a function of both particle Reynolds number and particle shape. Herein, a total of 828 settling data are collected from the literatures, which cover a wide range of particle Reynolds number (0.008–10000). To characterize the influence of particle shapes, sphericity is adopted as the general shape factor, which varies from 0.421 to 1.0. By comparing the measured drag with the standard drag curve of spheres, we modify the spherical drag law with three shape-dependent functions to develop a new drag law for non-spherical particles. Combined with an iterative procedure, a new model is thus obtained to predict the settling velocity of non-spherical particles of various shapes and materials. Further applications in hydrochorous propagule dispersal and sediment transport are projected based on deeper understanding of the settling process.


Introduction
In riverine ecosystems, hydrochory plays a major role in transporting and depositing freshly produced plants propagules (predominantly seeds) along the river corridors (Merritt and Wohl 2002;Yoshikawa et al. 2013). In the way of becoming an essential part of the ecosystem, the nonbuoyant seeds have to undergo a long period of settling, dispersal, germination, and then gradually contribute to the colonization of the riparian zone (Gurnell et al. 2008;Chambert and James 2009;Koch et al. 2010). Another factor that greatly affects the hydrodynamic process of rivers is the natural sediment. As mentioned by Meier et al. (2013), sediments can promote vegetation growth due to the transportation of nutrients and other organic matters. Moreover, by creating accretional structures in rivers, sediments can also facilitate the development of other species in the new habitats (Gurnell et al. 2012). Thus, as a starting point of the hydrodynamic process, the settlement of both hydrochorous seeds and natural sediments should be paid more attention since the construction of aquatic ecosystem grows more important.
Natural particles are usually generalized as spheres, which simplifies the characterization of particle shapes. Hence, many spherical drag laws, whether explicit or implicit, are proposed by previous studies (Clift and Gauvin 1971;Haider and Levenspiel 1989;Brown and Lawler 2003;Cheng 2009;Terfous et al. 2013). These drag laws reveal that the flow around spherical particles is strongly dependent on the particle Reynolds number. To be more specific, the flow at Re p ≪0.1 is called the Stokes' regime (or creeping flow), where flow remains attached to the sphere with no wakes behind (White 1998). The flow at 0.1 < Re p < 1000 is at the intermediate regime, where the spherical drag coefficient continues to decrease as Re p increases. At 1000 ≤ Re p < 3 × 10 5 , wakes behind the sphere grow fully turbulent while flow in front of the sphere remains laminar. The flow in this range of Reynolds number is called the Newton's regime (Clift et al. 2005). However, in practical applications, the most commonly encountered particles (e.g., seeds, pebbles, and gravels) are nonspherical and even irregular. Thus, the knowledge of drag coefficient and settling velocity of non-spherical particles is essential for solving the settling problem.
Based on numerous experimental and numerical settling data, many models have been proposed to predict the settling velocities of non-spherical particles. In these models, particles are usually separated into several categories, such as regularshaped particles (Komar 1980;Haider and Levenspiel 1989;Ganser 1993;Gogus et al. 2001;Wang et al. 2011;Lau and Chuah 2013;Song et al. 2017), natural sediments and crushed rock fragments (Komar and Reimers 1978;Hallermeier 1981;Dietrich 1982;Swamee and Ojha 1991;Chien 1994;Cheng 1997;Alcerreca et al. 2013;Wang et al. 2017), and conglomerates of particles (Tran-Cong et al. 2004). Hence, the performances of most models are restricted in certain types of particles. Moreover, compared to spherical drag coefficients, the drag coefficient of non-spherical particles are determined by parameters other than Re p , such as particle shape and orientation (Bagheri and Bonadonna 2016). To scale the influence of non-spherical particle shapes, the shape descriptor, aspect ratio (E) is especially designed for the axisymmetric particles, such as cylinders, spheroids, and ellipsoids (Trans-Cong et al. 2004;Loth 2008;Wang et al. 2011). The term S, which is defined as the ratio between equivalent sphere area and the projected area of particle settling direction, is also proposed to describe the effect of settling orientation (Song et al. 2017). However, Bagheri and Bonadonna (2016) found that the impact of both shape and orientation on the non-spherical drag coefficient increases with Re p , which can be reflected by neither of the two descriptors. They further illustrated that the particle orientation is random in the Strokes' regime, while in the Newton's regime, particle secondary motions and orientation are functions of the particle-to-fluid density ratio, which again proves the insufficiency of the term S. Therefore, it is difficult to generalize the effect of shape and orientation with just one shape descriptor. Another problem for some models is that only part of the flow regimes is covered due to the limitation of experimental conditions. For instance, the model of Wang et al. (2017) is applicable for particle Reynolds numbers in the range of 0.01 to 3700, which covers Strokes', intermediate, and early stage of Newton's flow regime. As for particles that settle in higher stage of Newton's regime, the prediction ability of the model is unknown.
To solve the above-mentioned problems, a reliable model is developed based on a large amount of settling data, including mineral sediments and artificial particles. Note that behaviors of non-buoyant seeds during entrainment and settling are consistent with that of mineral sediments, and the settling process of such artificial particles is analogous to that of natural sediments . Herein, the shape descriptor, sphericity, is used as the general parameter to show the influence of particle shapes. Note that the sphericity of these particles varies from 0.421 to 1.0. In addition, the particle Reynolds number ranges from 0.008 to 10000, which almost covers all regimes that can be encountered in natural processes. Herein, we modify the spherical drag law of Clift and Gauvin (1971) with shape-related functions to produce a new drag law for non-spherical particles. Subsequently, with an iterative procedure, a new model is developed to predict the settling velocity for different types of particles, which can eventually be applied in understanding the deposition, transport, and dispersal of seeds and natural sediments.

In situ settling velocity
For particles settling through a static fluid, the balance between surface (drag) and body forces acting on the particle is expressed as follows: Therefore, the settling velocity of the particle can be obtained through the following formula: where ρ f is the fluid density, ρ s is the particle density, C d is the drag coefficient depending on properties of particle and fluid, A p is the projected area of the particle perpendicular to the settling direction, W s is the settling velocity of the particle, g is the gravitational acceleration, and V is the volume of the particle. The particle Reynolds number (R ep ) is defined as: where d n is the diameter of the volume equivalent sphere, and μ is the dynamic viscosity of the fluid.

Particle shape characterization
The nominal diameter of non-spherical particles (d n ) can be calculated as the diameter of the sphere whose volume is equivalent to the non-spherical particle's volume (Wadell 1932).
To describe the shape of non-spherical particles, multiple shape descriptors are proposed, which can be divided in 1D, 2D, and 3D shape descriptors. 1D descriptors are called form factors, which are based on particle lengths in three dimensions (i.e., d s , d m , d l , which are the short, medium, and long principle axes of the particle, respectively). Form factors include flatness (f = d s /d m ) and elongation (e = d m /d l ) (Loth 2008;Bagheri et al. 2015). The Corey shape factor ) is the most commonly used 1D descriptor that describes particles of relatively smooth surfaces (Corey 1963;Swamee and Ojha 1991). As a 2D descriptor, circularity (X) is based on 2D variables obtained by image analysis, which is suitable for particles with sharp corners and large obtuse angles (Büttner et al. 2002). Circularity is defined as the ratio between the maximum projection perimeter P mp and that of the perimeter of the circle equivalent to the maximum projection area P eq : where P mp and P eq are calculated by approximating the maximum projective regions as ellipses, with the former determined by π ffiffiffiffiffiffiffiffiffi ffi d l d m p . The latter P eq can be calculated by the first approximation by Ramanujan (1914), which is expressed as: Sphericity (ϕ), which is related to the surface area and particle volume, is considered as a 3D shape descriptor. Applied in settling models where particles are of various shapes and materials, sphericity is proved to be the best parameter to describe the influence of particle shape, and thus is chosen in this study (Haider and Levenspiel 1989;Ganser 1993;Chien 1994). Sphericity is defined as the ratio between the surface area of the equivalent sphere A sph and the particle surface area A s : where A sph is calculated by A sph = 4π(d n /2) 2 . For particles that are similar to scalene ellipsoids, the surface area of particle A s can be approximately calculated as follows (Taylor et al. 2006): where a, b, and c are semi-axes of a tri-axial ellipsoid, λ=1.5349, k=0.0942. Except for ϕ, another frequently used 3D shape descriptor is ψ, which is defined as the ratio between sphericity and circularity (ψ = ϕ/X). This descriptor is first introduced by Dellino et al. (2005) for highly irregular particles, and its validity has been verified by Dioguardi and Mele (2015) and Dioguardi et al. (2018).

Formulation
To develop a new shape-dependent drag law, we collect 828 sets of settling data from previous studies (Komar and Reimers 1978;Komar 1980;Baba and Komar 1981;Chambert and James 2009;Koch et al. 2010;Wang et al. 2011;Dioguardi and Mele 2015;Song et al. 2017;Zhu et al. 2017). Details of the dataset are listed in Table 1. Note that within the database, the range of ϕ is 0.421-1.0, and the value of R ep varies from 0.008 to 10000.
Settling data of regular particles, such as cylinders, cuboids, cubes, and disks, are collected from Komar (1980), Wang et al. (2017), and Song et al. (2017). In these cases, the geometric parameters of particles are provided, which means both circularity and sphericity can be calculated through the definitions of P mp , P eq , A sph , and A s . In contrast, data from Komar and Reimers (1978) and Baba and Komar (1981) are aimed at irregular-shaped particles, such as pebbles and glasses. Herein, weight and tri-axial lengths of the particle (d s , d m , d l ) are given, as well as particle density (ρ s ). Thus, the corresponding circularity and sphericity can be obtained using the approximated equations (Eq. (5)~Eq. (8)). As for volcanic particles in Dioguardi and Mele (2015), three shape descriptors (X, ϕ, ψ) are provided directly, as well as d s , d m , d l . In light of the hydrochorous seeds in Chambert and James (2009), Koch et al. (2010), and Zhu et al. (2017), the values of sphericity are listed directly, along with the volume equivalent spherical diameter (d n ). Thus, in the case of seeds, both X and ψ cannot be obtained without more information of d s , d m , d l .
Enlightened by the idea of Dioguardi et al. (2018), we construct a new drag law for non-spherical particles based on the spherical drag law of Clift and Gauvin (1971). This spherical drag law is chosen since it can effectively cover the entire R ep range of the standard drag curve.
By comparing the difference between the measured drag coefficient (C d, meas ) in our database and the drag of a sphere (C d, sphere ) at the same R ep , we separate Eq. (7) into the sum of three terms: In Fig. 1, we plot the measured drag coefficient of particles C d, meas versus the drag curve for spheres. The drag of nonspherical particles is observed to be larger than C d, sphere in most cases. Specifically, the measured drag (black dots) is close to C d, sphere (solid line) for R ep <10. As R ep grows larger than 10, the bias between the measured drag and the standard drag curve also becomes larger, since the influence of particle orientation and irregularity on the non-spherical drag coefficient increases with R ep (Bagheri and Bonadonna 2016). In addition, for irregular particles whose sphericities are near 1.0, their corresponding black dots on the figure are also closely related to the standard spherical drag curve. From these observations, we hypothesize that the influence of particle shape can be separately added to the three parts of Eq. (10), as has been done by Dioguardi et al. (2018). Note that the shape descriptors (ϕ) in these functions are different from that (ψ) in the functions of Dioguardi et al. (2018).
For the shape-modified functions, the following constraints must be satisfied.
Therefore, when the shape of a particle is close to a sphere (ϕ ≈ 1), Eq. (11) can turn back to the spherical drag law (Eq.  (10)). Based on the collected database, the three functions are obtained by comparing the difference between C d, meas and C d, sphere focusing on the three parts separately. Following the assumption of Dioguardi et al. (2018), we also assume that the ratio between each terms of C d, sphere and the total C d, sphere is equal to the ratio between each part of C d, meas and the total drag of non-spherical particles (see Eq. (13)).
With this assumption, the three functions are separately searched by correlating each part of C d, calc with ϕ and R ep . The best function is the one with the minimum error when predicting the settling velocity W s, meas . Finally, the three functions satisfying the constraints and assumptions are: The values of three exponents a 1 , a 2 , and a 3 are obtained by iteratively searching for the values that can present the best fit with the measured data. The iterative searching process is conducted with the Matlab script. The best values of a 1 , a 2 , and a 3 are 1.29, 0.134, and 1.43, respectively.
Thus, Eq. (11) turns to the new drag law: As an implicit drag law depending on R ep , an iterative procedure is usually adopted for calculating C d , since both C d and R ep are relevant to W s . The trial-and-error procedure is presented in the flow chart (Fig. 2). Details of the new drag law can be found in the Online Resource (sheet of "Drag law").

Models for comparison
To evaluate the ability of the present model to predict the settling velocity of non-spherical particles, we compare it with previous models (Chien 1994;Alcerreca et al. 2013;Dioguardi and Mele 2015;Dioguardi et al. 2018). Details of comparison with other models are given in the Online Resource (sheet of "Model comparison"). Chien (1994) derived the drag law for irregular particles settling in Newtonian and non-Newtonian fluids. Since this drag law is also implicit, a numerical iteration method is required to predict the settling velocity. The model of Chien (1994) covers a ϕ range of 0.2-1.0, and a R ep range of 0.001-10000.
The drag law of Alcerreca et al. (2013) is derived based on a large amount of settling data of calcareous sand particles. The drag law is explicit since C d can be expressed as a function of the dimensionless particle diameter D * : Fig. 2 Flow chart of the trial-and-error procedure for the calculation of C d where d N is the nominal particle diameter which can be calculated by d N = (d l d m d s ) 1/3 , and υ is the kinematic viscosity of the fluid. Consequently, the settling velocity can be calculated directly from basic properties of particles and fluids (d l , d m , d s , ρ s , ρ f , υ).
According to the detailed dataset description in the "Formulation" section, the law of Alcerreca et al. (2013) can be applied directly in volcanic materials, pebbles, and glass particles, the basic properties of which have all been provided. For regular particles involved in the dataset, their form dimensions (d l , d m , d s ) are shown in Table 2, which enable the application of Alcerreca et al. (2013). As for hydrochorous seeds, d N can be directly replaced by the volume equivalent spherical diameter d n .Thus, the drag law of Alcerreca et al. (2013) can be applied in the whole dataset. Dioguardi and Mele (2015) applied the shape descriptor ψ to describe the irregularity of particles. Their drag law is based on the 340 settling data of volcanic particles, which are in a wide range of D * (0.03-10000). Developed from the drag law of Dellino et al. (2005), the form of this law is segmented and simple:  (9)). An iterative procedure is needed to construct the final model to predict the terminal settling velocity. As a development of Dioguardi and Mele (2015), the model of Dioguardi et al. (2018) also used the shape descriptor ψ. Their drag law is obtained by 304 settling velocity measurements, which are part of the 340 settling data mentioned in

Comparison results
To compare the accuracy of predicting the settling velocity for each model, we first plot the C d, calc and C d, meas versus R ep in Fig. 3. After calculating the C d, calc with the aforementioned laws, we iteratively obtain the terminal settling velocity W s, calc and compare it with the measured values W s, meas . In Fig.  4, the comparison results are displayed, along with the correlation coefficient of each model. As shown in Fig. 4, most models have similar correlation coefficients, which are approximately 0.94. The performance of Alcerecca et al. (2013) is relatively weak, which may due to the fact that their model is based on settling data of calcareous sand. Thus, for non-spherical particles of other shapes and materials, the ability of this model to predict the settling velocity is limited. Both Fig. 3 and Fig. 4 are presented following the idea of Dioguardi et al. (2018). Based on the correlations between W s, calc and W s, meas , three statistical parameters are used as indicators to assess the ability of different models to predict the settling velocity of non-spherical particles. The first parameter is the R-squared (R 2 ), which is obtained by linear fitting and presented in Fig. 4 for each model. The second parameter is average absolute error (|err%|), which is calculated as follows: We also evaluate each model with the root-mean-square error (RMSE) by applying the following formula: where N=828 is the number of settling data in our database. Error analyses of the three indicators for different models are summarized in Table 3. From R 2 listed in Table 3, the performance of the present model is slightly weaker than the model of Dioguardi et al. (2018). However, when evaluating the performance of models by |err%| and RMSE, the present model predicts the settling velocity better than all the other models.
To further explore the differences among these models, we compare the ability of each model to predict settling velocity of particles from different sources. The comparisons of |err%| are given in Table 4 while the results of RMSE are summarized in Table 5. About the two tables, two notations are Fig. 3 a-e Plot of C d versus R ep for previous and present models. The empty circles represent C d,meas and the solid circles represent C d,calc clarified. Firstly, since the number of settling data of hydrochorous seeds is low, we summarize the settling data of seeds from three sources in one row, which is simply marked as "hydrochorous seeds" (Chambert and James 2009;Koch et al. 2010;Zhu et al. 2017). Secondly, since the data of tri-axial lengths are missing for hydrochorous seeds, the shape descriptor ψ mentioned in the models of Dioguardi and Mele (2015) and Dioguardi et al. (2018) cannot be calculated without approximated values of circularity. Thus, in the last row of both tables, the error analyses for hydrochorous seeds are not given for the two models.
From Tables 4 and 5, the present model performs quite well for data collected from Song et al. (2017), Komar and Reimers (1978), and Baba and Komar (1981). Combined with particle shapes and materials listed in Table 1, the present model is found to be especially suitable for regular particles (e.g., cubes and Fig. 4 a-e Plot of W s,calc versus W s,meas for previous and present models. The solid line is the best fit line of type y=ax. The line relations are also reported with R 2 cylinders) and particles with relatively smooth surfaces (e.g., pebbles and irregular glass particles). Nevertheless, the accuracy of the present model to predict the settling velocities for data from Wang et al. (2011), Komar (1980, and hydrochorous seeds is only acceptable. For data from Wang et al. (2011) and Komar (1980), the present model performs better than the other models except for the model of Dioguardi and Mele (2015). The prediction for settling velocity of hydrochorous seeds, though not accurate enough, is the best among these models. A common reason for the deviations is that all the settling data from the three sources are no more than 50 data points, and thus the errors may be enlarged. At last, the abilities of the models to predict settling velocity of volcanic particles are compared. Since the models of both Dioguardi and Mele (2015) and Dioguardi et al. (2018) are developed from the settling data of volcanic materials, they naturally show smaller errors than the other models. Note that the performance of the present model is just slightly weaker than the two models. To summarize from the above analysis, it can be concluded that the present model shows the best performance when predicting settling velocity for particles of various shapes and materials.

Discussion
Among these models for comparison, the differences between the model of Dioguardi et al. (2018) and the present model need to be further discussed, since they are based on similar ideas. Herein, four major differences are clarified. Firstly, as shown in Table 1, the present model contains a larger database, which includes particles of various shapes and materials. On the other hand, the model of Dioguardi et al. (2018) is based on only part of the 340 data points of volcanic particles. Secondly, the shape descriptor applied by Dioguardi et al. (2018) is ψ rather than ϕ. As mentioned in the "Particle shape characterization" section, the calculation of ψ requires the value of circularity (X) in addition to sphericity (ϕ). Thus, the model of Dioguardi et al. (2018) has more difficulty in application. Thirdly, the first and last shape-modified functions of the present model (f 1 (ϕ) and f 3 (ϕ) in Eq. (15)) have different forms and positions with that of the model of Dioguardi et al. (2018) (Eq. (19)). Lastly, the model of Dioguardi et al. (2018) is based on the spherical drag law of Haider and Levenspiel (1989), which is suitable for R ep < 2.6 × 10 5 .
Developed from the idea of Dioguardi et al. (2018), the present model enhances the applicability in predicting the terminal settling velocity for particles of various shapes and materials, and thus can be considered as an improvement.

Conclusion
We propose a new drag law for non-spherical particles based on a total of 828 settling data collected from previous studies (Komar and Reimers 1978;Komar 1980;Baba and Komar 1981;Chambert and James 2009;Koch et al. 2010;Wang et al. 2011;Dioguardi and Mele 2015;Song et al. 2017;Zhu et al. 2017). This new database cover a R ep range of 0.008~10000 and a ϕ range of 0.421~1.0. Thus, the new model is suitable for the settlement of non-spherical particles in both laminar and turbulent flows. In addition, since this new model is developed from settling data from various sources, it can be applied in predicting settling velocity of regular particles (e.g., cubes, cylinders, disks, and cuboids), natural particles (e.g., volcanic particles, pebbles, and glass particles of various shapes), and even non-spherical hydrochorous seeds.  Following the idea of Dioguardi et al. (2018), the new model adopts three shaped-modified functions in the spherical drag law of Clift and Gauvin (1971). With these functions based on R ep and ϕ, the new model exhibits better performance than the other models. With the wide range of applicability in both flow regimes and particle shapes, the present model may be practical in solving problems concerning the riverine ecosystem. On the one hand, by relating the settling process of seeds with the dispersal process, more detailed information can be obtained about the germination of seeds and establishment of aquatic vegetation, which is crucial for flow characteristics in river systems (Meier et al. 2013). On the other hand, the advanced settling model can estimate the rate of sediment transport for various particles and flow regimes by being applied in predicting the distribution of sediment concentration (Fu et al. 2005;Graf and Cellino 2002). Such application may be verified when provided with further sediment concentration profile measurements of irregular particles. In addition to its application in the riverine ecosystems, the new settling model may be further improved by extending from the drag of single irregular particle to conglomerates of particles.
Author contribution All authors contributed to the model conception and derivation. F. Yang: data curation, methodology, formal analysis, writing-original draft; Y.H. Zeng: conceptualization, funding acquisition, supervision, writing-review and editing; W.X. Huai: conceptualization, writing-review and editing. All authors have read and approved the final manuscript. Availability of data and materials All data generated or analyzed during this study are included in the supplementary information files, which can be found in the online version of this article.

Declarations
Ethical approval and consent to participate Not applicable

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Competing interests The authors declare no competing interests.