Effect of thermo‐solutal stratification on recirculation flow patterns in a backward‐facing step channel flow

This paper presents results on the combined effect of thermo‐solutal buoyancy forces on the recirculatory flow behavior in a horizontal channel with backward‐facing step and the ensuing impact on heat and mass transfer phenomena. The governing equations for double diffusive mixed convection are represented in velocity–vorticity form of momentum equations, velocity Poisson equations, energy and concentration equations. Galerkin's finite‐element method has been employed to solve the governing equations. Recirculatory flow fields with heat and mass transfer are simulated for opposing and aiding thermo‐solutal buoyancy forces by assuming suitable boundary conditions for energy and concentration equations. The effect of Richardson number (0.1?Ri?10) and buoyancy ratio (?10?N?10) on the recirculation bubble and Nusselt and Sherwood numbers are studied in detail. For Richardson number greater than unity, distinct variations in the gradients of Nusselt number and Sherwood number with buoyancy ratio are observed for flow regimes with opposing and aiding buoyancy forces. Copyright © 2009 John Wiley & Sons, Ltd.     

Modelling the attenuation of laminar fluid transients in piping systems

The damping of laminar fluid transients in piping systems is studied numerically using a two-dimensional water hammer model. The numerical scheme is based on the classical fourth order Runge–Kutta method for time integration and central difference expressions for the spatial terms. The results of the present method show that the damping of transients in piping systems is governed by a non-dimensional parameter representing the ratio of the Joukowsky pressure force to the viscous force. In terms of time scales, this non-dimensional parameter represents the ratio of the viscous diffusion time scale to the pipe period. For small values of this parameter, the damping of the fluid transient becomes more pronounced while for large values, the fluid transient is subjected to insignificant damping. Moreover, the non-dimensional parameter is shown to influence other important transient phenomena such as line packing, instantaneous wall shear stress values and the Richardson annular effect.     

Analysis of numerical stability of algebraic oceanic turbulent mixing layer models

In this paper, we study the stability of oceanic turbulent mixing layers by the finite element method with respect to perturbations of the data. We prove that the equilibria states depend continuously on the data, and that they are asymptotically stable in time, when approximated by standard numerical schemes. We also perform some numerical tests for realistic initial conditions, that also show that the mixing-layer configurations are stable under perturbations of the data, in addition to confirm the theoretical expectations of our analysis.     

Simplified immersed interface methods for elliptic interface problems with straight interfaces

In this article, we propose simplified immersed interface methods for elliptic partial/ordinary differential equations with discontinuous coefficients across interfaces that are few isolated points in 1D, and straight lines in 2D. For one‐dimensional problems or two‐dimensional problems with circular interfaces, we propose a conservative second‐order finite difference scheme whose coefficient matrix is symmetric and definite. For two‐dimensional problems with straight interfaces, we first propose a conservative first‐order finite difference scheme, then use the Richardson extrapolation technique to get a second‐order method. In both cases, the finite difference coefficients are almost the same as those for regular problems. Error analysis is given along with numerical example. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 188–203, 2012     

Discretization errors in molecular dynamics simulations with deterministic and stochastic thermostats

We investigate the influence of numerical discretization errors on computed averages in a molecular dynamics simulation of TIP4P liquid water at 300 K coupled to different deterministic (Nosé–Hoover and Nosé–Poincaré) and stochastic (Langevin) thermostats. We propose a couple of simple practical approaches to estimating such errors and taking them into account when computing the averages. We show that it is possible to obtain accurate measurements of various system quantities using step sizes of up to 70% of the stability threshold of the integrator, which for the system of TIP4P liquid water at 300 K corresponds to the step size of about 7 fs.     

非广延统计力学中的理查逊公式

目的探讨非广延统计力学框架下的理查逊(Richardson)公式。方法采用分解近似的方法。结果与结论发现经典理查逊公式中的e指数分布被q幂函数分布代替,并非广延指标q→1时,非广延统计力学中的理查逊公式就会回到经典的表达式。     

A finite volume‐based high‐order,Cartesian cut‐cell method for wave propagation

Computational aeroacoustics requires numerical techniques capable of yielding low artificial dispersion and dissipation to preserve the amplitude and the frequency characteristics of the physical processes. Furthermore, for engineering applications, the techniques need to handle irregular geometries associated with realistic configurations. We address these issues by developing an optimized prefactored compact finite volume (OPC‐fv) scheme along with a Cartesian cut‐cell technique. The OPC‐fv scheme seeks to minimize numerical dispersion and dissipation while satisfying the conservation laws. The cut‐cell approach treats irregularly shaped boundaries using divide‐and‐merge procedures for the Cartesian cells while maintaining a desirable level of accuracy. We assess these techniques using several canonical test problems, involving different levels of physical and geometric complexities. Richardson extrapolation is an effective tool for evaluating solutions of no high gradients or discontinuities, and is used to evaluate the performance of the solution technique. It is demonstrated that while the cut‐cell method has a modest effect on the order of accuracy, it is a robust method. The combined OPC‐fv scheme and the Cartesian cut‐cell technique offer good accuracy as well as geometric flexibility. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive Wavelet Solution to the Stokes Problem

This paper deals with the design and analysis of adaptive wavelet method for the Stokes problem. First, the limitation of Richardson iteration is explained and the multiplied matrix M0 in the paper of Bramble and Pasciak is proved to be the simplest possible in an appropiate sense. Similar to the divergence operator, an exact application of its dual is shown; Second, based on these above observations, an adaptive wavelet algorithm for the Stokes problem is designed. Error analysis and computational complexity are given; Finally, since our algorithm is mainly to deal with an elliptic and positive definite operator equation, the last section is devoted to the Galerkin solution of an elliptic and positive definite equation. It turns out that the upper bound for error estimation may be improved.     

A least square extrapolation method for the a posteriori error estimate of the incompressible Navier Stokes problem

A posteriori error estimators are fundamental tools for providing confidence in the numerical computation of PDEs. To date, the main theories of a posteriori estimators have been developed largely in the finite element framework, for either linear elliptic operators or non‐linear PDEs in the absence of disparate length scales. On the other hand, there is a strong interest in using grid refinement combined with Richardson extrapolation to produce CFD solutions with improved accuracy and, therefore, a posteriori error estimates. But in practice, the effective order of a numerical method often depends on space location and is not uniform, rendering the Richardson extrapolation method unreliable. We have recently introduced (Garbey, 13th International Conference on Domain Decomposition, Barcelona, 2002; 379–386; Garbey and Shyy, J. Comput. Phys. 2003; 186 :1–23) a new method which estimates the order of convergence of a computation as the solution of a least square minimization problem on the residual. This method, called least square extrapolation, introduces a framework facilitating multi‐level extrapolation, improves accuracy and provides a posteriori error estimate. This method can accommodate different grid arrangements. The goal of this paper is to investigate the power and limits of this method via incompressible Navier Stokes flow computations. Copyright © 2005 John Wiley & Sons, Ltd.     

A high‐order finite difference discretization strategy based on extrapolation for convection diffusion equations

We propose a new high‐order finite difference discretization strategy, which is based on the Richardson extrapolation technique and an operator interpolation scheme, to solve convection diffusion equations. For a particular implementation, we solve a fine grid equation and a coarse grid equation by using a fourth‐order compact difference scheme. Then we combine the two approximate solutions and use the Richardson extrapolation to compute a sixth‐order accuracy coarse grid solution. A sixth‐order accuracy fine grid solution is obtained by interpolating the sixth‐order coarse grid solution using an operator interpolation scheme. Numerical results are presented to demonstrate the accuracy and efficacy of the proposed finite difference discretization strategy, compared to the sixth‐order combined compact difference (CCD) scheme, and the standard fourth‐order compact difference (FOC) scheme. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 18–32, 2004.