Fold‐2‐covering triangular embeddings

For a graph G and a positive integer m, G_(m) is the graph obtained from G by replacing every vertex by an independent set of size m and every edge by m² edges joining all possible new pairs of ends. If G triangulates a surface, then it is easy to see from Euler's formula that G_(m) can, in principle, triangulate a surface. For m prime and at least 7, it has previously been shown that in fact G_(m) does triangulate a surface, and in fact does so as a “covering with folds” of the original triangulation. For m = 5, this would be a consequence of Tutte's 5‐Flow Conjecture. In this work, we investigate the case m = 2 and describe simple classes of triangulations G for which G₍₂₎ does have a triangulation that covers G “with folds,” as well as providing a simple infinite class of triangulations G of the sphere for which G₍₂₎ does not triangulate any surface. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 79–92, 2003     

Equivalence of u–p and ζ–ψ formulations of the time-dependent Navier–Stokes equations

This paper deals with the non-stationary incompressible Navier--Stokes equations for two-dimensional flows expressed in terms of the velocity and pressure and of the vorticity and streamfunction. The equivalence of the two formulations is demonstrated, both formally and rigorously, by virtue of a condition of compatibility between the boundary and initial values of the normal component of velocity. This condition is shown to be the only compatibility condition necessary to allow for solutions of a minimal regularity, namely H¹ for the velocity, as in most current numerical schemes relying on spatial discretizations of local type.     

Analysis of plane and axisymmetric flows of incompressible fluids with the stream tube method: Numerical simulation by trust-region optimization algorithm

New concepts for the study of incompressible plane or axisymmetric flows are analysed by the stream tube method. Flows without eddies and pure vortex flows are considered in a transformed domain where the mapped streamlines are rectilinear or circular. The transformation between the physical domain and the computational domain is an unknown of the problem. In order to solve the non-linear set of relevant equations, we present a new algorithm based on a trust region technique which is effective for non-convex optimization problems. Experimental results show that the new algorithm is more robust compared to the Newton-Raphson method.     

A hybrid finite-element–volume-of-fluid method for simulating free surface flows and interfaces

A numerical technique is developed for the simulation of free surface flows and interfaces. This technique combines the strength on the finite element method (FEM) in calculating the field variables for a deforming boundary and the versatility of the volume-of-fluid (VOF) technique in advection of the fluid interfaces. The advantage of the VOF technique is that it allows the simulation of interfaces with large deformations, including surface merging and breaking. However, its disadantage is that is solving the flow equations, it cannot resolve interfaces smaller than the cell size, since information on the subgrid scale is lost. Therefore the accuracy of the interface reconstruction and the treatment of the boundary conditions (i.e. viscous stresses and surface tension forces) become grid-size-dependent. On the other hand, the FEM with deforming interface mesh allows accurate implementation of the boundary conditions, but it cannot handle large surface deformations occurring in breaking and merging of liquid regions. Combining the two methods into a hybrid FEM-VOF method eliminates the major shortcomings of both. The outcome is a technique which can handle large surface deformations with accurate treatment of the boundary conditions. For illustration, two computational examples are presented, namely the instability and break-up of a capillary jet and the coalescence collision of two liquid drops.     

Air Flow Computation around an Automated Guided Vehicle

An upwind finite element scheme for the incompressible viscous flow at a high Reynolds number was proposed by the fourth and fifth authors. The scheme has the potential to approximate the advection term in third-order accuracy. We apply it to a two-dimensional non-stationary analysis of airflows around an Automated Guided Vehicle (AGV), which starts with constant acceleration, runs at a constant speed and stops with constant deceleration. The results are at least qualitatively good and compatible with experimental ones.     

First-order system least squares and the energetic variational approach for two-phase flow

This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid, and adaptive local refinement, can be used to solve these types of complex fluid flow problems. In addition, from an energetic variational approach, it can be shown that an important quantity to preserve in a given simulation is the energy law. We discuss the energy law and inherent structure for two-phase flow using the Allen–Cahn interface model and indicate how it is related to other complex fluid models, such as magnetohydrodynamics. Finally, we show that, using the FOSLS framework, one can still satisfy the appropriate energy law globally while using well-known numerical techniques.     

Lattice-BGK approach to simulating granular flows

Many continuum theories for granular flow produce an equation of motion for the fluctuating kinetic energy density (granular temperature) that accounts for the energy lost in inelastic collisions. Apart from the presence of an extra dissipative term, this equation is very similar in form to the usual temperature equation in hydrodynamics. It is shown how a lattice-kinetic model based on the Bhatnagar-Gross-Krook (BGK) equation that was previously derived for a miscible two-component fluid may be modified to model the continuum equations for granular flow. This is done by noting that the variable corresponding to the concentration of one species follows an equation that is essentially analogous to the granular temperature equation. A simulation of an unforced granular fluid using the modified model reproduces the phenomenon of clustering instability, namely the spontaneous agglomeration of particles into dense clusters, which occurs generically in all granular flows. The success of the continuum theory in capturing the gross features of this basic phenomenon is discussed. Some shear flow simulations are also presented.     

Viscous potential flow analysis of Kelvin-Helmholtz instability with mass transfer and vaporization

Viscous potential flow analysis of Kelvin-Helmholtz instability with heat and mass transfer has been studied. A dispersion relation has been obtained. Stability criterion is given by a critical value of relative velocity. It has been found that heat and mass transfer has destabilizing effect on relative velocity when lower fluid viscosity is low while it has stabilizing effect when lower fluid viscosity is high. Various graphs have been plotted for relative velocity and growth rate. In statically unstable situation viscosity has stabilizing effect while in statically stable situation it has destabilizing effect.     

A Monte Carlo model for seeded atomic flows in the transition regime

A simple model for the numerical determination of separation effects in seeded atomic gas flows is presented. The model is based on the known possibility to provide a statistically convergent estimate of the exact solution for a linear transport equation using the test particle Monte Carlo method. Accordingly, the flow field of the main gas is preliminary calculated and as a second step the linear transport equations obtained by fixing the target distribution in the collision term of the Boltzmann equation for both main and minority components are solved. Both solutions are based on appropriately devised test particle Monte Carlo methods. The second step, the critical one in evaluating the separation effects, is exact and thereby completely free of numerical diffusion. The model is described in details and illustrated by 2D test cases of atomic separation in shock fronts.