Computation of turbulent-flow in a square duct - aspects of the secondary flow and its origin

A numerical study of turbulent flow in a straight duct of square cross-section is made. An order-of-magnitude analysis of the 3-D, time-averaged Navier-Stokes equations resulted in a parabolic form of the Navier-Stokes equations. The governing equations, expressed in terms of a new vector-potential formulation, are expanded as a multi-deck structure with each deck characterized by its dominant physical forces. The resulting equations are solved using a finite-element approach with a bicubic element representation on each cross-sectional plane. The numerical integration along the streamwise direction is carried out with finite-difference approximations until a fully-developed state is reached. The computed results agree well with other numerical studies and compare very favorably with the available experimental data. One important outcome of the current investigation is the interpretation analytically that the driving force of the secondary flow in a square duct comes mainly from the second-order terms of the difference in the gradients of the normal and transverse Reynolds stresses in the axial vorticity equation.

Character of simplified Navier-Stokes (SNS) equations near separation point for two-dimensional laminar flow over a flat plate

It is proved that the simplified Navier-Stokes (SNS) equations presented by Gao Zhi[1], Davis and Golowachof-Kuzbmin-Popof (GKP)[3] are respectively regular and singular near a separation point for a two-dimensional laminar flow over a flat plate. The order of the algebraic singularity of Davis and GKP equation[2,3] near the separation point is indicated. A comparison among the classical boundary layer (CBL) equations, Davis and GKP equations, Gao Zhi equations and the complete Navier-Stokes (NS) equations near the separation point is given.

Some theoretical aspects of the simplified Navier-Stokes(SNS) equations

This study deals with the formulation, mathematical property and physical meaning of the simplified Navier-Stokes (SNS) equations. The tensorial SNS equations proposed is the simplest in form and is applicable to flow fields with arbitrary body boundaries. The zones of influence and dependence of the SNS equations, which are of primary importance to numerical solutions, are expounded for the first time from the viewpoint of subcharacteristics. Besides, a detailed analysis of the diffusion process in flow fields shows that the diffusion effect has an influence zone globally windward and an upwind propagation greatly depressed by convection. The maximum upwind influential distance of the viscous effect and the relative importance of the viscous effect in the flow direction to that in the direction normal to the flow are represented by the Reynolds number, which illustrates the conversion of the complete Navier-Stokes (NS) equations to the SNS equations for flows with large Reynolds number.

Hierarchal structure of the simplified Navier-Stokes equations (SNSE) and its mechanical connotation and application

The hierarchial structure and mathematical property of the simplified Navier-Stokesequations (SNSE) are studied for viscous flow over a sphere and a jet of compressible flu-id. All kinds of the hierarchial SNSE can be divided into three types according to theirmathematical property and also into five groups according to their physical content. Amultilayers structure model for viscous shear flow with a main stream direction is pre-sented. For the example of viscous incompressible flow over a flat plate there existthree layers for both the separated flow and the attached flow; the character of thetransition from the three layers of attached flow to those of separated flow is elucidated.A concept of transition layer being situated between the viscous layer and inviscidlayer is introduced. The transition layer features the interaction between viscous flow andinviscid flow. The inner-outer-layers-matched SNSE proposed by the present author inthe past is developed into the layers matched (LsM)-SNSE.

Simplified Navier-Stokes equations (SNSE)

Ten kinds of the simplified Navier-Stokes equations (SNSE) are reviewed and also used to calculate the Jeffery-Hamel flow as well as to analyze briefly the seven kinds of flows to which the exact solutions of the complete Navier-Stokes equations (CNSE) have been found. Analysis shows that the actual differences among the solutions of the different SNSE can go beyond the range of the order of magnitude of Re-1/2 and result even in different flow patterns, therefore, how to choose the viscous terms included in the SNSE is worthy of notice where Re=S∞u∞ L/μ∞ is the Reynolds numbers. For the aforesaid eight kinds of flows, the solutions to the inner-outer-layer-matched SNSE and to the thin-layer-2-order SNSE agree completely with the exact solutions to CNSE. But the solutions to all the other SNSE are not completely consistent with the exact solutions to CNSE and not a few of them are actually the solutions of the classical boundary layer theory. The innerouter-layer-matched SNSE contains the shear stress causing angular displacement of the inormal axis with respect to the streamwise axis and the normal stress causing expansion-contraction in the direction of the normal axis and the viscous terms being of the order of magnitude of the normal stress; and it can also reasonably treat the inertial terms as well as the relation between the viscous and inertial terms. Therefore, it seems promising in respects of both mechanics and mathematics.

同位非结构和结构网格摄动有限体积法(PFV)求解二维Navier-Stokes方程组

根据NS方程组的一阶迎风和二阶中心有限体积(UFV和CFV)格式,导出NS方程组迎风和中心摄动有限体积(UPFV和CPFV)格式.该格式通过把控制体界面质量通量摄动展开成网格间距的幂级数,并由守恒方程本身求得幂级数系数而获得.迎风和中心摄动有限体积格式使用了与一阶迎风和二阶中心格式相同的基点数和相同的表达形式,宜于计算机编程.顶盖驱动方腔流和驻点流标量输运的数值实验证明,迎风PFV格式比一阶UFV、二阶CFV格式有更高的精度,更高的分辨率.尤其是良好的鲁棒特性.对顶盖驱动方腔流,在Re数从102到104范围内,亚松弛系数可在0.3～0.8任取,收敛性能良好.

Perturbational finite volume scheme for the one-dimensional Navier-Stokes equations

Starting from the second-order finite volume scheme,though numerical value perturbation of the cell facial fluxes, the perturbational finite volume (PFV) scheme of the Navier-Stokes (NS) equations for compressible flow is developed in this paper. The central PFV scheme is used to compute the one-dimensional NS equations with shock wave.Numerical results show that the PFV scheme can obtain essentially non-oscillatory solution.

Navier-Stokes(NS)方程组差分计算中的物理和网格尺度效应及NS方程组的简化

对于高Re数流动计算,在通常二阶精度NS差分格式和网格数条件下,存在某些粘性项落入修正微分方程截断误差项的问题。这类NS方程组计算实际是计算某种简化NS方程组,而且重复计算误差物理粘性项既浪费机时和内存,误差积累又会对数值解产生不可预测的影响。避免上述缺陷的办法一个是提高NS差分格式的精度,另一个是丢掉可能落入截断误差项的物理粘性项,把NS方程组简化为广义NS方程组。广义NS计算避免了误差物理粘性项误差积累对数值解的不可知影响,又可节省内存和机时,对高Re数流体工程计算很有好处。利用广义NS方程组计算超声速绕前向和后向台阶流动的结果表明:广义NS方程组与NS方程组的数值结果很好相符。

A block SPP algorithm for multidimensional tridiagonal equations with optimal message vector length

A parallel strategy for solving multidimensional tridiagonal equations is investigated in this paper. We present in detail an improved version of single parallel partition (SPP) algorithm in conjunction with message vectorization, which aggregates several communication messages into one to reduce the communication cost. We show the resulting block SPP can achieve good speedup for a wide range of message vector length (MVL), especially when the number of grid points in the divided direction is large. Instead of only using the largest possible MVL, we adopt numerical tests and modeling analysis to determine an optimal MVL so that significant improvement in speedup can be obtained.

On optimal message vector length for block single parallel partition algorithm in a three-dimensional ADI solver

It has long been recognized that many direct parallel tridiagonal solvers are only efficient for solving a single tridiagonal equation of large sizes, and they become inefficient when naively used in a three-dimensional ADI solver. In order to improve the parallel efficiency of an ADI solver using a direct parallel solver, we implement the single parallel partition (SPP) algorithm in conjunction with message vectorization, which aggregates several communication messages into one to reduce the communication costs. The measured performances show that the longest allowable message vector length (MVL) is not necessarily the best choice. To understand this observation and optimize the performance, we propose an improved model that takes the cache effect into consideration. The optimal MVL for achieving the best performance is shown to depend on number of processors and grid sizes. Similar dependence of the optimal MVL is also found for the popular block pipelined method.

Imaging properties of coherent anti-Stokes Raman scattering microscope

The coherent anti-Stokes Raman scattering (CARS) microscope with the combination of confocal and CARS techniques is a remarkable alternative for imaging chemical or biological specimens that neither fluoresce nor tolerate labelling. CARS is a nonlinear optical process, the imaging properties of CARS microscopy will be very different from the conventional confocal microscope. In this paper, the intensity distribution and the polarization property of the optical field near the focus was calculated. By using the Green function, the precise analytic solution to the wave equation of a Hertzian dipole source was obtained. We found that the intensity distributions vary considerably with the different experimental configurations and the different specimen shapes. So the conventional description of microscope (e.g. the point spread function) will fail to describe the imaging properties of the CARS microscope.

The intensity distributions of collected signals in coherent anti-Stokes Raman scattering microscopy

Coherent anti-Stokes Raman scattering (CARS) microscopy with the combining of confocal and CARS techniques is a remarkable alternative for imaging chemical or biological specimens that neither fluoresce nor tolerate labeling. The CARS is a nonlinear optical process, the imaging properties of CARS microscopy will be very different from the conventional confocal microscopy. In this paper, we calculated the propagation of CARS signals by using the wave equation in medium and the slowly varying envelope approximation (SVEA), and find that the intensity angular distributions vary considerably with the different experimental configurations and the different specimen shapes. So the conventional description of microscopy (e.g.. the point spread function) will fail to descript the imaging properties of CARS microscopy. (c) 2004 Elsevier B.V. All rights reserved.