Reduction of parasitic currents in the DNS VOF code FS3D

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1 M. Boger a J. Schlottke b C.-D. Munz a B. Weigand b Reduction of parasitic currents in the DNS VOF code FS3D Stuttgart, March 2010 a Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, Pfaffenwaldring 21, Stuttgart/ Germany {boger,munz}@iag.uni-stuttgart.de b Institut für Thermodynamik der Luft- und Raumfahrt, Universität Stuttgart, Pfaffenwaldring 31, Stuttgart/ Germany {jan.schlottke,bernhard.weigand}@itlr.uni-stuttgart.de Abstract In this contribution we consider the occurrence and the origin of parasitic currents in the DNS (direct numerical simulation) VOF (volume of fluid) code Free Surface 3D (FS3D) for isothermal, incompressible two-phase flows in 3D. Up to now, the presence of parasitic currents has limited the range of phenomena accessible by FS3D. We consider the CSF (continuum surface force) model for the surface tension and take into account the discretization of the surface force as well as the approximation of curvature. We show that the spurious currents are strongly reduced by locally discretizing the surface force at each cell face. In a second step we additionally improve the curvature estimation of FS3D by implementing a height function method and local fitting of a paraboloid. The effects of the changes in discretization and curvature calculation are clearly visible as the simulation results show a reduction of the parasitic currents by two orders of magnitude. The implemented scheme is validated considering a static and an oscillating droplet. Keywords Parasitic currents VOF DNS Balanced-force algorithm Height function method Incompressible multiphase flow Preprint Series Issue No Stuttgart Research Centre for Simulation Technology (SRC SimTech) SimTech Cluster of Excellence Pfaffenwaldring 7a Stuttgart publications@simtech.uni-stuttgart.de

2 2 M. Boger et al. 1 Introduction The treatment of fluid interfaces is a crucial factor for the direct numerical simulation (DNS) of twophase flow. As the interface represents a discontinuity separating the two fluids, jump conditions have to be applied. Using the so called one-fluid formulation, one is able to work with a single set of equations for the whole flow domain. The coupling of the fluids at the interface is taken into account by the use of variable material properties. Applying the one-fluid approach, the change in material properties has to happen in a smooth regularized way, in order to prevent numerically induced oscillations in the vicinity of the interface. Moreover, the Navier-Stokes equations contain an additional volume force term for the surface tension, which is directly linked to the interface location via a delta function concentrated on the surface. From a physical point of view the surface force is balanced by the pressure jump across the interface. As the volume force is only different from zero at the interface, no additional jump conditions with regard to momentum conservation have to be applied. In general, surface tension is numerically approximated by a volume force using a continuum model. Applying the widespread continuum surface force (CSF, [1]) or continuum surface stress (CSS, [9]) model, many DNS codes suffer from unphysical parasitic currents in the vicinity of the surface. Up to now, the presence of parasitic currents has limited the range of phenomena accessible by such codes, including FS3D. Especially the simulation of droplets or bubbles with small radii at low speed was not possible. In these cases, the numerically induced parasitic currents were dominant and therefore led to unphysical simulation results. In order to overcome this phenomenon, many efforts have been taken by various researchers. For two dimensional computations [11] proposed a method to determine curvature more accurately using an estimator function, tuned with a least-squares-fit against precomputed reference data. An approximation of the surface tension based on spline interpolants was presented by [5]. For 3D calculations, [13] developed the PROST algorithm (parabolic reconstruction of surface tension) and [8] presented the PCIL method (pressure calculation based on the interface location) where the pressure forces at the interfacial cell faces are calculated according to the pressure imposed by each fluid on the portion of the cell face that is occupied by that fluid. This list is far from being complete as there are many other approaches aiming at the reduction of parasitic currents. While all of the mentioned methods are based on the development of completely new models, we are presenting in the following an approach to reduce spurious velocities that is based on the principle of a balanced-force discretization introduced by [4]. Starting from the original CSF model of [1] only minor changes are needed, mainly linked to curvature calculation which we choose to be evaluated according to [12]. The outline of the paper is as follows. We firstly introduce the governing equations for the DNS of two-phase flows, as they are used in the framework of the FS3D code. The following section is dedicated to the calculation of surface tension. In this context the origin of parasitic currents is discussed. This includes on the one hand the correct discretization of the surface tension terms in order to have a so called balanced-force algorithm. On the other hand, the issue of curvature estimation has also to be taken into account and for that we are using a height function method. The last section includes the numerical test cases of a static and an oscillating droplet, showing the improvement of the surface tension calculation. 2 Governing equations 2.1 Navier-Stokes equations The incompressible two phase flow is described by the Navier-Stokes equations. Using the one-fluid formulation, the continuity equation and the momentum equations have the following form (ρu) t u = 0, (1) + (ρu) u = p + ρk + µ[( u + ( u) T )] + f γ δ S. (2)

3 Reduction of parasitic currents in the DNS VOF code FS3D 3 Here u represents the velocity, ρ denotes the density, p is the pressure, t is the time, µ is the dynamic viscosity, ρk takes into account body forces, the term f γ is a body force that represents the influence of surface tension in the vicinity of the interface and δ S is a delta function concentrated on the surface. These equations resemble the Navier-Stokes equations for one phase flows with the exception of the appearance of the volume force f γ on the right-hand side of the momentum equations in Eq. (2). This approach is called the one-fluid formulation and it allows the use of a single set of equations in the whole flow domain. In the absence of mass transfer, there is no need of establishing any additional jump conditions at the fluid interface as they are implicitly taken into account by this formulation. The coupling of the two fluids is provided by variable material properties ρ and µ. They are chosen according to the fluid occupying a grid cell. At the interface the volume force f γ comes into play for taking into account the effects of surface tension. The velocity update at each time step is obtained by applying a projection method according to [2] to the momentum equation Eq. (2). This leads to a Poisson equation coupling pressure and velocity terms. Corresponding to the projection method, based on the velocity at the old time level u n one first computes a temporary velocity ũ already taking into account the influences of accelerations due to viscosity, convection, surface tension and other forces excluding the influence of pressure. This leads in the following step to the solution of the Poisson equation [ 1 ρ(f n+1 ) pn+1 ] = ũ t. (3) Combining temporary velocity ũ and pressure p n+1 from the Poisson equation, one obtains the divergence free velocity field at the new time level u n+1 = ũ t ρ(f n+1 ) pn+1. (4) 2.2 Interface tracking by the VOF method The interface tracking method implemented in the FS3D code is based on the volume of fluid (VOF) method developed by [7]. For the representation of the phases, an additional variable f is introduced 0 in the gaseous phase, f = 0 < f < 1 in cells containing a part of the interface, (5) 1 in the liquid phase, which represents the volume fraction of the liquid phase. A two dimensional example is given by Fig. 1. According to the one-fluid formulation, the above mentioned change in material properties is based y x Fig. 1 Fluid representation by the volume fraction f variable. 0.9 on the volume fraction variable f ρ(x, t) = ρ g + (ρ l ρ g ) f(x, t), (6) µ(x, t) = µ g + (µ l µ g ) f(x, t), (7)

4 4 M. Boger et al. where the indices l and g denote the liquid and the gaseous phase, respectively. The movement of the interface is tracked via the transport of the volume fraction f by f t + (fu) = 0. (8) While advecting the volume fraction f, one is interested in keeping the interface sharply resolved. Therefore at every time step the interface is reconstructed based on a PLIC (piecewise linear interface calculation) algorithm (see e.g. [14]). This geometrical reconstruction is used to determine the liquid and gaseous fluxes across the cell faces and it prevents the interface from being smeared across several grid cells during the advection. 3 Surface tension At an interface separating two immiscible fluids a pressure jump p 2 p 1 p = σκ, (9) appears according to the Young-Laplace equation. Here σ is the surface tension coefficient and κ represents the curvature. Hence, surface tension is directly proportional to the curvature κ. This jump in pressure has to be considered when simulating two phase flows. Therefore, different models have been developed in order to include surface tension at the interface. As it is obvious from Eq. (2), surface tension is accounted for as a volume force in the momentum equations. The corresponding force f γ on the right-hand side of the equation is only present at the interface while it vanishes in grid cells away from the interface. In principle one could use directly Eq. (9) in order to derive the volume force f γ f γ = σκˆn γ, (10) where ˆn γ is the interface unit normal vector. Though, as we are dealing with a diffuse interface formulation, we are in need of some numerical smoothing operations near the interface. Therefore, surface tension has to be calculated based on a numerical model. 3.1 The continuum surface stress (CSS) model A widespread model for surface tension is the continuum surface stress (CSS) model of [9]. By writing the Navier-Stokes equations in a momentum conserving form, they introduce the capillary pressure tensor T = σ(i ˆn γ ˆn γ )δ S, (11) where I is the unit tensor. Moreover, the capillary force can be written as the divergence of T f γ = T. (12) Supposing a smooth transition from one fluid to the other, normal vectors and delta function δ S are calculated based on the gradients of the smoothed volume fraction variable.

5 Reduction of parasitic currents in the DNS VOF code FS3D The continuum surface force (CSF) model The basic idea of the continuum surface force (CSF) model introduced by [1] can be described as follows. Instead of considering the fluidic interface as a sharp discontinuity, one supposes a smooth transition from one fluid to another. The interface is considered to have a finite thickness of O(h), corresponding to the smallest length scale resolvable by the computational mesh. Consequently, surface tension is also considered to be of continuous nature and it acts everywhere within the transition region. Brackbill proposes to replace Eq. (10) by f γ = σκ f. (13) This corresponds to a dispersion of the surface tension across the transition region, using the gradient of the volume fraction variable f to weight the dispersed volume force from Eq. (10). Implementing this approach in a CFD code, one has to consider the 1. spatial discretization of Eq. (13), 2. estimation of curvature κ, in order to prevent parasitic currents. The following sections will provide a discussion on both aspects. 3.3 Balanced-force algorithm The FS3D code uses a spatial discretization based on a Cartesian grid. The variables are stored on a staggered grid arrangement according to [6]. On such a MAC (marker-and-cell) grid, the scalar variables (f, p) are stored at the cell centers, while the velocities are stored at the centers of the cell faces. In order to guarantee an accurate, balanced-force discretization, according to [4], the surface tension terms (cf. Eq. (13)) have to be calculated at the center of the cell faces. Furthermore, it is of crucial importance that pressure and surface tension are discretized in the same way. Moreover, special care has to be taken in order to evaluate the gradient f. In the FS3D code, based on [1], the evaluation of the gradient has been performed on a stencil of 18 cells for 3D calculations up to now. For explanation purposes Fig. 2 is illustrating a two dimensional example. Here, the gradients at the cell face centers of row (j) are given by the corresponding values in red. Taking the face at (i+1/2) as an example, the gradient f xi+1/2,j is based on the gradients in the rows (j 1), (j), (j +1) that are calculated on the basis of the cells adjacent to the face (i + 1/2) respectively, e.g. f xi+1/2,j = f (i+1,j) f (i,j) x for row (j). Here, designates the local gradient with respect to the cell face. Afterwards, the gradient at the position x i+1/2,j is obtained as f xi+1/2,j = 1 4 ( fxi+1/2,j f xi+1/2,j + f xi+1/2,j+1 ). (14) This leads to a total stencil for cell (i, j) that is surrounded by the dashed lines in Fig. 2. It is obvious that the above discretization implies a coupling of the rows (j 1), (j), (j + 1) for the surface tension calculation via the evaluation of the gradients. Having a closer look at the discretization of the Poisson equation, Eq. (3), in x direction for the cell (i, j), one finds ( ) p(i+1,j) p (i,j) 1 x ρ (i+1/2,j) p (i,j) p (i 1,j) ρ (i 1/2,j) = ũ(i+1/2,j) ũ (i 1/2,j). (15) t Here it is clearly visible that all pressure values are taken from row (j). As for the above mentioned discretization of the surface tension used to evaluate the velocity ũ, the rows (j 1), (j + 1) are also included via the gradient evaluation. We found this coupling to be one of the causes for the parasitic currents in the FS3D code. Therefore we changed the gradient evaluation to a more local formulation only taking into account direct neighbors of the cell faces. This leads for the cell face (i + 1/2, j) to f xi+1/2,j = f xi+1/2,j = f (i+1,j) f (i,j). (16) x

6 6 M. Boger et al. Fig. 2 Evaluation of f based on a stencil of 6 cells (red) and on the direct neighbors of the cell face (blue). The box surrounded by the dashed lines marks the 6 cell stencil for the cell face (i + 1/2, j) Returning to Fig. 2, the two approaches can be compared directly. According to the two methods, the gradients are given for the previous method (red) and the local approach (blue) in row (j). While the transition from one fluid to the other is spread over four cell faces with the previous approach, the local approach only uses two cell faces to disperse the jump in pressure. In 3D, the gradients for the different space directions are evaluated in an analogous way, only taking into account direct neighbors of the respective cell face. Besides the discretization of the surface tension force (Eq. (13)), the correct estimation of surface curvature is very important and shall be discussed in the following section. 3.4 Curvature estimation In the context of the VOF method, difficulties in determining topology information like normal vectors and curvature are due to the discrete nature of the volume fractions. Contrary to a level-set function, the VOF field is not smooth and thus the correct curvature as the second derivative is hard to obtain. Direct calculation of curvature from the given volume fractions leads to high frequency (order of the grid) errors [aliasing error, 3]. This is why most CSF implementations do a smoothing of the VOF field prior to the calculation of curvature. In addition, this error does not vanish with grid refinement. For the method presented here, the curvature is calculated based on a height function approach, coupled to a local paraboloid fitting if the grid resolution is not sufficient. The procedure is inspired by the article of [12]. The height function approach is a geometrical one. The stencil used to determine the local height is not fixed [as used by 4, e.g.], but adapts itself to capture the local topology in an optimal way. As already mentioned, grid resolution does not always permit the determination of curvature via the height function approach. This is the case for highly curved topology, e.g. during breakup, or in case of very coarse grids. Local curvature is then determined by fitting a paraboloid to known points on the surface. These points can either originate from local heights or the barycenters of the reconstructed surface (using PLIC) are used for this purpose. Once the paraboloid, given by (in 3D) f(a i, x) = a 0 x 2 + a 1 y 2 + a 2 xy + a 3 x + a 4 y + a 5, (17)

7 Reduction of parasitic currents in the DNS VOF code FS3D 7 is fitted (via least squares fit) to these points, the curvature can easily be calculated as the second derivative, κ = 2 a 0(1 + a 2 4) + a 1 (1 + a 2 3) a 2 a 3 a 4 (1 + a a2 4 )3/2. (18) 4 Numerical examples In the following, two numerical test cases are performed in order to validate the changes in the FS3D code. First, the simulation of a static spherical droplet is performed. Afterwards surface tension driven oscillations are simulated. 4.1 Static drop in equilibrium For this test we consider a static drop in equilibrium without gravity. In the absence of any outer y 0.2 y x (a) Previous CSF model v max = cm s x (b) New approach CSF model v max = cm s (vectors scaled by factor 100). Fig. 3 Velocity field around the droplet in the xy plane at t = 0.03s (cut through the center of the droplet). forces, the drop should maintain its initial shape and velocities should be zero throughout the whole computation. Moreover, for a spherical droplet the curvature κ can be determined analytically as κ = 2 R, (19) where R is the radius. Therefore Eq. (9) can be used to derive the pressure jump across the interface. We simulate a spherical water drop with radius R = 10 3 m that is surrounded by air. The drop is placed at the center of a cubic computational domain with an edge length of m. The domain is discretized by 64 equidistant grid cells in each space direction. The resulting velocities for both the standard CSF implementation and our new approach are shown in Fig. 3. The vectors in Fig. 3(b) are scaled by a factor 100 compared to the length of those in Fig. 3(a). There is a dramatic reduction of spurious currents in the whole computational domain.

8 8 M. Boger et al v max [cm/s] 10-1 CSS CSF CSF (balanced,hf) e kin [g/(cm s²)] CSS CSF CSF (balanced,hf) time [s] time [s] (a) Maximum velocity v max. (b) Kinetic energy e kin. Fig. 4 Comparison of maximum velocity and kinetic energy in the static droplet case for the different surface tension models. (a) Previous CSF model. (b) New approach CSF model. Fig. 5 Pressure around the droplet in the xy plane at t = 0.03s (cut through the center of the droplet). In order to get a more quantitative evaluation, the temporal evolution of the maximum velocity and the kinetic energy is depicted in Fig. 4. Compared to the standard CSS and CSF models, our new method shows a maximum velocity which is reduced by two orders of magnitude. While this maximum velocity only gives information on one location in the domain, the kinetic energy represents all spurious currents. Again, a strong reduction with the new model can be observed (five orders of magnitude). As tests with prescribed, constant curvature and completely vanishing velocities have shown, the remaining spurious currents are due to the still not perfect calculation of curvature. Finally, the pressure jump across the interface is predicted quite well by the new method as can be seen in Fig. 5. The Young-Laplace equation Eq. (9) predicts a pressure jump p = N/m that is accurately reproduced by the new method. One also clearly notifies the sharper interface resolution due to the presented, more local discretization of the surface tension force.

9 Reduction of parasitic currents in the DNS VOF code FS3D Droplet oscillations For this test case we simulate surface tension driven oscillations. Therefore we start the simulation with a non-moving ellipsoidal droplet with the polar radius being bigger than the equatorial radii. For small deformations there exists an analytical approach by [10]. Neglecting the density of the surrounding fluid, he derived the following equation for the angular frequency ω 2 σ = l(l 1)(l + 2) ρ 1 R 3. (20) The parameter l characterizes the oscillation mode. For ellipsoidal deformations l = 2. Here R represents the radius of the equivalent spherical droplet corresponding to the mass of the ellipsoidal droplet. Deviation [%] Mass [g] T (Eq. (20)) [s] CSS CSF CSF (balanced,hf) Table 1 Droplet oscillations, polar radius 10% bigger than the equatorial radii; Comparison of the surface tension models CSS CSF CSF (balanced,hf) surface area [cm²] time [s] Fig. 6 Droplet oscillations, polar radius 10% bigger than the equatorial radii, mass m = g. Comparison of the surface tension models. The deviation of the resulting oscillation periods compared to the analytical value from Eq. (20) for various test cases with droplets of different sizes is given in Table 1. While the results for the standard CSF and CSS models are acceptable, one can observe an improvement using the new surface tension method. This improvement gets more evident by looking at the amplitude of the oscillation, given in Fig. 6 by the temporal evolution of the surface area. While the periodic deformation of the droplet is described nicely with a constant minimum and a slight damping by the new method, the standard CSF method shows an erratic development of the droplet s surface area. Looking at the CSS method, things get

10 10 M. Boger et al. worse as the amplitude of the droplet oscillation increases in time, thus there is a generation of kinetic energy due to the spurious currents. 5 Conclusion A new method for the calculation of surface tension within the framework of the VOF code FS3D has been presented. The main features are a balanced discretization of the volume force at the interface as well as an improved curvature calculation using height functions and a local paraboloid fitting. The new method shows a strong reduction of spurious currents for the calculation of a drop in equilibrium. Here the maximum velocity of the spurious currents could be reduced by two orders of magnitude. Looking at the case of an oscillating droplet, there is a clear improvement with regard to oscillation frequency and the temporal evolution of the oscillation amplitude. Future investigations will be amongst others the calculation of the rise behavior of small bubbles or Marangoni force driven phenomena, where it is crucial to have spurious currents limited. Acknowledgment The authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart and within the SFB-TR 40. References 1. Brackbill, J. U., Kothe, D. B., and Zemach, C. (1992). A continuum method for modeling surface tension. Journal of Computational Physics, 100(2): Chorin, A. J. (1968). Numerical Solution of the Navier-Stokes Equations. Mathematics of Computation, 22(104): Cummins, S. J., Francois, M. M., and Kothe, D. B. (2005). Estimating curvature from volume fractions. Computers & Structures, 83(6-7): Francois, M. M., Cummins, S. J., Dendy, E. D., Kothe, D. B., Sicilian, J. M., and Williams, M. W. (2006). A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework. Journal of Computational Physics, 213(1): Ginzburg, I. and Wittum, G. (2001). Two-Phase Flows on Interface Refined Grids Modeled with VOF, Staggered Finite Volumes, and Spline Interpolants. Journal of Computational Physics, 166: (34). 6. Harlow, F. H. and Welch, J. E. (1965). Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface. Physics of Fluids, 8(12): Hirt, C. W. and Nichols, B. D. (1981). Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics, 39(1): Jafari, A. (2007). An improved three-dimensional model for interface pressure calculations in freesurface flows. International Journal of Computational Fluid Dynamics, 21:87 97(11). 9. Lafaurie, B., Nardone, C., Scardovelli, R., Zaleski, S., and Zanetti, G. (1994). Modelling Merging and Fragmentation in Multiphase Flows with SURFER. Journal of Computational Physics, 113(1): Lamb, H. (1957). Hydrodynamics. Cambridge University Press. 11. Meier, M., Yadigaroglu, G., and Smith, B. L. (2002). A novel technique for including surface tension in PLIC-VOF methods. European Journal of Mechanics - B/Fluids, 21(1): Popinet, S. (2009). An accurate adaptive solver for surface-tension-driven interfacial flows. Journal of Computational Physics, 228(16):

11 Reduction of parasitic currents in the DNS VOF code FS3D Renardy, Y. and Renardy, M. (2002). PROST: A Parabolic Reconstruction of Surface Tension for the Volume-of-Fluid Method. Journal of Computational Physics, 183(2): Rider, W. J. and Kothe, D. B. (1998). Reconstructing Volume Tracking. Journal of Computational Physics, 141(2):

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