Olivier Desjardins

High order conservative finite difference scheme for variable density low Mach number turbulent flows

The high order conservative finite difference scheme of Morinishi et al. Y. Morinishi, O.V. Vasilyev, T. Ogi, Fully conservative finite difference scheme in cylindrical coordinates for incompressible flow simulations, J. Comput. Phys. 197 (2004) 686] is extended to simulate variable density flows in complex geometries with cylindrical or cartesian non-uniform meshes. The formulation discretely conserves mass, momentum, and kinetic energy in a periodic domain. In the presence of walls, boundary conditions that ensure primary conservation have been derived, while secondary conservation is shown to remain satisfactory. In the case of cylindrical coordinates, it is desirable to increase the order of accuracy of the convective term in the radial direction, where most gradients are often found. A straightforward centerline treatment is employed, leading to good accuracy as well as satisfactory robustness. A similar strategy is introduced to increase the order of accuracy of the viscous terms. The overall numerical scheme obtained is highly suitable for the simulation of reactive turbulent flows in realistic geometries, for it combines arbitrarily high order of accuracy, discrete conservation of mass, momentum, and energy with consistent boundary conditions. This numerical methodology is used to simulate a series of canonical turbulent flows ranging from isotropic turbulence to a variable density round jet. Both direct numerical simulation (DNS) and large eddy simulation (LES) results are presented. It is observed that higher order spatial accuracy can improve significantly the quality of the results. The error to cost ratio is analyzed in details for a few cases. The results suggest that high order schemes can be more computationally efficient than low order schemes.

An accurate conservative level set/ghost fluid method for simulating turbulent atomization

This paper presents a novel methodology for simulating incompressible two-phase flows by combining an improved version of the conservative level set technique introduced in [E. Olsson, G. Kreiss, A conservative level set method for two phase flow, J. Comput. Phys. 210 (2005) 225-246] with a ghost fluid approach. By employing a hyperbolic tangent level set function that is transported and re-initialized using fully conservative numerical schemes, mass conservation issues that are known to affect level set methods are greatly reduced. In order to improve the accuracy of the conservative level set method, high order numerical schemes are used. The overall robustness of the numerical approach is increased by computing the interface normals from a signed distance function reconstructed from the hyperbolic tangent level set by a fast marching method. The convergence of the curvature calculation is ensured by using a least squares reconstruction. The ghost fluid technique provides a way of handling the interfacial forces and large density jumps associated with two-phase flows with good accuracy, while avoiding artificial spreading of the interface. Since the proposed approach relies on partial differential equations, its implementation is straightforward in all coordinate systems, and it benefits from high parallel efficiency. The robustness and efficiency of the approach is further improved by using implicit schemes for the interface transport and re-initialization equations, as well as for the momentum solver. The performance of the method is assessed through both classical level set transport tests and simple two-phase flow examples including topology changes. It is then applied to simulate turbulent atomization of a liquid Diesel jet at Re=3000. The conservation errors associated with the accurate conservative level set technique are shown to remain small even for this complex case.

An Euler-Lagrange strategy for simulating particle-laden flows

In this work, a strategy capable of simulating polydisperse flows in complex geometries is employed where the fluid transport equations are solved in an Eulerian framework and the dispersed phase is represented as Lagrangian particles. Volume filtered equations for the carrier phase are derived in detail for variable density flows, and all unclosed terms are discussed. Special care is given to the interphase coupling terms that arise, in order to ensure that they are implemented consistently and that they converge under mesh refinement. This provides the flexibility of using cell sizes that are smaller than the particle diameter if necessary. Particle collisions are handled using a soft-sphere model that has been modified for parallel efficiency. Simulations are carried out for a number of laboratory-scale configurations, showing excellent agreement with experiments.

A spectrally refined interface approach for simulating multiphase flows

This paper presents a novel approach to phase-interface transport based on pseudo-spectral sub-grid refinement of a level set function. In each flow solver grid cell, a set of quadrature points is introduced on which the value of the level set function is known. This methodology allows to define a polynomial reconstruction of the level set function in each cell. The transport is performed using a semi-Lagrangian technique, removing all constraints on the time step size. Such an approach provides sub-cell resolution of the phase-interface and leads to excellent accuracy in the transport, while a reasonable cost is obtained by pre-computing some of the metrics associated with the polynomials. To couple this approach with a flow solver, an converging curvature computation is introduced. First, a second order explicit distance to the sub-grid interface is reconstructed on the flow solver mesh. Then, a least squares approach is employed to extract the curvature from this distance function. This technique is found to combine the high accuracy and good conservation found in the particle level set method with the converging curvature usually obtained with classical high order PDE transport of the level set function. Tests are presented for both transport as well as two-phase flows, that suggest that this technique is capable of retaining the thin liquid structures that are expected in turbulent atomization of liquids.

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 ghost fluid, level set methodology for simulating multiphase electrohydrodynamic flows with application to liquid fuel injection

In this paper, we present the development of a sharp numerical scheme for multiphase electrohydrodynamic (EHD) flows for a high electric Reynolds number regime. The electric potential Poisson equation contains EHD interface boundary conditions, which are implemented using the ghost fluid method (GFM). The GFM is also used to solve the pressure Poisson equation. The methods detailed here are integrated with state-of-the-art interface transport techniques and coupled to a robust, high order fully conservative finite difference Navier-Stokes solver. Test cases with exact or approximate analytic solutions are used to assess the robustness and accuracy of the EHD numerical scheme. The method is then applied to simulate a charged liquid kerosene jet.

A localized re-initialization equation for the conservative level set method

The conservative level set methodology for interface transport is modified to allow for localized level set re-initialization. This approach is suitable to applications in which there is a significant amount of spatial variability in level set transport. The steady-state solution of the modified re-initialization equation matches that of the original conservative level set provided an additional Eikonal equation is solved, which can be done efficiently through a fast marching method (FMM). Implemented within the context of the accurate conservative level set method (ACLS) (Desjardins et al., 2008, 6]), the FMM solution of this Eikonal equation comes at no additional cost. A metric for the appropriate amount of local re-initialization is proposed based on estimates of local flow deformation and numerical diffusion. The method is compared to standard global re-initialization for two test cases, yielding the expected results that minor differences are observed for Zalesak?s disk, and improvements in both mass conservation and interface topology are seen for a drop deforming in a vortex. Finally, the method is applied to simulation of a viscously damped standing wave and a three-dimensional drop impacting on a shallow pool. Negligible differences are observed for the standing wave, as expected. For the last case, results suggest that spatially varying re-initialization provides a reduction in spurious interfacial corrugations, improvements in the prediction of radial growth of the splashing lamella, and a reduction in conservation errors, as well as a reduction in overall computational cost that comes from improved conditioning of the pressure Poisson equation due to the removal of spurious corrugations.

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.

Strongly coupled fluid-particle flows in vertical channels. II. Turbulence modeling

In Part I, simulations of strongly coupled fluid-particle flow in a vertical channel were performed with the purpose of understanding, in general, the fundamental physics of wall-bounded multiphase turbulence and, in particular, the roles of the spatially correlated and uncorrelated components of the particle velocity. The exact Reynolds-averaged (RA) equations for high-mass-loading suspensions were presented, and the unclosed terms that are retained in the context of fully developed channel flow were evaluated in an Eulerian–Lagrangian (EL) framework. Here, data from the EL simulations are used to validate a multiphase Reynolds-stress model (RSM) that predicts the wall-normal distribution of the two-phase, one-point turbulence statistics up to second order. It is shown that the anisotropy of the Reynolds stresses both near the wall and far away is a crucial component for predicting the distribution of the RA particle-phase volume fraction. Moreover, the decomposition of the phase-average (PA) particle-ph...