The Arbitrary Lagrangian–Eulerian (ALE) framework coupled with a direct boundary tracking technique is proven as an effective method for simulation of capillary jet breakup. In this paper, an alternative derivation of ALE is proposed by introducing instantaneous inertial coordinates. Several improvements for simulations are made, including a linear implicit time scheme, a compact scheme for dynamic boundary conditions, and an iterative direct boundary tracking technique for kinematic boundary conditions. Numerical results are verified and improvements are demonstrated.

Juxtaposition-based domain decomposition requires complicated pre-processing and communications of pre-consolidated data, and is restricted to field problems. In this paper, we propose a superposition-based domain decomposition parallelization, which employs element-by-element construction, processor-level assembling, and condensed random data structure. Superposition-based parallelization shows great flexibility in partitioning the computational domains, communicates more consolidated data, and can be applied beyond field problems. Moreover, superposition-based parallelization can, as an option, follow the same numerical process as its serial counterpart to produce digit-by-digit identically the same result, which makes code development and debugging much easier. Solving large scale indefinite systems continues to pose as a challenging issue for incompressible flows. In this paper, we propose the discrete operator splitting (DOS) technique to break the original ill-natured large indefinite system into two smaller well-natured definite systems coupled through source terms. The underpinning idea of the technique is to seamlessly combine the splitting and iterations together. Equipped with the parallelization and DOS, we present in details the superposition-based parallel discrete operator splitting finite element method and apply it to incompressible Navier–Stokes flows. Backward-facing step flow, cavity flow, and pipe flow are simulated to demonstrate the success of the method.

The collocated grid and the staggered grid have been heavily implemented for calculations of Navier-Stokes equations. In this work, we present a new conservative lattice grid for solving incompressible flows with the finite-volume method. The name "lattice'' is to reflect the pattern of this grid. For example, in the case of 2-D incompressible flows, every node for a velocity vector is surrounded by four pressure nodes, and every pressure node is surrounded by four nodes for a velocity vector. As in the staggered grid, local conservations can be conveniently imposed on the lattice grid. This is illustrated through a simulation of the shear-driven cavity flow with an exact factorization technique on the new grid. In the same numerical example, the compact velocity cell is introduced to simplify the application of boundary conditions. In addition, the developing channel flow, a transient flow of the modified Stokes first problem with the exact solution, and a transient pulsatile channel flow with the exact solution are calculated. Good results are obtained, based on a compressed general-purpose Krylov-subspace iterative solver, GPBiCG(m,l).

The main aim of this work is to investigate how reconstructing instantaneous carrier-phase velocities from filtered ones through defiltering improves the prediction of particle concentration in large-eddy simulation (LES) of particle-laden turbulent flows. A particle-laden homogeneous turbulent shear flow is simulated by LES employing the approximate deconvolution method (ADM) to approximate the instantaneous velocities of the carrier phase at the location of particles to solve their Lagrangian momentum equations. The carrier phase is simulated using a Fourier pseudo-spectral method with dynamic Smagorinsky model for subgrid-scale closures. The level of particle concentration is measured by the radial distribution function of particles and the probability density function of the particle number density. Particles with various time constants and terminal velocities are considered and it is shown that employing ADM highly improves the prediction of particle concentration by LES as results are compared against the results obtained by the direct numerical simulation performing a posteriori test.

Splitting techniques break an ill-conditioned indefinite system resulting from incompressible Navier–Stokes equations into well-conditioned subsystems, which can be solved reliably and efficiently. Apart from the ambiguity regarding numerical boundary conditions for the pressure (and for intermediate velocities, whenever introduced), splitting techniques usually incur splitting errors which reduce time accuracy. The discrete approach of approximate factorization techniques eliminates the need of numerical boundary conditions and restores time accuracy by an approximate inversion of some matrix in the case of semi-implicit time schemes. For linear implicit, non-linear implicit, and higher-order semi-implicit time schemes, however, approximate factorization techniques are laborious. In this paper, we systematically present a new and straightforward exact factorization technique. The main contributions of this work include: (1) the idea of removing the splitting error or the idea of restoring time accuracy for fully discrete systems, (2) the introduction of the pressure-update type and the pressure-correction type of exact factorization techniques for any time schemes, and (3) an analysis of several established techniques and their relations to the exact factorization technique. The exact factorization technique is implemented with a standard second-order finite volume method and is verified numerically.

Compact finite difference methods feature high‐order accuracy with smaller stencils and easier application of boundary conditions, and have been employed as an alternative to spectral methods in direct numerical simulation and large eddy simulation of turbulence. The underpinning idea of the method is to cancel lower‐order errors by treating spatial Taylor expansions implicitly. Recently, some attention has been paid to conservative compact finite volume methods on staggered grid, but there is a concern about the order of accuracy after replacing cell surface integrals by average values calculated at centres of cell surfaces. Here we introduce a high‐order compact finite difference method on staggered grid, without taking integration by parts. The method is implemented and assessed for an incompressible shear‐driven cavity flow at Re = 103, a temporally periodic flow at Re = 104, and a spatially periodic flow at Re = 104. The results demonstrate the success of the method. Copyright © 2006 John Wiley & Sons, Ltd.

The presence of the pressure and the convection terms in incompressible Navier-Stokes equations makes their numerical simulation a challenging task. The indefinite system as a consequence of the absence of the pressure in continuity equation is ill-conditioned. This difficulty has been overcome by various splitting techniques, but these techniques incur the ambiguity of numerical boundary conditions for the pressure as well as for the intermediate velocity (whenever introduced). We present a new and straightforward discrete splitting technique which never resorts to numerical boundary conditions. The non-linear convection term can be treated by four different approaches, and here we present a new linear implicit time scheme. These two new techniques are implemented with a finite element method and numerical verifications are made. Copyright (c) 2005 John Wiley & Sons, Ltd.