A Volume of Fluid Dual Scale Approach for Modeling Turbulent Liquid/Gas Phase Interfaces

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1 ILASS-Americas 29th Annual Conference on Liquid Atomization and Spray Systems, Atlanta, GA, May 217 A Volume of Fluid Dual Scale Approach for Modeling Turbulent Liquid/Gas Phase Interfaces D. Kedelty, J. Uglietta, and M. Herrmann School for Engineering of Matter, Transport and Energy Arizona State University Tempe, AZ USA Abstract In atomizing flows, the properties of the resulting liquid spray are determined by the interplay of fluid and surface tension forces. The resulting flow and interface dynamics typically span 4-6 orders of magnitude in length scales, making direct numerical simulations exceedingly expensive. This motivates the need for modeling approaches based on spatial filtering or ensemble averaging. In this contribution, advances to a dual-scale modeling approach (Herrmann, 213) are presented to describe turbulent two-phase interface dynamics in a large-eddy-simulation-type spatial filtering context. Spatial filtering of the governing equations introduces several sub-filter terms that require modeling (Toutant et al., 28). These include terms associated with sub-filter acceleration, transport, viscous effects, and surface tension. Instead of developing individual closure-models for each of the terms associated with the phase interface, the proposed dual-scale approach uses an exact closure by explicitly filtering a fully resolved realization of the phase interface. This resolved realization is maintained on a high-resolution over-set mesh using a Refined Local Surface Grid approach (Herrmann, 28) employing an un-split, geometric, bounded, and conservative Volume-of-Fluid method (Owkes and Desjardins, 214). The advection equation for the phase interface on this DNS scale requires a model for the fully resolved interface advection velocity. This velocity is the sum of the LES filtered velocity, readily available from the LES approach solving the filtered Navier Stokes equation, and the sub-filter velocity fluctuation that has two contributions. The first is due to sub-filter turbulent eddies, the second is due to sub-filter surface tension forces. We propose four different methods to reconstruct the sub-filter turbulent eddy fluctuation velocity using the fractal interpolation approach of Scotti & Meneveau (1999). Results of the dual-scale model are compared to recent direct numerical simulations of a phase interface in homogeneous isotropic turbulence (McCaslin and Desjardins, 215). Corresponding Author: marcus.herrmann@asu.edu

2 Introduction Atomization in turbulent environments involves a vast range of length and time scales. Predictive simulations aiming to resolve all relevant scales thus require enormous computational resources, taxing even the most powerful computers available today [1]. Since primary atomization is governed by the dynamics of the interface, a need therefore exists for appropriate interface models that make the computational cost of predicting the atomization outcome more tractable. A wide range of phenomenological models aiming to represent statistically the essential features of atomization have been proposed in the past. Although these aim to introduce the dominant mechanisms for breakup, they often use round blobs injected from the nozzle exit and hence neglect all details of the interface dynamics. Other modeling approaches to atomization include stochastic models [2, 3] representing the interface by constituent stochastic particles and the mean interface density transport equation model for Reynolds-Averaged Navier-Stokes (RANS) approaches [4, 5]. The former treats the interface dynamics in a stochastic sense but requires the a priori knowledge of the breakup mechanism, whereas the latter is affected by the drawbacks of the RANS approach: the transport of the mean interface density is modeled by a diffusion-like hypothesis, thereby neglecting the spatial grouping effects of liquid elements [1]. In the context of Large Eddy Simulations (LES), [6, 7, 8, 9] have proposed models to close the unclosed terms arising from the introduction of spatial filtering into the governing equations. However, these models typically neglect the contribution of the sub-filter surface tension term and are based on a cascade process hypothesis that may be questionable in the context of surface tension-driven atomization. In [1, 11], a dual-scale approach for LES of interface dynamics was proposed and a model for the sub-filter surface tension induced motion of phase interfaces was developed. The purpose of this contribution is to develop a model for the sub-filter phase interface motion induced by sub-filter turbulent velocity fluctuations. Combining such a model with the surface tension model proposed in [1, 11] will result in a LES model applicable to atomizing flows. Governing equations The equations governing the fully resolved motion of an unsteady, incompressible, immiscible, twofluid system in the absence of surface tension are the Navier-Stokes equations, ρu t + (ρu u) = p + (µ ( u + T u )), (1) where u is the velocity, ρ the density, p the pressure, and µ the dynamic viscosity. Here, we neglect surface tension to solely focus on the turbulence induced dynamics of phase interfaces. Furthermore, the continuity equation results in a divergence-free constraint on the velocity field u =. (2) Assuming ρ and µ are constant within each fluid, density and viscosity can be calculated from ρ = ψρ l + (1 ψ)ρ g (3) µ = ψµ l + (1 ψ)µ g, (4) where indices l and g denote values in liquid and gas, respectively, and ψ is a volume-of-fluid scalar that is ψ = in the gas and ψ = 1 in the liquid with ψ t = u ψ = (uψ) + ψ u. (5) Here, the last term on the right-hand side is zero for incompressible flows, see Eq. (2). Filtered governing equations Introducing spatial filtering into Eqs. (1) and (2) and assuming that the filter commutes with both the time and spatial derivatives, the filtered governing equations can be derived [7], ρ u t + (ρū u) = p + (6) (µ( u + T u)) + τ 1 + (τ 2 + τ 3 ), ū =, (7) where indicates spatial filtering, and τ 1 = ρ u ρu (8) t t τ 2 = ρ u u ρu u (9) τ 3 = µ( u + T u) µ( u + T u), (1) where τ 1, τ 2, and τ 3 are associated, respectively, with acceleration, advection, and viscous effects [7]. Using Eqs. (3) and (4), the filtered density and viscosity in Eq. (6) are ρ = ρ l ψ + ρ g (1 ψ) (11) µ = µ l ψ + µ g (1 ψ), (12) 2

3 where ψ = G(x)ψdx, (13) and G is a normalized spatial filter function. The dual-scale approach to modeling subfilter interface dynamics Instead of relying on a cascade process by which dynamics on a sub-filter scale can be inferred from the dynamics on the resolved scale, the dual-scale approach proposed in [11] aims to maintain a fully resolved realization of the immiscible interface geometry at all times, expressed, for example, in terms of a volume-of-fluids scalar ψ. Then ψ can be calculated exactly by explicit filtering using Eq. (13). Although this is an exact closure, the problem of modeling is of course simply shifted to the problem of maintaining a fully resolved realization of the interface geometry, i.e., describing the fully resolved motion of the interface, Eq. (5). Since the fully resolved velocity is the sum of the filtered velocity and the sub-grid velocity, u = u + u sg, this results in ψ t = ((u + u sg) ψ) + ψ (u + u sg ), (14) where the only term requiring modeling is u sg. In [11], a model for u sg is proposed consisting of three contributions, u sg = u + δu + u σ, (15) where u is due to sub-filter turbulent eddies, δu is attributed to the interface velocity increment due to relative sub-filter motion between the two immiscible fluids, and u σ is due to sub-filter velocities induced by sub-filter surface tension forces. The focus of the current contribution is on the first term; for a modeling outline of the second term, the reader is referred to [11, 14], and for modeling of the last term, the reader is referred to [1]. Sub-filter turbulent fluctuation velocity models We propose to reconstruct the sub-filter turbulent fluctuation velocity u using fractal interpolation [15]. To demonstrate fractal interpolation in one dimension, consider 3 adjacent LES scale nodes x, x 1, and x 2 with velocities u, u 1, and u 2. Following [15, 16] the fractal interpolation operator W F I can be written as W F I (x) = u + u 1 u x 1 x (x x ) + d 1 (u(2x x ) u (16) ) u 2 u (2x x ) if x [x, x 1 ] x 2 x W F I (x) = u 1 + u 2 u 1 x 2 x 1 (x x 1 ) + d 2 (u(2x x ) u (17) ) u 2 u (2x x ) if x [x 1, x 2 ] x 2 x Here d 1 < 1 and d 2 < 1 are stretching factors making W F I a contractive mapping [15]. Successively applying the fractal interpolation operator W F I starting with the LES filter velocities, generates the fully resolved turbulent fluctuation velocity. Two different approaches are pursued to extend the above formulation to three dimensions. In the first, fractal interpolation is first performed in one spatial direction only, followed by separate 1D fractal interpolations in the other two directions [15]. The second approach replaces the one-dimensional linear interpolation operators in Eqs. (16) and (17) with three-dimensional trilinear interpolation operators and performs the fractal interpolation directly in three dimensions [17]. Furthermore, two different approaches are used to determine the values of the stretching factors d 1 and d 2. The first, follows the so-called ZE1 model of [15] by using d 1 = d 2 = ±2 1/3 with the sign chosen randomly with equal probability. This choice of d i generates a velocity signal that satisfies the 5/3 kinetic energy spectrum of turbulence. The second approach chooses d i with random sign and absolute value determined from a log- Poisson distribution [16, 17], P with ( d i = ( ) γ ) 1 β n = e 2 λ λn, n =, 1,... (18) n! λ = 1 3γ ln 2 (19) 1 β3 and γ = 1/9 and β = (2/3) 1/3 [17]. Finally, two different approaches are analyzed concerning the spatial location of the LES velocities u, u 1, and u 2. If the LES flow solver utilizes a staggered mesh layout, face normal velocities are not colocated and hence the fractal interpolation has to be performed for different locations depending on the spatial component of the velocity vector. However, if one first interpolates the staggered face velocities 3

$to the cell corners (nodes), then the velocity components are co-located and fractal interpolation for all components can be performed at the same location.$ Numerical methods Equation (14) is solved using an unsplit geometric transport scheme for volume-of-fluid scalars that ensures both discrete volume conservation of each fluid and boundedness of the

Numerical methods Equation (14) is solved using an unsplit geometric transport scheme for volume-of-fluid scalars that ensures both discrete volume conservation of each fluid and boundedness of the

4 to the cell corners (nodes), then the velocity components are co-located and fractal interpolation for all components can be performed at the same location. However, this interpolation step is in essence an additional spatial filter of the LES velocity before fractal interpolation is performed. Numerical methods Equation (14) is solved using an unsplit geometric transport scheme for volume-of-fluid scalars that ensures both discrete volume conservation of each fluid and boundedness of the volume-of-fluid scalar ψ [18]. Geometric reconstruction of the interface within each computational cell is done using PLIC reconstruction, employing analytical formulas [19] using ELVIRA estimated normals [2, 21]. To efficiently solve Eq. (14) for the fully resolved immiscible interface, the RLSG method [12] is employed. By design, it solves the interface capturing advection equation on a separate, highly resolved Cartesian overset grid of mesh spacing h G, independent of the underlying LES flow solver grid of mesh spacing h. In the dual scale LES approach, h G needs to be chosen sufficiently small to maintain a fully resolved realization of the phase interface. The unsplit, geometric advection scheme of [18] requires face-centered velocities that are discretely divergence-free to ensure both conservation and boundedness. While discretely divergence free filtered velocities u are available on the flow solver mesh due to the application of a projection step of the velocities in a standard fractional step method, such velocities u are not directly available on the fine overset mesh. Since u = u + u, both u and u need to be discretely divergence free on the fine overset mesh. To ensure hg u = if h u =, we employ the optimal constrained approach for divergence-free velocity interpolation [22], recursively applying the interpolation technique for nested staggered meshes up to the refinement level of the overset mesh. The sub-filter fluctuation velocity u is calculated from u = WF k Iu u, (2) where the superscript k indicates k-times application of the fractal interpolation operator W F I, with ( ) h log h G k =. (21) log(2) u is then projected into the subspace of solenoidal velocity fields using the projection/correction step of a standard fractional step method. Although the fractal interpolation can be applied to the temporally evolving LES velocity u to obtain a time de- Figure 1. PLIC geometry on LES mesh cell (left), fully resolved PLIC geometry (center), difference used to calculate C mix (right). pendent u, here, we use a single snapshot of the LES field only (the initial velocity field), and thus u is frozen in time. Finally, to calculate ψ, Eq. (13) is evaluated by setting the filter size to the local flow solver mesh spacing h and evaluating the integral by explicitly summing the VoF scalar ψ of those overset-mesh cells that are contained within a given LES flow solver cell. Comparison Metrics Assuming that the phase interface is initially planar with normal orientation in the y-direction, α(x, z, t) defines the liquid volume fraction that is contained within a square column normal to the planar interface and with cross sectional area equal to h 2 G. The corresponding quantity on the LES mesh, α(x, z, t), can be calculated using a column with cross sectional area h 2. Then, the sub-filter liquid column height RMS can be defined as α 1 (t) = L 2 ((α(x, z, t) α(x, z, t)) 2 dxdz. L L (22) To take into account that even on an LES mesh scale the PLIC reconstruction of the filtered phase interface geometry provides a level of sub-filter geometry resolution beyond the mere liquid column height α, we define a local quantity C mix that measures the sub-filter variation of the phase interface on the overset mesh compared to the LES mesh. C mix is calculated by first performing a PLIC reconstruction on the LES mesh using the filtered volume fraction ψ, see Fig. 1. The reconstructed planar representation of the phase interface is then used to calculate liquid volume fractions on the overset mesh ψ hg. The root mean square of the difference of these volume fractions and the fully resolved volume frac- 4

$E(k) 1 4 1 2 1 1 1 1 1 2 k Figure 2. Comparison of LES filtered velocity (left) and fractal interpolated velocity using method 3D- ZE1-N. tions ψ, see Fig.$

5 Method interpolation scaling factors velocity location 3D-ZE1-N 3D ZE1 node 1 8 3D-P-N 3D Poisson node 3D-P-F 3D Poisson face 1D-P-F 1D Poisson face 1 6 Table 1. Fractal interpolation methods used. E(k) k Figure 2. Comparison of LES filtered velocity (left) and fractal interpolated velocity using method 3D- ZE1-N. tions ψ, see Fig. 1, is C mix, 1 ( ) 2 C mix (t) = ψ L 3 ψhg dxdydz. L L L (23) It should be pointed out that without a dual scale model, both α and C mix remain zero for all time. Results An initially flat interface is placed inside a box of fully developed isotropic turbulence. Both density and viscosity ratio are unity, and no surface tension forces are present with a Reynolds number of Re λ = 313 and W e =. Direct numerical simulation results using a mesh for this case are reported in [23]. Here, we present LES results using the dual-scale approach employing a LES mesh resolution of 32 3 and an overset mesh resolution of To study the impact of the sub-filter mesh resolution, an overset mesh resolution of is also studied. The sub-filter velocity u is obtained from fractal interpolation using the combination of approaches summarized in Table 1. Figure 2 shows as an example a comparison of the LES filtered velocity field and the fractal interpolated velocity using method 3D-ZE1-N. Noticeably more small scale structure is visible in the fractal interpolated velocity compared to the filter scale LES velocity. This is corroborated by the kinetic energy Figure 3. Kinetic energy spectrum for overset mesh using only trilinear interpolation (blue), fractal interpolation 3D-ZE1-N (red), sub-filter velocity u only (green), and 5/3 reference line (dashed). spectrum shown in Fig. 3 that compares the fractal interpolated velocity spectrum to one obtained using only trilinear interpolation. Note that the fractal interpolated spectrum agrees well with the theoretical 5/3 spectrum, but does not show any viscous dissipation range. The velocity spectrum using trilinear interpolation only, shows significantly lower kinetic energy on the sub-filter scale with incorrect scaling. Figure 4 compares realizations of the phase interface geometry using fractal interpolation 3D-ZE1- N and the interface geometry that would be obtained without any dual scale model. Including the fractal interpolated sub-filter velocity in the advection velocity of the resolved realization of the interface clearly leads to significantly more surface corrugations, as expected. To enable a quantitative comparison of the different fractal interpolation methods, Figs. 5 to 8 show comparisons using the two comparison metrics α and C mix. Figure 5 shows the sub-filter liquid column height RMS α using the four methods listed in Table 1 compared to the result of the DNS [23]. No significant difference between the four fractal interpolation methods can be discerned. Compared to the DNS, the dual scale LES slightly over-predicts α. This is due to the fact that the dual scale model uses a frozen fluctuation velocity field, whereas the DNS is simulated in decaying turbulence. Interactive coupling of the dual scale approach to the time evolving LES field, to be done in future work, should eliminate this discrepancy. Figure 6 analyzes the impact of the overset 5

(red), 3D-P-F-256 (green), 3D-P-N-512 (black), 3D-P-F-512

$model (left) and fractal interpolation velocity method$ 3D-ZE1-N (right)..1.2.3.4 t/t Figure 7.

(black), 3D-P-N-256 (red), 3D-P-F256 (green), and

3D-ZE1-N-256 (black), 3D-P-N-256 (red), 3D-P-F-256

6 .15 ` t/t Figure 6. Impact of overset mesh resolution on α : 3D-P-N-256 (red), 3D-P-F-256 (green), 3D-P-N-512 (black), 3D-P-F-512 (blue), and DNS (symbol)..2 Cmix Figure 4. Comparison of phase interface geometry: no dual scale model (left) and fractal interpolation velocity method 3D-ZE1-N (right) t/t Figure 7. Comparison of sub-filter variation Cmix : 3D-ZE1-N-256 (black), 3D-P-N-256 (red), 3D-P-F256 (green), and 1D-P-F-256 (blue) Cmix ` t/t t/t Figure 5. Comparison of sub-filter liquid column height RMS α : 3D-ZE1-N-256 (black), 3D-P-N-256 (red), 3D-P-F-256 (green), 1D-P-F-256 (blue), and DNS-124 (symbol). Figure 8. Impact of overset mesh resolution on Cmix : 3D-P-N-256 (red), 3D-P-F-256 (green), 3DP-N-512 (black), and 3D-P-F-512 (blue). 6

7 mesh resolution, comparing results obtained using the overset mesh to the mesh. No significant differences are present, indicating that in this metric, the phase interface is well resolved even on the overset mesh. Figure 7 compares the sub-filter variation C mix for the four different fractal interpolation methods. Unlike α, small differences between the methods are discernible. The successive application of 1D interpolations results in slightly higher sub-filter variation than the 3D trilinear interpolation methods, indicating that the trilinear interpolation introduces some minor dissipation in the fractal interpolated velocities. Furthermore, using filter scale face velocities instead of interpolated node velocities also results in higher C mix. This can be explained by the fact that the interpolation to the nodes is in effect an additional filter operation reducing slightly the kinetic energy at the LES filter scale before fractal interpolation is performed. While this does not impact α, C mix seems to be sensitive enough to pick up this effect. Finally, using stretching factors obtained by either method ZE1 or the more complex log-poisson distribution seems to have no significant impact on C mix. Finally, Fig. 8 analyzes the impact of the overset mesh resolution on C mix, comparing results obtained using the overset mesh to the mesh. Unlike α, increasing the overset mesh resolution increases C mix noticeably, because a higher overset mesh resolution enables the resolution of smaller scales in the resolved realization, and hence potentially larger C mix. Unfortunately, DNS results for comparison are not yet available. However, proper model verification would require that the metric C mix converges under overset mesh refinement. This will be confirmed in future work. Conclusions A dual-scale modeling approach for phase interface dynamics in turbulent flows is presented that is based on a volume-of-fluid approach. The method uses overset high-resolution meshes to capture a resolved realization of the phase interface geometry that can be explicitly filtered to close the terms that require modeling in the filtered Navier-Stokes equations. This contribution focuses on different fractal interpolation techniques to generate sub-filter velocity fluctuations that, as shown, have a significant impact on interface metrics as compared to the case of no dual-scale model, where these metrics remain zero for all time. Results show that no differences between the different fractal interpolation methods can be discerned using the sub-filter column height RMS, however, small differences are present using the more sensitive sub-filter variation metric. Necessary future work will include a more detailed comparison to DNS data both for planar interfaces and parallel liquid sheets in homogenous isotropic turbulence. Acknowledgements The support of NASA TTT grant NNX16AB7A is gratefully acknowledged. References [1] M Gorokhovski and M Herrmann. Annu. Rev. Fluid Mech., 4(1): , 28. [2] M Gorokhovski, J Jouanguy, and A Chtab. Proc. Int. Conf. Liq. Atom. Spray Syst., 26. [3] M Gorokhovski, J Jouanguy, and A Chtab. Int. Conf. Multiph. Flow, Leipzig, Germany, 27. [4] A Vallet and R Borghi. Proc. Int. Conf. Multiph. Flow, Lyon, France, [5] A Vallet, A Burluka, and R Borghi. Atom. Sprays, 11: , 21. [6] E Labourasse, D Lacanette, A Toutant, P Lubin, S Vincent, O Lebaigue, J P Caltagirone, and P Sagaut. Int. J. Multiphase Flow, 33(1):1 39, January 27. [7] A Toutant, E Labourasse, O Lebaigue, and O Simonin. Comput. Fluids, 37(7): , August 28. [8] A Toutant, M Chandesris, D Jamet, and O Lebaigue. Int. J. Multiphase Flow, 35(12): , December 29. [9] A Toutant, M Chandesris, D Jamet, and O Lebaigue. Int. J. Multiphase Flow, 35(12): , December 29. [1] M Herrmann. Comput. Fluids, 87():92 11, October 213. [11] M Herrmann and M Gorokhovski. ICLASS 29, 11th Triennial International Annual Conference on Liquid Atomization and Spray Systems, Vail, CO, 29. [12] M Herrmann. J. Comput. Phys., 227(4): , 28. [13] M Herrmann. ICLASS-215, Tainan, Taiwan, August

8 [14] M Herrmann and M Gorokhovski. Proceedings of the 28 CTR Summer Program, pp , Stanford, CA, 28. Center for Turbulence Research, Stanford University. [15] A Scotti and C Meneveau. Physica D: Nonlinear Phenomena, 127(3-4): , March [16] Ke-Qi Ding, Zhi-Xiong Zhang, Yi-Peng Shi, and Zhen-Su She. Phys. Rev. E, 82(3):36311, September 21. [17] Z X Zhang, K Q Ding, Y P Shi, and Z S She. Physical Review E, 211. [18] Mark Owkes and Olivier Desjardins. J. Comput. Phys., 27: , August 214. [19] Ruben Scardovelli and Stéphane Zaleski. J. Comput. Phys., 164(1): , October 2. [2] J E Pilliod. An analysis of piecewise linear interface reconstruction algorithms for volume-offluid methods. Master s thesis, University of California Davis, [21] James Edward Pilliod Jr and Elbridge Gerry Puckett. J. Comput. Phys., 199(2):465 52, September 24. [22] Antonio Cervone, Sandro Manservisi, and Ruben Scardovelli. Comput. Fluids, 47(1):11 114, August 211. [23] J McCaslin and O Desjardins. Proceedings of the 214 CTR Summer Program, Stanford, CA, 214. Center for Turbulence Research, Stanford University. 8

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