A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

A review of level-set methods and some recent applications

We present an adaptive algorithm for low Mach number reacting flows with complex chemistry. Our approach uses a form of the low Mach number equations that discretely conserves both mass and energy. The discretization methodology is based on a robust projection formulation that accommodates large density contrasts. The algorithm uses an operator-split treatment of stiff reaction terms and includes effects of differential diffusion. The basic computational approach is embedded in an adaptive projection framework that uses structured hierarchical grids with subcycling in time that preserves the discrete conservation properties of the underlying single-grid algorithm. We present numerical examples illustrating the performance of the method on both premixed and non-premixed flames.

Numerical simulation of laminar reacting flows with complex chemistry

We present and study a new algorithm for simulating the N-phase mean curvature motion for an arbitrary set of (isotropic) surface tensions. The departure point is the threshold dynamics algorithm of Merriman, Bence, and Osher for the two-phase case.



A new energetic interpretation of this algorithm allows us to extend it in a natural way to the case of N phases, for arbitrary surface tensions and arbitrary (isotropic) mobilities. For a large class of surface tensions, the algorithm is shown to be consistent in the sense that at every time step, it decreases an energy functional that is an approximation (in the sense of Gamma convergence) of the interfacial energy. A broad range of numerical tests shows good convergence properties.



An important application is the motion of grain boundaries in polycrystalline materials: It is also established that the above-mentioned large class of surface tensions contains the Read-Shockley surface tensions, both in the two-dimensional and three-dimensional settings.© 2015 Wiley Periodicals, Inc.

https://www.mis.mpg.de/preprints/2013/preprint2013_2.pdf

Threshold dynamics for networks with arbitrary surface tensions

Next Generation Drying Technologies for Pharmaceutical Applications

A high-order accurate numerical quadrature algorithm is presented for the evaluation of integrals over curved surfaces and volumes which are defined implicitly via a fixed isosurface of a given function restricted to a given hyperrectangle. By converting the implicitly defined geometry into the graph of an implicitly defined height function, the approach leads to a recursive algorithm on the number of spatial dimensions which requires only one-dimensional root finding and one-dimensional Gaussian quadrature. The computed quadrature scheme yields strictly positive quadrature weights and inherits the high-order accuracy of Gaussian quadrature: a range of different convergence tests demonstrate orders of accuracy up to 20th order. Also presented is an application of the quadrature algorithm to a high-order embedded boundary discontinuous Galerkin method for solving partial differential equations on curved domains.

/pdf/high-order-quadrature-methods-for-implicitly-defined-2tb8g7dxv0.pdf

High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles ∗

We introduce a numerical framework, the Voronoi Implicit Interface Method for tracking multiple interacting and evolving regions (phases) whose motion is determined by complex physics (fluids, mechanics, elasticity, etc.), intricate jump conditions, internal constraints, and boundary conditions. The method works in two and three dimensions, handles tens of thousands of interfaces and separate phases, and easily and automatically handles multiple junctions, triple points, and quadruple points in two dimensions, as well as triple lines, etc., in higher dimensions. Topological changes occur naturally, with no surgery required. The method is first-order accurate at junction points/lines, and of arbitrarily high-order accuracy away from such degeneracies. The method uses a single function to describe all phases simultaneously, represented on a fixed Eulerian mesh. We test the method’s accuracy through convergence tests, and demonstrate its applications to geometric flows, accurate prediction of von Neumann’s law for multiphase curvature flow, and robustness under complex fluid flow with surface tension and large shearing forces.

/pdf/the-voronoi-implicit-interface-method-for-computing-esxf1551zr.pdf

The Voronoi Implicit Interface Method for computing multiphase physics

Modeling the physics of foams and foamlike materials, such as soapy froths, fire retardants, and lightweight crash-absorbent structures, presents challenges, because of the vastly different time and space scales involved By separating and coupling these disparate scales, we have designed a multiscale framework to model dry foam dynamics This leads to a predictive and flexible computational methodology linking, with a few simplifying assumptions, foam drainage, rupture, and topological rearrangement, to coupled interface-fluid motion under surface tension, gravity, and incompressible fluid dynamics Our computed results match theoretical analyses and experimentally observed physical effects, including thin-film drainage and interference, and are used to study bubble rupture cascades and macroscopic rearrangement The developed multiscale model allows quantitative computation of complex foam evolution phenomena

/pdf/multiscale-modeling-of-membrane-rearrangement-drainage-and-2jq6d14h0k.pdf

Multiscale modeling of membrane rearrangement, drainage, and rupture in evolving foams.

https://escholarship.org/content/qt87p9f9rp/qt87p9f9rp.pdf?t=ou80ie

Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part I

https://escholarship.org/content/qt8rb3v3gh/qt8rb3v3gh.pdf?t=ou80ys

Robert Saye

Papers

High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles ∗

The Voronoi Implicit Interface Method for computing multiphase physics

Multiscale modeling of membrane rearrangement, drainage, and rupture in evolving foams.

Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part I

Analysis and applications of the Voronoi Implicit Interface Method