Correlations for aerodynamic coefficients for prolate spheroids in the free molecular regime
Introduction
Modeling of multi-phase flows consisting of gas and particles involves the prediction of particle motion and its effect on the gas phase. The assumption of particles as spheres is commonly used in the multi-phase flow modeling to avoid the numerical complexity associated with non-spherical particles; however, at some cost on the accuracy of the computation. The major numerical difficulty involves the calculation of aerodynamic parameters at various speed ratios and angles of attack for non-spherical geometries. There are no established correlations available in the literature, which can give such variations of aerodynamic properties, such as coefficients of lift, drag, etc. This is particularly true for the free molecular flow regime. The key to solve multi-phase flows, involving gas and grain phase with non-spherical particles, is to find such correlations for the variation of aerodynamic coefficients with flow and particle characteristics, which can be used to model two-phase flow in an effective manner.
Several studies have been performed to find aerodynamic correlations of non-spherical particles such as cylinder [1], ellipsoid [2], [3] and disc [4] shaped particles in the continuum regime. The experimental study was performed by Tran-Cong et al. [5] for six different shaped non-spherical particles and proposed an empirical correlation for the drag coefficient in the continuum regime. Waddel [6] proposed the “sphericity” factor to define non-spherical particles and the results showed that the drag coefficient decreased with sphericity. A linear relationship between drag coefficient and sphericity was established at high turbulent regime, for a range of sphericity between 0.6 to 1.0. The “cross-wise sphericity” was introduced by Hölzer et al. [7] and established a drag correlation for a wide range of Reynolds number. The mean deviation between the proposed correlation and the experimental data from the literature was found to be 14%. Yow et al. [8] found a new set of drag correlations for a wide range of Reynolds number from used data available in the literature. This study was performed for different sphericities, however, the orientation of the particle was not taken into account. Ganser [9] introduced a new relation for drag predictions based on Stokes and Newton’s shape factors. Rosendahl [10] introduced a shape parameter for non-spherical particles and used it to find correlations for non-spherical particles in a swirling flow. Zastawny et al. [11] performed DNS simulations to obtain aerodynamic correlations for four different shaped non-spherical particles at a wide range of Reynolds numbers, and the drag values were compared with those obtained by Happel and Brenner [12] at low Reynolds numbers. Loth [13] studied numerical and experimental data from literature, and for different shapes proposed the best shape parameters, such as, aspect ratio for ellipsoids, surface area ratio for regular particles, and minimum, maximum and averaged area ratio for irregular particles. A detailed review of the aerodynamic correlations for non-spherical particles is given in Ref. [14]. There are several other studies that were performed with different numerical techniques to find aerodynamic correlations for non-spherical particles [15], [16], [17], [18].
All of the aforementioned correlation studies on non-spherical particles were performed in the continuum regime [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], and particularly for low speed flows, to address several practical applications, such as, pulp/paper production, coal combustion, red blood cells in plasma, etc. However, the above-mentioned correlations cannot be applied to particles moving at high speed. At the same time, the results of these studies cannot be extended to the free molecular regime due to failure of continuum models [19]. However, the aerodynamic correlations for non-spherical particles in the non-continuum regime are essential for multi-phase flow modeling in order to address situations, such as, dust dispersion during lunar/planetary landings [20], solid-propellant rocket nozzle design [21], aircraft flight through sand storm [22], etc. One of the critical aerodynamic problems in lunar landings is that the lunar particles get displaced due to the plume impingement from lander nozzle, which could cause several problems such as structural damage to the lander, obstruction to the pilot vision, etc[20]. Since the lunar environment has a very low density, the particles are locally in the free molecular regime. It has also been reported that the particles are highly ellipsoidal and prismatic in shape [23].
There are a few studies reported in particular to the non-spherical particle movement in the free molecular regime [24], [25], [26], [27]. Dahneke [24] developed free molecule drag equations for spheres, cylinders, discs and spheroids, with the help of slip correction factors. Cheng [25] developed theoretical drag equations for spheroids, rectangular prisms, aggregates, and cylinders in the free molecular regime in terms of slip correction factor, dynamic shape factor that accounts for non-spherical effect, and non-Stokesian drag for increased drag at high Reynolds numbers. Li et al. [26] proposed an approach to obtain the mobility of non-spherical particles by averaging their drag forces at different orientations. Moreover, they have extended two other existing approaches, the averaged-collision-integral and averaged-drift-velocity methods, to obtain mobility of non-spherical particles. Ivanovski et al. [27] analyzed the shape effects of particles by considering the dynamics of spherical and ellipsoidal particles in a spherically symmetric expanding gas flow. They observed that the difference in particle shape and its orientation lead to velocity dispersion in space and change in terminal velocity. In the above mentioned theoretical works in the free molecular regime [24], [25], the expressions for drag for different non-spherical particles were analyzed, while the lift and torque expressions were not reported.
In this work, we have attempted to find aerodynamic correlations for ellipsoidal (prolate spheroids) particles in the free molecular regime using the Direct Simulation Monte Carlo (DSMC) method [28]. To the best of our knowledge, this is the first work to find aerodynamic correlations including drag, lift and torque for an ellipsoidal particle in the free molecular regime, even though many studies have been performed to find drag coefficient for spheres [29], [30], [31] and ellipsoids [24], [25], [26], [27] in this regime. To this end, we have developed an in-house collisionless DSMC solver to study free molecular flows. The solver is used to find aerodynamic coefficients of ellipsoidal particles at different aspect ratios, angles of attack and speed ratios. The aspect ratios of ellipsoidal particles are chosen in such a way that the volume of an ellipsoidal particle is the same as that of the volume of a sphere having a diameter of .
The remainder of this paper is arranged as follows: The computational model used in this work is presented in Section 2, while the validation study for the same is presented in Section 3. The results and discussions are given in Section 4, followed by conclusions in Section 5.
Section snippets
Computational model: the collisionless DSMC method
The Direct Simulation Monte-Carlo (DSMC) method was introduced by Bird [28] to study non-equilibrium dilute gas flows. An in-house parallel multi-species Non-equilibrium Flow Solver (NFS) used in this work was developed based on Bird’s methodology and validated for several numerical and experimental results [32], [33]. The NFS DSMC solver is modified in this work to study free molecular flows in order to be consistent with the free molecular flow theory and also to reduce the computational
Validation study
The modified DSMC collisionless solver, as discussed above, has been validated against the free molecular theory for flow over a sphere, flat plate and spherical particle relaxation at low speed ratios in one of our earlier works [34]. The properties such as drag and lift are compared with the free molecular theory, and are found to be in good agreement with the theory [34]. Additionally, we have presented two validation case studies, viz., flow over a sphere over a range of speed ratios, and
Results and discussion
Flow of argon gas over an ellipsoidal particle is considered as the main case study in order to develop correlations for the variation of aerodynamic coefficients at various conditions. To this end, a prolate ellipsoidal geometry is considered for the non-spherical particle, as shown in Fig. 4. The aspect ratio of an ellipsoidal particle is defined as where and are semi-major and semi-minor axes, respectively. The radius in the third dimension, is equal to the semi-minor axis as
Conclusions
A new set of correlations are obtained for ellipsoidal particles in the free molecular regime. The in-house DSMC solver is modified to be consistent with the free molecular theory, as well as to reduce the associated computational time. The solver is verified against theory and an open source DSMC solver, SPARTA. The correlations are obtained for aerodynamic coefficients, such as drag, lift and torque. The proposed correlations are validated against the in-house DSMC simulation results and
CRediT authorship contribution statement
Arun K. Chinnappan: Methodology, Formal analysis, Software, Writing - original draft. Rakesh Kumar: Conceptualization, Methodology, Writing - original draft, Supervision. Vaibhav K. Arghode: Methodology, Writing - original draft. Kishore K. Kammara: Methodology, Writing - original draft. Deborah A. Levin: Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
We acknowledge the financial support provided by the Science and Engineering research Board (SERB) through the Grant Number MTR/2019/000041, and by the Indian Space Research organization through the Grant Number STC/AE/2018033. We also acknowledge the support provided by the Ministry of Human Resource Development (MHRD) through the Grant Number SPARC/2018-2019/P1103/SL, using which the collaborative work between RK and DAL was carried out. The authors also acknowledge the use of computing
References (37)
- et al.
Drag coefficient and settling velocity for particles of cylindrical shape
Powder Technol
(2008) - et al.
New correlations for heat and fluid flow past ellipsoidal and cubic particles at different angles of attack
Powder Technol
(2013) - et al.
Drag coefficient and terminal velocity of spherical and nonspherical particles
Powder Technol
(1989) - et al.
Drag coefficients of irregularly shaped particles
Powder Technol
(2004) The coefficient of resistance as a function of reynolds number for solids of various shapes
J Franklin Inst
(1934)- et al.
New simple correlation formula for the drag coefficient of non-spherical particles
Powder Technol
(2008) - et al.
Drag correlations for particles of regular shape
Adv Powder Technol
(2005) A rational approach to drag prediction of spherical and nonspherical particles
Powder Technol
(1993)Using a multi-parameter particle shape description to predict the motion of non-spherical particle shapes in swirling flow
Appl Math Model
(2000)- et al.
Derivation of drag and lift force and torque coefficients for non-spherical particles in flows
Int J Multiphase Flow
(2012)