Least Square Curvature Calculation Method for VOF Schemes

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1 ILASS Americas 28th Annual Conference on Liquid Atomization and Spray Systems, Dearborn, MI, May 2016 Least Square Curvature Calculation Method for VOF Schemes E. Cauble, M. Owkes Department of Mechanical & Industrial Engineering Montana State University Bozeman, MT USA Abstract A new method is proposed to compute the curvature of the gas-liquid interface present in a multiphase flow such as an atomizing jet. The curvature is proportional to the surface tension force thus an accurate evaluation of the curvature is necessary for predictive simulations of these flows. Particularly, the small scales present in the atomization of a coherent liquid structure into droplets will be controlled by surface tension forces. The proposed scheme employs a least square method to fit an implicit polynomial function to a cloud of points created at the interface. The points are constructed from the volume of fluid (VOF) representation of the phase interface. The curvature is computed explicitly from this polynomial function. The purposed curvature method is tested extensively in order to determine the optimal stencil size and the degree of the polynomial function in order to reduce computational cost and increase numerical accuracy. The results are then compared against the standard height function method. Corresponding Author: eric.cauble@msu.montana.edu

2 Introduction An accurate representation of gas-liquid interfaces in a multiphase flow simulation is important to predict the flow dynamics of an atomization simulation. Surface forces affect: droplet and fluid velocities, droplet size distributions [1, 2], interface deformations, and spray characterization [3], thus understanding how surface forces are defined in a simulation and their limitations is critical to being able to predict and understand the characteristics of a multiphase flow simulation. The characteristics of droplets are dominated by surface tension and the greater the surface tension, the smaller the droplet sizes [4]. As the liquid structures disperse into droplets, surface tension forces become dominate. Therefore, an accurate representation of the interface and the curvature is critical in flows involving the breakup of droplets on a microscale. The volume-of-fluid (VOF) method [5, 6] is a common interface capturing scheme. VOF methods capture the interface of a fluid by computing the liquid volume fraction contained within a computational cell. VOF methods differ in how they transport the interface. But a challenge common to all VOF schemes is computing the interface curvature from the liquid volume fractions. The height function method is a popular method used to compute the curvature in VOF schemes, [7, 8]. The height function method determines local height functions by integrating the liquid volume either vertically or horizontally depending on the direction of the interface normal, Fig. 1. Most commonly, a 7x3x3 stencil is used to compute the heights. The corresponding curvature is evaluated by: κ = h xx (1 + h 2 x) 3/2 (1) j + 4 j + 3 j + 2 j + 1 j where h x and h xx is the first and second derivative of the height function, respectively. It has been shown that the height function method, using a 7x3x3 stencil, is second-order accurate [9] when exact liquid volume fractions are available. However, even with recent advancements to the height function method [10], the computed curvature fails to converge as the mesh is refined when secondorder liquid volume fractions are used. Second-order liquid volume fractions are obtained with many VOF schemes [11] and a method to compute converging curvatures with this data is desirable. Our work focuses on developing a method to compute the curvature of an interface based on the reconstruction of the liquid interface using an imj 1 h i 1 h i h i+1 Liquid Gas i 1 i i + 1 i + 2 i + 3 i + 4 Figure 1: Example of a height function stencil used to compute interface curvatures. plicit polynomial function fitted to a cloud of points obtained from VOF. Using the polynomial reconstruction of the interface, we are then able to easily compute the corresponding curvature of the interface. The purposed method provides a converging curvature even when only a second-order VOF representation is available. Numerical Methodology Mathematical Model The Navier-Stokes s equations are used to describe the motion of a fluid in a domain based on conservation principles. For an incompressible flow the Navier-Stokes equation can be written as: ρ i u i + (ρ i u i u i ) = p i t ( + µ i ui + u T ) (2) i + f i where u = [u, v, w] is the velocity vector in three dimensions, t is time, ρ is density, p is pressure, µ is the dynamic viscosity, and f includes surface and body forces. The subscript i designates the phase of the fluid and takes values of g or l for gas and liquid, respectively. Conservation of mass for an incompressible fluid is enforced as: α i t + (u iα i ) = 0 (3) where α is the volume fraction which tracks the amount of the liquid phase present in a computational cell and stores the ratio of liquid volume to cell volume. In a gas-liquid interfacial flow, α is 2

3 unity if the cell is composed entirely of liquid and zero if the cell is composed entirely of gas. If α is greater than zero and less than unity, an interface exists which separates the phases within the cell. The equations above have been written in both the gas and liquid phases. They are connected through jump conditions at the phase interface. For example, the jumps in density and viscosity at the interface Γ are written as: [ρ] Γ = ρ l ρ g (4) [µ] Γ = µ l µ g, (5) respectively. In the absence of a phase change the velocity field is continuous [u] Γ = 0. (6) The pressure is discontinuous due to contributions from surface tension and the normal component of the viscous stress, i.e., [p] Γ = σκ + 2[µ] Γ n T u n, (7) where σ is the surface tension coefficient and κ is the interface curvature. The proposed model provides an accurate and converging curvature that is used in the pressure discontinuity. An accurate curvature is critical for predicting many multiphase flows such as atomization wherein surface tension forces dominate at an important length scale. Interface Reconstruction Interface reconstruction of the fluid within a cell containing an interface is done by using the piecewise linear interface calculation method (PLIC) [5, 12], which is second-order accurate. The interface of the fluid within the cell is approximated by a straight line (2D) or plane (3D) and the position of this segment is dependent on the volume fraction of the fluid. The orientation of the interface is determined based on the approximated unit normal, Fig. 2. In this work, the normal is determined by the ELVIRA method [13]. Curvature Scheme In this section, we present details of the new curvature scheme. Utilizing the methods discussed above, the interface in each cell is represented by a straight line or a plane (PLIC). In order to compute the interface curvature of the liquid, a polynomial function is determined that best fits a set of points defined on the PLIC reconstruction. This function is then used to determine the corresponding curvature of the interface Figure 2: Example of PLIC reconstruction of the interface in 2D. For each cell that contains an interface, a set of points are assigned to the PLIC one third the distance to the centroid from the points where the PLIC intersects the cells edges. Figure 3 shows the points distributed on a triangulated PLIC reconstruction. Similar definitions are used if the PLIC has more intersections with the cell. These points along with the points located in the adjacent neighboring cells in each direction are combined to create a cloud of points on the interface in the vicinity of the cell of interest. C A O Figure 3: Example of the point distribution on a triangular PLIC. The cloud of points is fitted with an implicit polynomial, such as the second order function defined in three dimensions: f(x, y, z) = a a 100 x + a 010 y + a 001 z + a 200 x 2 + a 110 xy + a 020 y 2 B + a 101 xz + a 011 yz + a 002 z 2 (8) 3

4 using the 3L algorithm [14]. The 3L algorithm was chosen because it is computationally efficient and does not require a high degree polynomial which may lead to oscillations for relatively simple surfaces such as a sphere while also providing accurate results to complex surfaces which may include singularities. The 3L algorithm works by defining additional points Γ +c and Γ c. These additional points are created by moving +c and c, respectively, in the normal direction from the original points Γ o. The enlarged cloud of points containing Γ o, Γ +c, and Γ c define three iso-surfaces on the implicit polynomial. The value of the iso-surfaces (values of f at Γ o, Γ +c, and Γ c ) are set to 0, +c, and c respectively to enforce a unity gradient near the interface. A value of c = x, the cell size, is used in this work. polynomial are found by solving: a = (M3L T M 3L) 1 M3L T b. (11) The implicit polynomial is only fitted to points within the specified stencil. An example of a point distribution along a circle fitted with a 2 nd order polynomial is shown in Fig. 5. The grey surface represents the PLIC, the dots represent the distribution of points using a three neighbor stencil, and the red surface denotes the 2 nd order fitted polynomial function. Figure 5 shows good correlation between the fitted function and the interface of the circle in the region of the stencil. The curvature determined by the fitted polynomial is computed only in the cell containing an interface and not its neighbors. The curvature in the remaining neighboring cells are then determined individually if an interface exists following the same procedure but using different neighbors. Γ c Γ 0 Γ +c c c Figure 4: Example of the 3L points used in the x-y plane. The implicit polynomial is fit using the least squares method to the enlarged cloud of points. The evaluated monomials for each level set are used to generate the 3L block matrix, Eq. (9), and the block vector b, Eq. (10), which specifies the distance to the inner and outer points from the zero level set: M Γ c [M 3L ] = M Γ0 (9) M Γ+c c [b] = 0 (10) +c The least-square solution for the coefficients of the Figure 5: Three neighbor stencil of points assigned to the PLIC (grey) and fitted with 2 nd order polynomial function (red). In order to determine the curvature of the liquid interface, the curvature is defined in terms of the computed polynomial function: ( ) f(x, y, z) κ l = f(x, y, z) (12) The corresponding curvature is then calculated at the average of the points on the PLIC in the cell containing an interface. A 2 nd and 3 rd order implicit polynomials were considered in the initial tests in addition to variating the number of neighbors used to form Γ 0. 4

5 Circle Test Case Several tests were conducted using a circle defined in two dimensions in order to verify the accuracy of the new curvature scheme and to gain a better understanding of how the size of the stencil and the order of the fitted polynomial affect the computed curvature. These results are compared to the results obtained using the mesh decoupled height function method using the exact analytical solution for the curvature a circle. For each test case, a circle was centered in a square unit domain and had a radius of 0.2. For a circle with radius R, the exact curvature is computed by: κ = 1 R. (13) Two tests were conducted. The first was performed by assigning points along the entire interface using the exact radius of a circle, Fig. 6. These points were used to compute the corresponding curvature to minimize any error that may occur by computing the curvature located on a point not directly on the circle. Finally, exact normals were used to reduce any sources of error outside of fitting the implicit polynomial and computing the corresponding curvature. Similarly, a perfect test case for the height function method was created by specifying exact liquid volume fractions to define the circle Figure 6: Exact circle test case. The next 2D test case was performed using a second-order VOF approximation of the interface. This is a common situation because most VOF transport schemes are second-order accurate. The volume fractions were computed using the trapezoidal rule to approximate the volume integrals. Points were then assigned to the PLIC and ELVIRA was used to specify the direction normal to the interface. In order to determine if the solution of the curvature converged, several tests were conducted on uniform meshes while increasing the mesh resolution. Two error metrics assessed the curvature: L 2 (κ) = L (κ) = Ncells i=1 (κ i κ E ) 2 Ncells max i=1...n cells i=1 κ 2 E κ i κ E κ E (14) (15) where κ i and κ E represent the computed and exact curvatures, respectively. For the second-order VOF tests, the exact circle location was randomly displaced over 50 simulations per each mesh level. The computed curvature error was determined by computing the average curvature error of all 50 simulations. This was done in order to reduce the noise caused by a perfect alignment of the interface with the cell boundary and to simulate a more realistic situation. Results and Discussion The results shown in figure 7 show the errors calculated for the exact circle test case when using a 1-3 neighbor stencil. The errors are plotted as a function of the number of cells (N) across the diameter of the droplet (D). The results show that both the 2 nd and 3 rd order polynomials are second-order accurate when using 1-2 neighbors. The 2 nd order polynomial fails to converge with second-order accuracy when using 3 neighbors. The 3 rd order polynomial becomes numerically unstable as the mesh is refined when using 1-2 neighbors. This is partly due to the characteristics of the 3 rd order polynomial and its inability to accurately fit a tight grouping of points due to the number of inflection points. When the number of neighbors increased, the span of the distribution of points increased which improved the accuracy of the results. The results of the height function method showed a second-order convergence using an exact VOF which was expected [9]. The 2 nd order polynomial produced the best results overall. Both the 2 nd and 3 rd order polynomials are seen to be converging for course meshes but both results display noise as the mesh is refined. The noise is reduced as the number of neighbors is increased. However, even with the extra noise, the results using the 2 nd order polynomial and 3 neighbors are slightly more accurate than the mesh decou- 5

6 pled height function method which fails to converge for relatively course meshes, Fiq. 8e. In summary, the purposed scheme provides a converging curvature on meshes as fine as 60. The height function only converges up to 15. This extra range of convergence will improve the accuracy of many engineering simulations that resolve liquid structures with between 15 and 60 points. Conclusion Many variables were considered in order to accurately determine the curvature along the interface of a droplet. Detailed tests were performed by altering these variables in order to determine the optimal settings that produced the best results. The results of these tests were then compared to the mesh decoupled height function method which has been shown to produce accurate results for well defined volume fractions. The height function method showed a strong second-order convergence when an exact liquid volume fraction was specified. However, the height function method failed to converge when the liquid volume fractions were second-order accurate. This becomes an issue for fine meshes and could have a large impact on the final solution especially when the behavior of a liquid is highly dependent on interfacial and surface tension forces. Our new curvature method showed second-order convergence using exact points and normals. This method worked very well for course meshes when the volume fractions were not well defined and showed convergence even when a second-order VOF was used out to 60. This improved convergence on realistic VOF fields will provide better simulation results for many engineering problems such as atomization. Acknowledgments The authors would like to acknowledge the Research Computing Group at Montana State University and HPC at Montana Tech of the University of Montana for support through computational resources. [4] a. Yu. Vasilev and a. I. Maiorova. High Temperature, 52(2): , [5] Roger B. DeBar. Technical Report, (March):UCID 17366, [6] C.W Hirt and B.D Nichols. Journal of Computational Physics, 39(1): , [7] Mayank Malik, Eric Sheung Chi Fan, and Markus Bussmann. International Journal for Numerical Methods in Fluids, 55(7): , [8] J. Hernández, J. López, P. Gómez, C. Zanzi, and F. Faura. International Journal for Numerical Methods in Fluids, 58(8): , [9] G. Bornia, a. Cervone, S. Manservisi, R. Scardovelli, and S. Zaleski. Journal of Computational Physics, 230(4): , [10] Mark Owkes and Olivier Desjardins. Journal of Computational Physics, 281: , [11] G Tryggvason and J Lu. Direct Numerical Simulations of Multiphase Flows [12] Grzegorz K. Karch, Filip Sadlo, Christian Meister, Philipp Rauschenberger, Kathrin Eisenschmidt, Bernhard Weigand, and Thomas Ertl. IEEE Pacific Visualization Symposium, pp , [13] James Edward Pilliod and Elbridge Gerry Puckett. Journal of Computational Physics, 199(2): , [14] Michael M. Blane. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(3): , References [1] G. Malkawi, a. L. Yarin, and F. Mashayek. Journal of Applied Physics, 108(6):1 6, [2] Zhen Tao Wang, Aleksandar M. Mitrašinović, and John Z. Wen. Energies, 5(11): , [3] a. Kourmatzis and J. S. Shrimpton. Journal of Electrostatics, 70(3): ,

7 (a) 1 neighbor L 2 error (b) 1 neighbor L error (c) 2 neighbors L 2 error (d) 2 neighbors L error (e) 3 neighbors L 2 error (f) 3 neighbors L error Figure 7: Exact circle curvature test case. The triangles and squares indicate a 2 nd and 3 rd order fitted polynomial functions, respectively. The stars represent the height function method and the dash-dotted and dashed-dashed lines show first and second order convergence for comparison. 7

8 10 1 (a) 1 neighbor1 L 2 error (b) 1 neighbor L error 10 1 (c) 2 neighbors L 2 error (d) 2 neighbors L error (e) 3 neighbors L 2 error (f) 3 neighbors L error Figure 8: Second-order VOF approximation test case. Results were averaged over 50 simulations per each mesh level. The triangles and squares indicate a 2 nd and 3 rd order fitted polynomial functions, respectively. The dash-dotted and dashed-dashed lines show first and second order convergence for comparison. 8

A 3D VOF model in cylindrical coordinates

A 3D VOF model in cylindrical coordinates Marmar Mehrabadi and Markus Bussmann Department of Mechanical and Industrial Engineering, University of Toronto Recently, volume of fluid (VOF) methods have improved