Modelling aerodynamic forces and torques of spheroid particles in compressible flows

In this subsection, we investigate the variation of the drag force with the four key parameters stated above. It is necessary to recall first the empirical formula of the drag coefficient for incompressible flows. Happel and Brenner (1983) provided an analytical drag expression for spheroidal particles in creeping flows, which was subsequently summarized by Ouchene (Ouchene et al., 2016; Ouchene, 2020). For prolate spheroidal particles with $w>1$, the drag coefficient $C_{D}$ at the attack angle of $\alpha=0^{\circ}$ and $\alpha=90^{\circ}$ are given by

while for oblate particles with $w<1$, the drag coefficient $C_{D}$ are expressed as follows,

At higher $Re_{p}$, Reynolds number effects should be considered. Our study primarily considers the formula proposed by Fröhlich et al. (2020), incorporating a correction factor $f_{d,\alpha}$ for prolate particles at $\alpha=0^{\circ}$ and $90^{\circ}$,

where the correction factor $f_{d,\alpha}$ for the drag coefficient at $\alpha=0^{\circ}$ and $90^{\circ}$ of prolate particles are

and for oblate particles, $w$ should be replaced by $1/w$. The drag coefficient exhibits a sinusoidal function squared trend for attack angles between $0^{\circ}$ and $90^{\circ}$,

which is found to have a wide range of applicability up to $Re_{p}=2000$ for prolate spheroid (Sanjeevi and Padding, 2017), and maintain a specific degree of precision for oblate particles as validated by the present simulations. As $w$ equals to 1, the formula above degenerates to the drag coefficients of spherical particles given by Schiller and Naumann (1933). The parameters in Eq. (11) and (12) are detailed in Table 4, which are obtained by data fitting using the present numerical simulation results. The relative errors are given in Table 5. In comparison to the predictions provided by previous formulas, the empirical law presented here exhibits significantly lower relative errors than those proposed by Ouchene et al. (2016); Ouchene (2020) and Zastawny et al. (2012) for all cases considered. The formula by Sanjeevi et al. (2018) shows lower mean and maximum relative errors for prolate particles, but higher errors for oblate particles, which should probably attributed to the fact that the formula for former was originally proposed only for $w=2.5$.

We now start to incorporate compressibility effects into this formula. One approach is to introduce an additional correlation factor into the existing formulas. Nevertheless, research by Loth et al. (2021) on compressible spherical particles reveals a coupling between the particle Reynolds number $Re_{p}$ and Mach number $M_{p}$. For the current spheroid particles under consideration, likely, $w$ is also linked to $M_{p}$. To clarify this relationship, we perform Spearman correlation analysis using all simulation data. The Spearman correlation coefficients demonstrate the consistent changes in drag coefficients with these flow parameters, where $\pm 1$ signifies a perfect monotonic correlation, whereas a value of 0 signifies the absence of a correlation. The findings, presented in Table 6, suggest that both $Re_{p}$ and $M_{p}$ are significant factors to be included in the empirical formula for drag force coefficients, while $w$ is comparatively less influential.

We analyze the influence factor mathematically by considering its relationship with various variables,

The drag coefficients $C_{D,\alpha=0^{\circ}}$ and $C_{D,\alpha=90^{\circ}}$ for spheroid particles in compressible flows are, therefore, expressed as

with $C_{D,\alpha=0^{\circ}}(Re_{p},w)$ and $C_{D,\alpha=90^{\circ}}(Re_{p},w)$ representing the empirical formulas for incompressible flows, as given in Eq. (9) and (10). For other attack angles, $C_{D}$ follows a sinusoidal function squared trend as detailed in Eq. (13). To sum up, $C_{D}$ formula can be written as

As $M_{p}$ approaches 0, the values of $g_{d,\alpha=0^{\circ}}$ and $g_{d,\alpha=90^{\circ}}$ converge to 1.0, leading to a transition to incompressible flow scenarios. The formula parameters are provided in Table 7.

In Figure 5, we illustrate the variation of drag coefficients with Mach numbers at $Re_{p}=10$ and 100 for particles of different aspect ratios $w$. Across all shapes, drag coefficients exhibit a monotonic increase at $M_{p}<1$ but experience a sudden decline at the critical point $M_{p}=1$. Beyond $M_{p}=1$, drag coefficients at $\alpha=0^{\circ}$ for prolate particles display a non-monotonic trend with $M_{p}$, initially decreasing and then increasing, while other drag coefficients decrease monotonically at $Re_{p}=10$. A similar trend of variation is observed for cases at $Re_{p}$ = 100. The proposed empirical formulas accurately capture these trends, with an average relative error of approximately 3%, indicating their capability to predict drag coefficients for particles of varying aspect ratios $w$ in terms of $M_{p}$.

Figure 6 illustrates the variation of the drag coefficient with the particle Reynolds number $Re_{p}$ for prolate and oblate particles with aspect ratios $w=2.5$ and 0.4, respectively, grouped by three $M_{p}$ values. Across all particle aspect ratios $w$ and $M_{p}$ considered, the drag coefficients decrease monotonically with $Re_{p}$, which is well captured by the proposed formula. This behavior holds true for particles with different aspect ratios, affirming the validity of the proposed formula concerning $Re_{p}$.

Lastly, we examine the impact of the angle of attack on the drag coefficient at various Mach numbers, shown in Figure 7. The trend in drag coefficient variation is consistent for both compressible and incompressible flows, suggesting that the attack angle $\alpha$ plays a minor role in drag coefficient modification. Overall, the predicted $C_{D}$ using the proposed formula aligns well with simulation results, validating the use of the sinusoidal function squared to derive $C_{D}$ at any arbitrary attack angle $\alpha$ from 0^∘ to 90^∘.

In this subsection, we focus on adjusting drag coefficients for spheroid particles in compressible flows. Through an examination of drag coefficient variations with $Re_{p}$, $M_{p}$, $w$, and $\alpha$, we demonstrate that the proposed empirical formulas closely match simulation results, with an average relative error of less than 3%.

We further consider the lift force in this subsection. Theoretical derivations and empirical formulas for the lift force coefficients are usually associated with the modeling of their drag forces (Ouchene et al., 2016; Ouchene, 2020; Fröhlich et al., 2020), Happel and Brenner (1983) derived lift coefficients for spheroidal particles in stokes flow as follows

coupling the drag coefficients $C_{D,Stokes,\alpha=0^{\circ}}$ and $C_{D,Stokes,\alpha=90^{\circ}}$ with trigonometric function $\sin(\alpha)\cos(\alpha)$ to determine the lift coefficients in creeping flows. To incorporate high Reynolds number effects, the maximum value of the lift coefficient reached at $\alpha=45^{\circ}$ should be modified by multiplying the correction factor (Fröhlich et al., 2020) as

with the correction factor

In creeping flows, the maximum lift coefficients are observed at $\alpha=45^{\circ}$ and exhibit a symmetrical distribution as the angle approaches $0^{\circ}$ and $90^{\circ}$. However, such a symmetrical variation is violated at higher Reynolds numbers. To account for the unsymmetrical shift, we propose to use the following expression,

which is based on the results of our simulation and previous studies in the field. Note that these formulas are valid for prolate particles. Those for the oblate particles can be obtained by replacing the $w$ as $1/w$, same as the procedure employed for drag force coefficients. The correlation for lift coefficient proposed above has the same form as the theoretical solution in Eq. (19) when the particle Reynolds number $Re_{p}$ approaches zero. The parameters in the formulas are listed in Table 8 obtained by the best fitting using the results at $M_{p}=0.1$.

In Figure 8, we present a comparison of lift coefficients obtained from numerical simulations with empirical formulas proposed by Zastawny et al. (2012); Ouchene et al. (2016); Ouchene (2020) and Sanjeevi et al. (2018), along with the revised formula given herein. At low Reynolds numbers, the lift coefficient exhibits symmetry around $\alpha=45^{\circ}$. However, for prolate spheroidal particles, increasing aspect ratios $w$ and Reynolds numbers $Re_{p}$ disrupt this symmetry, particularly pronounced at $w=5$ and $Re_{p}=100$. The variations of $C_{L}$ against $Re_{p}$ at $\alpha=45^{\circ}$ are depicted in Figure 9. Additionally, lift coefficients decrease with the Reynolds number. The prolate particles, especially those with $w=5$, are more significantly affected. Both prolate and oblate particles experience enhanced lift coefficients as $w$ deviates from unity, and $C_{L,\alpha=45^{\circ}}/C_{L,\alpha=45^{\circ},Re_{p}=10}$ increases with $w$. The proposed empirical formula accurately predicts lift coefficients with an average error of less than 3%, outperforming previous studies that exhibit larger discrepancies across different flow scenarios.

We further investigate the impact of Mach numbers on the lift force coefficient. Following our analysis of the drag coefficient modifications, we initially conducted Spearman correlation analysis for lift force coefficient. The results, detailed in Table 9, indicate a strong dependence of lift force coefficients on $Re_{p}$ and $M_{p}$, but a much weaker correlation with $w$.

Similar to drag coefficients, lift coefficients exhibit significant variation at $M_{p}=1$. Considering the dependancy of lift coefficients on $Re_{p}$ and $M_{p}$, the compressible impact factor is structured as

Henceforth, the lift coefficients of the spheroid particles in compressible flows can be empirically expressed as

The parameters in the aforementioned formulas, as given in Table 9, are obtained by data fitting utilizing the numerical simulation results. The average relative error of the proposed formula is less than $4\%$.

Figures 10 and 11 illustrate the variation of the maximum lift coefficients $C_{L,\alpha=45^{\circ}}$ with Mach number $M_{p}$ and Reynolds number $Re_{p}$. When $Re_{p}$ is held constant, the maximum lift coefficients initially rise at $M_{p}<1$, peak around $M_{p}\approx 1$, and then decline beyond $M_{p}=1$, partially mirroring the behavior of drag coefficients discussed earlier. At a fixed Mach number $M_{p}$, the maximum lift coefficients $C_{L,\alpha=45^{\circ}}$ exhibit a monotonic decrease with $Re_{p}$ for prolate particles. This trend is prominent in subsonic scenarios but diminishes in supersonic flows. Conversely, for oblate particles, the maximum lift coefficient $C_{L,\alpha=45^{\circ}}$ decreases solely with $Re_{p}$ in subsonic conditions at $M_{p}=0.3$, while this decreasing trend disappears in transonic and supersonic cases. The proposed formula provides reasonably accurate predictions in terms of $Re_{p}$ and $M_{p}$, except for extreme cases with low $Re_{p}$ but high $M_{p}$, as depicted in Figure 11. In such instances, rarefaction effects cannot be overlooked, rendering both simulations and empirical formulas unsuitable. However, in point-particle simulations within turbulence, such scenarios are infrequent, as $Re_{p}$ and $M_{p}$ typically increase concurrently. This observation holds for drag forces as well. The exploration of rarefaction effects falls outside the scope of this paper and will be considered in our future investigation.

In Figure 12, we present the variation of the lift coefficients with the angle of attack $\alpha$ at Mach numbers $M_{p}=0.3$, 1.0, and 2.0, and Reynolds numbers $Re_{p}=10$ and 100. The trend of variation resembles that of incompressible flow. The lift coefficient curves exhibit a slight shift to the right for prolate particles and to the left for oblate particles, a pattern well-captured by the proposed empirical formula. Particularly, the relative error for the lift coefficients of oblate particles at $Re_{p}=10$ and $M_{p}=2.0$ is comparatively high. These discrepancies can be neglected, for as we have stated in the previous subsection, the high $M_{p}$ and low $Re_{p}$ correspond to the high particle Knudsen number in such cases, rendering the results meaningless.

This analysis demonstrates that the variation of lift coefficients with angle of attack at different Mach numbers is similar to that of incompressible flow. The lift force coefficients for prolate spheroidal particles exhibits a slight rightward shift, whereas that for oblate spheroidal particles displays a slight leftward shift. When considering Reynolds numbers at 10 and 100 for analysis and comparison, at higher Reynolds numbers, the lift coefficients are notably lower than the lift coefficients at lower $Re_{p}$. This discrepancy diminishes gradually with an increase in Mach number.

In this section, we initially refined the lift coefficient for incompressible flows by analyzing cases at low Mach numbers and incorporating asymmetrical distributions with respect to the attack angle. Subsequently, we considered the effects of compressibility for higher Mach number cases and elucidated the formulation of the compressibility factor as a function of both $Re_{p}$ and $M_{p}$. Through a comparison of numerical simulation databases, we have demonstrated that the current proposed formula for the lift coefficient provides accurate results, except in scenarios of high $M_{p}$ and low $Re_{p}$ where rarefaction effects become significant.

In this subsection, we consider the pitching torque coefficients. Unlike drag and lift forces, pitching torques in creeping flows lack exact solutions, with low Reynolds numbers suggesting zero pitching torque. Empirical formulas are scarce at higher Reynolds numbers. Zastawny et al. (2012) highlighted the similarity between pitching torque coefficients and lift coefficients, leading to the modeling of pitching torque coefficients based on a formula akin to lift coefficients. Ouchene et al. (2016) refined the formula to accommodate higher Reynolds numbers and broader aspect ratios, focusing solely on the prolate spheroid formulation in this study. Subsequently, Ouchene (2020) extended this analysis to oblate spheroidal particles. Building on prior research, the expression for pitching torque coefficients will be determined in conjunction with the lift coefficient expression in the current study, with that of prolate particles as

with the factor $f_{t,\alpha=45^{\circ}}$ cast as

The pitching torque coefficient can be obtained at varying attack angles by employing the following formula

The $w$ should be replaced with $1/w$ for oblate particles. The variation with the attack angle is expressed using a combination of trigonometric functions. The pitching torque is zero for spherical particles in a uniform flow at relatively low Reynolds numbers, which results in a zero value for $C_{T}$ as $Re_{p}$ approaches zero. The parameters included in the equations above are listed in Table 11, calculated using the results of cases at $M_{p}=0.1$.

Figure 13 demonstrates that pitching torque coefficients $C_{T}$ decrease monotonically with Reynolds number at an attack angle of $\alpha=45^{\circ}$, similar to the trend observed for lift coefficients. At a given $Re_{p}$, $C_{T}$ is higher as the aspect ratio deviates from 1.0. Comparative analysis with empirical formulas from previous studies (Ouchene et al., 2016; Zastawny et al., 2012; Sanjeevi et al., 2018) reveals that the currently proposed formula exhibits the lowest relative errors. In Figure 14, we further explore the variation of $C_{T}$ with the attack angle $\alpha$, comparing predictions from the proposed formula in this study with those of Ouchene et al. (2016); Ouchene (2020); Zastawny et al. (2012) and Sanjeevi et al. (2018). Overall, the variation of $C_{T}$ follows a similar pattern to that of lift coefficients, with asymmetrical distributions, particularly notable for oblate particles. The proposed formula provides the most precise predictions, with an average relative error of approximately 1.39%.

The compressibility impact factor for the $C_{T}$ is similar to the drag and lift coefficients in compressible flow, in that the Spearman correlation analysis also show that the $C_{T}$ is monotonically correlated with $Re_{p}$ and $M_{p}$ (not shown here for brevity). However, it is noteworthy that the $C_{T}$ peaks around $M_{p}\approx 0.8$, and thus the segmentation was determined to be at $M_{p}=0.8$.

With such refinement, the pitching torque coefficient for spheroidal particles in compressible flows can be written as

The parameters in the equation above is reported in Table 12. Compared with the numerical simulation databases, the presently proposed formula predicts $C_{T}$ with high accuracy, showing a relative error of approximately 4.6%.

The details of the comparison are given as follows. Figures 15 and 16 illustrate the variation of pitching torque coefficients $C_{T}$ with Mach number $M_{p}$ and Reynolds number $Re_{p}$, respectively. For a constant Reynolds number $Re_{p}=10$, the $C_{T}$ of prolate and oblate particles decreases monotonically until $M_{p}=1.2$, beyond which the variation becomes insignificant across all six groups with different aspect ratios. However, at $Re_{p}=100$, the $C_{T}$ increases, peaks at $M_{p}=0.8$, and then decreases at higher $M_{p}$. Conversely, for a fixed $M_{p}=0.3$, the $C_{T}$ decreases monotonically with Reynolds number $Re_{p}$. In transonic cases at $M_{p}=0.8$, the $C_{T}$ initially increases and then decreases, peaking at $Re_{p}=30$. In supersonic cases at $M_{p}=2.0$, the $C_{T}$ are minimally affected by Reynolds number, indicating the significant influence of detached shock waves in these scenarios. Figure 17 showcases the variation of the pitching torque coefficient with angle of attack at different Mach numbers, demonstrating a consistent trend across higher Mach numbers, with a peak at $\alpha=45^{\circ}$ and slight asymmetrical distributions. The empirical formula proposed herein effectively captures the trend of variation and provides relatively accurate results for all cases.