Ellipsoids evaluated by Zastawny et al. [20].
Abstract
In this study, the responsiveness of nonspherical particles, specifically ellipsoids and cylinders, in homogeneous and isotropic turbulence is investigated through kinematic simulations of the fluid velocity field. Particle tracking in such flow field includes not only the translational and rotational components but also the orientation through the Euler angles and parameters. Correlations for the flow coefficients, forces and torques, of the nonspherical particles in the range of intermediate Reynolds number are obtained from the literature. The Lagrangian time autocorrelation function, the translational and rotational particle response, and preferential orientation of the nonspherical particles in the turbulent flow are studied as function of their shape and inertia. As a result, particle autocorrelation functions, translational and rotational, decrease with aspect ratio, and particle linear root mean square velocity increases with aspect ratio, while rotational root mean square velocity first increases, reaches a maximum around aspect ratio 2, and then decreases again. Finally, cylinders do not present any preferential orientation in homogeneous isotropic turbulence, but ellipsoids do, resulting in preferred orientations that maximize the cross section exposed to the flow.
Keywords
- kinematic simulations
- Lagrangian tracking
- nonspherical particles
- response behavior
- preferential orientation
1. Introduction
Nowadays, the use of numerical simulation techniques to assist the development and optimization of industrial processes dealing with turbulent multiphase flow has been included as one more step in their layout. Examples of them include pneumatic conveying, fluidized bed reactors, cyclones, classifiers, or flow mixers. Industrial sectors where such processes are important are the chemical, food, or paper industries as well as electric energy production. Due to the complexity of the involved flow, a great majority of simulations are carried out under Reynolds-averaged Navier-Stokes (RANS) in connection with an appropriate turbulence model to describe the turbulent dynamics of the carrier phase.
Two main frames are employed for the description of complex multiphase flows: the two-fluid model or Euler-Euler and the discrete particle models or Euler-Lagrange. In both of them, particles are approximated as point masses being transported in the carrier phase flow field; the solution of the flow around individual elements is usually too expensive and cannot be afforded. In the two-fluid model, both phases are conceived as two interpenetrating continua [1] whose properties are described by sets of partial differential equations. In the Euler-Lagrange approach, the discrete elements are considered as individual objects whose dynamics is governed by a Lagrangian motion equation. Therefore, to obtain the discrete phase variables in the computational domain, a large enough number of discrete element trajectories must be computed. On each particle, appropriate forces act reflecting the various microprocesses taking place at the element scale such as fluid-particle turbulent interaction, particle-(rough) wall interactions, and interparticle collisions [2]. Such technique is especially appropriate for the description of disperse multiphase flow, where usually particles have a size distribution, in confined domains where particle-wall collisions play a predominant role as pneumatic conveying, separation, and classification processes. Both techniques, two-fluid model and Euler-Lagrange, have been applied mainly considering spherical particles. This means that the forces due to the flow (drag and lift) as well as the microprocesses modeling, wall-particle and inter-particle interactions, are assumed to be for spherical-shaped elements [3]. In practical situations, however, nonspherical particles are encountered, either of irregular shape, either with well-defined shapes (fibers or granulates). For example, the paper industry uses large amounts of turbulent liquid to handle and transport the fibers that compose the paper pulp. Besides, such particles in the flow experience particle Reynolds numbers larger than one,
With the objective of performing the numerical simulation of turbulent flows laden with nonspherical particles, additional information about the forces and torques due to flow (drag and lift forces and pitching and rotational torques due to the shear flow and particle rotation) is needed. It is known that for regular nonspherical particles, that is, ellipsoids or fibers, such forces depend on particle orientation with respect to the flow. For instance, fiber orientation plays a major role in chemical processes as injection, compression molding, or extrusion in which the mechanical properties of the suspensions are determined by the orientation distribution.
For the Stokes regime, particle Reynolds number much lower than 1, the behavior of the nonspherical particles can be determined by analytical methods. Forces and torques acting on an ellipsoidal particle were analytically computed by Jeffery [4]. In a series of papers, Brenner determined the forces due to the flow acting on arbitrary-shaped nonspherical particles in the Stokes regime under different flow configuration by means of theoretical methods [5]. In the creeping flow regime, also with particle Reynolds number much lower than 1, Bläser [6] computed the forces acting of the surface on an ellipsoid in free motion for different flow situations, which allow him to suggest a simple criterion for particle breakup.
The drag coefficient for particle Reynolds number higher than 1 must be obtained by experiments, physical or numerical, as the analytical methods are not applicable any more.
The experimental studies to determine the drag coefficients for nonspherical particles employ wind tunnels or sedimentation vessels. For a moderately wide particle Reynolds number range, there exist results for thin discs [7], isometric irregular particles [8], cylinders and plates [9], discs [10], and discs and cylinders [11]. Drag coefficients were developed in all cases only for certain particle orientations. Compiling such results, different correlations have been developed in terms of particle shape [12, 13, 14]. As representative parameter of the particle shape, two options appear to be dominant: the spherical particle equivalent diameter
The first numerical computations of very small nonspherical particles in pseudoturbulent flow were performed by Fan and Ahmadi [22] and Olson [23]. The hydrodynamic forces and torques were computed by the theoretical coefficients of the Stokes regime. Olson [23] estimated the time step for the translation and rotation motions in function of the fiber length, obtaining the corresponding dispersion coefficients. Fan and Ahmadi [22] showed that the dispersion of both, translation and rotation, was reduced with the fiber length. However, Olson [23] found a different result in the case of ellipsoidal particles. Lin et al. [24] investigated numerically the distribution of the orientation of the fibers in a developing mixing layer, comparing the obtained results with experiments. The fiber length was smaller than the Kolmogorov scale, so they employed the forces due to the flow of the Stokes regime. Zhang et al. [25], Mortensen et al. [26], and Marchioli et al. [27] studied the transport and deposition of ellipsoidal particles in a turbulent channel flow using direct numerical simulation (DNS). Again the hydrodynamic forces and torques were computed with Stokes regime expressions. Beyond the Stokes regime, van Wachem et al. [28] and Ouchene et al. [29] studied a turbulent channel flow laden with ellipsoidal particles using LES and DNS, respectively, employing the flow coefficients developed by themselves in previous works.
Rosendahl group developed a model for the numerical computation of cylindrical and superellipsoidal particles in laminar and turbulent flows in the intermediate Reynolds numbers regime [30, 31, 32]. Particle angular velocity and orientation were computed by means of the Euler parameters. Using a linear relationship between the drag coefficient and the ellipsoid parameters, it was possible to establish a correlation valid up to Reynolds numbers of 1000. To estimate the influence of the orientation, a correlation between the maximum (90°) and minimum (0°) drag was employed. Drag force was calculated using the projected area perpendicularly to the flow. The lift force was expressed in function of the drag coefficient and particle orientation. Other lift forces, such as those due to the fluid velocity gradients or particle rotation, were not considered. The study case was a combustion chamber with straw particles, which were quite well approximated by cylinders. It was found that straw particles were better dispersed than spheres [30], a fact that properly illustrates the importance of the correct modeling of nonspherical particles motion. In a later work [32], other forces such as added mass and pressure force were also included. Drag coefficient was computed using the Ganser [15] correlation, making it possible to numerically compute the biomass combustion chamber.
This contribution aims to study the motion of nonspherical particles immersed in homogeneous isotropic turbulent (HIT) velocity fields built from kinematic simulation at moderate Reynolds numbers. Computations were performed in a tailored in-house code. Properties analyzed include the Lagrangian time autocorrelation function, the translational and rotational particle response, and preferential orientation of the nonspherical particles in the turbulent flow, all of them in terms of particle aspect ratio and inertia.
2. Governing equations
2.1 Coordinate systems
To build the trajectory of a regular nonspherical particle, it is necessary to solve for its translational as well as rotational motion. However, whereas translation is solved in an inertial frame, rotation is solved referred to the so-called particle frame. Thus, the relevant coordinate frames and the transformations between them have to be introduced.
Figure 1 illustrates, in the case of a cylindrical particle, the employed coordinate systems:
Goldstein [33] gives the transformation between the comoving and particle coordinate systems, which is frequently employed in regular nonspherical particle tracking [31].
The time evolution of such Euler angles depends on the particle angular velocity regarding the particle frame axes. However, there is a difficulty in the sense that such time evolution equations present an unavoidable singularity. Therefore, instead of the Euler angles, the Euler parameters
And the transformation matrix
In the present study, the initial particle orientations are assigned by means of the Euler angles. From them, the corresponding Euler parameters are computed by Eq. (3), and with them, the initial transformation matrix is evaluated using Eq. (4). The Euler parameters evolve in time following Eq. (5), where the particle angular velocities are expressed in the particle frame of reference
2.2 Particle motion equations
The nonspherical particle motion equations in a general fluid flow [34] are written as:
Translational motion:
Rotational motion:
Here,
In addition to a sphere, the four ellipsoids of Zawstawny et al. [20] have been chosen. They have different sphericities and aspect ratio (see Table 1). In Table 1,
Shape | Aspect ratio | Sphericity |
---|---|---|
Ellipsoid 1 (prolate) | 0.88 | |
Ellipsoid 2 (prolate) | 0.99 | |
Disc (oblate) | 0.62 | |
Fiber (prolate) | 0.73 |
Using DNS for ellipsoidal particles immersed in a uniform flow, Zastawny et al. [20] determined correlations for the flow coefficients (drag
Here,
The developed correlations depend not only on particle Reynolds number
Moreover, for the cylinders, Vakil and Green [19] developed correlations for drag and lift coefficients depending on orientation, Reynolds number based on its diameter
The correlations are expressed as:
Coefficients
can be found in Vakil and Green [19].
However, expressions for the pitching and rotational torque coefficients are not provided in [19]. Therefore, for the cylinders, the approach of [31] has been assumed. In [31], the distance between the center of mass and the center or pressure in a cylinder,
Then, the pitching torque
Particle motion equations and correlations for cylinders and ellipsoids presented in this section have been implemented in an in-house code. The numerical integration of the ordinary differential equations that govern the motion of nonspherical particles has been performed by a fourth-order Runge-Kutta method, with small enough time steps to avoid numerical instabilities [35, 36]. The fluid velocity field in which particles are immersed has been built by the kinematic simulation technique described in the next section. It is known that Runge-Kutta methods do not satisfy the time-reversal property, a fact that makes such methods inappropriate for integrating energy-conserving systems, for instance. However, particle equations are dissipative systems (as they include viscous drag forces) and, for them, Runge-Kutta algorithms can be used [37] provided that the time step is small enough to keep the errors bounded.
3. Kinematic simulation
There exist different options to calculate the Lagrangian properties in a turbulent flow. The starting point is the trajectory equation in which the position
Here,
The three-dimensional Eulerian velocity field to be employed in Eq. (18) is built as a series of random Fourier modes. The velocity field is solenoidal at each realization by construction. Moreover, the energy spectrum of the Fourier modes is prescribed, for example, by a power law, so the effects of small flow scales on Lagrangian statistics are directly included. Such KS velocity field is written as [39]:
Here,
Because of the construction of the velocity field given by Eq. (19),
To validate the spherical particles tracking in the KS velocity field, the values of particle Reynolds stresses (RS) in HIT have been selected. In this configuration, Hyland et al. [41] demonstrated that, as the fluid turbulence is homogeneous, particle RS only depend on time and they can be written as
4. Numerical simulation
Computations were performed in a tailored in-house code. The turbulent velocity field generated with KS resembles one of the fields worked in [39]. Such velocity field is characterized by a fluctuating velocity
The regular nonspherical particles studied have been the ellipsoids in [20] and the cylinders in [19]. In all cases, particles have the same particle volume equivalent diameter
If the Kolmogorov time scale
Simulations proceed in the following way: for each KS realization of HIT fluid velocity field, a particle is located in the center of the domain with zero initial velocity; particle translational and rotational motion is computed from Eqs. (6) and (7), its orientation is calculated from Eq. (1), and its trajectory is built; particle tracking lasts for around 10 fluid integral time scales; and particle properties are stored every second for evaluation. Such process is carried out a sufficient number of times to reach significant statistical results. In this study, statistics has been performed based on 105 KS realizations.
In the following section, the results of the particle Lagrangian time autocorrelation function, the translational and rotational particle response, and preferential orientation of the nonspherical particles in the turbulent flow are analyzed as function of their shape and effective Stokes number.
5. Results and discussion
The Lagrangian autocorrelation function
As it can be readily seen from Figure 3, higher inertia particles are characterized by larger integral Lagrangian time scales (ILTSs) (defined as the integral up to infinity of
Moreover, for intermediate inertia, the curves for all particle shapes nearly collapse in a single curve. On the other hand, a shape effect is noticeable for the heaviest particles, where the various shapes present differences in their curves. It is interesting to realize that ILTSs of higher aspect ratio (
In an analogous way to translational LAF, a rotational autocorrelation function (RAF)
where the particle angular velocity is denoted by
The obtained results for the Lagrangian rotational autocorrelation function of the ellipsoids of Zastawny et al. [20] are shown in the left part of Figure 4, whereas those of the cylinders of Vakil and Green [19] are in the right side of such figure. Again, horizontal axis is the nondimensional time delay,
In the case of cylinders (Figure 4, right), the RAF curves for all
Next, the response of the nonspherical particles to the fluid fluctuating velocities is analyzed for both translation and rotation motions. Figure 5(a) shows the behavior of the particle’s relative linear root mean square (rms) velocity, that is,
As inertia increases,
Figure 5(b) presents the behavior of the particle angular rms velocity,
In the following, the correlation relative velocity direction-particle orientation is analyzed depending on inertia and aspect ratio. A well-known fact is that regular nonspherical particles falling through a still liquid at intermediate Reynolds numbers tend to be oriented in a determined direction. Cylinders and prolate ellipsoids are prone to keep their symmetry axis (
Newsom and Bruce [44] analyzed the influence of turbulence on preferential alignment of quite elongated fibers with
Previous reasoning is valid too for another kind of nonspherical shapes as disc-like, cylindrical, or ellipsoidal [45]. For high Reynolds numbers, that is,
Let now
As it is observed in Figure 6(a), it is found that prolate ellipsoids do manifest preferential orientation with respect to the relative velocity. Of course, spherical particles do not have a preferred orientation and the corresponding Pdf is a horizontal line (black color). Prolate ellipsoids have a preference for orientating its symmetry axis orthogonal to the relative flow, tending to maximize the drag, similar to what occurs in particle sedimentation studies. The orientation preference increases with inertia, which is quite similar for all aspect ratios considered in this study.
On the other hand, as it is presented in Figure 6(b), cylinders seem not to have any preferred orientation in the HIT KS velocity field, being all the curves pretty flat. Only for the case of higher
For the discs, Figure 7 shows that there is a clear trend of the particle symmetry axis to be aligned with the relative flow, again maximizing drag, similar to the results obtained for sedimenting particles in stagnant fluid. Such trend is more marked when particle inertia increases.
6. Conclusions
In this study, regular nonspherical particle responsiveness to HIT flows has been investigated in combination with KS of fluid velocity field. The main results obtained are the following: the particle LAF reduces when particle
Acknowledgments
The financial support of Universidad Autónoma de Occidente is gratefully acknowledged.
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