AUTOMATIC MESH MOTION FOR THE UNSTRUCTURED FINITE VOLUME METHOD. Summary

Transcription

1 Hrvoje Jasak Željko Tuković AUTOMATIC MESH MOTION FOR THE UNSTRUCTURED FINITE VOLUME METHOD Summary ISSN UDK 532.5:519.6 Moving-mesh unstructured Finite Volume Method (FVM) provides a capability of tackling flow simulations where the spatial domain shape changes during the simulation. In such cases, the computational mesh needs to adapt to the time-varying shape of the domain and preserve its validity and quality. In this paper, we present a vertex-based unstructured mesh motion solver with polyhedral cell support which calculates internal point motion based on the prescribed motion of the boundary. Performance of the method is preserved through the choice of decomposition of polyhedral cells, bounded discretisation and use of iterative solvers. A mechanism for minimising mesh distortion through variable stiffness is proposed and tested on a simple deformation case, showing marked improvement over previous attempts. Finally, the moving mesh solver is used with an unstructured moving mesh FVM algorithm to simulate free-rising air bubble in water. Key words: Moving mesh, vertex motion, motion solver, arbitrarily unstructured mesh, polyhedral mesh, finite volume, free-surface, surface tracking 1. Introduction There exists a number of physical phenomena in which the continuum solution couples with additional equations influencing the shape of the domain or position of an internal interface. Examples of such cases include prescribed boundary motion in pumps and internal combustion engines; free-surface flows, where the interface between the phases is a part of the solution; fluid-structure interaction, where the deformation of a solid changes the shape of the fluid domain etc. Numerical simulation techniques for such cases track the interface either by using marker particles, an indicator variable (e.g. [1, 2, 3]) or a distance function, or adjust the computational mesh to accommodate interface motion. In the deforming mesh method, the computational mesh is moved to follow the changing shape of the boundary by moving its points in every step of the transient simulation. The main difficulty in this case is maintaining mesh validity and quality without user interaction. Several deforming mesh algorithms have been presented in literature, with various approaches to defining mesh motion. The most popular method to date is the spring analogy [4]. Here, all pointto-point connections within the mesh are replaced by linear springs and point motion is obtained as a response to boundary displacement. However, this approach proved to lack robustness, particularly for arbitrarily unstructured (polyhedral) meshes. A review of merits and limitations of spring analogy and its variants is given by Blom [5]. In an effort to improve the robustness of the method, Degan and Farhat [6] propose addition of torsional springs to control all mechanisms of invalidating a tetrahedral cell. Other approaches to creating a robust mesh motion solver include the use of Laplacian smoothing [7] with constant and variable diffusivity and the pseudo-solid equation (static equilibrium equation for small deformations of a linear elastic solid) [8] in Arbitrary Lagrangian-Eulerian (ALE) FEM codes. In an effort to simultaneously control the position of a moving boundary and mesh spacing next to it, Helenbrook [9] proposes the use of a biharmonic equation to govern mesh motion. 1

2 H. Jasak, Ž. Tuković Automatic mesh motion for In this paper we present a general-purpose moving mesh algorithm for deforming mesh simulations compatible with arbitrarily unstructured FV solvers. A new second-order polyhedral minielement consistent with the FV mesh handling has been developed as a part of a vertex-based solution method. Two choices for a governing equation for mesh motion are examined: the Laplace and the pseudo-solid equation. In an attempt to control mesh quality by redistributing boundary deformation and limiting cell distortion, a variable coefficient version of both equations has also been implemented. A crucial requirement is that algorithmic efficiency of the mesh motion solver matches the segregated FV flow solver, both is terms of storage and CPU time requirements. The rest of this paper is organised as follows. In Section 2 the FV method for arbitrary moving volumes is summarised and requirements on an automatic mesh motion system are given. Based on deficiencies of previous efforts, Section 3 lays the foundation for a novel automatic mesh motion method, starting from the requirements on a robust motion system, choice of motion equation, solution variable, appropriate polyhedral cell decomposition and control of mesh quality through variable diffusion in the motion equation. The new method is tested on two sample mesh deformation problems in Section 4. The paper is completed in Section 5 with a simulation of a free-rising air bubble in water in 2- and 3-D and closed with a short conclusion. 2. Finite volume method on moving meshes A static mesh FVM is based on the integral form of the governing equation over a control volume (CV) fixed in space. More generally, the integral form of the conservation equation for a tensorial property φ defined per unit mass in an arbitrary moving volume V bounded by a closed surface S states: d dt V ρφ dv + S ρn (v v s )φ ds S ργ φ n φ ds = V s φ dv, (1) where ρ is the density, n is the outward pointing unit normal vector on the boundary surface, v is the fluid velocity, v s is the velocity of the boundary surface, Γ φ is the diffusion coefficient and s φ is the volume source/sink of φ. The relationship between the rate of change of the volume V and the velocity v s of the boundary surface S is defined by the so called space conservation law (SCL): d dt V dv S n v s ds = 0. (2) Unstructured FVM discretises the computational space by splitting it into a finite number of convex polyhedral cells bounded by convex polygons. The cells do not overlap and completely fill the domain. The temporal dimension is split into a finite number of time-steps and equations are solved in a time-marching manner. A sample cell around the computational point P located in its centroid, a face f, the face area S f, the face unit normal vector n f and the neighbouring computational point N are shown in Fig. 1. Second-order FV discretisation of Eqn. (1) transforms the surface integrals into sums of face integrals and approximates them and the volume integrals to second order using the mid-point rule. If a fully implicit second-order accurate tree time levels scheme is used for temporal discretisation, the discretised form of Eqn. (1) for the control volume V P reads: 3ρ n P φn P V P n 4ρo P φo P V P o + ρoo P φoo P V P oo + 2 t f = f (ργ φ ) n f S n f n n f ( φ)n f + s n φv n P, (ṁ n f ρ n f V n f )φ n f (3) 2

3 Automatic mesh motion for H. Jasak, Ž. Tuković S f n f d f N f P r P z y V P x Fig. 1 Polyhedral control volume (cell). where the subscript P represents the cell values, f the face values and superscripts n and o the new and old time level, t = t n t o = t o t oo is the time step size, ṁ f = n f v f S f is the fluid mass flux and V f = n f v sf S f is the cell face volume flux. The fluid mass flux ṁ f is usually obtained as a part of the solution algorithm and satisfies the conservation requirements (if any). In order to prevent the introduction of errors in the form of artificial mass sources, the cell face volume flux V f must satisfy the discretised SCL where temporal discretisation scheme used for SCL should be the same as that for the other conservation equations in the considered mathematical model (Demirdžić and Perić [10]) Finite volume mesh definition Traditional points-and-cells mesh definition consists of a list of points (vertices) and a list of cells defined in terms of point labels. Additionally, a vertex ordering pattern is pre-defined for each cell shape and allows cell faces to be assembled. This approach limits the number of available cell shapes which, while acceptable in the FEM (due to the fact that a shape function needs to be defined a-priori), is unnecessarily limiting for the face-addressed FVM [11]. In the face-addressed mesh definition, a polyhedral mesh for the FVM is defined by the following components: A list of point co-ordinates; point label is implied from its location in the list; A list of polygonal faces, where a face is defined as an ordered list of point labels. Faces can be separated into internal (between two cells) and boundary faces; A list of cells defined in terms of face labels. Note that the cell shape is unknown and irrelevant for discretisation; Boundary faces are grouped into patches, according to the boundary condition; With face addressing, each cell is only known in terms of its faces. This format, however, poses additional requirements in examining mesh validity. In general, one can distinguish between topological and geometrical validity tests. During deformation, mesh topology remains unchanged and an initially valid topology will be preserved. Geometrical tests deal with the positivity of face areas and cell volumes, convexness and orientation requirements. Geometrical measures (cell and face centroid, face area, cell volume etc.) are calculated by decomposing the face into triangles and the cell into tetrahedra. Based on that, geometrical validity criteria can be summarised as follows: 3

4 H. Jasak, Ž. Tuković Automatic mesh motion for All faces and cells must be weakly convex, i.e. all triangle normals in face decomposition must point in the same direction as the resultant face normal; for a cell, all pyramids constructed with a face base and cell centroid apex must have positive volume; For all internal faces, the dot-product of the face normal vector n f and the d f = PN, Fig. 1, must be positive. This is usually termed the orthogonality test: d f n f > 0. (4) We shall assume the existence of an initial topologically and geometrically valid mesh. 3. Polyhedral vertex-based motion solver The defining feature of a moving mesh simulation is temporal variation of the external shape of the domain. Thus, one can distinguish between boundary motion and internal point motion. Boundary motion can be considered as given, either prescribed by external factors or a part of the solution. The role of internal point motion is to accommodate boundary motion and preserve the validity and quality of the mesh. It influences the solution only through mesh-induced discretisation errors [12] and is detached from the remainder of the problem, owing to ALE formulation of the conservation equations. Consequently, internal point motion can be specified in a number of ways, ideally without user interaction Mesh deformation problem The mesh deformation problem can be stated as follows. Let D represent a domain configuration at a given time t with its bounding surface B and a valid computational mesh, Fig. 2. During a time interval t, D changes shape into a new configuration D. A mapping between D and D is sought such that the mesh on D forms a valid mesh on D with minimal distortion of control volumes. Initial configuration Final configuration t D t + t D B f B f B m u : r r B m y r y r x B = B m B f x B = B m B f Fig. 2 Mesh deformation problem. In this study, the displacement vector u is chosen as the dependent variable in the mesh motion problem. Thus, point position in the deformed configuration is calculated as: r = r + u, (5) where r D and r D are point position vectors. 4

5 Automatic mesh motion for H. Jasak, Ž. Tuković 3.2. Choice of motion equation It remains to consider the choice of equation to govern mesh deformation. Mesh validity constraints indicate that a domain could be considered as a solid body under large deformation, governed by the Piola-Kirchoff stress-strain formulation. This is a non-linear equation and thus expensive to solve; as stresses are of no interest, a similar and numerically cheaper approach along the same lines is sought. Two obvious choices are the pseudo-solid equation [8] and the Laplace equation [7]. While the Laplace equation only allows direction-decoupled transfinite mapping, the pseudosolid equation also allows rotation. However, this comes at a relatively high price: the pseudo-solid equation couples the components of the motion vector due to rotation, [13]. The choice here is either an increase in storage associated with the block solution of all displacement components or an iterative segregated solution method Solution method for motion equation The simplest idea for automatic mesh motion in the FV framework would be to re-use the available numerical machinery and solve an equation to provide point motion. However, as the FVM provides the solution in cell centres and motion is required on the points, this necessarily leads to interpolation. Experience shows it is extremely difficult to construct an interpolation practice which stops the cells from degenerating even if the cell-centred motion field is bounded. Moreover, motion of corner points (belonging to only one cell) and intersections of free-moving boundaries cannot be reliably reconstructed. Finally, while the FVM is unconditionally bounded for the convection operator, on badly distorted meshes one needs to sacrifice either the second-order accuracy or boundedness in the Laplacian, due to the explicit and potentially unbounded nature of non-orthogonal correction [14]. A combination of the two has forced us to quickly abandon this approach with the lesson that a point-based solution for the motion equation is essential. We can state the following requirements on the mesh motion solver for the selected mesh motion equation: A vertex-based solution method, avoiding the need for interpolation; Use of iterative solvers for efficiency, implying positive definite matrices resulting from discretisation; No triangles or tetrahedra in the cell decomposition should be inverted. Use of a classical FEM solver for mesh motion is unsatisfactory: to the authors knowledge, definition of shape functions for arbitrary polyhedra does not exist. Also, it is unclear whether such a shape function would produce a positive definite matrix we are seeking for efficiency reasons. Tetrahedral finite elements for a Laplacian produce a symmetric positive definite matrix and second-order discretisation. Also, matrix coefficients tend to infinity when a tetrahedral element approaches a degenerate state. We can illustrate this by analysing the expression for calculation of matrix coefficients for a linear tetrahedron in real space, Fig. 3. The coefficient contribution for a point pair (i, j) and the Laplacian operator can be calculated as [15]: a ij = VT N i N j dv = Sτ i Sτ j 9 V τ, (6) where N i is the element shape function, S τ i is the surface-area vector on the triangle opposite point i and V τ is the volume of the tetrahedron. Consider a case where a tetrahedron approaches a degenerate state, either by a point approaching another point or the opposite face. The face area vectors S i and S j remain finite (and come close to being parallel), whereas the volume approaches zero. Thus, with the finite numerator and the denominator approaching zero in Eqn. (6), a ij tends to infinity. As a result, the tetrahedral FEM discretisation will remain bounded irrespective of mesh quality. 5

6 H. Jasak, Ž. Tuković Automatic mesh motion for 1 S τ S τ 1 Fig. 3 Calculating matrix coefficients for a tetrahedron in real space Cell decomposition It remains to choose an appropriate decomposition of a polyhedron into tetrahedra; two methods used in this study are shown in Fig. 4. (a) Cell split. (b) Cell-and-face split. Fig. 4 Decomposing a polyhedral cell into tetrahedra. A cell is decomposed by introducing a point in its centroid and building tetrahedra above the triangular decomposition of a face. The two methods proposed here are the cell decomposition, Fig. 4(a), where additional points are introduced only in cell centres; and cell-and-face decomposition, Fig. 4(b), where points are introduced in both face and cell centres. In the first, the number of algebraic equations in the matrix equals the sum of cell and point count, while the second adds an equation for each face, giving a considerable increase in the number of unknowns Controlling mesh distortion When the Laplace equation governs mesh motion, the prescribed boundary deformation is not uniformly distributed through the domain. The nature of the equation is such that point movement is largest adjacent to the moving boundary, potentially leading to local deterioration in mesh quality. Ideally, largest deformation should be confined to the internal part of the mesh, where it causes less distortion. This can be achieved by prescribing variable diffusivity in the Laplacian. It is unclear how to formulate variable diffusivity in general and two ideas are examined in this study: 1. Distance-based method. Here, the diffusion coefficient γ is a function of cell distance to the nearest moving boundary l: γ(l) = 1 l m, (7) 6

7 Automatic mesh motion for H. Jasak, Ž. Tuković where m is a positive exponent. Alternatively, exponential equation has also been tested: γ(l) = e l. 2. Distortion energy method. Based on total mesh displacement u tot = u dt, (8) (9) the distortion energy U d for a fictitious elastic body is calculated: U d = 1 2 ε dev(σ), (10) [ where ε = 1 utot + ( u 2 tot ) T], σ = 2µε + λ tr(ε)i, and Lamé coefficients are taken to be constant: µ = λ = 1. This corresponds to the Poisson s ratio ν = The diffusion coefficient is calculated from the distortion energy: γ(u d ) = U m d + ǫ, (11) where ǫ is a small constant. In both cases, integer values of m, 1 m 3 have been tested Efficiency concerns A critical requirement on the motion solver is that it should match the FVM flow solver in efficiency and storage requirements. Unlike the face-addressed FV solver [12], where the matrix is assembled by looping over mesh faces, the FEM assembles the matrix by looping over all elements. The number of tetrahedra in the cell decomposition is considerably higher than the number of cells: for efficiency, cell decomposition is done on-the-fly for each visited cell and poses only limited storage overheads. On the linear equation solver side, the matrix structure from the motion equation closely mimics the FVM matrices: it is symmetric positive definite and well suited for iterative solvers. Thus, the same solver can be used for both purposes, providing equivalent performance and possibility of parallelisation in the domain decomposition mode [15]. One should note that in segregated FVM fluid flow solvers, the memory peak occurs during the pressure-velocity solution (using SIMPLE [16] or PISO [17]), when the momentum and pressure matrices are stored simultaneously. The mesh motion solver operates either before or after the pressure-velocity block and the released storage can be re-used. This somewhat decreases the perceived storage peak of the motion solver relative to the FVM part of the algorithm Final form of the motion solver In summary, the polyhedral mesh motion solver is constructed as follows: 1. Every polyhedral cell is split into tetrahedra by splitting its faces into triangles and introducing a point in cell centroid and optionally in face centres. 2. For cases with variable diffusivity, γ is calculated from the current state 1. 1 In some cases, it is beneficial to preserve the diffusivity field from the undeformed configuration. This may help to minimise possible hysteresis effects, especially for cases where γ is a function of the distance to the moving wall. 7

8 H. Jasak, Ž. Tuković Automatic mesh motion for 3. The Laplace equation: (γ u) = 0 (12) and Eqn. (5) are used to determine new point positions. Eqn. (12) is discretised on the tetrahedral decomposition using a second-order finite element method and produces a symmetric positive definite matrix. For efficiency reasons, matrix coefficients are calculated in real space using Eqn. (6). The pseudo-solid equation: [ ] µ u + µ ( u) T + λ tr( u)i = 0, (13) will also be used for comparison purposes. 4. Boundary conditions for the motion equation are enforced from the known boundary motion; this may include free boundaries, symmetry planes, prescribed motion boundary etc. 5. The matrix is solved using an iterative linear equation solver; here the choice falls on the Incomplete Cholesky preconditioned Conjugate Gradient (ICCG) solver [18], also used by the FVM solver. One could consider it an overkill to implement a fully-fledged FEM solver to move the mesh in an existing FVM code. In this study, the motion solver is implemented in OpenFOAM [19, 20], an object-oriented C++ computational continuum mechanics library. The software is constructed to allow extensive code re-use, typically impossible in more traditional designs. OpenFOAM currently implements a second-order collocated FVM on arbitrarily unstructured meshes. It is written in operator form and has a class hierarchy designed to be shared between various discretisation practices. Lower level objects, including mesh, matrix, field, boundary conditions, linear solvers etc. are re-used without change. The actual motion solver is implemented by using the discretisation operators in the FEM library and packed for ease of use in a separate module, together with the necessary boundary condition handling, mesh checking and setup tools. Further details on code organisation, boundary conditions and parallelisation in the domain decomposition mode are given in [15]. 4. Examples of mesh motion We shall now apply the novel motion algorithm on two test problems and examine the mesh quality for various definitions of non-constant diffusion fields. The first example is set up to test the limits of applicability of the new mesh deformation solver, while the second illustrates a typical application of interest Motion of a cylinder The case consists of a circle moving in a channel in 2-D 2. Identical setup and a triangular mesh has been used by Baker [21] with the pseudo-solid equation, and Helenbrook [9] on the biharmonic equation. Figure 5 shows the polygonal mesh used for the test, where D is the cylinder diameter, the height of the channel is 2 D and average mesh size is 0.15 D. The polygonal mesh is generated using the algorithm proposed by Virag and Džijan [22, 23]. The first test consists determining the maximum displacement of the cylinder in one step without mesh inversion when the outside boundary remains fixed. Mesh quality is determined in terms of the non-orthogonality angle α f between d f and n f, Fig. 1. For reference, on the initial polygonal mesh α f,max = and α f,mean = The deformed meshes obtained using the 2 In reality, the mesh is 3-D and consists of prismatic elements, as the software only operates on 3-D meshes. 8

9 Automatic mesh motion for H. Jasak, Ž. Tuković Fig. 5 Cylinder motion in 2-D: Initial polygonal mesh. Laplace and pseudo-solid mesh motion equations for one step maximum cylinder displacement are shown in Fig. 6. Maximum achievable single-step cylinder displacement is max = D for the Laplace equation and max = D for the pseudo-solid equation. (a) Laplace equation: max = 0.65 D, α f,max = 83.4, α f,mean = 4.1. (b) Pseudo-solid equation: max = 0.97 D α f,max = 88.14, α f,mean = Fig. 6 Cylinder motion in 2-D: Single-step mesh deformation. In transient simulations, the mesh is moved in a number of time-steps. This situation will be examined by repeating the above test, but with prescribed cylinder motion of 0.15 D per time-step until the mesh becomes invalid. This equates to the effective Courant number of unity, based on the boundary motion velocity. Fig. 7 shows that this approach allows considerably higher deformation, because it handles inherent non-linearity of the mesh motion problem. It is interesting to notice that the Laplace and pseudo-solid equations allows the same cylinder displacement max = 1.2 D, contrary to the previous test. On the other hand, increased cost of solving the pseudo-solid equation compared to the Laplace equation does not seem to be justified with the higher allowed single-step mesh deformation [15]. For this reason, the Laplace equation will be used in the rest of this study. Remains to show how the proposed methods for defining the variable diffusivity in the Laplace equation influence the mesh motion performance. Fig. 8 shows the deformed meshes for maximum single-step cylinder displacement obtained with various non-constant diffusivity fields in the Laplace mesh motion equation. One can see that all proposed variable diffusivity fields allow considerably higher single-step displacement of the cylinder. Maximum single-step displacement max = 1.39 D is achieved using the diffusivity proportional to the distortion energy, γ(u d ) = U d. For distance based methods maximum single-step displacement max = 1.1 D is obtained with the diffusivity field inversely proportional to the distance from the moving boundary, γ(l) = l Pitching airfoil The second test case consists of a pitching airfoil with a 2-D hybrid mesh. The airfoil of chord length c moves according to the sinusoidal law, including both translation and pitching. Rotation with the amplitude of 10 is centred 0.3 c downstream of the leading edge and superimposed by translation in the y-direction of 0.5 c amplitude. The period of motion for both components is 2 s, which is discretised with the time step size t = s. Figure 9 shows the initial mesh around the airfoil. Of particular interest is the mesh around the trailing edge. Maximum and mean non-orthogonality angles of the initial mesh are α f,max =

10 H. Jasak, Ž. Tuković Automatic mesh motion for (a) Laplace equation: max = 1.2 D, α f,max = 86.8, α f,mean = 6.6. (b) Pseudo-solid equation: max = 1.2 D, α f,max = 86.5, α f,mean = 8.1. Fig. 7 Cylinder motion in 2-D: Mesh deformation with time stepping. (a) γ(l) = l 1, max = 1.1 D, α f,max = 80, α f,mean = 7.3. (b) γ(l) = l 2, max = 0.95 D, α f,max = 82, α f,mean = 7. (c) γ(l) = e l, max = 0.94 D, α f,max = 85.1, α f,mean = 6.3. (d) γ(u d ) = U d, max = 1.39 D, α f,max = 85, α f,mean = (e) γ(u d ) = U 2 d : max = 1.19 D, α f,max = 78.5, α f,mean = 10. (f) γ(u d ) = U 3 d, max = 0.67 D, α f,max = 68, α f,mean = 6.2. Fig. 8 Deformed mesh for the limiting cylinder displacement with various non-constant diffusivity fields in the Laplace mesh motion equation. Fig. 9 Oscillatory motion of a NACA airfoil, initial mesh. 10

11 Automatic mesh motion for H. Jasak, Ž. Tuković (a) γ(l) = const., α f,max = 78.8, α f,mean = (b) γ(l) = l 1, α f,max = 35.4, α f,mean = 2.3. (c) γ(l) = l 2, α f,max = 34.5, α f,mean = 2.2. (d) γ(u d ) = U d, α f,max = 39.2, α f,mean = 2.5. (e) γ(l) = U 2 d, α f,max = 37.94, α f,mean = 2.4. (f) γ(u d ) = U 3 d, α f,max = 37, α f,mean = 2.3. Fig. 10 Oscillatory motion of a NACA airfoil, mesh quality around the trailing edge. 11

12 H. Jasak, Ž. Tuković Automatic mesh motion for and α f,mean = 1.4. Figure 10 shows how various diffusivity fields influence mesh quality in the region of interest when airfoil is in maximum pitch position. A quantitative confirmation of improved mesh quality in respect to maximum and mean non-orthogonality angles of the mesh is also given in Fig. 10. Distance-based methods produce meshes with lowest non-orthogonality, while the constant diffusivity method considerably distorts the mesh. However, in all cases the resulting mesh remains valid. Numerical experiments show that the cell decomposition is sufficiently robust for 2-D and trivial 3-D meshes. Comparing the performance of two polyhedral decompositions on complex 3-D meshes, it has been noted that the cost of solution of the cell-and-face decomposition in total execution time becomes substantially lower than that for the cell decomposition. This is counter-intuitive, as the number of unknowns for the cell-and-face decomposition is considerably larger. However, higher quality of tetrahedra in the cell-and-face decomposition results in a betterconditioned matrix which is considerably cheaper to solve using ICCG, thus counterbalancing the cost of solving a larger linear system. 5. Free-surface flow simulations This study concludes with a two-phase interface tracking simulations. Here, the fluid flow equations are solved in both phases and coupled across the interface through boundary conditions [15]. The interface is represented as a mesh interface whose motion depends on the flow solution. A schematic representation of the computational domain and interface conditions is given in Fig. 11. Second-order FVM is used for the fluid flow [15]. On the interface, a double boundary condition is imposed: the dynamic condition (equilibrium of forces) and kinematic condition (zero net mass flux) need to be satisfied simultaneously. The fluid flow equations are solved using a segregated SIMPLE procedure. The zero net mass flux condition is satisfied in an iterative sequence [15, 24]. First, the dynamic condition, including the surface tension [15], is enforced on the interface and consequently a non-zero net mass flux is obtained. Position of the faces in the interface patch is adjusted such that the face volume flux V f equals the fluid volume flux for the face (ṁ f /ρ f ) and the automatic mesh motion solver described above adjusts the mesh to interface motion. Clearly, the change in domain shape influences the fluid flow solution and the procedure is repeated in an iterative manner for every time-step until convergence. Details of the interface tracking algorithm and validation on cases with analytical solutions have been presented by Tuković [15]; here, we shall limit ourselves to examining the performance of the mesh motion solver Free-rising air bubbles in water The driving force behind this study is a desire to assemble a tool for Direct Numerical Simulation (DNS) of air bubbles in water, with the aim of providing lift and drag data needed for two-phase Eulerian modelling [15, 25]. Here, we shall report some initial results for free rising air bubbles in water in 2- and 3-D. The bubble is located in a large box, Fig. 12, and the flow boundary conditions are adjusted such that it remains centred in the domain [15]. The bubble rises through a quiescent fluid and the material properties of air and water are used, including surface tension effects. Note that, unlike in the surface capturing methods, the strength of surface tension or the jump in the density and viscosity do not pose a problem due to a more robust handling of inter-phase coupling and accurate curvature calculation. We shall first examine a 2-D problem: while this is not physically realistic, it contains all relevant modes of interaction between the two phases, as well as the coupling with the moving mesh solver. 12

13 Automatic mesh motion for H. Jasak, Ž. Tuković Fluid B, ρ B, µ B Mesh B σ n n σ A n σ B g Interface Kinematic condition: v A v B = 0 Dynamic condition: n σ A n σ B = σκn Fluid A, ρ A, µ A Mesh A Fig. 11 Definition of the spatial domain and interface conditions for the moving mesh interface tracking method, where subscripts A and B represent the values at the two sides of the interface, σ is the surface tension, κ is twice the mean curvature of the interface, σ is the stress tensor, µ is the dynamic viscosity and g is the gravity. Inflow part of the outer boundary v b = v F v F y a F y o r F x o x Bubble path Interface Outflow part of the outer boundary Fig. 12 Moving reference frame setup for free-rising bubble simulation, where v F and a F are the velocity and acceleration of the moving reference frame, r F is the position vector of the moving reference frame in respect to fixed reference frame. 13

14 H. Jasak, Z. Tukovic Automatic mesh motion for Fig. 13 shows the mesh deformation and the pressure field around a 2-D air bubble of 1.5 mm diameter freely rising in water. After the initial transient, the bubble reaches terminal velocity and shape. The mesh in this simulation consists of CVs in two disconnected regions and captures the interface through the coupled free surface condition. (a) t = 0 s, αf,max = 26.1, αf,mean = 1.1. (b) t = s, αf,max = 32.41, αf,mean = 3.8. (c) t = s, αf,max = 37.9, αf,mean = 5.5. (d) t = 0.25 s, αf,max = 39.9, αf,mean = 6.1. Fig. 13 Free-rising air bubble in water in 2-D: pressure field and interface deformation. Finally, a simulation of a free rising air bubble of the 2 mm diameter in 3-D will be presented. The mesh consists of cells with sufficient near-surface resolution as described by Blanco and Magnaudet [26]. A detailed breakdown of the timing for a single SIMPLE iteration of the simulation is given in Table 1. The simulation is performed on a Linux computer with a 2 GHz Intel Pentium IV processor and 1 GB of memory. However, the cost of s per iteration on a single CPU computer is too high for the available resources and a coarser mesh with cells will be used instead. The maximum and mean non-orthogonality angles of the initial mesh are αf,max = 25.5, αf,mean = 2.8. Fig. 14 shows the deformed 3-D bubble and the velocity field on the coarse mesh for several snap-shots in time, corresponding to zero and maximum recorded lift coefficient. As in the 2-D case, the quality of the mesh is preserved during the simulation. In Fig. 15, the bubble centre velocity variation in time is shown. One can see that bubble moves along the zig-zag trajectory. The simulated mean rising velocity vz = m/s is in good agreement with the results by de Vries et al. [27, 28] of vz = m/s. Some of the error can also be associated with insufficient mesh resolution. Fig. 16 shows the visualisation of massless particle streaks, indicating a symmetrical doublethreaded wake behind the bubble. This feature of the flow is also in close agreement with the 14

15 Automatic mesh motion for H. Jasak, Ž. Tuković Table 1 Timing breakdown for a 3-D bubble simulation. Operation Time (s) Cumulative (s) Building momentum matrix Solving momentum equation Building pressure matrix Solving pressure equation Building motion matrix Solving motion equation Z Z Y X Y X (a) t = 0.56 s, α f,max = 38.5, α f,mean = 10. (b) t = s, α f,max = 43.3, α f,mean = Z Z Y X Y X (c) t = 0.63 s, α f,max = 41.4, α f,mean = (d) t = s, α f,max = 42, α f,mean = 10. Fig. 14 Free-rising air bubble in water in 3-D: Free surface and velocity vectors in the central x z plane. 15

16 H. Jasak, Ž. Tuković Automatic mesh motion for vx, vy, vz, m/s t, s v x v y v z de Vries [27], v z Fig. 15 Rising velocity components for a 3-D bubble. reported flow visualisation experiments [27]. Z Y X Fig. 16 Particle streaks for a 3-D bubble, t = 0.56 s. The cost associated with the mesh motion solver is 50 60% of the complete cost of simulation, which is high but acceptable. However, the selected mesh motion algorithm is inherently parallel, both in terms of selected discretisation and choice of linear equation solvers. A combination of a massively parallel FVM flow solver already available in OpenFOAM and a parallel motion solver working on the identical mesh decomposition offers considerable scope in reducing execution time. Good parallel efficiency seems to be the best way to handle the cost of long transient runs needed to accumulate sufficient DNS statistics. 6. Conclusion This study describes a novel a vertex-based automatic mesh motion algorithm. Its purpose is to determine the point positions based on the prescribed boundary motion which can be prescribed by 16

17 Automatic mesh motion for H. Jasak, Ž. Tuković external events or calculated as a part of the solution. Validity and quality of the mesh is preserved through vertex-based discretisation and automatic tetrahedral decomposition of polyhedral cells. Having analysed several popular automatic mesh motion approaches and their advantages and drawbacks, we have settled on a second-order quasi-tetrahedral Finite Element method and the Laplace operator to govern the motion. Support for polyhedral cells is provided using the minielement technique, where each cell splits into tetrahedra on-the-fly and a linear shape function is used. The chosen method of discretisation produces a symmetric positive definite matrix ideal for iterative linear equation solvers. The quality of the mesh in motion is preserved by prescribing non-constant diffusion field in the Laplace operator. Several techniques have been tested, most notably the distance-based diffusion, where the coefficient depends on the distance between the cell centre and the nearest boundary of interest, as well as one based on cell distortion. A combination of the above components with a second-order FV flow solver creates a robust and efficient dynamic mesh motion solver capable of handling free surface flows using a surfacetracking algorithm. The solver has been tested on simulation of free rising air bubbles in water. Overall, the cost of the automatic motion solver is about 50 60% of the overall cost of simulation. In future work, the flow solver will be used in DNS simulations of gas bubbles in liquids, establishing a base for phase interaction modelling in Eulerian multi-phase simulations. Extensions allowing topological changes for the volume and surface mesh are also being considered. REFERENCE [1] Harlow, F. H., Welch, J. E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluds 8 (12) (1965), pp [2] Hirt, C. W., Nichols, B. D., Volume of fluid (VOF) method for the dynamics of free boundaries, Journal of computational physics 39 (1981), pp [3] Ubbink, O., Issa, R., A method for capturing sharp fluid interfaces on arbitrary meshes, Journal of computational physics 153 (1999), pp [4] Batina, J. T., Unsteady Euler airfoil solutions using unstructured dynamic meshes, AIAA Journal 28 (8) (1990), pp [5] Blom, F. J., Considertions on the spring analogy, International journal for numerical methods in fluids 32 (2000), pp [6] Degan, C., Farhat, C., A three-dimensional torsional spring analogy method for unstructured dynamic meshes, Computers and structures 80 (2002), pp [7] Löhner, R., Yang, C., Improved ALE mesh velocities for moving bodies, Communications in numerical methods in engineering 12 (1996), pp [8] Johnson, A. A., Tezduyar, T. E., Mesh update stategies in parallel finite element computations of flow problems with moving boundaries and interfaces, Computer methods in applied mechanics and engineering 119 (1994), pp [9] Helenbrook, B. T., Mesh deformation using the biharmonic operator, International journal for numerical methods in engineering 56 (2003), pp [10] Demirdžić, I., Perić, M., Space conservation law in finite volume calculations of fluid flow, International journal for numerical methods in fluids 8 (1988), pp [11] Jasak, H., Weller, H. G., Gosman, A. D., High resolution NVD differencing scheme for arbitrarily unstructured meshes, International journal for numerical methods in fluids 31 (1999), pp [12] Jasak, H., Error analysis and estimation for finite volume method with applications to fluid flows, Ph.D. thesis, Imperial College, University of London,

18 H. Jasak, Ž. Tuković Automatic mesh motion for [13] Jasak, H., Weller, H. G., Application of the finite volume method and unstructured meshes to linear elasticity, International journal for numerical methods in engineering 48 (2000), pp [14] Jasak, H., Gosman, A. D., Automatic resolution control for the finite volume method. Part 1: A-posteriori error estimates, Numerical heat transfer, Part B 38 (2000), pp [15] Tuković, Ž., Finite volume method on domains of varying shape (in Croatian), Ph.D. thesis, Faculty of mechanical engineering and naval architecture, University of Zagreb, [16] Patankar, S. V., Numerical heat transfer and fluid flow, Hemispher Publiching Corporation, [17] Issa, R. I., Solution of the implicitly discretised fluid flow equations by operator-splitting, Journal of computational physics 62 (1) (1986), pp [18] Hestens, H. R., Steifel, E. L., Method of conjugate gradients for solving linear systems, Journal of research 29 (1952), pp [19] Weller, H. G., Tabor, G., Jasak, H., Fureby, C., A tensorial approach to computational continuum mechanics using object orientated techniques, Computers in physics 12 (6) (1998), pp [20] Jasak, H., Weller, H., Nordin, N., In-cylinder CFD simulation using a C++ object-oriented toolkit, SAE 2004 World Congress & Exhibition, SAE Technical Paper , [21] Baker, T. J., Mesh modification for solution adaptation and time evolving domains, 7th International Conference on Numerical Grid Generation in Computational Field Simulations, Whistler, British Columbia, Canada, [22] Virag, Z., Džijan, I., Šavar, M., Unstructured grid solver for the convection-diffusion equation, S. Atluri (Editor) Proceedings of the 2003 International Conference on Computational & Experimental Engineering & Science, Tech Science Press, Corfu, Greece, [23] Džijan, I., Numerical method for fluid flow analysis on unstructured grid (in Croatian), Ph.D. thesis, Faculty of mechanical engineering and naval architecture, University of Zagreb, [24] Muzaferija, S., Perić, M., Computation of free-surface flows using the finite-volume method and moving grids, Numerical heat transfer, Part B 32 (1997), pp [25] Rusche, H., Computational fluid dynamics of dispersed two-phase flows at high phase fractions, Ph.D. thesis, Imperial College, University of London, [26] Blanco, A., Magnaudet, J., The structure of the axisymmetric high-reynolds number flow around an ellipsoidal bubble of fixed shape, Physics of fluids 7 (6) (1995), pp [27] de Vries, A. W. G., Path and wake of a rising bubble., Ph.D. thesis, University of Twente, [28] de Vries, A. W. G., Biesheuvel, A., van Wijngaarden, L., Notes on the path and wake of a gas bubbles rising in pure water, International journal of multiphase flow 28 (2002), pp Submitted: Accepted: Doc. dr. sc. Hrvoje Jasak h.jasak@wikki.co.uk Wikki Ltd. 10 Palmerston House, 60 Kensington Place, London W8 7PU, England Dr. sc. Željko Tuković zeljko.tukovic@fsb.hr Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lučića 5, Zagreb, Croatia 18