Pairwise Interaction Extended Point-Particle (PIEP) Model for droplet-laden flows: Towards application to the mid-field of a spray Georges Akiki, Kai Liu and S. Balachandar * Department of Mechanical & Aerospace Engineering University of Florida Gainesville, FL Euler-Lagrange (EL) methodology is the common approach used to simulate the collective dynamics of a very large number of dispersed droplets in the mid region of a spray. Traditional point-particle models used in these EL simulations assume the force on a droplets to depend only on mean Reynolds number and volume fraction. This approximation ignores interactions among the droplets at the microscale and therefore significantly under-predicts fluctuations in droplet velocity and feed-back forces, resulting in incorrect mesoscale structures and macroscale dispersion. This study introduces a new extended point-particle force model that attempts to rigorously account for the hydrodynamic influence of the neighboring droplets. The model computes the drag and lift forces, and heat/mass transfer on each droplet by accounting for the precise location of few surrounding neighbors. A pairwise interaction is assumed to superpose the perturbation fields induced by each neighbor and the Faxén form is used to obtain the forces and heat transfer. The model is tested for the cases of a random array of stationary and freely sedimenting particles. The ultimate objective if our ONR Multi-University Research Initiative (MURI) project is to demonstrate multiphysics control of liquid sprays. The talk will also present on-going effort at simulation of mid-field region of a spray where a turbulent inflow of gas with a dispersed distribution of droplets is specified. While first generation simulations will employ standard point-particle models, the goal is to employ the PIEP model and to firmly establish the role of droplet-droplet interaction in such dense droplet-laden flows. * Corresponding author, bala1s@ufl.edu
Introduction Efficient and effective control of droplet spray is a problem of great interest in many engineering applications. The present work is part of a Multidisciplinary University Research Initiative (MURI) that has been funded by the Office of Naval Research to explore active control of the atomization process with an integrated simulation and experimental approach. The coordinated effort is further divided into near-field and mid-field control of the spray. This short paper is to provide an update on the progress made towards largescale simulations of mid-field control of spray droplets and their dispersion using acoustic and electrostatic forcing. We define the near-field as the region where all the primary breakup of the liquid jet into droplets, and most of the subsequent breakups into a fine droplet spray is complete. Thus, control of droplet size spectra is primarily in the near-field. Control of the spatial distribution of the droplet spray in the radial direction, using active acoustic and electrostatic forcing is the primary goal of the mid-field simulation. With this in mind, we pursue a Euler-Lagrange (EL) dispersed turbulent multiphase flow simulation strategy for the mid-field. In the EL methodology, the location of all the droplets, or a subset of computational droplets that accurately represent the actual distribution in a statistical sense, is tracked. Since the flow around the individual droplets is not resolved, a point-particle (PP) model is used to track the trajectory of the computed droplets by solving their equations of motion. The accuracy of the Eulerian-Lagrangian point-particle (EL-PP) technique is dependent on the fidelity of the force, thermal and mass transfer couplings laws used for representing the fluiddroplet mass, momentum and energy exchange that occurs at the microscale. These closure models must account for the net effect of the droplet-flow interactions that happen at the microscale, which are unresolved in the EL-PP approach. In the dilute limit when the droplet size is smaller than the ambient flow scales, force models, such as the standard drag law, provide an excellent closure [1]. At intermediate droplet volume fraction, the effects of indirect interaction between the droplets as mediated by the surrounding gas flow starts to become important. With further increase in volume fraction, the probability of direct collisions between the droplets increases, which further can lead to collision-induced agglomeration and breakup of droplets. Recent EL-DEM (discrete element method) simulations of dispersed multiphase flow partially account for these effects through volume fraction dependent drag coefficient and collision detection [2-5]. Such models however do not make use of information on the location of neighboring droplets on the motion of each droplet, except abruptly when they collide. The purpose of this paper is to briefly describe the recently developed pairwise interaction expended pointparticle (PIEP) model that systematically accounts for the effect of neighbors on the motion of individual particles. The PIEP model includes lubrication and direct collisional effects. The model as described below has been developed and tested for forces and torque on rigid spheres and is currently being extended for droplets and bubbles and also for including other effects such as heat and mass transfer. We will also briefly present our initial progress towards accurate simulation of the mid-field of a spray. In the future these simulations will be performed with the inclusion of the PIEP model. PIEP Model The PIEP model is built upon two basic ideas. First, the undisturbed flow at each droplet location, defined as the flow that would exist at a droplet location in the absence of that droplet but with all other droplets present, is separated into two parts: a macroscale flow that accounts for the collective action of all the droplets and a microscale flow that accounts for the presence of all the neighbors taken one at a time (this is the pairwise interaction approximation). Second, the above defined undisturbed flow is used to calculate the net aerodynamic force and torque on the droplet using the Faxén form of the quasi-steady, added-mass, Basset history, and vorticity-induced (lift) forces relation (and similarly for the torque), since the undisturbed flow obtained from the first step is non-uniform. The important aspect of the PIEP model is it attempts to systematically account for the microscale flow induced by the neighbors by making use of their precise location, an information which is readily available in EL simulations. By accounting for the precise location of neighbors, the PIEP model goes beyond the mean neighborhood information of local droplet volume fraction, and distinguishes the influence of upstream, downstream and laterally located neighbors. Such droplet-droplet interaction information is critical in order to capture phenomenon such as collision and close-range interaction of droplets. It is important to note that PIEP model retains the computational efficiency of the standard Euler-Lagrange PP approach. The microscale perturbation field induced by a neighbor can be easily computed for a range of Reynolds number and ambient conditions. Which can then be used to pre-compute the pairwise interactions and store as PIEP maps. These maps are then used repeatedly to calculate the microscale contribution to hydrodynamic force and torque. The implementation of the complete PIEP model requires the use of seven force maps. There maps correspond to (i) streamwise velocity perturbation, (ii)
transverse velocity perturbation, (iii) streamwise component of perturbation stress-gradient force, (iv) transverse component of perturbation stress-gradient force, (v) perturbation vorticity, (vi) streamwise twoparticle correction, and (vii) transverse two-particle correction. Here streamwise direction is along the macroscale ambient flow seen by the moving particle and transverse direction is perpendicular in the plane formed with the neighbor (see [6-7] for details of the model and notations). Each map is a function of streamwise and transverse distance between the reference particle and the neighbor, whose effect on the reference particle is being computed. Figure 2 shows a plot of the streamwise component of the total force on the reference particle due to the presence of a neighbor (marked as the black circle) at four different Reynolds numbers. Both particles are held fixed in a uniform ambient flow directed along the positive x direction. Results As an example of the force map, Figure 1 shows contours of the streamwise and transverse components of the pressure gradient force on the reference particle due to the presence of a neighbor (pressure component of maps 3 and 4) at one Reynolds number. Since the center of the reference particle cannot get closer than one diameter the contours are absent in the annular region around the neighbor. Figure 2: Contour plot of the non-dimensional streamwise force on a particle whose center is located at (x, y) in the presence of a second particle located at the origin (marked by the dark circle) predicted by the PIEP model. Results for four different Reynolds numbers are shown. Figure 1: Mapping of the pressure gradient force on a reference sphere whose center is a distance (x,y) from the neighboring sphere (which is marked by the black circle). a) streamwise and b) spanwise components for Re i = 87. We present results from a test where eighty spheres are randomly released to settle in a large periodic computational box. This problem was simulated using fully resolved DNS using immersed boundary method [8,9] using a grid resolution of 20 points per particle diameter. The problem is also simulated using EL methodology with PIEP model where we only solved for the translational and rotational motion of the 80 spheres. As a result the PIEP model simulation is orders of magnitude faster than the DNS. In the PIEP model, we limit the number of neighbors used to 30. Figure 3 shows a comparison of the spatial dispersion of the 80 spheres at time t = 46. In comparison EL simulation using the standard drag would predict all the spheres settling at the same rate without any interaction or dispersion. We can observe very good comparison between the DNS and PIEP simulations.
Mid-Field Simulations As a preliminary result we finally show results from an initial mid-field simulation of a gas jet of inlet Reynolds number of 2000. An inlet Gaussian velocity distribution with random initial perturbation was applied. Although only axial gas velocity is shown in Figure 4, the present one-way coupled simulations included ability to inject at the inlet a specified inlet distribution of droplet spectra. We envision performing two-way coupled turbulent simulations with PIEP model implemented to account for both gas-droplet and droplet-droplet interactions. Nek5000, the code used for the above simulation, employs a higher-order accurate spectral element methodology to which Lagrangian droplet tracking capability has recently been added by us. The preliminary simulation shown in figure 4 employs a modest grid of about 3 million grid points. Acknowledgements This work was sponsored by the Office of Naval Research (ONR) as part of the Multidisciplinary University Research Initiatives (MURI) Program, under grant number N00014-16-1-2617. The views and conclusions contained herein are those of the authors only and should not be interpreted as representing those of ONR, the U.S. Navy or the U.S. Government. References 1. Crowe, C. T., Schwarzkopf, J. D., Sommerfeld, M., & Tsuji, Y. (2011).Multiphase flows with droplets and particles. CRC press. 2. Beetstra, R., Van der Hoef, M. A., & Kuipers, J. A. M. (2007). Drag force of intermediate Reynolds number flow past mono and bidisperse arrays of spheres. AIChE Journal, 53(2), 489-501. 3. Tenneti, S., Garg, R., & Subramaniam, S. (2011). Drag law for monodisperse gas solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres. International journal of multiphase flow, 37(9), 1072-1092. 4. Zaidi, A. A., Tsuji, T., & Tanaka, T. (2014). A new relation of drag force for high Stokes number monodisperse spheres by direct numerical simulation. Advanced Powder Technology, 25(6), 1860-1871. 5. Deen, N. G., et al. "Review of discrete particle modeling of fluidized beds." Chemical Engineering Science 62.1 (2007): 28-44. 6. Akiki, G., Jackson, T. L., & Balachandar, S. (2017). Pairwise interaction extended point-particle model for a random array of monodisperse spheres. Journal of Fluid Mechanics, 813, 882-928. 7. Akiki, G., Jackson, T., & Balachandar, S. (2016, November). Pairwise Interaction Extended Point Particle (PIEP) Model for a Random Array of Spheres. In APS Meeting Abstracts. 8. Uhlmann, M. (2005). An immersed boundary method with direct forcing for the simulation of particulate flows. Journal of Computational Physics, 209(2), 448-476. 9. Akiki, G., & Balachandar, S. (2016). Immersed boundary method with non-uniform distribution of Lagrangian markers for a non-uniform Eulerian mesh. Journal of Computational Physics, 307, 34-59.
Figure 3: Side view of 80 sedimenting spheres starting from a random initial distribution at t = 38.6. Both the DNS results as well the PIEP model prediction are shown. Figure 4: Streamwise velocity contours on the mid-plane passing through a cylindrical turbulent jet. A Gaussian inflow with random perturbation is specified as the inlet on the left.