简化Navier-Stokes方程的层次结构及其力学内涵和应用

本文在文献[1]的基础上,按照流场中长度尺度分布,惯性项与粘性项相对大小及数量级简化基本方程和划分流动区域的原则,给出:(1)可压缩绕球粘性流和射流的简化Navier-Stokes(NS)方程的层次结构和诸简化NS方程(SNSE),表明从边界层方程到NS方程和从Euler方程到NS方程的层次结构均包含十多种SNSE,但就SNSE的数学特征而言证明只有椭圆型,扩散抛物化和抛物型三类;(2)扩散抛物化方程(DPE)的数学特征与Euler方程一致,力学上表示扰动通过“压力梯度项”向上游传播,高阶扩散项“规定的”椭圆型下游效应可以忽略,故判断诸DPE优劣的标准应看能否准确计算压力场。(3)提出粘性流的多层结构模型,对绕固壁附近的流动为三层,即粘性层、过渡层和无粘层,给出了分层的准则;适用于三层的最简单和最重要的SNSE分别为边界层方程、诸层匹配(LsM)-SNSE和Euler方程;LsM-SNSE同时适用于三层、即适用于全流场,并可准确计算压力场。LsM-SNSE把两层、即内外层匹配SNSE推广为多层。(4)对平板绕流,给出附着流及分离流的新的三层结构,阐明了附着流三层向分离流三层过渡的力学特征。

论简化Navier-Stokes方程组(SNSE)

本文论述简化 Navier-Stokes 方程组(SNSE),利用十种 SNSE分析Jeffery-Hamel流动并简要分析已知完全 Navier-Stokes 方程组(CNSE)精确解的八类流动。表明:不同SNSE结果之间的实际差异能够大大超出O(Re~(-1/2))量级的理论误差范围,甚至给出不同的流动图案。因此,SNSE 的粘性项如何取舍值得重视。内外层匹配SNSE和薄层二阶SNSE的解在八类流动情况下均与CNSE的精确解完全一致;而所有其它SNSE 的解则与CNSE的精确解不完全一致,它们的解在不少情况下实际就是经典边界层理论的解。内外层匹配SNSE包含了法向轴相对流向轴剪切的剪应力项和法向轴伸缩的法应力项以及与该法应力项同量级的粘性项,且对惯性项和粘性-惯性项相互关系的处理较合理,故在力学上和数学上都比较可取。

简化Navier-Stokes方程组(SNSE)的几个精确解

对不可压二维驻点流、三维驻点流和旋转圆盘附近的流动等三种流动情况,本文给出简化Navier-Stokes方程组(SNSE)及其精确解。表明:文献[1]理论的SNSE的精确解,在三种流动情况下均与完全Navier-Stokes方程组(NSE)的精确解完全一致;文献[3]SNSE的精确解的速度解与完全NSE精确解的速度解一致,但压力解在三种流动情况下均与完全NSE精确解的压力解不同。文献[3]SNSE精确解给出的压力分布相对与完全NSE精确解给出的压力分布的最大相对误差为100％。

三维简化Navier-Stokes方程的最优形式

<正> 引言 最近十多年,简化NS方程(以下记为SNS)的研究和计算有长足进展。由于在NS方程组中对粘性项的取舍不同,因而有几种不同的简化NS方程组,究竟哪种形式更合理,是需进一步探讨的一个问题。文献[1]利用原始NS方程及三种不同的简化NS方程组,对球的超音速绕流数值试验表明,其效果是不一样的。文献[3]也指出,如果SNS方程组的形式选择不当,会带来不可忽略的误差。从二维研究不难看出,目前广泛采用的三维SNS方程即粘性激波层方程组(VSL)及抛物化NS方程组(PNS),都不是最合理的简化形式。本文提出三维NS方程组的一种最好形式,称为修正的PNS方程组(记为MPNS),并论证它的合理性及精确度。

ADI方法求解Navier-Stokes方程的一个改进格式

ADI方法常被用来计算不可压缩Navier-Stokes方程。在处理涡度方程的非线性项和涡度在壁面上的条件时,通常采用滞后的方法对涡度方程和流函数方程分别求解。然而,非线性项的滞后破坏了ADI方法的完全二阶精度;涡度方程和流函数方程分别求解减弱了两个方程的耦合性;涡度壁面条件的滞后则破坏了方法的完全隐式。本文在应用ADI方法求解涡度方程和流函数方程时应用了一种交替线性化的技术,对涡度方程和流函数方程耦合求解,内点和边界点上的涡度和流函数值同时求出。因此,ADI方法保持了完全的二阶精度,避免了上面所提到的问题。作者应用这一方法计算了雷诺数R_θ等于1,10,100,500,1000时的二维方腔流动(空间步长h=1/20)。计算结果表明:这一方法保持了通常ADI方法的优点,可以应用大的时间步长。最后补充计算了雷诺数R_θ=2000的二维方腔流动。

简化Navier-Stokes方程组及无粘和粘性边界层联合求解

<正> 简化N-S方程组具有抛物-双曲方程组的特性,对定常情况可用向前推进的计算方法,要比数值求解椭圆型完全N-S方程组简单得多;求解简化N-S方程组能够同时算出无粘外部流和粘性边界层流,理论上要比先算无粘流、然后再算粘性边界层流的常规方法

简化Navier-Stokes方程组及其数学性质

本文对二绝简化Navier-stokes方程组作了定性分忻,作者认为当流动的切向速度分量u

Compact finite difference - Fourier spectral method for three-dimensional incompressible Navier-Stokes equations

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.

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。