A Finite Particle Approach for the Simulation of Multiphase Flows

Transcription

1 ILASS-Americas 29th Annual Conference on Liquid Atomization and Spray Systems, Atlanta, GA, May 2017 A Finite Particle Approach for the Simulation of Multiphase Flows E.A. Wenzel and S.C. Garrick Department of Mechanical Engineering University of Minnesota Minneapolis, MN USA Abstract The simulation of multiphase flows presents challenges related to accuracy and affordability, particularly for practitioners in industrial settings where computational time is a significant constraint. In response to the need for improved multiphase models, we present a coupled Eulerian-Lagrangian multiphase methodology called the Lagrangian volume of fluid. In this method, Lagrangian phase-identifying particles seeded throughout the computational domain define the local phase, interface geometry, and surface tension force, while the Navier-Stokes equations are solved on an Eulerian mesh. The finite particle method is used to compute the interfacial geometry and surface tension on the Lagrangian particles as a function of their volume of fluid values and spatial distribution. Unlike other methods, the approach is free of remeshing and retains identical Lagrangian domain mass conservation (independent of resolution). Breakup and coalescence occur without explicit modeling, allowing for the capture of primary and secondary atomization. We present the method and apply it to the simulation of multiphase mixing layers and the capillary breakup of droplets to illustrate physical realizability and to demonstrate its sensitivity to numerical parameters including particle number density. Eulerian mass conservation errors are shown to be less than 0.02% for temporal mixing layers undergoing sheet breakup. Conservation errors in laminar capillary breakup are greater, ranging between 1.5% and 2.5%, which is attributable to interpolation errors near merging characteristics. The inherent Lagrangian domain mass conservation and sub-grid scale resolution provided by the phase-identifying particles makes the approach a promising foundation for large eddy simulation. Corresponding Author: wenz0081@umn.edu

2 Introduction Successful control of sprays requires a thorough understanding of primary atomization, but spray dynamics in general and primary atomization in particular are still not fully understood. Experimental techniques have difficulty examining the dense, three-dimensional, highly turbulent zone where primary atomization occurs. Numerical simulation offers the ability to elucidate the underlying processes, but further developments are required to improve fidelity and computational affordability. The Lagrangian volume of fluid (LVOF) method is a coupled Eulerian-Lagrangian simulation approach that has been designed to achieve superior mass conservation and is poised for extension to large eddy simulation (LES). These features address both shortcomings in the current state of the art: affordability and accuracy. In the LVOF method, Lagrangian particles are seeded coincident with an Eulerian mesh and carry a phase-identifying variable called the volume of fluid. These particles are associated with a volume and a mass. The surface tension corresponding to the particle field is computed in the framework of the continuum body force (CBF) [1] via the finite particle method (FPM) [2]. The particles communicate the local phase and surface tension to the Eulerian mesh, and the Eulerian mesh communicates the local velocity to the particles. Because the FPM maintains first order consistency for disorganized particle fields, the Lagrangian remeshing required by other schemes [3] is not required, which allows for identical Lagrangian domain mass conservation. Forthcoming work has demonstrated superior Eulerian mass conservation via the single vortex problem and the dynamic accuracy has been shown to be comparable with the point-set method [4] for a two-dimensional oscillating droplet. The method is more expensive than other direct numerical simulation approaches, but extension to large eddy simulation will dramatically reduce the expense and fully leverage the multiscale nature of the method. This paper presents the first application of the LVOF-FPM method to more general, three dimensional flows with sheet and capillary breakup. The intent of the simulations is to demonstrate mass conservation behavior in nontrivial flows. We also seek to determine the sensitivity of performance on the particle number density. With these considerations, the layout of the paper is as follows: the mathematical structure of the method is briefly presented; temporal mixing layers are then simulated with Weber numbers of 45 and 90, with the results discussed from the perspective of breakup dynamics and mass conservation; next, the sensitivity of capillary breakup to Lagrangian particle number density is qualitatively evaluated and discussed; lastly, conclusions are made and future work is discussed. Methodology The system is initialized by seeding the domain with LVOF particles carrying a volume V i, such that the sum of the particle volumes equals the volume of the system. The subscript i refers to a particular Lagrangian element. Particles seeded in one phase carry a VOF value φ i = 0 and in the other phase φ i = 1. The surface tension is computed as a function of the Lagrangian elements in the context of the continuum body force model [1], F i = δ i σ i κ iˆn i, (1) where σ i is the surface tension coefficient, κ i is the interfacial curvature, and ˆn i = n i / n i is the inward pointing unit normal vector. The interface delta function δ i is defined as δ i = n i. The normal vector and curvature are computed from a smooth definition of the interface called the color function c i. We consider standard expressions for the normal vector n i = c i (2) and curvature where κ i = c ih i c i T c i 2 Trace (H i ) c i 3, (3) H i = ( c i ). (4) The color function c i is obtained from the Lagrangian VOF φ i via the FPM. The FPM relates the VOF φ i to the color function c i by constructing a weighted Taylor series associating nearby particles. The weighting is determined by a sampling basis ζ defined by the spatial derivatives of a compact weight function W of nonzero extent r e [2], ζ = W x W y W z x 2 y 2 z 2 x y x z y z. (5) 2

3 The basis ζ is used to construct a linear system for each Lagrangian element i within r e of the interface by sampling its N p neighbors within the nonzero extent of ζ, N p j=1 (φ j φ i ) ζ k V j = c i x α 2 c i x α x β N p j=1 N p ( x α j x α ) i ζ k V j + j=1 1 2 (xα j x α i )(x β j xβ i )ζk V j, (6) where the free index k assumes values 1 through 9. The notation in Eq. (6) is taken from [2], where Greek superscripts denote spatial dimensionality with implied summation. Solving the system of equations defined by Eq. (6) returns c i / x α and 2 c i / x α x β. The normal vectors and curvature are computed with these derivatives according to Eq. (2) and Eq. (3). Lagrangian methods for computing curvature tend to be error prone far from the interface (where the normal vector magnitude is small), so a cutoff method similar to that implemented in [1] is used here, { n i, n i > ɛ, δ i = (7) 0, n i ɛ. The cutoff value is selected to be ɛ = 0.05 where is the Eulerian mesh spacing. After performing the cutoff correction to the curvature field, the Lagrangian surface tension is computed according to Eq. (1) on N a particles in each interfacial Eulerian cell. Interfacial cells are defined as those with particles carrying non-zero normal vector magnitudes. The particle surface tension vectors F i are transferred to the Eulerian grid by a volume-averaging operation, Na F s i=1 = F iv i Na i=1 V, (8) i where F s is the Eulerian surface tension. The Eulerian mean volume of fluid (used to inform fluid properties) is similarly computed, φ = Na i=1 φ iv i Na i=1 V. (9) i The total number of particles in an Eulerian cell is defined as the particle number density N pc. It is not necessary for N a = N pc. Selecting N a < N pc is referred to as under-sampling and can be performed to reduce computational time. The particle positions x i are updated according to Dx i Dt = up i (10) in conjunction with the Eulerian time integration scheme. The particle velocity vector u p i is obtained from the Eulerian velocity, N g u p i = W ( x g x i, r ev )u g V g, (11) g=1 where N g is the number of Eulerian grid cells within r ev of particle i, u g is the Eulerian velocity at cell g, and V g is the volume of cell g. Governing equations In all of the simulations presented we solve the compressible Navier-Stokes equations. The system is made approximately incompressible by implementing a stiffened equation of state for the pressure P [5, 2], ( ( ) β ρ P = B 1), (12) ρ o where B = 100/β, β = 7, ρ o = 1, and ρ is the density. The Eulerian system includes conservation of mass, ρ + (ρu) = 0, (13) t and conservation of momentum, t (ρu) + (ρu u + P I) = τ + Fs, (14) where τ is the viscous stress tensor. The Eulerian equations are solved with a predictor-corrector finite-difference scheme with second order accuracy [6]. The following simulations use a quartic weight to define both the sampling basis ζ, and the weight function used for the velocity transfer in Eq. (11), W (R, r e ) = { α d ( 3R 4 + 8R 3 6R ), 0 R 1, 0, R > 1, (15) where R = x j x i /r e and α d = 105/16πr 3 e in three dimensions [7]. Case W e Re r e N pc Table 1. Parameters for the temporal mixing layers. 3

4 where m o is the initial mass and m t is the instantaneous mass obtained by integrating the Eulerian VOF φ over the entire domain. This is a conservation measure for the phase with φ i = 1. Numerical parameters The computational domain is shown in Fig. 1. The top and bottom of the domain are initialized as different fluids, { 1 y i 0, φ i (y) = (18) 0 y i > 0, Figure 1. Initial condition and geometry for the temporal mixing layers. Temporal mixing layers Three-dimensional temporal mixing layers demonstrate sheet breakup and provide a platform to explore mass conservation. Mixing layers are characterized by the counterflow of two fluid streams with a relative velocity γ = 2U o and an initial vorticity thickness δ ωo. Instability at the interface produces vortices that grow, pair, and merge [8, 9, 10, 11, 12]. These dynamics are modified when the two streams are composed of different fluids with an interfacial tension. In this section we explore these dynamics for two different Weber numbers defined by W e = ρδ ω o U 2 o σ, (16) where ρ is the fluid density and σ is the surface tension coefficient. Other expressions for the Weber number are often used, but we use Eq. (16) to ensure the same scaling appears in the Weber and Reynolds numbers. The purpose of this demonstration is two-fold: first, we seek to demonstrate qualitatively the ability of the method to capture sheet breakup and present Weber number-dependent dynamics; second, we wish to examine mass conservation behavior in a nontrivial flow configuration. The change in mass is presented as a signed percent change m c = 100 (m t /m o 1), (17) with the top fluid moving to the right and the bottom fluid moving to the left (transitioning over a thickness of δ ωo ). Perturbations are added to the initial velocity field to expedite the growth instabilities to reduce the simulation time [13, 9, 10, 14]. The wavelength of the initial instability is set to λ = πδ ωo. Both phases have equal fluid properties and the flow is defined to have a Reynolds number of Re = U o δ ωo /ν = 400, where ν is the kinematic viscosity. Weber numbers of W e = 90 and W e = 45 are considered. These simulation parameters are summarized in Table 1. Time is nondimensionalized as t = U o t/δ ωo. The domain size is 2πδ ωo 2πδ ωo πδ ωo and is covered by a uniform grid comprised of computational nodes. The boundary conditions in the cross-stream direction are zero-derivative and the stream-wise direction is periodic. Each Eulerian cell is initialized with N pc = 8 particles and the influence radius for ζ is chosen to be r e = 3. The particles are seeded on a uniform cartesian mesh and then perturbed by a small random number. The surface tension is computed with N a = 1 and the particle velocities are computed with r ev = 2. Results An exhaustive discussion of the sheet breakup dynamic is beyond the scope of this brief presentation. Instead, we discuss two instantaneous fields produced slightly after the sheet breakup event and then proceed to the discussion of mass conservation. The instantaneous fields appear in Fig. 2 for case 1 and case 2. Figure 2 panel (a) shows the case 1 mixing layer at t = Ligaments have formed from the merging of ruptured holes in the thin sheet. Some of the ligaments have already undergone capillary breakup and have produced satellite droplets. The same flow is shown at a slightly later time of t = 11.5 in Fig. 2 panel (b). At this later time, many more droplets have been produced and some of the ligaments that did not rupture have receded into the bulk. These 4

5 (a) (b) (c) (d) Figure 2. Comparison of the case 1 and case 2 temporal mixing layers at two times shortly after sheet breakup: (a) case 1 at t? = 10.5; (b) case 1 at t? = 11.5; (c) case 2 at t? = 10.5; (d) case 2 at t? = breakup dynamics are a function of Weber number, as made apparent by comparing case 1 with case 2 in panels (c) and (d). The behavior of case 2 is similar to case 1, aside from clear differences due to the change in Weber number. At time t? = 10.5 in panel (c), the case 2 mixing layer also presents a number of ligaments resulting from a sheet breakup event, but there are fewer, larger ligaments than in case 1. At the later time of t? = 11.5 in panel (d), the droplets resulting from capillary breakup of the ligaments are considerably larger on average and fewer in number than for case 1. This is consistent with the lower Weber number corresponding to stronger surface tension. The Eulerian mass conservation behavior of case 1 and case 2 is presented in Fig. 3. Both cases show an initial increase of 0.06% followed by scatter of less than 0.015% until t? = 12. The initial increase is likely due to a relaxation of particles near the lower boundary into cells that did not initially hold particles, which would artificially increase the volume of φ = 1 cells by a small amount; this is not considered to be characteristic of the method. The case 1 simulation was continued to a time of 5

6 t = 16 (nearly three flow through times) and at this time m c = %. These results demonstrate superior mass conservation, especially considering the complexity of the flow and the large increase in interfacial area due to breakup and stretching. One must also note that the Lagrangian mass has gone unchanged. Each Lagrangian particle maintains its initial mass for all time. m c Case 1 Case 2 Weber number and Reynolds number scalings are the same as the previous section. For each case W e = 8 and Re = 100. In all four cases the particles are placed in a pseudorandom fashion (as was done for the temporal mixing layer cases). The particle volumes for cases I III are globally uniform (all particles in the domain have the same volume and mass). In case IV the particle volume and mass are set inversely proportional to the local particle number density, computed by sampling the local particle field with W and r e = 2. Case W e Re r e N pc r ev I II III IV Table 2. cases. Parameters for the capillary breakup t Figure 3. Percent change in mass for the temporal mixing layer cases. Capillary breakup We proceed to examine the dependency of capillary breakup on the Lagrangian particle number density N pc. Particle number density determines how well the initial geometry is resolved, it affects the accuracy of the Lagrangian surface tension, and it determines the minimum resolvable sub-grid scale features. As a secondary test we also present a single case where r ev is increased and the particle volume initialization is performed differently from the other cases. The general magnitude of influence these parameters have on the solution is examined by suspending a spherical droplet of radius r o = 0.75δ ωo in the center of the mixing layer used in the previous section. The domain size, Eulerian mesh, and influence radius used in ζ are all adopted from the previous section. Again, N a = 1, and the perturbed initial condition is used, but this time the instability wavelength is set to the entire length of the domain λ = 2πδ ωo. The parametric definitions for four test cases (cases I, II, III, and IV ) appear in Table 2. The Results We do not present the full evolution of the droplet field, but consider only the period encompassing the breakup event. We are interested in the differences between the results, rather than the individual results themselves. This period is provided in Fig. 4 for cases I III. Case I appears on the left, case II is in the center, and case III appears on the right. The first image in each series corresponds to t 5.7, where we observe the droplet has been deformed into an elongated shape with a nodule on each end connected by a ligament. The case III droplet filament appears more smooth than the other two cases, perhaps suggesting the other cases have insufficient particle resolution. At the next time level the ligament has thinned at the edges for all three cases. The case III droplet is shown at a slightly later time than the other two cases because the thinning progresses more slowly. The third time level shows the production of an oscillatory satellite droplet, which becomes a steady sphere by the fourth time level. The three results are similar but not identical. Case III, which has the highest particle number density of N pc = 24, undergoes a longer breakup process: the ligament separates from the two nodules at a later time. This increased thinning time results in a slightly smaller satellite drop than cases I and II. In general, the dynamic performance does not appear to be a strong function of N pc. The mass conservation behavior, again evalu- 6

7 (a) t*=5.718 (e) t*=5.718 (i) t*=5.719 (b) t*=6.259 (f) t*=6.258 (j) t*=6.531 (c) t*=6.523 (g) t*=6.519 (k) t*=6.795 (d) t*=7.866 (h) t*=7.865 (l) t*=7.871 Figure 4. Comparison of the droplet interface for the capillary breakup tests: (a)-(d) case I; (e)-(h) case II; (i)-(l) case III. ated by m?c, appears in Fig. 5. Unlike the temporal mixing layer, there is notable Eulerian mass loss for the droplet stretching problem. By t? = 8, cases I III have lost approximately 2.5% of the initial Eulerian mass, while case IV has lost approximately 1.5%. The mass loss is effectively independent of particle number density. It is presently unknown if the improvement in case IV is due to the increase in rev or the difference in initial condition. This is an area of future work. The degradation of Eulerian conservation in this configuration (as compared to the temporal mixing layer) is due to the challenge of interpolating velocities near merging characteristics [15]. This geometry is particularly challenging because it is laminar and characteristics merge at both ends of the droplet for the entire simulation. The interpolation errors result in particles being transported from the center of the droplet to both ends where they remain for the duration of the simulation. This is a problem for laminar flows but is essentially nonexistent in turbulent flows as demonstrated by the temporal mixing layer problems. It should be noted that the Lagrangian domain mass is still available, even when errors arise in the Eulerian field. Summary and Conclusions We have presented a finite particle methodbased Lagrangian volume of fluid method (LVOFFPM). This coupled Eulerian-Lagrangian approach computes the interfacial tension and local phase on Lagrangian particles via the finite particle method and then transfers them to an Eulerian solver for solution of the Navier-Stokes equations. Use of the FPM is advantageous because it eliminates the need for Lagrangian remeshing required by other methods [3], and therefore allows for identical Lagrangian domain mass conservation. The mathematical framework of the method was briefly presented, followed by the simulation of a series of three-dimensional flows undergoing different breakup mechanisms. We first considered sheet breakup as a function of Weber number in temporal mixing layers, followed by capillary breakup as a function of particle number density. Eulerian conservation errors in the temporal mixing layers undergoing sheet breakup were approximately 0.015%. Expectedly, the droplets pro7

8 m c Case I Case II Case III Case IV t References [1] J. P. Morris. Int. J. Numer. Meth. Fl., 33(3): , [2] M. B. Liu, W. P. Xie, and G. R. Liu. Appl. Math. Model., 29(12): , [3] S. E. Hieber and P. Koumoutsakos. J. Comp. Phys., 210(1): , November [4] D. J. Torres and J. U. Brackbill. J. Comp. Phys., 165: , [5] J. J. Monaghan. J. Comp. Phys., 159: , [6] C. A. Kennedy and M. H. Carpenter. Appl. Num. Math., 14: , [7] L.B. Lucy. Astron. J., 82: , Figure 5. Percent change in mass for the capillary breakup cases. duced in the W e = 45 mixing layer were larger and fewer in number than the droplets produced in the W e = 90 mixing layer. Conservation errors in the capillary breakup test cases were greater than for the mixing layers, with losses ranging from 1.5% to 2.5%. These errors were independent of particle number density N pc, but dependent on either r ev or the particle initialization. Future work must determine which of these parameters influenced the result. In general, these conservation errors are due to interpolation errors near merging characteristics, which should not be a prominent concern for turbulent flows. The capillary breakup dynamic in general was observed to be a relatively weak function of N pc. The LVOF-FPM is a promising simulation tool because of its conservation properties, its sub-grid scale phase resolution, and the Lagrangian nature of the multiphase description. These properties make the LVOF-FPM robust for direct numerical simulation, and also make it a strong candidate for large eddy simulation. The Lagrangian description of phase allows for direct extensions of the LVOF-FPM to existing multiphase LES formulations. It is also possible to develop new LES formulations to leverage the properties of the LVOF. The development of LES extensions is an immediate area of future work. [8] P. E. Dimotakis. AIAA J., 24: , [9] R. W. Metcalfe, S. A. Orszag, M. E. Brachet, S. Menon, and J. J. Riley. J. Fluid Mech., 184: , [10] M. M. Rogers and R. D. Moser. 7th Symp. on Turbulent Shear Flows, Stanford, CA, [11] S. Modem and S. C. Garrick. J. Visualization, 6(3): , [12] G. Wang and S. C. Garrick. J. Aerosol Sci., 37(4): , [13] R. T. Pierrehumbert and S. E. Widnall. J. Fluid Mech., 114:59 82, [14] S. A. Ragab and J. L. Wu. Phys. Fluids A, 1(6): , [15] D. Enright, R. Fedkiw, J. Ferziger, and I. Mitchell. J. Comp. Phys., 183(1):83 116, Acknowledgements All computational resources were provided by the Minnesota Supercomputing Institute. URL: 8