Mark Owkes

A discontinuous Galerkin conservative level set scheme for interface capturing in multiphase flows

The accurate conservative level set (ACLS) method of Desjardins et al. O. Desjardins, V. Moureau, H. Pitsch, An accurate conservative level set/ghost fluid method for simulating turbulent atomization, J. Comput. Phys. 227 (18) (2008) 8395-8416] is extended by using a discontinuous Galerkin (DG) discretization. DG allows for the scheme to have an arbitrarily high order of accuracy with the smallest possible computational stencil resulting in an accurate method with good parallel scaling. This work includes a DG implementation of the level set transport equation, which moves the level set with the flow field velocity, and a DG implementation of the reinitialization equation, which is used to maintain the shape of the level set profile to promote good mass conservation. A near second order converging interface curvature is obtained by following a height function methodology (common amongst volume of fluid schemes) in the context of the conservative level set. Various numerical experiments are conducted to test the properties of the method and show excellent results, even on coarse meshes. The tests include Zalesaks disk, two-dimensional deformation of a circle, time evolution of a standing wave, and a study of the Kelvin-Helmholtz instability. Finally, this novel methodology is employed to simulate the break-up of a turbulent liquid jet.

A computational framework for conservative, three-dimensional, unsplit, geometric transport with application to the volume-of-fluid (VOF) method

Abstract In this work, a novel computational framework for calculating convection fluxes is developed and employed in the context of the piecewise linear interface calculation (PLIC) volume-of-fluid (VOF) method. The scheme is three-dimensional, unsplit, discretely conservative, and bounded. The scheme leverages the idea of semi-Lagrangian transport to estimate the amount of liquid that is fluxed through each face during a time-step. The present work can be seen as an extension of the two-dimensional EMFPA method of Lopez et al. (2004) [16] to three dimensions and an improvement of the three-dimensional FMFPA-3D method of Hernandez et al. (2008) [17] with the addition of discrete conservation. In FMFPA-3D, fluxes of liquid volume fraction are calculated by transporting a cell face back in time with a semi-Lagrangian method that uses cell face edge velocities to produce a flux hexahedron with flat faces. The flux hexahedron may overlap with neighboring fluxes hindering the conservation properties of the method. The proposed method computes the fluxes by transporting the cell face back in time using a semi-Lagrangian step based on the cell face corner velocities, which results in a three-dimensional, generalized flux hexahedron that does not typically have flat faces. However, the flux volumes do not overlap and discrete conservation can be achieved. The complex flux volume is partitioned into a collection of simplices and a simple sign convention allows the calculation of the flux to be reduced to a straightforward and systematic algorithm. The proposed VOF scheme is tested on multiple benchmark cases including Zalesaks disk, two- and three-dimensional deformation tests, and the evolution of a droplet in homogeneous isotropic turbulence.

A mesh-decoupled height function method for computing interface curvature

In this paper, a mesh-decoupled height function method is proposed and tested. The method is based on computing height functions within columns that are not aligned with the underlying mesh and have variable dimensions. Because they are decoupled from the computational mesh, the columns can be aligned with the interface normal vector, which is found to improve the curvature calculation for under-resolved interfaces where the standard height function method often fails. A computational geometry toolbox is used to compute the heights in the complex geometry that is formed at the intersection of the computational mesh and the columns. The toolbox reduces the complexity of the problem to a series of straightforward geometric operations using simplices. The proposed scheme is shown to compute more accurate curvatures than the standard height function method on coarse meshes. A combined method that uses the standard height function where it is well defined and the proposed scheme in under-resolved regions is tested. This approach achieves accurate and robust curvatures for under-resolved interface features and second-order converging curvatures for well-resolved interfaces.

A finite-volume HLLC-based scheme for compressible interfacial flows with surface tension

Shock waves are often used in experiments to create a shear flow across liquid droplets to study secondary atomization. Similar behavior occurs inside of supersonic combustors (scramjets) under startup conditions, but it is challenging to study these conditions experimentally. In order to investigate this phenomenon further, a numerical approach is developed to simulate compressible multiphase flows under the effects of surface tension forces. The flow field is solved via the compressible multicomponent Euler equations (i.e., the five equation model) discretized with the finite volume method on a uniform Cartesian grid. The solver utilizes a total variation diminishing (TVD) third-order RungeKutta method for time-marching and second order TVD spatial reconstruction. Surface tension is incorporated using the Continuum Surface Force (CSF) model. Fluxes are upwinded with a modified HartenLaxvan Leer Contact (HLLC) approximate Riemann solver. An interface compression scheme is employed to counter numerical diffusion of the interface. The present work includes modifications to both the HLLC solver and the interface compression scheme to account for capillary force terms and the associated pressure jump across the gasliquid interface. A simple method for numerically computing the interface curvature is developed and an acoustic scaling of the surface tension coefficient is proposed for the non-dimensionalization of the model. The model captures the surface tension induced pressure jump exactly if the exact curvature is known and is further verified with an oscillating elliptical droplet and Mach 1.47 and 3 shock-droplet interaction problems. The general characteristics of secondary atomization at a range of Weber numbers are also captured in a series of simulations.

A mass and momentum conserving unsplit semi-Lagrangian framework for simulating multiphase flows

In this work, we present a computational methodology for convection and advection that handles discontinuities with second order accuracy and maintains conservation to machine precision. This method can transport a variety of discontinuous quantities and is used in the context of an incompressible gas–liquid flow to transport the phase interface, momentum, and scalars. The proposed method provides a modification to the three-dimensional, unsplit, second-order semi-Lagrangian flux method of Owkes & Desjardins (JCP, 2014). The modification adds a refined grid that provides consistent fluxes of mass and momentum defined on a staggered grid and discrete conservation of mass and momentum, even for flows with large density ratios. Additionally, the refined grid doubles the resolution of the interface without significantly increasing the computational cost over previous non-conservative schemes. This is possible due to a novel partitioning of the semi-Lagrangian fluxes into a small number of simplices. The proposed scheme is tested using canonical verification tests, rising bubbles, and an atomizing liquid jet.

Importance of curvature evaluation scale for predictive simulations of dynamic gas–liquid interfaces

Abstract The effect of the scale used to compute the interfacial curvature on the prediction of dynamic gas–liquid interfaces is investigated. A new interface curvature calculation methodology referred to herein as the Adjustable Curvature Evaluation Scale (ACES) is proposed. ACES leverages a weighted least squares regression to fit a polynomial through points computed on the volume-of-fluid representation of the gas–liquid interface. The interface curvature is evaluated from this polynomial. Varying the least squares weight with distance from the location where the curvature is being computed, adjusts the scale the curvature is evaluated on. ACES is verified using canonical static test cases and compared against second- and fourth-order height function methods. Simulations of dynamic interfaces, including a standing wave and oscillating droplet, are performed to assess the impact of the curvature evaluation scale for predicting interface motions. ACES and the height function methods are combined with two different unsplit geometric volume-of-fluid (VoF) schemes that define the interface on meshes with different levels of refinement. We find that the results depend significantly on curvature evaluation scale. Particularly, the ACES scheme with a properly chosen weight function is accurate, but fails when the scale is too small or large. Surprisingly, the second-order height function method is more accurate than the fourth-order variant for the dynamic tests even though the fourth-order method performs better for static interfaces. Comparing the curvature evaluation scale of the second- and fourth-order height function methods, we find the second-order method is closer to the optimum scale identified with ACES. This result suggests that the curvature scale is driving the accuracy of the dynamics. This work highlights the importance of studying numerical methods with realistic (dynamic) test cases and that the interactions of the various discretizations is as important as the accuracy of one part of the discretization.

Large-eddy simulation study of injector geometry on liquid jet in cross-flow and validation with experiments

In this work, large-eddy simulation of the atomization of a liquid jet in cross-flow is performed. Two different injector geometries are investigated that result in significantly different liquid jets. One of the injectors, referred to as the round-edged injector, produces a laminar flow at the exit plane. The other injector, known as the sharp-edged injector, produces a turbulent flow that enhances the atomization of the liquid jet. The jet penetration, mean droplet size spatial distribution, and mean droplet velocity spatial distribution are compared to experimental results by Gopala, and good agreement is observed. To perform the simulations, we employ a computational methodology that is accurate and robust even when large density ratios and turbulent flows are present. The accurate conservative level set is used to transport the gas-liquid interface. A density correction formulation is used to ensure consistency between the interface transport and momentum transport steps, making a robust scheme for simulating high density ratio flows. A conservative immersed boundary method is used to simulate the injector geometries, which avoids the complexity of generating a body-fitted mesh.