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ABSTRACT：In this study，Particle-Resolved Simulations（PRS）are performed for suspensions of different ellipsoids that are randomly oriented and the drag forces are calculated for each individual particle in the flow. The investigated conditions include Reynolds number from 10 to 200，solid volume fraction from low（0.1）to intermediate（around 0.3）with ellipsoid aspect ratios（AR）of 2.5，5 and 10.
Keywords：Particle suspensions;Particle-Resolved Simulation;prolate ellipsoids;drag force;statistical analysis.
1.INTRODUCTION
Fluid drag forces acting on particles in suspension are of critical importance for capturing the dynamics of fluid-particle systems accurately. In this paper，particle-resolved simulations are performed for suspensions of ellipsoids with aspect ratio（defined as the ratio between length of the particle along rotational symmetry axis and the diameter of the equatorial plane）of 2.5，5 and 10. Reynolds number range of [10，200] with solid fraction of [0.1，0.35]，[0.1，0.3] and [0.1，0.2] for these three ellipsoids，respectively are investigated with derivation of drag force for each individual particle in the suspensions. To get a better understanding of the drag force distribution. Statistical analysis are carried out including the PDF，standard deviation to provide more insight in such problem aside from the ensemble mean drag force.
2.METHODS
Detailed explanation about the computational algorithm and validation can be found in [1] The particle suspensions are generated using a physical simulation engine PhysX developed by Nvidia [2] which has been implemented in the previous works [3，4]. Detailed procedure can be found in the work of He et al. [3]，which contains the process of particle collision detection，resolving the post-collision translational and rotational velocity and kinetic energy decaying to finally reach a stationary state. Examples of the particle suspension for AR5 ellipsoid under solid fraction of 0.1 are shown in Figure 1. The total amount of particle required for a specific solid fraction is calculated by：
Where represents the amount of particles needed under. is the volume of the domain that contains the particles with a dimension of .
3.RESULTS
3.1Drag Force Distribution in the Whole Suspension
As is demonstrated in the study of Rong et al. [5]，drag force of spheres in suspension follow an approximate Gaussian distribution. In this work，the geometry is extended to ellipsoids of aspect ratio of 2.5，5 and 10. The drag force at Reynolds number of 10 with solid fraction of 0.1 is shown in Figure 2. Figures on the right share the same y-axis title as the figures on the left. The left three plots present the drag force distribution with respect to the particle inclination angle. Considering ellipsoids with aspect ratio of 2.5，the analysis of He and Tafti [4] indicates that increase of pressure force and decrease of viscous force results in an almost constant total drag with respect to. Besides，it can be observed that the data has insignificant variation of spread under different，which is due to the almost constant dispersity of viscous drag under varying and its dominant effect on the total drag force. However，for data of ellipsoids with aspect ratio of 5 and 10，a significant increasing trend in drag force can be observed with respect to. Unlike spherical particle suspensions that show Gaussian distribution of the drag force [5]，the current results for ellipsoids exhibit varying degrees of skewness at values of 0.50，0.92 and 0.86 for ellipsoids with aspect ratio of 2.5，5 and 10，respectively. As discussed previously，the drag of AR2.5 ellipsoid doesn’t show significant variation with respect to ，however the PDF indicates that the drag shows preference for less than the average. As the aspect ratio increases to AR5 and AR10，the drag data exhibits larger skewness. This is because for small ，the spread of drag is small. As inclination angle increases，dispersity of drag increases to the extent that a particle at inclination angle of 90°can exhibit the same drag force as a particle at inclination angle of 0°. The propensity for low drag values that appears at almost all ，results in an overall low drag preference in the suspensions.
Results at Reynolds number of 200 and the highest solid fraction for each aspect ratio are also analyzed and the results are shown in Figure 3. The results for AR2.5 ellipsoids at Re=200，=0.35 show an increasing trend in magnitude and dispersity with respect to. The PDF plot shows a similar trend as the results for Re=10，=0.1 with a skewness of 0.45. Therefore，this indicates that the change in Re and does not have a significant influence on the drag force PDF distribution in the suspensions of AR2.5 ellipsoids. For the suspension of AR5 ellipsoids at Re=200，=0.3，the PDF also shows a trend similar to the results at Re=10，=0.1，but with a lower skewness value of 0.69. The lower skewness indicates that the drag tends towards a more uniform distribution. Actually，after investigating the results at different Re and，a decreasing trend in skewness with respect to Re can always be observed at different. This is attributed to the observation that as Re increases，a particle in the suspension will have a stronger influence on the surrounding fluid. As a result，the mixing effect of the flow becomes more prominent and the drag force tends to a more uniform distribution. For AR10 ellipsoids，the drag has a skewness of 1.04，which is slightly larger than the case with Re=10，=0.1.
4.CONCLUSIONS
This paper investigates the statistical properties of drag force variation in suspensions of AR2.5，AR5 and AR10 ellipsoid. The large range of calculated drag force indicates that even using the average drag confined in a small range based on aspect ratio，Reynolds number，solid fraction，and inclination angle is a rough approximation of the actual drag force experienced by different particles. Thus the statistical nature of drag force distribution at different inclination angles is investigated. Based on the results，we can draw the conclusion as：unlike the spherical suspensions that show an approximate Gaussian distribution，drag force on ellipsoids shows a biased distribution with positive skewness. This indicates that more particles tend to have drag below the average value. Larger aspect ratio ellipsoids tend to have higher skewness. Both Re and do not have much effect on the skewness of AR2.5 ellipsoid suspensions. However，skewness decreases with increasing Re for AR5 and AR10 ellipsoids. This can be related to the stronger mixing effect of the fluid at higher Reynolds numbers.
REFERENCES
[1]Cao，Z.，Tafti，D. K.，and Shahnam，M.，2020，“Development of Drag Correlation for Suspensions of Ellipsoidal Particles，” Powder Technol.，369，pp. 298–310.
[2]Nvidia，“NVIDIA PhysX SDK 3.3.4 Documentation.”
[3]He，L.，Tafti，D. K.，and Nagendra，K.，2017，“Evaluation of Drag Correlations Using Particle Resolved Simulations of Spheres and Ellipsoids in Assembly，” Powder Technol.，313，pp. 332–343.
[4]He，L.，and Tafti，D.，2018，“Variation of Drag，Lift and Torque in a Suspension of Ellipsoidal Particles，” Powder Technol.，335，pp. 409–426.
[5]Rong，L. W.，Dong，K. J.，and Yu，A. B.，2013，“Lattice-Boltzmann Simulation of Fluid Flow through Packed Beds of Uniform Spheres：Effect of Porosity，” Chem. Eng. Sci.，99，pp. 44–58.
作者簡介：
曹泽（Ze Cao），1994年10月1日，男，汉，河南平舆，博士研究生，研究方向：气固两相流体仿真。
（作者单位：1山东大学;2山东大学能源与动力工程学院）
Keywords：Particle suspensions;Particle-Resolved Simulation;prolate ellipsoids;drag force;statistical analysis.
1.INTRODUCTION
Fluid drag forces acting on particles in suspension are of critical importance for capturing the dynamics of fluid-particle systems accurately. In this paper，particle-resolved simulations are performed for suspensions of ellipsoids with aspect ratio（defined as the ratio between length of the particle along rotational symmetry axis and the diameter of the equatorial plane）of 2.5，5 and 10. Reynolds number range of [10，200] with solid fraction of [0.1，0.35]，[0.1，0.3] and [0.1，0.2] for these three ellipsoids，respectively are investigated with derivation of drag force for each individual particle in the suspensions. To get a better understanding of the drag force distribution. Statistical analysis are carried out including the PDF，standard deviation to provide more insight in such problem aside from the ensemble mean drag force.
2.METHODS
Detailed explanation about the computational algorithm and validation can be found in [1] The particle suspensions are generated using a physical simulation engine PhysX developed by Nvidia [2] which has been implemented in the previous works [3，4]. Detailed procedure can be found in the work of He et al. [3]，which contains the process of particle collision detection，resolving the post-collision translational and rotational velocity and kinetic energy decaying to finally reach a stationary state. Examples of the particle suspension for AR5 ellipsoid under solid fraction of 0.1 are shown in Figure 1. The total amount of particle required for a specific solid fraction is calculated by：
Where represents the amount of particles needed under. is the volume of the domain that contains the particles with a dimension of .
3.RESULTS
3.1Drag Force Distribution in the Whole Suspension
As is demonstrated in the study of Rong et al. [5]，drag force of spheres in suspension follow an approximate Gaussian distribution. In this work，the geometry is extended to ellipsoids of aspect ratio of 2.5，5 and 10. The drag force at Reynolds number of 10 with solid fraction of 0.1 is shown in Figure 2. Figures on the right share the same y-axis title as the figures on the left. The left three plots present the drag force distribution with respect to the particle inclination angle. Considering ellipsoids with aspect ratio of 2.5，the analysis of He and Tafti [4] indicates that increase of pressure force and decrease of viscous force results in an almost constant total drag with respect to. Besides，it can be observed that the data has insignificant variation of spread under different，which is due to the almost constant dispersity of viscous drag under varying and its dominant effect on the total drag force. However，for data of ellipsoids with aspect ratio of 5 and 10，a significant increasing trend in drag force can be observed with respect to. Unlike spherical particle suspensions that show Gaussian distribution of the drag force [5]，the current results for ellipsoids exhibit varying degrees of skewness at values of 0.50，0.92 and 0.86 for ellipsoids with aspect ratio of 2.5，5 and 10，respectively. As discussed previously，the drag of AR2.5 ellipsoid doesn’t show significant variation with respect to ，however the PDF indicates that the drag shows preference for less than the average. As the aspect ratio increases to AR5 and AR10，the drag data exhibits larger skewness. This is because for small ，the spread of drag is small. As inclination angle increases，dispersity of drag increases to the extent that a particle at inclination angle of 90°can exhibit the same drag force as a particle at inclination angle of 0°. The propensity for low drag values that appears at almost all ，results in an overall low drag preference in the suspensions.
Results at Reynolds number of 200 and the highest solid fraction for each aspect ratio are also analyzed and the results are shown in Figure 3. The results for AR2.5 ellipsoids at Re=200，=0.35 show an increasing trend in magnitude and dispersity with respect to. The PDF plot shows a similar trend as the results for Re=10，=0.1 with a skewness of 0.45. Therefore，this indicates that the change in Re and does not have a significant influence on the drag force PDF distribution in the suspensions of AR2.5 ellipsoids. For the suspension of AR5 ellipsoids at Re=200，=0.3，the PDF also shows a trend similar to the results at Re=10，=0.1，but with a lower skewness value of 0.69. The lower skewness indicates that the drag tends towards a more uniform distribution. Actually，after investigating the results at different Re and，a decreasing trend in skewness with respect to Re can always be observed at different. This is attributed to the observation that as Re increases，a particle in the suspension will have a stronger influence on the surrounding fluid. As a result，the mixing effect of the flow becomes more prominent and the drag force tends to a more uniform distribution. For AR10 ellipsoids，the drag has a skewness of 1.04，which is slightly larger than the case with Re=10，=0.1.
4.CONCLUSIONS
This paper investigates the statistical properties of drag force variation in suspensions of AR2.5，AR5 and AR10 ellipsoid. The large range of calculated drag force indicates that even using the average drag confined in a small range based on aspect ratio，Reynolds number，solid fraction，and inclination angle is a rough approximation of the actual drag force experienced by different particles. Thus the statistical nature of drag force distribution at different inclination angles is investigated. Based on the results，we can draw the conclusion as：unlike the spherical suspensions that show an approximate Gaussian distribution，drag force on ellipsoids shows a biased distribution with positive skewness. This indicates that more particles tend to have drag below the average value. Larger aspect ratio ellipsoids tend to have higher skewness. Both Re and do not have much effect on the skewness of AR2.5 ellipsoid suspensions. However，skewness decreases with increasing Re for AR5 and AR10 ellipsoids. This can be related to the stronger mixing effect of the fluid at higher Reynolds numbers.
REFERENCES
[1]Cao，Z.，Tafti，D. K.，and Shahnam，M.，2020，“Development of Drag Correlation for Suspensions of Ellipsoidal Particles，” Powder Technol.，369，pp. 298–310.
[2]Nvidia，“NVIDIA PhysX SDK 3.3.4 Documentation.”
[3]He，L.，Tafti，D. K.，and Nagendra，K.，2017，“Evaluation of Drag Correlations Using Particle Resolved Simulations of Spheres and Ellipsoids in Assembly，” Powder Technol.，313，pp. 332–343.
[4]He，L.，and Tafti，D.，2018，“Variation of Drag，Lift and Torque in a Suspension of Ellipsoidal Particles，” Powder Technol.，335，pp. 409–426.
[5]Rong，L. W.，Dong，K. J.，and Yu，A. B.，2013，“Lattice-Boltzmann Simulation of Fluid Flow through Packed Beds of Uniform Spheres：Effect of Porosity，” Chem. Eng. Sci.，99，pp. 44–58.
作者簡介：
曹泽（Ze Cao），1994年10月1日，男，汉，河南平舆，博士研究生，研究方向：气固两相流体仿真。
（作者单位：1山东大学;2山东大学能源与动力工程学院）