Efficient discretization of pressure‐correction equations on non‐orthogonal grids

An alternative discretization of pressure‐correction equations within pressure‐correction schemes for the solution of the incompressible Navier–Stokes equations is introduced, which improves the convergence and robustness properties of such schemes for non‐orthogonal grids. As against standard approaches, where the non‐orthogonal terms usually are just neglected, the approach allows for a simplification of the pressure‐correction equation to correspond to 5‐point or 7‐point computational molecules in two or three dimensions, respectively, but still incorporates the effects of non‐orthogonality. As a result a wide range (including rather high values) of underrelaxation factors can be used, resulting in an increased overall performance of the underlying pressure‐correction schemes. Within this context, a second issue of the paper is the investigation of the accuracy to which the pressure‐correction equation should be solved in each pressure‐correction iteration. The scheme is investigated for standard test cases and, in order to show its applicability to practical flow problems, for a more complex configuration of a micro heat exchanger. Copyright © 2003 John Wiley & Sons, Ltd.     

气-固流化床反应器内双流体力学模型及其验证 Ⅱ.单组分两维射流床内流场特性分析

由以质量、动量和能量三大守恒定律为基础的湍流两相流理论出发,建立了描述气固流化床内两相流动的数学模型;在微型计算机上编写了相应的数值计算和图形处理程序;为了验证模型的可靠性,本文着手模拟与分析了单组分颗粒体系两维射流流化床内气、固相速度场、空隙度和压力场随时间、空间变化规律;得到了与前人实验结果相吻合的结论。     

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

This paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first‐order upwind approximation for the viscoelastic stress. A non‐uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non‐linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss–Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd‐B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright © 2001 John Wiley & Sons, Ltd.     

基于非结构化同位网格的SIMPLE算法

通过基于非结构化网格的有限体积法对二维稳态Navier—Stokes方程进行了数值求解。其中对流项采用延迟修正的二阶格式进行离散；扩散项的离散采用二阶中心差分格式；对于压力-速度耦合利用SIMPLE算法进行处理；计算节点的布置采用同位网格技术，界面流速通过动量插值确定。本文对方腔驱动流、倾斜腔驱动流和圆柱外部绕流问题进行了计算，讨论了非结构化同位网格有限体积法在实现SIMPLE算法时，迭代次数与欠松弛系数的关系、不同网格情况的收敛性、同结构化网格的对比以及流场尾迹结构。通过和以往结果比较可知，本文的方法是准确和可信的。     

The effect of the VOF–CSF static contact angle boundary condition on the dynamics of sliding and bouncing ellipsoidal bubbles

The static contact angle is the only empiricism introduced in a Volume of Fluid–Continuum Surface Force (VOF–CSF) model of bubbly flow. Although it has previously been shown to have a relatively limited effect on the accuracy of velocity and shape predictions in the case of large gas bubbles sliding under inclined walls (e.g. Cook and Behnia, 2001), it may have a more determining influence on the numerical prediction of the dynamics of smaller ellipsoidal bubbles which were shown by Tsao and Koch (1997) to bounce repeatedly when sliding under inclined walls at certain wall inclinations. The present paper reports on the influence of surface tension and the static contact angle on the dynamics of an ellipsoidal air bubble of equivalent diameter D_e = 3.4 mm. The bubble Eötvös and Morton numbers are Eo = 1.56 and Mo = 2 × 10⁻¹¹ respectively. The computational results are achieved with a Piecewise Linear Construction (PLIC) of the interface and are reviewed with reference to experimental measurements of bubble velocity and interface shape oscillations recorded using a high speed digital camera. Tests are performed at plate inclination angles θ ∈ {10°, 20°, 30°, 45°} to the horizontal and computational models consider three static contact angles θ_c ∈ {10°, 20°, 30°}. The static contact angle has been found to have a significant effect on the bubble dynamics but to varying degree depending on the plate inclination. It is shown to promote lift off and bouncing when the plate inclination angle reaches 30° in a way that does not necessarily reflect experimental observations.     

Numerical simulation of fluid flows using an unstructured finite volume method with adaptive tri‐tree grids

A tri‐tree grid generation procedure is developed together with a finite volume method on the unstructured grid for solving the Navier–Stokes equations. A hierarchic numbering system for the data structure is used. The grid is adapted by adding and removing cell elements dependent on the vorticity magnitude. A special treatment is developed to ensure good quality triangular elements around the cylinder boundary. The adopted finite volume method is based on the cell‐centred scheme. The pressure–velocity coupling is treated using the SIMPLE algorithm. A modified QUICK scheme for unstructured grids is derived. The developed method is used to simulate the flow past a single and multiple cylinders at low Reynolds number. The obtained results are in good agreement with the published data. Copyright © 2002 John Wiley & Sons, Ltd.     

Three-dimensional unsteady flow simulations: Alternative strategies for a volume-averaged calculation

This work builds on a SIMPLE-type code to produce two numerical codes of greatly improved speed and accuracy for solution of the Navier–Stokes equations. Both implicit and explicit codes employ an improved QUICK (quadratic upstream interpolation for convective kinematics) scheme to finite difference convective terms for non-uniform grids. The PRIME (update pressure implicit, momentum explicit) algorithm is used as the computational procedure for the implicit code. Use of both the ICCG (incomplete Cholesky decomposition, conjugate gradient) method and the MG (multigrid) technique to enhance solution execution speed is illustrated. While the implicit code is first-order in time, the explicit is second-order accurate. Two- and three-dimensional forced convection and sidewall-heated natural convection flows in a cavity are chosen as test cases. Predictions with the new schemes show substantial computational savings and very good agreement when compared to previous simulations and experimental data.     

Orthogonal arrays: An introduction and their application in optimizing underrelaxation factors in ‘simple’-based algorithm

The primary aim of this paper is to demonstrate how the ‘design-of-experiments’ techniques which are successful in physical experiments could also be adapted to a numerical simulation code. As an example this technique is applied to a general finite difference code used for predicting three-dimensional turbulent recirculating flows. Here the equations for velocities and continuity are solved using the algorithm called SIMPLE, which stands for semi-implicit method for pressure-linked equations. Physical modelling of turbulence is taken care of by means of kinetic energy and turbulence dissipation rate equations. The objective is to optimize the underrelaxation factors of primary and secondary flow variables so that the number of iterations required for convergence is minimum. This is done by the orthogonal array technique (a particular type of design-of-experiment technique). The geometry considered for this purpose is that of a simple gas turbine can combustor and the study is restricted to the isothermal non-reacting condition. Tests are carried out on three different grid configurations. In each case the underrelaxation factor for velocities contributed most to speed up the rate of convergence. Also, for each grid configuration the underrelaxation factor settings for minimum iterations for convergence was found to be same. Hence it is proposed that when doing grid independence tests for any similar flow situation, all the underrelaxation factors could be optimized on coarse grids.     

A multigrid method with higher‐order discretization schemes

The implementation of the multigrid method into the SIMPLE algorithm presents interesting aspects concerning the mass fluxes conservation on coarser grids, the k–ε turbulence model and the higher‐order discretization schemes. Higher‐order discretization schemes for the convection terms are increasingly used in order to guarantee accuracy in demanding engineering applications. However, when used in single‐grid algorithms, their convergence is considerably slower compared with the first‐order schemes. Unbounded higher‐order schemes offer maximum accuracy, but quite often they do not converge due to their oscillatory behaviour. This paper demonstrates the dual function of the multigrid method: reduction of CPU time and stabilization of the iterating procedure, making it possible to perform computations with the third‐order accurate QUICK scheme in all cases. The method is applied to the calculation of two‐ and three‐dimensional flows with or without turbulence modelling. The results show that the convergence rate of the present algorithm does not deteriorate when QUICK is used and that, if applied on complex engineering cases, large gains in computational time can be achieved. Copyright © 2001 John Wiley & Sons, Ltd.     

热声波数值模拟的虚假振荡研究

采用可压缩流动的SIMPLE算法对一维封闭空腔内由边界突然加热所引起的非稳态热声波进行了数值模拟,对流-扩散项采用了中心差分、一阶迎风差分、QUICK、及MIJSCL等不同格式。计算表明各种格式均存在不同程度的虚假振荡现象,其大小与热声波的强度及离散格式的形式等多种因素有关。这些结果对热声波的进一步研究及高效可靠的对流差分格式的开发具有重要意义。