Available online at ScienceDirect. Procedia IUTAM 15 (2015 )

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1 Available online at ScienceDirect Procedia IUTAM 15 (2015 ) IUTAM Symposium on Multiphase flows with phase change: challenges and opportunities, Hyderabad, India (December 08 December 11, 2014) Recent Numerical and Algorithmic Advances within the Volume Tracking Framework for Modeling Interfacial Flows Marianne M. François Los Alamos National Laboratory,Los Alamos, NM 87545, USA LA-UR Abstract A review of recent advances made in numerical methods and algorithms within the volume tracking framework is presented. The volume tracking method, also known as the volume-of-fluid method has become an established numerical approach to model and simulate interfacial flows. Its advantage is its strict mass conservation. However, because the interface is not explicitly tracked but captured via the material volume fraction on a fixed mesh, accurate estimation of the interface position, its geometric properties and modeling of interfacial physics in the volume tracking framework remain difficult. Several improvements have been made over the last decade to address these challenges. In this paper, the multimaterial interface reconstruction method via power diagram, curvature estimation via heights and mean values and the balanced-force algorithm for surface tension are highlighted The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license ( Peer-review Peer-review under under responsibility responsibility of Indian of Indian Institute Institute of Technology, of Technology, Hyderabad. Hyderabad. Keywords: volume tracking, volume-of-fluid, curvature, interface reconstruction, surface tension 1. Introduction Interfacial flows are multi-material flows comprised of two or more immiscible materials demarcated by interfaces. They are ubiquitous in nature and are encountered in many industrial applications. A prototypical example of such flows with phase change occurs in materials processing (casting, welding, additive manufacturing, etc.), where liquid-solid phase change (solidification/melting) takes place. An illustration of a simulation of laser spot welding using the Truchas code 1,2 is shown in Figure 1. High-fidelity simulations of interfacial flows require development of physical models and development of accurate numerical methods and robust algorithms. The position of the interface between the fluids and the interface physics needs to be predicted as part of the solution of The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of Indian Institute of Technology, Hyderabad. doi: /j.piutam

2 Marianne M. François / Procedia IUTAM 15 ( 2015 ) the flow equations. In addition, the discontinuity in fluid properties at the interface and interfacial physics (such as surface tension, phase change) present additional numerical challenges. There is a range of numerical techniques to model interfacial flows at the continuum level. They can be classified into three categories: (1) Lagrangian method, (2) mixed Eulerian-Lagrangian and (3) Eulerian method. In Lagrangian methods, the mesh on which the governing equations are solved adapts and moves with the interface. Lagrangian methods are particularly well suited and accurate for problems with small deformation (e.g. for fluidstructure interaction problem). In mixed Eulerian-Lagrangian methods, the governing equations are solved on a fixed background mesh and the interface is represented by marker points and moved in a Lagrangian way. Front-tracking methods 3 and immersed boundary methods 4 belong to the mixed Eulerian-Lagrangian class of method. These methods allow accurate representation of the interface, however they rely on ad-hoc algorithms to handle change in interface topology and ensure mass conservation. In Eulerian methods, the governing equations are solved on a fixed mesh and the interface is represented through an indicator function that is defined on each computational cell of the mesh and is advected along with the fluid velocity through an advection equation. Level-set 5,6 and volume tracking 7,8,9 (also known as Volume-Of-Fluid, VOF) methods belong to this category. These methods can handle large deformation and account for change in interface topology. However, because the interface is not explicitly tracked but captured, accurate estimation of the interface geometric properties, its advection and modeling of interfacial physics such as surface tension are challenging. In the level-set method, the indicator function is the distance function from the interface. In volume tracking method, the indicator function is the material volume fractions. The values of the volume fractions are between 0 and 1, as they represent the amount of a given material volume with respect to the computational cell volume. By its nature/design, volume tracking methods are conservative, whereas level-set methods are not and require correction algorithm. In this paper, we focus on the presentation of recent improvements that we have proposed in the numerical methods and algorithms within the context of the volume tracking method. Specifically, we highlight multimaterial interface reconstruction via power diagram, curvature estimation via heights and mean values and surface tension modelling via balanced-force algorithm. (a) (b) Fig. 1. Simulation of laser spot welding of 304 stainless steel with Truchas 1. (a) Temperature contours of a moving heat source on stainless steel piece; (b) Weld pool with flow circulation shown by velocity vector. Nomenclature Subscripts 1 fluid 1 2 fluid 2 c cell centered f face centered k material/fluid index x derivative with respect to x (horizontal direction) Superscripts n time level (N) normal component

3 272 Marianne M. François / Procedia IUTAM 15 ( 2015 ) (T) δ γ κ μ ρ σ A f, F n P t T Y tangential component delta function surface tension gradient interface curvature fluid dynamic viscosity fluid density surface tension computational cell face area volume fraction surface force gravitational acceleration unit interface normal computational cell face normal pressure time temperature fluid velocity vector height function in vertical direction 2. Multimaterial interface reconstruction via power diagram In volume tracking method, the interface position is not known, since the interface is not explicitly tracked by marker points, but is captured through the volume fraction. Knowing the interface position accurately is of primary importance in volume tracking as it is used to estimate the material fluxes geometrically to avoid numerical diffusion. These material fluxes are then used to estimate consistent mass and momentum advection terms that is of primary importance for simulating large density ratio 8. Finding the interface position given the volume fraction is commonly named interface reconstruction. Given the volume fractions in a computational cell, the goal is to find the interface position that matches the volume fractions assuming the shape of the interface. Early volume tracking method 10,11 used a simple interface reconstruction by assuming that the interface is aligned with one of the coordinate axis. This is referred to as simple line interface calculation (SLIC). In 1982, Youngs 12 extended the method to allow the interface to have an orientation within the cell based on the interface normal. The interface normal is calculated as the gradient of the volume fraction using finite-difference formulas, Green Gauss formula or least-square technique. This interface representation is referred to as piecewise linear interface calculation (PLIC). A parabolic interface reconstruction was proposed by Renardy and Renardy 13, but is much more complex to implement. Several studies have been conducted in PLIC area to improve the interface normal calculation 14, to generalize the method to unstructured meshes 7, to seek analytical solution on arbitrary mesh 15 and to develop smoothing algorithm to avoid discontinuity of interface planes across cells 16,20. While many improvements have been made in PLIC method, they have mostly focused on single interface reconstruction, i.e. interface between two materials. Reconstructing multiple interfaces within a computational cell (i.e. where more than two materials are present) has received little attention. The most common approach for multimaterial interface reconstruction has been to treat material one-by-one leading to a reconstruction that is strongly dependent on the order in which the materials are treated. The onionskin approach 12,17 consists of considering each material in a specific order, by cumulating the volume fractions in order to reconstruct the material interface sequentially. For the first interface, the interface is reconstructed using the volume fraction from the first material. Then, the volume fraction of the first material is added to the volume fraction of the second material and the interface between the first two materials and the remainder ones is reconstructed. This procedure is repeated until all interface planes have been reconstructed, which corresponds to

4 Marianne M. François / Procedia IUTAM 15 ( 2015 ) the number of material present in the cell minus one. The onion-skin technique works well for layer structures (for interface planes that are close of being parallel) but can result in overlapping layers for other cases. A more general and correct approach is the nested dissection method 18, where each material is separated from the others in a specified order. In the method, a pure polygon (or polyhedron) representing the first material is marked out from the cell, leaving a mixed polygon for the remaining materials. Then, a polygon representing the second material is marked out from the mixed polygon and the process continues until the last material is treated. With the correct material ordering, the interface reconstructed by one of the above methods is close to the correct configuration. However an incorrect ordering results in substantial degradation of the interface during advection. Motivated by the need to develop a material-order independent multimaterial interface reconstruction, Schofield et al 19,20 proposed a unique method that employs power diagram (weighted Voronoi diagram) to generate the multiple interfaces. In this method, the positions of materials in the computational cells are first approximated (either using particles or a slope-limited linear reconstruction of volume fraction). These approximate positions then serve as the generators to build the weighted Voronoi diagram. The weights are then adjusted iteratively in order to match the volume fractions of each material. The power diagram then results in a unique partition of the computational cells into multi-material regions. All materials are treated simultaneously and, as such, have no material order dependency. An illustration of a triple point configuration and its reconstruction using power diagram is shown in Figure 2. (a) (b) Fig. 2. Triple point configuration. (a) Exact interface; (b) Material independent interface reconstruction via power diagram. 3. Curvature estimation via heights and mean values Another challenge in the volume tracking framework, is the calculation of interface curvature. Estimating curvatures, involve taking spatial second-order derivatives of abruptly varying volume fractions. Various methods are presented and compared in Cummins et al 21. In this section, we review how to calculate interface curvature via heights and mean values. This is also referred to as the height function method. The height function technique 6,9,21 consists of computing curvature to second-order accuracy given integral mean values (or columns heights) associated with three adjacent columns on uniform or nonuniform orthogonal grid and to fourth-order accuracy given mean values associated with five adjacent columns 22. In multi-dimension, the mean values are computed in the dominant direction of the interface normal component. Once the mean values are computed, the curvature is calculated by differentiating the mean values (Y) as: (1) In two-dimension, the stencil used to compute the height function is 3x7, and in three-dimension, the stencil is 3x3x7. An illustration of the two-dimensional case is shown in Figure 3. Recent studies on the height function technique have focused on improving the estimation of the height function itself by choosing the best local rectangular set of mesh squares 23,24,25,26.

5 274 Marianne M. François / Procedia IUTAM 15 ( 2015 ) Fig. 3. Illustration of the height function method in three vertical columns with volume fractions shown in each computational cell. 4. Balanced-Force Algorithm for Modeling Surface Tension Modeling surface tension force, an interfacial force, in the volume tracking framework represents some challenges, since the interface is embedded within the computational mesh and not explicitly tracked. In this section, we present the balanced forced-algorithm of Francois et al 9 to model surface tension driven flow for the case of two fluids. We first review the governing equations for the fluid flow and volume tracking, and then present the formulation for surface tension force and the balanced-force algorithm Governing Equations The governing equations consist of the volume tracking equation, also known as volume fraction advection equation and of the mass and momentum conservation equations. The flow is assumed to be incompressible and governed by a single set of mass and momentum conservation equations on a fixed grid. The fluids are assumed to be Newtonian and immiscible and denoted by subscripts 1 and 2. In computational cells occupied with fluid 1, the volume fraction, f is unity, and in cells occupied with fluid 2, f is zero. For cells containing the interface bounding fluid 1 and 2, f lies between zero and unity. The volume fractions of fluid 1 and 2 sum to unity in every computational cell. The governing equations are the equation for the advection of the volume fraction f : f t + u f = 0 (2) and the mass and momentum conservation equations: u = 0 (3)

6 ( ) ρu t + ρu u Marianne M. François / Procedia IUTAM 15 ( 2015 ) ( ) = P + μ ( ( u + T u ))+ ρg + F S (4) where ρ and μ are the fluid density and viscosity, respectively, defined as: ρ = ρ 1 f + ρ 2 1 μ = μ 1 f + μ 2 1 ( f ) ( f ) (5) (6) The expression for density in Eq. (5) results from mass conservation, whereas the expression for mixture viscosity in Eq. (6) is a model. The system of equations (Eq. 3-4) is solved using a pressure correction projection method as described in Francois et al 9 and the volume fraction advection equation (Eq. 2) is solved by first geometrically reconstructing the interface by piecewise linear interface planes (PLIC) and secondly by geometrically calculating the advection fluxes to keep a sharp representation of the interface Surface Tension Models The Continuum Surface Force (CSF) method 27 is widely used in VOF, level-set, front-tracking methods. It consists of transforming the interfacial force to volume force via delta function. This method allows a smooth and continuous representation of the interfacial jump condition. Another method that has mainly been used in the context of level-set function but was extended by Francois et al 9 for volume tracking method is the Ghost Fluid Method (GFM) 28,29. In this method, the interfacial jump conditions due to surface tension are imposed via ghost values. This method allows a sharp representation of the interfacial jumps. Both methods are known to generate unphysical flow near the interface spurious currents particularly when surface tension forces are dominant. We have identified the source for these spurious currents (numerical errors) to several main factors. The first factor is the coupling of the surface tension force with flow algorithm. The formulation should seek proper balance of forces in the discretized momentum equation. If the flow algorithm ensures proper balance of forces, there should be no spurious current, as shown in Francois et al 9 when exact interface curvature is prescribed. The second factor is accurate estimations of curvature from volume fractions. Using the second-order accurate height function method in conjunction with the balanced-force algorithm resulted in drastic reduction of spurious currents. We now detail the continuum surface force (CSF) model of Brackbill et al 27 and its implementation using the balanced-force algorithm of Francois et al 9. The surface tension force is given as a volumetric force: F S = ( σκ ˆn + s σ )δ where σ is the surface tension, κ is the interface curvature, n ˆ is the interface unit normal vector, s is the surface (N) gradient, and δ is the delta function. The surface tension force is decomposed into a normal F S and tangential F S (T) volumetric force: where F S = F S (N) + F S (T) F S (N) = σκ ˆ n δ = σκ f F (T) S = ( s σ )δ = ( σ ˆn ( ˆn. )σ )δ (10) (7) (8) (9)

7 276 Marianne M. François / Procedia IUTAM 15 ( 2015 ) If the surface tension coefficient varies with temperature, i.e. σ = σ ref + γ ( T T ref ) with γ = σ T (11) the tangential component will be non-zero and it will generate thermo-capillary forces along the interface also known as Marangoni effects. In our balanced-force algorithm 9, both the normal and tangential surface tension force terms appear in the predictor step of the flow algorithm: ρ n+1 uc * = ρ n uc n Δt f u c n k ρ k f k u n f. n n+1 ( f )A f Δtρ f P c + Δt μ f A f nf u + T u + ΔtF (T ) Sc f ( ) ρ f (N ) F Sf ρ f n f c (12) The normal component of surface tension needs to balance the pressure jump across the interface. Therefore it is computed on the computational cell faces since the pressure gradient is evaluated at faces. This is to ensure proper balance of forces. Similarly, the tangential component of surface tension needs to balance the viscous stresses. They are both computed at cell centers. Verification and validation test cases of our surface tension force implementation can be found in Francois et al 9 for the normal component of surface tension and in Francois et al 30 for the implementation of the tangential surface tension component. Our surface tension model is implemented within the flow solver of the Truchas code. 4. Summary Over the last decade, substantial improvements have been made in the volume tracking method. In this paper, we have highlighted three recent advances: (1) a material-order independent multimaterial interface reconstruction, (2) the height function method to calculate interface curvature from volume fractions information and (3) the balancedforce algorithm for surface tension. These improvements have resulted in substantial reduction of numerical errors (drastic reduction of spurious currents in surface tension dominated flow) and reduction in the number of numerical input parameters from the user (material ordering and smoothing length to compute curvature). More accurate efficient, robust computational methods are always sought as well as modeling additional physics. Acknowledgements At LANL, I have had the privilege to work with outstanding colleagues and develop collaborations. In particular, I would like to thank Sharen Cummins, Edward Dendy, Rao Garimella, Douglas Kothe, Kin Lam, Raphael Loubere, Lucie Parietti, Sam Schofield, Jim Sicilian, Blair Swartz, Matthew Williams. I would also like to thank the organizers of the IUTAM symposium for their invitation. This work was performed under the auspices of the National Nuclear Security Administration of the US Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA References 1. Parietti L, Lam K, Validation of a multiphysics software package for simulations of pulsed laser welding at Los Alamos National Laboratory, Technical Report LA-UR , Los Alamos National Laboratory, The Telluride Team, Truchas: Physics and Algorithms, Technical Report LA-UR , Los Alamos National Laboratory 2003.

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