Finite volume method for two-phase flows using level set formulation

Transcription

1 Finite volume method for two-phase flows using level set formulation Peter Frolkovič,a,1, Dmitry Logashenko b, Christian Wehner c, Gabriel Wittum c a Department of Mathematics and Descriptive Geometry, Slovak University of Technology, Radlinskeho 11, SK Bratislava, Slovakia b Steinbeis-Forschungszentrum 936, Schillerstr. 21, D Ölbronn-Dürrn, Germany c Goethe-Zentrum für Wissenschaftliches Rechnen, Goethe-Universität Frankfurt am Main; Kettenhofweg 139; D Frankfurt am Main, Germany Abstract We introduce a novel finite volume method for the problem of two-phase incompressible flows using a level set formulation to represent the interface between phases. The method profits from using unified discrete local balance formulations for the approximation of the conservation laws for all related values the momentum, the mass and the level set function. Possible jumps of the pressure and the directional derivative of velocity at the interface are modeled directly within the method using the approach of extended approximation spaces. For the chosen numerical tests, we have found very good conservation properties of the proposed method, negligible parasite currents and good overall performance. These results are obtained without artificial post-processing steps often required in numerical methods for the two-phase flows based on the level set formulation. Key words: method incompressible two-phase flow, level set method, finite volume 1. Introduction One of the most active research fields in computational fluid dynamics is the numerical simulation of relevant examples for immiscible incompressible two- This work was supported by the German Research Foundation (DFG) under the project Wi 1037/11-3, the Federal Ministry of Education and Research (BMBF) under the project FKZ 02E10326 and 03SF0349D, and the state of Hessen under the program LOEWE (Hic4Fair). Corresponding author addresses: peter.frolkovic@stuba.sk (Peter Frolkovič), dimitriy.logashenko@gcsc.uni-frankfurt.de (Dmitry Logashenko), christian.wehner@gcsc.uni-frankfurt.de (Christian Wehner), wittum@gcsc.uni-frankfurt.de (Gabriel Wittum) 1 The first author was supported by the grants APVV and VEGA 1/0269/09. Preprint submitted to Elsevier September 18, 2009

2 2 phase flow. This great interest has contributed to the development of many, often substantially different numerical methods for this kind of problems. One of them is based on the level set formulation [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] in which the interface between two phases is represented implicitly by the zero level set of a smooth function. This approach has important advantages over the other methods: It is simpler than the direct representation of the dynamic interface using Lagrangian-type methods and moving grids. Besides that, it allows straightforward computation of geometric quantities of the interface and handling of topological changes during tracking of its position. Nevertheless, many variants of this approach have substantial drawbacks. In particular, it is a popular belief that the level set formulation suffers from inaccurate mass conservation [12, 3, 8, 6, 13, 9, 14, 15, 11] and from the necessity of postprocessing steps like the reinitialization of the level set function [12, 3, 2, 6, 9, 15] or additional smoothing after the tracking or reinitialization [3, 7, 14, 9, 16, 15]. Furthermore, in the implicit treatment of the interface condition, e.g. as in the approach of the localized surface tension force, nontrivial algorithms of the smoothed Dirac function are often used whereas the accuracy of this approach has certain open issues [3, 2, 17]. Finally, the flow field obtained using the level set formulation may contain so-called parasite currents arising due to inappropriate choice of the approximation spaces for the pressure [16, 18, 15]. In this paper we present an attempt to develop a pure finite volume discretization of the immiscible incompressible two-phase flow that overcomes these difficulties when applying the level set formulation. Our method requires no artificial post-processing steps for stable behavior and good conservation properties, it minimizes the parasite currents and uses a sharp treatment of the numerical interface. The idea is to apply analogous discretizations of the local balance formulations for the conservation of momentum, mass and level sets on identical control volumes. The discretization fits into a general class of finite volume methods based on a dual mesh with respect to the primary grid consisting of triangles (2D case) or tetrahedra (3D case). The method exploits substantially the related piecewise linear interpolation of nodal values. To enable the sharp treatment of the interface condition in the method, we extend the approximation spaces for the pressure and the velocity by additional degrees of freedom only for the elements intersected by the interface. Such a construction of extended approximation spaces is inspired by [18] where it is used to construct the extended pressure finite element space, see also related references there. To avoid any smoothing of discontinuous properties over the interface, similarly to the ghost fluid method [19, 2], we extend continuous fluid properties across the interface and use the same finite volume mesh independently on the position of the interface. The approximation of the curvature is computed using the quadratic interpolation of the level set function on a refined grid. In each time step, we refine only the elements that are near or at interface and use this locally refined grid only in the numerical solution of the level set equation.

3 3 We present relevant numerical tests with the simulation of a rising bubble that confirm the desired properties of our method. Note that in this paper we aim to introduce the method itself in detail, therefore we study extensively its most important aspects (like the role of the reinitialization and of the character of the grids) for two representative numerical examples. At this stage of the development, the solver for the advection equation and the Navier-Stokes solver are fully decoupled. Such splitting of two problems is often used in the numerical simulation of two-phase flows [8, 6, 9, 16, 15] and requires some restrictions on the choice of discretization parameters [16, 20, 15, 21]. The paper is organized as follows. In Section 2, we present the system of the partial differential equations describing the flow of two immiscible incompressible fluids. In Section 3, we introduce the principles of the flux-based finite volume method that is used to discretize each partial differential equation. Particularly, in Section 3.1, we apply the method to find the numerical solution of the advection equation and, in Section 3.2, to discetize the single-phase Navier- Stokes equations. The extensions of the method for the numerical solution of the two-phase flow problem are described in Section 4. The computation of the curvature is given in Section 4.1, and the extended approximation spaces for pressure and velocity are introduced in Section 4.2. The simulation method is summarized in Section 4.3 and the reinitialization of the level set function is discussed in Section 4.4. Finally, in Section 5, the results of the numerical experiments are presented. 2. Mathematical models Let Ω R d denote a fixed domain that contains two different immiscible incompressible fluids. Let the first fluid occupy the time-dependent subdomain Ω 1 = Ω 1 (t) Ω with the interface Γ = Γ(t) := Ω 1 (t) that does not touch the boundary of Ω: Γ Ω =. The second fluid occupies subdomain Ω 2 = Ω 2 (t) := Ω \ Ω 1 (t), e.g., the first fluid constitutes a bubble in the second one. We denote the constant densities and dynamic viscosities of these fluids by ρ (k) and µ (k), k = 1, 2. The globally defined pressure p = p(t, x) : [0, + ) Ω R and velocity u = u(t, x) : [0, + ) Ω R d satisfy the systems of the Navier- Stokes equations: } ρ (k) ( t u + u( u) T ) = T + ρ (k) g, x Ω k, k = 1, 2, t > 0, (1) u = 0, where g is the gravity acceleration vector and T = pi + µ (k) ( u + ( u) T ) is the stress tensor. An initial condition for u is required at t = 0 and some appropriate boundary conditions must be supplied for x Ω. The two systems (1) are coupled by the inner boundary conditions at Γ that can be formulated as two jump conditions: [u] = 0, [T] n = σκn, x Γ, t > 0, (2)

4 4 where [ ] denotes the jump when crossing Γ from Ω 1 to Ω 2, n is the outer unit normal vector with respect to Ω 1, σ is the surface tension factor and κ is the sum of principle curvatures for Γ, κ = n (so that κ > 0 if Γ is a sphere). Due to the first condition in (2), u is continuous at Γ. In general, p and directional derivatives of u may have a jump across the interface. To define later the approximation spaces for p and u that resolve these jumps, we use the approach from [2] and reformulate (2) into two equations that express the jumps of p and u explicitly in the terms of the continuous quantities. We restrict our presentation to the two-dimensional case, the details for the general situation can be found in [2]. Choose time t and a fixed point x Γ(t). Let n and t be the normal and tangential unit vectors to Γ(t) at x. Then for the expansion u = un + vt, one can show [2] that only the component v n exhibits a jump across Γ and all other derivatives are continuous: [ u n] = [ u t] = [ v t] = 0. Then the second equality in (2) yields: [p] = σκ + 2 [µ] u n, x Γ, t > 0, (3) [µ v n] = [µ] u t, x Γ, t > 0, (4) where µ = µ(t, x) : [0, + ) Ω R is the viscosity defined in the whole domain Ω, i.e., µ(t, x) = µ (k), if x Ω k (t). Note that if µ is continuous across the interface, then all derivatives of u are continuous at Γ, and (3) can be reduced to a much simpler condition [p] = σκ, x Γ, t (0, T ). (5) For this purpose the piecewise constant form of µ is sometimes smoothed over the interface, but this approximation is not used in our method. The interface Γ is initially given by Γ(0) and moves in time with the velocity u(t, x), x Γ. To model its dynamic position Γ(t) = Ω 1 (t) and to compute its geometric quantities n(x) and κ(x), x Γ(t), we use the level set formulation. Let φ 0 = φ 0 (x) be a smooth function such that Γ(0) = {x Ω, φ 0 (x) = 0}, Ω 1 = {x Ω, φ 0 (x) < 0} and Ω 2 = {x Ω, φ 0 (x) > 0}. Then the solution φ = φ(t, x) of the following advection equation t φ + u φ = 0, φ(0, x) = φ 0 (x), x Ω, t > 0, (6) retains in time the properties of the initial function φ 0 (x): the zero level set of φ coincides with Γ(t), φ(t, x) is negative for x Ω 1 (t) and positive for x Ω 2 (t). Furthermore, if φ is sufficiently smooth then n = φ φ, κ = ( ) φ. (7) φ Boundary conditions must be supplied for (6). We consider them in Section 3.1 below. The system (1), (4) and (6) constitutes a closed system of equations for the two-phase flow problem.

5 5 (a) (b) (c) Figure 1: Implicit representation of the interface: (a) the distance function for the initial condition, (b) the level set function for some ˆt > 0 with no reinitialization, (c) the reinitialized function. In general, the level set function φ(t, x) may get very steep or flat gradient in some regions near Γ(t) for t > 0 (cf. Fig. 1(b) for some illustration). This causes essential difficulties in the numerical computations, e.g., inaccurate computation of the curvature. These problems do not arise in some time interval, say t [0, ˆt], if an appropriate initial level set function φ 0 like the so-called signed distance function is chosen (as in Fig. 1(a)). In general, it is reasonable to replace (reinitialize) φ at t = ˆt with a different level set function with more favorable properties (cf. Fig. 1(c)). To this end, the following nonlinear advection equation can be solved for ϕ = ϕ(s, x) : [0, ) Ω R: s ϕ + sign(φ(ˆt, x))) ϕ ϕ ϕ = sign(φ(ˆt, x)), x Ω, s > 0, (8) with the initial condition ϕ(0, x) = φ(ˆt, x). One can show [1] that the stationary solution of (8) is the signed distance function (in a weak sense) to the zero level set of φ(ˆt, x) = 0 and is reached at some finite time S > 0. Since the characteristic curves of (8) spread out from the zero level set, any solution ϕ(ŝ, x), 0 ŝ < S is an appropriate choice for the reinitialization of φ(ˆt, x) in the computation of (1), (4) and (6) for t > ˆt until at some further time point, the gradients of the analytical solution of (6) become too steep or flat again. 3. Finite volume discretization In this work, we discretize all partial differential equations presented in Section 2 using the same finite volume discretization method. The main advantage

6 6 T e x i ω i Figure 2: The primal/dual mesh and numerical interface. of such an approach is the unified treatment of all the presented balance laws with similar representations of important quantities in the different equations. In the literature, one can find many variants of the finite volume method. Our choice is related to the so-called control volume [22] or finite volume [23] element methods. The method was used in [24, 25] to solve the advection equation and in [26, 22, 27] to solve the single-phase Navier-Stokes equations. We use the following notation. The time interval is discretized by 0 = t 0 < t 1 <... < t n <..., and τ n := t n+1 t n. The domain Ω is supposed to be polygonal and covered by a conformal triangulation T that consists of triangles if d = 2 and tetrahedra if d = 3. Particular elements of such a triangulation are denoted by T e T, e = 1, 2,..., E and the grid nodes by x i T, i = 1, 2,..., I. If x i T e, we say that i Λ e, and, analogously, e Λ i. We associate a computation cell ( control volume ) with each vertex x i by constructing a conformal dual mesh of finite volumes ω i, i = 1, 2,..., I. Our choice are the so-called barycenter based control volumes. To construct ω i, the closed boundary ω i is created by connecting certain line segments (d = 2) or planar segments (d = 3). The segments are denoted by γij e, and they are given by γe ij = T e ω i ω j or γi0 e = T e ω i Ω, see Fig. 2 for some illustration. The unit normal vector n e ij to γe ij is constant for all x γij e and points from ω i to ω j, i.e. n e ji = ne ij. Finally, xe ij denotes the barycenter of γij e that can be used in numerical integration over ω i. Furthermore, we use the analogous notation to define some numerical approximations. The simplest quadrature rule is given by f(x) dx ω i f i, (9) ω i where f i = f(x i ). The integral of f over ω i is approximated by f(γ) dγ γij f e ij e, (10) ω i e,j

7 7 where the sum is realized for e Λ i and j Λ e, j i. To compute fij e, some specific interpolation of nodal values for f must be used, see the discretization schemes in particular cases later. All the finite volume schemes in our method are obtained by integrating the partial differential equations (1), (6), and (8) over ω i at time t = t n. After using integration by parts (when possible), the quadrature rules (9) and (10) are applied to approximate the involved integrals. Finally, some time discretization is applied to obtain fully discrete systems, therefore, the upper index n for time is added in (9) and (10). All unknown functions p, u, φ, (and ϕ if the reinitialization is used) are approximated by their values at grid nodes x i T. The key role in our method is played by the level set function for which the following notation is introduced. Nevertheless, similar treatment will be used for other functions. The nodal values φ 0 i are given by evaluating the initial condition in grid points x i, i = 1,..., I. The discrete equations obtained by finite volume discretization are used to determine the unknown values φ n i, i = 1,..., I, and n = 1, 2,.... If Dirichlet boundary conditions are available for x i Ω, they are used to determine φ n i, n 0. Very often, the usual finite element interpolation φ n (x) := φ n i N i (x), x T e, (11) i Λ e is used in our method. The local basis functions N i (x) are the piecewise linear functions such that N i (x j ) = δ ij. Consequently, the notation e is used to denote the constant approximation of the gradient inside of T e, i.e., e φ n = φ n (x), x T e. In Sections 3.1 and 3.2, we describe particular details of our finite volume method when applied separately to partial differential equations (6) and (1) Discretization of advection equation To apply our finite volume method for the solution of (6), we follow the so-called high-resolution flux-based level set method as described in [24]. To introduce it independently of (1), we suppose that the velocity field, say v = v(x), is known and fixed in time. The idea is to write t φ + v φ = 0 in the conservative form t φ + (vφ) = φ v (12) with the source term on the right hand side due to v. The essential part of the method is the approximation of v at x i T. To define ( v) i v(x i ) in a way consistent with our finite volume method, we integrate v over ω i by parts and use (10) to obtain ( v) i = 1 ω i γij e (v n) e ij 1 (v n) (γ) dγ. (13) ω e,j i ω i

8 8 In general, we do not suppose that ( v) i = 0. Applying the finite volume method described in the beginning of Section 3 for (12) at t n+1/2 = t n τ n, we obtain the semi-discrete equations ω i t φ n+1/2 i + e,j γij φ e e,n+1/2 ij (v n) e ij = ω i φ n+1/2 i ( v) i. Finally, using the second order accurate time discretization and (13), we obtain after a simple algebraic manipulation φ n+1 i = φ n i τ n e,j γ e ij ω i ( φ e,n+1/2 ij ) φ n+1/2 i (v n) e ij, (14) where n = 1, 2,..., and i = 1,..., I. To complete scheme (14), we have to define the values φ e,n+1/2 ij and φ n+1/2 i. For this, the reconstructed gradients i φ n φ(t n, x i ) are introduced by i φ n = 1 ω i e Λ i T e ω i e φ n. (15) By the usual upwind arguments together with the finite Taylor series approximation and t φ(t n, x i ) v i i φ n we obtain { φ e,n+1/2 φ n ij = i + i φ n (x e ij 1 2 τ n v i ), (v n) e ij > 0, φ n j + jφ n (x e ij 1 2 τ n v j ), (v n) e ij < 0. (16) For the boundary flux we define φ e,n+1/2 i0 = φ n i + i φ n (x e i0 1 2 τ n v n i ). (17) Note that (17) is identical to (16) for the outflow boundary. Following [24], the value φ n+1/2 i is defined as the arithmetic mean of the averaged values of φ e,n+1/2 ij at inflow and outflow part of ω i, φ n+1/2 i = 1 [(v n) e ij ]+ φ e,n+1/2 ij [ (v n) e ij ]+ φ e,n+1/2 ij e,j e,j 2 [(v n) e + ij ]+ [ (v n) e (18) ij ]+ e,j where [a] + = a, if a > 0, and [a] + = 0 otherwise. If one of the denominators in (18) vanishes, the first order accurate approximation φ n+1/2 i = φ n i instead of (18) is used [24, 28]. Other useful replacements of (15) and (18) that can improve the precision of the method in the case of structured grid, can be found in [25]. The numerical scheme (14) is fully explicit in time. The following restriction on τ n is used in all our numerical experiments: τ n 1 min i ω i e,j [ (v n) e ij ]+ 1, (19) e,j

9 9 where the left hand side can be viewed as the grid Courant number [28, 24]. Numerical results that were obtained using (14) for several well-known benchmarks with externally given velocity field like the single vortex example or the Zalesak s disc were published in [24]. These results confirm the second order accuracy of the method in the case of smooth solutions Discretization of Navier-Stokes equations To apply the finite volume method described in the beginning of Section 3 to the Navier-Stokes equations, we follow [26, 22, 27] and refer to these works for the details. In this section, we give only a brief overview of the discretization to prepare for the extension to two-phase flow problem in Section 4.2. Assume that there is no interface in the domain. We denote by ρ e and µ e the avarages of the density and the viscosity over grid element T e. Furthermore, let ρ i := 1 ω i ω i T e ρ e. e Λ i Then the discretization of (1) at t = t n and x = x i has the form ω i ρ i t u i (t n ) + e,j γij e ( ρ e (uu T ) e ij(t n ) T e ij(t n ) ) n e ij = ω i ρ i g, (20) γij u e e ij(t n ) n e ij = 0 (21) e,j where the stress tensor T is evaluated at integration points x e ij γe ij : T e ij(t n ) := p n (x e ij) + µ e ( e u n + ( e u n ) T ). (22) The time derivative in (20) is approximated by the implicit Euler method. This leads to the fully implicit discretization that is unconditionally stable. It is important to note that if the linear interpolation u n (x) is used in (21), i.e., u e ij (tn ) = u n (x e ij ), the discretization (20 21) becomes unstable. For the stabilization, a special interpolation for u e ij (tn ) that is based on the nodal values of the velocity and the pressure is introduced, see [27] for the details. In this work, we use the so-called FLOW stabilization, see [22]. Consequently, for velocity u n in the solution of (20 21), the approximation ( u n ) i defined in (13) can be non-zero. For the numerical solution of the discretized system (20 21), we use a fixed point iteration as nonlinear solver. For the solution of the sparse linear systems arising in the nonlinear solver, we apply the BiCGStab-iteration with a geometrical multigrid preconditioner (V-cycle), see [29, 30]. As smoother, we use the point-block ILU β -decomposition that proved to be very efficient for this type of discretizations, see [31].

10 10 4. Discretization of two-phase flow problem Before describing the finite volume method for two-phase flow problem (1), (2) and (6) in its complete form, we introduce some important parts of the method. In Section 4.1, we derive a piecewise constant approximation κ n (x) of the curvature at each time level t n for intersected elements T e T, i.e., κ n (x) = κ e,n, x T e. In Section 4.2, we define the extended approximations p n (x) and ũ n that allow jumps of the pressure and the directional derivative of velocity in some elements T e. After that, in Section 4.3, we summarize our method for the numerical solution of the two-phase flow problem. Finally, some important details on the numerical solution of reinitialization equation (8) are given in Section Approximation of the curvature Let the interface Γ be implicitly given as the zero level set of some function Φ = Φ(x). The exact curvature κ(x) of Γ at x Γ satisfies with κ(x) = Φ(x)H(Φ)( Φ(x))T Φ(x) 3, (23) ( ) Φx2x H(Φ) = 2 (x) Φ x1x 2 (x), (24) Φ x1x 2 (x) Φ x1x 1 (x) see, e.g., [32], where the 2D and 3D cases of (23) can be found. In our method, the curvature is needed only in the elements intersected by the interface. In what follows, we consider T e to be intersected, if φ n i φn j 0 for some i, j Λ e. Let T e T n be such an element. To use (23), we need to approximate the second derivatives of Φ. For this, additionally to piecewise linear approximation φ n, we introduce in T e a quadratic function Φ n : T e R that satisfies the relation Φ n (x i ) = φ n i, i Λ e. (25) Then the piecewise constant approximation κ n (x) = κ e,n, x T e can be introduced by κ e,n = φn (x)h(φ n (x))( φ n (x)) T φ n (x) 3. (26) To construct Φ n (x), we refine T to obtain a (locally) refined triangulation T n. This approach is illustrated in Fig. 3, see also the related Fig. 2. Intersected triangles T e T are divided regularly into four triangles that belong to T n. The quadratic function Φ n (x), x T e is then obtained by interpolating the nodal values φ n i of some piecewise linear function φ n : Ω R defined on T n. Using the six values φ n i = φ n ( x i ) associated with T e, the quadratic interpolation can be constructed in a straightforward way by requiring that Φ n ( x i ) = φ n i. To be consistent with (25), we require that φ n i = φ n (x i ) for all x i T, i.e., that φ n (x) is the canonical restriction of φ n (x).

11 11 T e x i Figure 3: Illustration of a locally refined triangulation T n. Note that to make T n conforming, some elements T e T that are not intersected by the interface must be also refined, see Fig. 3 for an illustration. In practice, to avoid some irregular refinement near the interface, the regular refinement is used in a band of the elements near the interface, see the corresponding numerical experiments in Section 5. Finally, we comment on how the approximations φ n (x) and φ n (x) are obtained in our method to compute (26). For n = 0, the values φ 0 i are defined by evaluating the initial condition at nodes x i T. These values are used to refine T locally to obtain T 0 as described before. Computing φ 0 i from the initial condition at nodes x i T 0 and reconstructing the quadratic interpolation Φ 0 (x) for intersected elements T e T, the values κ e,0 can be obtained from (26). For n > 0, we suppose that the locally refined triangulation T n and the corresponding approximation φ n (x) are available after solving the advection equation, see Section 4.3 later. The approach, as described here, allows a clear extension for the 3D case Extended approximation spaces In this section (including Sections and 4.2.2), we suppose that the interface is approximated by the zero level set of φ n 1 (x). Our extension of the finite volume method for the single-phase Navier-Stokes equations described in Section 3.2 to the two-phase case is inspired by the idea of the extended finite element spaces from [18]. To realize it in our finite volume context, we make the numerical solution in the intersected elements T e to be based not only on nodal values p n i and un i, but also on additional values P e,n and V e,n. For these elements, the piecewise linear interpolations p n (x) and u n (x) from (11) are enriched by special test functions that are defined in a piecewise manner in T e separately for x T e where φ n 1 (x) 0 and for x T e with φ n 1 (x) > 0. The resulted approximations p n (x) and ũ n (x) are then linear (the first order polynomial) in each of these parts of T e.

12 12 In the derivations of the discrete equations from the numerical integration over ω i or ω i, we extend (extrapolate) this definition to avoid any evaluation of discontinuous properties, see below for particular cases. Finally, the unknowns P e,n and V e,n are eliminated in the discretized equations by taking into account some approximations of interface conditions (3 4) that must hold for p n (x) and ũ n (x). This allows us to express P e,n and V e,n in the terms of nodal values. In Section and 4.2.2, we describe the details of the extended approximation spaces separately for the cases of the continuous viscosity and the piecewise constant viscosity. We suppose here that the piecewise constant approximation κ n 1 (x) is available on the intersected elements of the fixed triangulation T The case of continuous viscosity If there is no jump in the viscosity, the interface condition takes the simple form (5) of the prescribed jump in pressure across the interface. This condition can be illustrated on the famous example of the so-called Laplace law for a circular bubble, where the stationary solution of two-phase flow in 2D case is given by { σ u 0, p(x) = R, x Ω 1, (27) 0, x Ω 2, where R is the radius of the bubble, so that the interface has the constant curvature κ = 1 R. Clearly, for intersected elements T e, the linear interpolation p n (x), x T e can approximate this jump only in some smoothed way that results in wellknown unphysical currents in the numerical simulation of (27), see [16, 18, 15]. To overcome this disadvantage of the standard interpolation, we introduce the additional test functions N e,n (p) (x) for the approximation of p at t = tn in intersected element T e, i.e., p n (x) = p n (x) + P e,n N e,n (p) (x), x T e, (28) with P e,n being the additional degree of freedom. As u and u are continuous, we use the standard interpolation for u n (x). To construct N e,n (p) (x), we require that this function has a jump equal to 1 across the numerical interface in T e, and that N e,n (p) (x) is constant and well defined in T e. Moreover, we require that N e,n (p) (x i) = 0, i Λ e, so that p e,n (x i ) = p n i. Thus, the nodal values of the extended approximation are identical with the standard approximation. The desired test functions are then given by N i (x), x T e, φ n 1 (x) 0 i Λ e N e,n (p) (x) = φ n 1 >0 i N i (x), x T e, φ n 1 (29) (x) > 0. i Λ e Therefore, we obtain e p n = e p n P e,n φ n 1 0 i i Λ e e N i = e p n + P e,n i Λ e e N i. (30) φ n 1 >0 i φ n 1 0 i

13 13 The additional degree of freedom P e,n in (30) can be eliminated directly using P e,n = σκ n 1 e. Consequently, one can implement the method using the usual approximation e p n of the pressure gradient (i.e., no changes are necessary for the Jacobian of the discretized system), whereas the additional correction term in (30) is moved to the right hand side of the discretized system. Although this might resemble the approach of localized surface tension force, we stress that in our method no approximation of any source term localized on the numerical interface is used, and e p n shall be viewed as the only correct approximation for the pressure gradient in the case of two-phase flow. Having the extended approximation p n (x), one has to use it carefully when approximating the integrals over ω i. To evaluate it, one can express it in the form p n (x) = p n i + e p n (x x i ). (31) The definition (31) is valid only in one of the two subsets of T e, namely, if φ n 1 i 0 then for x T e φ n 1 (x) 0, or if φ n 1 i > 0 then for x T e φ n 1 (x) > 0. When integrating p n (x) over ω i, we use the approach of ghost fluid method [19] and consider (31) to be valid for x T e ω i. In such a way, when applying our finite volume method, we use the same mesh of control volumes independently on the position of interface, and we avoid any evaluation of discontinuous functions The case of piecewise constant viscosity If the viscosity function µ(t, x) is discontinuous across the interface, the jump condition for the pressure takes the more complex form (3), and, moreover, one component of the velocity gradient exhibits a jump given by (4). To include this feature into our approximation spaces, additionally to the extended pressure approximation p n (x) (with a different definition for P e,n than in Section 4.2.1), we have to construct an extended approximation ũ n (x). To do so, we follow the analogous derivation as for (4) in Section 2. Let n e,n and t e,n denote the normal and tangential vector to the interface located in T e, so that {t e,n, n e,n } is a right basis. By (7), n e,n = computed from n e,n using the orthogonality of these vectors. Let u e,n (x) = u n (x) n e,n and v e,n (x) = u n (x) t e,n so that u n (x) = u e,n (x)n e,n + v e,n (x)t e,n, x T e. e φ n 1 e φ n 1 and te,n can be If we denote u e,n i := u e,n (x i ) and v e,n i := v e,n (x i ), one obtains u e,n (x) = i N i (x), v e,n (x) = i N i (x). (32) i Λ e u e,n i Λ e v e,n The basis {t e,n, n e,n } allows us to introduce the extended approximation ũ n (x). Note that no extension is necessary for u e,n (x). To extend v e,n (x) to some ṽ e,n (x) that has a jump of the directional derivative, we introduce the additional degree of freedom V e,n associated with a test function N e,n (v) (x), i.e., ṽ e,n (x) = v e,n (x) + V e,n N e,n (v) (x), x T e. (33)

14 14 Before introducing N e,n (v) (x), we define a function αe,n (x) that is continuous in T e with a piecewise constant gradient that has a jump across the interface so that [ α e,n n e,n ] = 1. Again, we avoid any explicit reconstruction of the zero level set for φ n 1 (x). Clearly, the following function has the desired properties: α e,n (x) = { 0, φ n 1 (x) 0, 1 e φ n 1 φn 1 (x), φ n 1 (x) > 0. (34) Finally, we use (34) to define the function N e,n (v) (x) with the same properties but N e,n (v) (x i) = 0: N e,n (v) (x) = αe,n (x) α e,n (x i )N i (x). (35) i Λ e To determine the value V e,n using the interface condition, we substitute (33) in (4): µ (2) ( e v e,n + V e,n ( n e,n i µ (1) ( e v e,n V e,n i α e,n (x i ) e N i )) n e,n After some algebraic manipulations, we obtain α e,n (x i ) e N i ) n e,n = [µ] e u e,n t e,n. (36) V e,n = [µ] µ e,n (te,n ) T ( e u n + ( e u n ) T ) n e,n (37) where µ e,n := µ (2) [µ] i α e,n (x i ) e N i n e,n. (38) The value µ e,n lies between µ (1) and µ (2). Analogously to (37), one can eliminate P e,n using P e,n = σκ e,n 1 + 2[µ] e u e,n n e,n, (39) see (3). Note that using (37) in (33) (and (39) in (28)), additional terms occur in the Jacobian of the discretized system when compared to the discretization using only p n (x) and u n (x). Similarly to (31), see the related discussion there, if T e ij (tn ) from (22) is computed using the extended approximations p n (x) and ũ n (x), one has to extrapolate their definitions for x ω i analogously to (31) depending on the position of x i. To simplify our method, we do not use ũ n (x) when approximating the convective term in (1), as we consider the approximation u n (x) to be appropriate in such case even if T e is intersected by the interface.

15 Computation of two-phase flow problem Using the basic building blocks of our method as described in previous sections, we can summarize the method when applied to two-phase flow problem (1), (4) and (6). To simplify the method, we choose the usual decoupling of the problem to two subproblems that are solved once in each time step when advancing the numerical solution from t n 1 to t n. We are aware of the fact that such a splitting brings additional restriction on the choice of time step τ n as discussed, e.g., in [16, 20, 15, 21]. To solve the first subproblem, the numerical solution u n and p n of Navier- Stokes equations (1) with interface condition (4) is computed on the chosen triangulation T using the finite volume discretization as described in Section 3.2 and 4.2. In this step, the approximations u n 1, φ n 1 and κ n 1 must be available. For n = 1, u 0 is computed from the initial condition and φ 0 and κ 0 are determined as described in Section 4.1. For n > 1 these approximations must be available from the computation of the previous time step. The output of this step for the second subproblem is u n defined on T. To solve the second subproblem, the numerical solution φ n of advection equation (6) is computed on the refined triangulation T n using the finite volume discretization as described in Section 3.1. The locally refined triangulation is obtained from T using the available approximation φ n 1 to refine the intersected elements as described in Section 4.1. The velocity v(x) in (12) is defined using simply the piecewise linear interpolation (11) of values u n ( x i ), x i T n. Once the numerical solution φ n is available, φ n defined on T is obtained using the interpolation (11) of restricted nodal values φ n i = φ n (x i ), x i T. The output of this step for the first subproblem in the next time step are the approximations φ n and κ n. Consequently, the numerical solution of the two-phase flow problem can be advanced to the next time step by returning to the solution of the first subproblem and so on. Finally, we comment on the role of reinitialization in our method. As already discussed in Section 2, the exact solution of the advection equation (6) can develop in time some regions with steep or flat gradient near its zero level set, see Fig. 1 for an illustration. The reason for such behavior is that the level set formulation of two-phase flows requires the velocity for each level set of φ, but only the movement of zero level set prescribed by u(t, x), x Γ is physically motivated. Of course, it is more difficult to approximate accurately the analytical solution with steep or flat gradient, especially concerning the precision of curvature approximation. We observe instabilities in numerical experiments after some time t 0 if a coarse mesh is used. To produce meaningful results also for coarse meshes and for long time intervals, we replace the approximation φ N (x) by the numerical solution of (8) in regular time intervals given, say, by T, e.g., N = T/τ, 2 T/τ,.... Note that, similarly to [16], T is chosen independently of τ and τ T, see also related numerical experiments in Section 5.

16 Reinitialization of level set function Let φ N (x) be the approximation of the level set function that should be replaced by the numerical solution ϕ M (x) of the reinitialization equation (8). We obtain ϕ M (x) by applying M time steps of the finite volume method from Section 3.1 with the pseudo time step s, where M and s are parameters of the reinitialization. To avoid any evaluation of discontinuous function sign in (8), we solve this problem in two subdomains, where sign takes only constant value 1 or 1. To do so, we fix the values of ϕ(s, x i ), s 0 for all grid points x i near the interface using the distance of x i to the zero level set of quadratic interpolation function Φ N (x). Such an approach is common in fast marching methods for the solution of eikonal equation [32, 33] and guarantees that the movement of the interface position remains small. Note that in solving the time-dependent reinitialization equation (8), only a few time steps are necessary to obtain a signed distance function close to the interface [34], whereas in fast marching methods the related eikonal equation has to be solved on the whole domain. We now describe the method in detail. First of all, we define the set of indices Λ N D such that i ΛN D if φn i = 0 or i Λ N D if there exists T e, i Λ e and j Λ e such that φ N i φn j < 0. Roughly speaking, Λ N D is the set of such indices i that nodes x i T lie near or on the interface. Let us suppose for a moment that all nodal values, say ϕ 0 i, i ΛN D, are known and fixed in time and have the same sign as φ N i. We split the reinitialization equation (8) into two decoupled subproblems where the method from Section 3.1 can be easily applied. The first subproblem is defined only on the union of T e T for which φ N j 0, j Λ e, and, analogously, the second subproblem only for T e T such that φ N j 0, j Λ e. Consequently, all elements T e where φ N (x) changes its sign are excluded from the method. Considering v = in (12), the scheme (14) then takes the form ϕ m+1 ψ ψ i = ϕ m i σ s γij e ω e,j i ( ϕ e,m+1/2 ij ) ϕ m+1/2 e ϕ m n e ij i e ϕ m 1, (40) where σ = 1 for the first subproblem and σ = 1 for the second one. For both subproblems, we initialize the nodes far away from the interface by ϕ 0 i = φn i, i ΛN D. At the interface, the first subproblem can use the Dirichlet boundary conditions ϕ m i = ϕ 0 i 0, the second one ϕm i = ϕ 0 i 0 for i ΛN D. No other boundary conditions are necessary for the second subproblem, the condition (17) is used for the first subproblem on Ω. The time step is chosen typically by setting the maximal grid Courant number as given by (19) to be slightly less than 1, see the numerical experiments in Section 5. Note that in our numerical experiments we apply the scheme (40) on the refined triangulation T N instead of T. Now we describe how to determine the fixed values ϕ 0 i, i ΛN D. The desired property is that the difference between the zero level set of φ N (x) and ϕ M (x) is negligible.

17 17 In our method, we fix the values ϕ 0 i, i ΛN D, by computing the exact distance of x i, i Λ N D to the zero level set of the piecewise quadratic function ΦN (x) used for the approximation of curvature (see Section 4.1). For each node x i, i Λ N D and for each intersected element T e, e Λ i, we search for the point yi e that solves the optimization problem min y x i 2 s.t. Φ e,n (y) = 0. (41) y R d Here, Φ e,n (x) is a quadratic function obtained by extending the definition of Φ N (x) from T e to R d. We solve (41) using the SQP method [35]. As the starting guess for yi e, we take x i. In our numerical experiments, we can report very good performance of this method. We compute yi e in a cycle for all intersected elements T e. The obtained distance yi e x i is then accepted as ϕ 0 i only if it fulfills some additional constraints. Firstly, it must be smaller than yēi x i, if i Λē for some element T ē for which (41) was solved earlier in the cycle. Secondly, it must be smaller than the distance of x i to each of the points of the intersection of the zero level set of φ N (x) and T e. Note that this is the only case in our method when some positional information is reconstructed for the interface. When an analogous initialization procedure for grid points close to the interface is performed in fast marching methods, usually a linear interpolation of the level set function is used [32, 9], but for our method we found that quadratic interpolation leads to much better results in the overall computation [33]. 5. Numerical experiments In this section, we present the results of numerical tests when using the proposed method. The computations were performed with the software tool UG (Unstructured Grids) that supports grid hierarchies created by the local refinement and coarsening of grids. Such multi-grids can be used in the solvers for the discrete algebraic systems [36, 31]. In the first series of the tests, we simulate a rising bubble in a two-dimensional domain [2, 37, 7, 16]. The aim of these experiments is to study the numerical convergence of the discretization with respect to the grid and the time step. Moreover, the independence of the computations on the character of the chosen grid is shown and the role of reinitialization is illustrated. The domain is Ω = [0, 0.5] [0, 1] cm 2. For the two fluids, we take the same parameters as used in [9]: ρ (1) = g cm 3, ρ (2) = g cm 3, 2 dyn s 2 dyn s µ (1) = cm, µ 2 (2) = cm, 2 σ = 2 dyn cm, cm g = (0, 980)T s. 2 The following boundary and initial conditions are used in all these experiments. For the flow, we impose the zero velocities on the left and the right side

18 18 (a) (b) (c) (d) Figure 4: Numerical solution of a rising bubble with the reinitialization on the regular grids: nodes (b), nodes (c) and nodes (d). Picture (a) presents the initial condition. In Pictures (b-d), the numerical flow field and the 0-th isoline of the level set function are shown at time t = 0.25s. of Ω. On the upper side, a parabolic profile of the inflow velocity is prescribed with the maximal velocity being 4 cm s. On the lower side, the zero-stress outflow boundary conditions are used. For the level set equation, the boundary conditions (17) are used. At time t = 0, the bubble Ω 1 is a circle with radius R = cm and the center at (0.25, 0.5) cm. The standard signed distance function to this circular interface is taken as the initial condition for the level set function. To define the initial condition for the velocity u, the given inflow profile is spread along the y-axis, see Fig. 4(a). From the analytical description, one expects that the bubble is pressed upwards due to the gravitation, but, at the same time, the incoming fluid hinders the bubble to move up. Although the interface is deformed, the area of the bubble remains constant. In our first numerical experiment, we investigate the grid convergence of the numerical solution for this example when the uniform grid refinement is used. All the grids are obtained from the initial regular 3 4 grid (12 triangles) called grid level 0. To obtain the grid level l + 1, each triangle of grid level l is subdivided regularly into 4 ones. The grid of level 1 is shown in Fig. 6(a). The numerical solutions at t = 0.25s for grid levels l = 4, 5, 6 are presented in Fig. 4. Note that to investigate the grid convergence of our method in this experiment, the advection equation is computed on the uniformly refined grid of level l + 1, l = 4, 5, 6. In Table 1, the number of grid points, the time steps and the absolute error of bubble area is presented for each grid level. To compute the last error, the area surrounded by the zero level set of the piecewise linear level set function is computed exactly.

19 19 (a) (b) (c) (d) Figure 5: Comparison of the numerical solutions for t = 0.05s and t = 0.115s with the reinitialization ((a) and (c)) and without it ((b) and (d)). The isolines of the level set function for values ±0.1i, i = 0, 1,... are shown. The zero level set is the thick line. Although the numerical velocity used for the tracking of the interface is not divergence free, see Section 3.2, the approximate area of the bubble converges to the exact value πr 2 with first order of accuracy. At the same time, starting with grid level 5, the numerical solution, according to Fig. 4, seems to be grid independent. In the previous experiments, the level set function is reinitialized after equal time intervals of T = 0.01s, see Section 4.4 for the notations. This means that the reinitialization is performed 25 times during the entire computations whatever the grid level l is. The pseudo-time step s is always chosen such that the grid Courant number in (19) is equal to For l = 4, the number of the steps in one reinitialization is 10. This number of steps is doubled when changing from the grid level l to level l + 1. Table 1: Error in the volume of the bubble at t = 0.25s w.r.t. the grid size. I τ [s] Time steps Area error [cm 2 ] To illustrate the role of reinitialization more profoundly, we present the results of previous experiment for l = 5 in Fig. 5(a) for t = 0.05s and in Fig. 5(c) for t = 0.115s. For comparison, the analogous results obtained in the next experiment with no reinitialization are shown in Fig. 5(b) and Fig. 5(d). The

20 20 (a) (b) (c) (d) Figure 6: Comparison of the simulation on the regular grid ((a) and (b)) with the simulation on a non-regular grid ((c) and (d)) at time t = 0.25s. zero level sets in the results for t = 0.05s are comparable, since the complicated shape of the level set function with no reinitialization can be still resolved by the grid. Continuing with computations without the reinitialization leads to very steep and flat gradient of the level set function. For this reason, the simulation without reinitialization at later times produces unacceptable result. In practice, one can not use always a structured grid as in the case of rectangular Ω. Moreover, the local refinement of the structured grid can lead to unstructured grids. Therefore, in the next numerical experiment we study the behaviour of our method on an unstructured grid, see Fig. 6. The left two pictures show the structured grid of level l = 1 and the corresponding numerical solution for l = 5 that is identical to Fig. 4(c). In Fig. 6(d), the unstructured grid is presented with approximately the same number of triangles as in Fig. 6(a). This grid was refined regularly 4 times and the corresponding numerical solution is presented in Fig. 6(c). The difference between Fig. 6(b) and Fig. 6(c) is negligible. Finally, we present a numerical experiment using grid adaptivity. In this experiment, the grid T n, used for the solution of the advection equation, is not obtained by the uniform refinement of T as in previous experiments, but only the elements in which the level set function has values between 0.03 and 0.03 are refined, see Section 4.3. The numerical solution and the corresponding adaptive grids are plotted in Fig. 7. One can see that the results obtained on the locally refined grids are analogous to those shown in Fig. 4. In the final series of numerical tests, we investigate the accuracy of our method for a simple, but important two-phase flow problem with a known analytical solution, the stationary bubble. The domain and the (discontinuous) parameters are identical to that in the previous experiments. Similarly, the

21 21 (a) (b) (c) (d) Figure 7: Results of the computations with the adaptive grid for the level set function: (a) and (b) presents the grid for t = 0 and t = 0.25s for l = 4; (c) shows the flow at t = 0.25s for l = 4; (d) presents the flow and the interface at t = 0.25s for l = 5. identical initial level set function (and, consequently, the bubble with the radius R = cm) is chosen. The velocity is set to zero at t = 0 and the Dirichlet boundary conditions u = 0 are imposed on the whole boundary. Furthermore, we set g = 0, so that no flow can be induced due to the gravitation. In the computations, we use the same regular grid as in the previous example (cf. Fig. 6(a) for grid level 1). In the analytical solution, functions u, p and φ do not depend on time. By the Laplace law, the pressure is { σ p(t, x) = p analytic (x) := R, x Ω 1, (42) 0, x Ω 2 (up to a constant). The velocity u is identically zero and the level set function φ is equal to the initial condition. It is important to note that the projection of these functions onto T would be the exact solution of our discretized system, if one sets κ e,n = R 1 instead of (26). Neverthelless, due to the approximation of the curvature by (26), this is not the case in the numerical solution. The error in velocity field is very often characterized as the parasite currents near the interface [16, 18, 15]. Fig. 8 illustrates these currents in the numerical solution obtain with our method on grid level l = 5 at t = s. In Table 2, we compare the numerical solution at time t N = s for successively refined (in space as well as in time) grids with the exact values. The spatial grid is regular as in the previous experiment. The L 2 -norm u N 2 and the maximum norm u N 0 are computed exactly. For this example, the pressure is defined only up to a constant, therefore, we present the norm of

22 22 x y Figure 8: Illustration of the parasite currents the error( of the ( projected pressures, e N (x) = p N (x) p analytic (x) C, where C := 1 Ω Ω p N (x) p analytic (x) ) dx ). In Table 2, the following norm of e N (x) is given: e N 2,FV := ( I i=1 ω i ( e N (x i ) ) ) Note that no reinitialization was applied in this example. The results show that the error decreases in both norms as the grid size is halved, and the error of the pressure decreases linearly. Table 2: Dependence of the parasite currents and the errors in the pressure on the grid size. Simulations with no reinitialization. I τ [s] N u N 0 u N 2 e N 2,FV Furthermore, Table 3 presents the results of the same experiment performed with the analogous reinitialization strategy as in previous numerical experiments. For each grid level, we reinitialize 10 times with T = For l = 4, we perform 4 steps of (40) with s chosen such that the grid Courant number in (19) is equal to This number of time steps is doubled with each grid refinement. In this experiment, one can observe a slightly better than the linear decrease of all presented errors. 6. Conclusions In this paper, we introduced a novel method for two-phase flow problems using the level set formulation. The method is based on the pure finite volume