Numerical simulation of axisymmetric unsteady incompressible flow by a vorticity-velocity method

A new numerical method for solving the axisymmetric unsteady incompressible Navier-Stokes equations using vorticity-velocity variables and a staggered grid is presented. The solution is advanced in time with an explicit two-stage Runge-Kutta method. At each stage a vector Poisson equation for velocity is solved. Some important aspects of staggering of the variable location, divergence-free correction to the velocity held by means of a suitably chosen scalar potential and numerical treatment of the vorticity boundary condition are examined. The axisymmetric spherical Couette flow between two concentric differentially rotating spheres is computed as an initial value problem. Comparison of the computational results using a staggered grid with those using a non-staggered grid shows that the staggered grid is superior to the non-staggered grid. The computed scenario of the transition from zero-vortex to two-vortex flow at moderate Reynolds number agrees with that simulated using a pseudospectral method, thus validating the temporal accuracy of our method.

Triple-region structure for turbulent-flow in a square duct - a finite-element approach

The Reynolds-averaged Navier-Stokes equations for describing the turbulent flow in a straight square duct are formulated with two different turbulence models. The governing equations are then expanded as a multi-deck structure in a plane perpendicular to the streamwise direction, with each deck characterized by its dominant physical forces as commonly carried out in analytical work using triple-deck expansion. The resulting equations are numerically integrated using higher polynomial (H-P) finite element technique for each cross-sectional plane to be followed by finite difference representation in the streamwise direction until a fully developed state is reached. The computed results using the two different turbulence models show fair agreement with each other, and concur with the vast body of available experimental data. There is also general agreement between our results and the recent numerical works anisotropic k-epsilon turbulence model.

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.

Upwind compact method and direct numerical-simulation of driven flow in a square cavity

A high-order accurate finite-difference scheme, the upwind compact method, is proposed. The 2-D unsteady incompressible Navier-Stokes equations are solved in primitive variables. The nonlinear convection terms in the governing equations are approximated by using upwind biased compact difference, and other spatial derivative terms are discretized by using the fourth-order compact difference. The upwind compact method is used to solve the driven flow in a square cavity. Solutions are obtained for Reynolds numbers as high as 10000. When Re less than or equal to 5000, the results agree well with those in literature. When Re = 7500 and Re = 10000, there is no convergence to a steady laminar solution, and the flow becomes unsteady and periodic.

Approximate theory of natural-convection with small grashof number

To gain some insight into the behaviour of low-gravity flows in the material processing in space, an approximate theory has been developed for the convective motion of fluids with a small Grashof number Gr. The expansion of the variables into a series of Gr reduces the Boussinesq equation to a system of weakly coupled linearly inhomogeneous equations. Moreover, the analogy concept is proposed and utilized in the study of the plate bending problems in solid mechanics. Two examples are investigated in detail, i. e. the 2-dimensional steady flows in either circular or square infinite closed cylinder, which is horizontally imposed at a specified temperature of linear distribution on the boundaries. The results for stream function ψ, velocity u and temperature T are provided. The analysis of the influences of some parameters such as the Grashof number Gr and the Prandtl number Pr, on motions will lead to several interesting conclusions. The theory seems to be useful for seeking for an analytical solutions. At least, it will greatly simplify the complicated problems originally governed by the Navier-Stokes equation including buoyancy. It is our hope that the theory might be applicable to unsteady or 3-dimensional cases in future.

Compressible laminar boundary-layer flows of a dusty gas over a semi-infinite flat-plate

The compressible laminar boundary-layer flows of a dilute gas-particle mixture over a semi-infinite flat plate are investigated analytically. The governing equations are presented in a general form where more reasonable relations for the two-phase interaction and the gas viscosity are included. The detailed flow structures of the gas and particle phases are given in three distinct regions : the large-slip region near the leading edge, the moderate-slip region and the small-slip region far downstream. The asymptotic solutions for the two limiting regions are obtained by using a seriesexpansion method. The finite-difference solutions along the whole length of the plate are obtained by using implicit four-point and six-point schemes. The results from these two methods are compared and very good agreement is achieved. The characteristic quantities of the boundary layer are calculated and the effects on the flow produced by the particles are discussed. It is found that in the case of laminar boundary-layer flows, the skin friction and wall heat-transfer are higher and the displacement thickness is lower than in the pure-gas case alone. The results indicate that the Stokes-interaction relation is reasonable qualitatively but not correct quantitatively and a relevant non-Stokes relation of the interaction between the two phases should be specified when the particle Reynolds number is higher than unity.

Theory of local mode excitation in polyatomics by frequency-modulated lasers

that the Stokes-interaction relation is reasonable qualitatively but not correct

Characteristics of a CW HF chemical-laser calculated from a simplified two-dimensional model

A two-dimensional simplified model of an HF chemical laser is introduced. Using an implicit finite difference scheme, the solution of two adjacent parallel streams with diffusion mixing and chemical reaction is generated. A contour of mixing and reaction boundary is obtained without presupposition. The distribution of the HF(v) concentrations, gas temperature and the optical small signal gain (alpha sub V, J) on the flowing plane (X, Y) are presented. Compared with the solution solved directly from a set of Navier-Stokes equations, the results of these two methods agree with each other qualitatively. The influences of the different velocity, temperature (T sub 0) and composition of the two streams on the small signal gain after the nozzle exit are investigated. It is interesting that for larger J with a fixed v, the peaks of alpha sub v-T sub 0 profiles move towards higher T sub 0. The computing method is simple and only a short computing time is needed.

Numerical experiments on one-dimensional model of turbulence

The initial-value problem of a forced Burgers equation is numerically solved by the Fourier expansion method. It is found that its solutions finally reach a steady state of 'laminar flow' which has no randomness and is stable to disturbances. Hence, strictly speaking, the so-called Burgers turbulence is not a turbulence. A new one-dimensional model is proposed to simulate the Navier-Stokes turbulence. A series of numerical experiments on this one-dimensional turbulence is made and is successful in obtaining Kolmogorov's (1941) k exp(-5/3) inertial-range spectrum. The (one-dimensional) Kolmogorov constant ranges from 0.5 to 0.65.

多孔介质中流体流动的细观-宏观模拟

阐述了如何用体积平均化方法推导具有非线性吸附效应的多孔介质中的 Forchhermer方程和对流 -扩散方程。这两个方程对分析多孔介质中质量输运有着极大的应用价值。在分析 Navier- Stokes方程中 ,我们发现非滑移条件在平均化过程中并不扮演重要的角色 ,而吸附边界条件对体积平均化输运方程作出了重要贡献。所以 ,在推导多相输运方程时对平均化方法和边界条件的处理必须非常小心。

激波-涡干扰声场的数值研究

采用高精度差分格式求解非定常可压缩Navier-Stokes方程,对激波-单涡/双涡相互干扰产生的声场进行了直接数值模拟。详细研究了波-涡干扰声场结构的早期发展阶段,将激波-单涡的计算结果和相应实验进行了对比,并给出近场声压的衰减规律。在此基础上模拟较为复杂的激波-双涡干扰,给出不同旋涡旋转方向下的声场结构。

可压剪切层线性稳定性特征直接数值模拟

基于伪随机数生成技术促生白噪声扰动，以高精度迎风/对称紧致混合差分算法求解二维/三维非定常可压Navier-Stokes方程，揭示了可压自由剪切层初始剪切过程中扰动的线性演化特征，以及该过程对扰动波数方向的内在选择性。验证了所用算法的有效性，表明线性理论同数值模拟相结合是可压剪切层研究的合理途径之一。

可压缩双组分混合层的直接数值模拟研究

通过数值求解可压缩Navier-Stokes方程模拟了三维可压缩双组分混合层问题,着重研究了双组分密度比和可压缩效应对混合层发展演化的影响.研究表明,随着密度比的增大,混合层扰动增长率降低.在混合层发展初期,可压缩性作用尤其是斜压效应对旋涡的发展起主导作用;在混合层发展后期,可压缩性影响减弱,旋涡的拉伸扭转效应占主导作用.同时,还分析讨论了可压缩混合层中湍流的转捩及拟序结构的演化过程.

可压涡卷空间演化的迎风紧致差分数值模拟

从数值算法的耗散和色散特征的时空全离散Fourier分析出发，通过直接求解二维非定常可压Navier-Stokes方程，将发展的5阶迎风紧致差分格式用于无约束可压平面受迫剪切层中基频涡卷空间演化过程的数值模拟。采用被动守恒标量等方法显示了基频涡卷的饱和、一次对并、二次对并等现象，据此探讨了入口来流亚谐扰动引起的初值效应问题，表明可压大尺度涡结构空间演化形态与受迫扰动方式之间存在关联。

弱粘性流体中垂直激励表面波的阻尼效应

在竖直振动的圆柱形容器中，将Navier-Stokes方程线性化，利用两时间尺度奇异摄动展开法研究了弱粘性流体的单一自由面驻波运动．整个流场被分为外部势流区和内部边界层区两部分，对两部分区域分别求解，得到包含阻尼项和外驱动影响的线性振幅方程．利用稳定性分析，得到形成稳定表面波的条件，给出了临界曲线．此外，还获得了阻尼系数的解析表达式．最后，将线性阻尼加到理想流体条件下所得到的色散关系中对其进行修正，理论结果证明修正后的驱动频率更加接近实验的结果．通过计算发现，当驱动的频率较低时，流体的粘性对表面波模式选择有重要影响，而表面张力的影响不明显；但当驱动频率较高时，流体的表面张力起主要作用，而流体的粘性影响甚小．