Gas-kinetic numerical method for solving mesoscopic velocity distribution function equation

A gas-kinetic numerical method for directly solving the mesoscopic velocity distribution function equation is presented and applied to the study of three-dimensional complex flows and micro-channel flows covering various flow regimes. The unified velocity distribution function equation describing gas transport phenomena from rarefied transition to continuum flow regimes can be presented on the basis of the kinetic Boltzmann–Shakhov model equation. The gas-kinetic finite-difference schemes for the velocity distribution function are constructed by developing a discrete velocity ordinate method of gas kinetic theory and an unsteady time-splitting technique from computational fluid dynamics. Gas-kinetic boundary conditions and numerical modeling can be established by directly manipulating on the mesoscopic velocity distribution function. A new Gauss-type discrete velocity numerical integration method can be developed and adopted to attack complex flows with different Mach numbers. HPF parallel strategy suitable for the gas-kinetic numerical method is investigated and adopted to solve three-dimensional complex problems. High Mach number flows around three-dimensional bodies are computed preliminarily with massive scale parallel. It is noteworthy and of practical importance that the HPF parallel algorithm for solving three-dimensional complex problems can be effectively developed to cover various flow regimes. On the other hand, the gas-kinetic numerical method is extended and used to study micro-channel gas flows including the classical Couette flow, the Poiseuille- channel flow and pressure-driven gas flows in two-dimensional short micro-channels. The numerical experience shows that the gas-kinetic algorithm may be a powerful tool in the numerical simulation of micro-scale gas flows occuring in the Micro-Electro-Mechanical System (MEMS). The project supported by the National Natural Science Foundation of China (90205009 and 10321002), and the National Parallel Computing Center in Beijing. The English text was polished by Yunming Chen.     

Numerical simulation of instabilities and secondary regimes in spherical couette flow

In all previous numerical investigations of spherical Couette flow only axisymmetric regimes were considered. At the same time, in experiments [1–4] it was found that when both spheres rotate and the layer is thin centrifugal instability of the main flow leads to the appearance of nonaxisymmetric secondary flows of the azimuthal traveling wave type. The results of an initial numerical investigation of these flows are presented below. Solving the linear problem of the stability of the main flow and simulating the secondary flows on the basis of the complete nonlinear Navier-Stokes equations has made it possible to supplement and explain many of the results obtained experimentally. The type of bifurcation and the structure of the disturbances whose growth leads to the appearance of three-dimensional nonstationary flows are determined, and the transitions between different secondary regimes in the region of weak supercriticality are described.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–15, January–February, 1995.     

An Euler solver for three-dimensional turbomachinery flows

A time-marching finite volume numerical procedure is presented for three-dimensional Euler analysis of turbomachinery flows. The proposed scheme is applied to the conservative form of the Euler equations written in general curvilinear co-ordinates. A simple but computationally efficient grid is constructed. Numerical solution results for three 3D turbine cascade flows have been presented and compared with available measurements as well as with another state-of-the-art 3D Euler analysis numerical solution in order to demonstrate the accuracy and computational efficiency of the analysis method. Also, the predicted results are compared with a 3D potential flow solver and comparison is made with the analytical solution. The proposed method is an accurate and reliable technique for solving the compressible flow equations in turbomachinery geometries.     

Direct numerical simulation of the laminar-turbulent transition in a thick spherical layer

The results of a numerical study of the laminar-turbulent transition in unsteady isothermal three-dimensional flows of viscous incompressible fluid in a thick spherical layer between counter-rotating spherical boundaries are presented. The calculations are performed for the governing parameters corresponding to the experimental data [1, 2]. The numerical investigations include both solving the complete system of Navier-Stokes equations and analyzing the linear stability of steady-state axisymmetric flows with respect to three-dimensional disturbances. A stochastic flow regime is calculated for the first time. The limits of existence of different flow regimes and the hysteresis regions are found. The spatial flow patterns and frequency characteristics are obtained, which makes it possible to extend and refine the existing experimental data.     

Three-dimensional turbulent boundary layers on bielliptic bodies in a stream of compressible gas at an angle of attack

The results are given of a numerical investigation of the three-dimensional turbulent boundary layer formed on bielliptic bodies in a stream of compressible gas at an angle of attack. The investigation was made on the basis of the finite-difference method of calculation. The influence of a number of determining parameters on the development of the three-dimensional flows is analyzed. The characteristic flow regions in the boundary layer are found: lines of flow divergence and convergence, the region of “separation,” and flow division surfaces. The positions of the maximal values of the heat flux and the friction on the surface are determined, and the behavior of the limiting streamlines on the body is described.     

Three-dimensional flow in a lid-driven cavity with width-to-height ratio of 1.6

The flow in a lid-driven cavity with width-to-height ratio of 1.6 is investigated numerically and experimentally. Experimental investigation use an apparatus with a spanwise length-to-height ratio of $\Uplambda = 10.85.$ Λ = 10.85 . Increasing the Reynolds number, we experimentally find a gradual change from the quasi-two-dimensional basic flow to a three-dimensional flow pattern. The three-dimensional flow has a significant amplitude at considerably low Reynolds numbers. Streak-line photographs and PIV vector maps are presented to illustrate the structure of the finite-amplitude flow pattern. The smooth transition is in contrast to the linear instability predicted by a linear-stability analysis for a cavity with infinite span. LDV measurements confirm the absence of a distinct threshold Reynolds number and indicate an imperfect bifurcation. The deviations between experimental observations and numerical critical Reynolds number for infinite span are explained by conducting three-dimensional simulations for a finite-span geometry. A good agreement between experimental and numerical simulation is obtained. The numerical and experimental data lead to the conjecture of a premature onset of the three-dimensional flow caused by strong secondary flows which are induced by the cavity end walls. Nevertheless, the flow structure in the finite-span cavity carries the same characteristic signatures as the nonlinear flow in the corresponding infinite-length cavity. We conclude that the observed flow can be identified as the continuation of the normal mode C _e ⁴ earlier identified in a linear-stability analysis.     

Spatial interaction of jets propagating in an accompanying supersonic flow

A numerical method of calculating a three-dimensional laminar supersonic underexpanded jet escaping into an accompanying supersonic flow is developed. The simplified Navier-Stokes equations for a steady-state three-dimensional flow are employed. Numerical calculations are carried out for several cases relating the outflow of jets from a four-nozzle assembly into an accompanying supersonic flow, and a number of the characteristics of three-dimensional flows of this kind are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 88–93, November–December, 1972.     

Flows in a lower half heated upper half cooled cylindrical model reactor loaded with porous media

This paper presents an experimental and numerical investigation on the natural convection flow in a cylindrical model hydrothermal reactor. The flow is visualized non-intrusively and simulated with a conjugate computational model. Results show that the flow structure consists of wall layers and core flows. In the lower half, the flows are steady due to the porous media. The three-dimensional unsteady upper core flow is driven by the streams originated from the wall layer collision. The thermal condition in the upper half core region is mainly determined by the total heat flow rate specified on the lower sidewall; while the variations of porous media parameters, in the normal range for hydrothermal crystal growth process, have minor effects.     

基于Boltzmann模型方程的气体运动论统一算法研究

模型方程出发,研究确立含流态控制参数可描述不同流域气体流动特征的气体分子速度分布函数方程; 研究发展气体运动论离散速度坐标法, 借助非定常时间分裂数值计算方法和NND差分格式, 结合DSMC方法关于分子运动与碰撞去耦技术, 发展直接求解速度分布函数的气体运动论耦合迭代数值格式; 研制可用于物理空间各点宏观流动取矩的离散速度数值积分方法, 由此提出一套能有效模拟稀薄流到连续流不同流域气体流动问题统一算法. 通过对不同Knudsen数下一维激波内流动、二维圆柱、三维球体绕流数值计算表明, 计算结果与有关实验数据及其它途径研究结果(如DSMC模拟值、N-S数值解)吻合较好, 证实气体运动论统一算法求解各流域气体流动问题的可行性. 尝试将统一算法进行HPF并行化程序设计, 基于对球体绕流及类``神舟'返回舱外形绕流问题进行HPF初步并行试算, 显示出统一算法具有很好的并行可扩展性, 可望建立起新型的能有效模拟各流域飞行器绕流HPF并行算法研究方向. 通过将气体运动论统一算法推广应用于微槽道流动计算研究, 已初步发展起可靠模拟二维短微槽道流动数值算法; 通过对Couette流、Poiseuille流、压力驱动的二维短槽道流数值模拟, 证实该算法对微槽道气体流动问题具有较强的模拟能力, 可望发展起基于Boltzmann模型方程能可靠模拟MEMS微流动问题气体运动论数值计算方法研究途径.      

A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

Both compressible and incompressible Navier-Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, M ≈0.1, in order to mimic incompressible flows. This strategy is widely used for high-order discontinuous Galerkin (DG) discretizations of the compressible Navier-Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to an incompressible formulation. Our contributions to the state of the art are twofold: Firstly, we present a high-performance DG solver for the compressible Navier-Stokes equations based on a highly efficient matrix-free implementation that targets modern cache-based multicore architectures with Flop/Byte ratios significantly larger than 1. The performance results presented in this work focus on the node-level performance, and our results suggest that there is great potential for further performance improvements for current state-of-the-art DG implementations of the compressible Navier-Stokes equations. Secondly, this compressible Navier-Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix-free implementation. We discuss algorithmic differences between both solution strategies and present an in-depth numerical investigation of the performance. The considered benchmark test cases are the three-dimensional Taylor-Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall-bounded turbulent flows. The results indicate a clear performance advantage of the incompressible formulation over the compressible one.     

Rheology of a dilute suspension of axisymmetric Brownian particles

Explicit results are presented for the complete rheological properties of dilute suspensions of rigid, axisymmetric Brownian particles possessing fore-aft symmetry, when suspended in a Newtonian liquid subjected to a general three-dimensional shearing flow, either steady or unsteady. It is demonstrated that these rheological properties can be expressed in terms of five fundamental material constants (exclusive of the solvent viscosity), which depend only upon the sizes and shapes of the suspended particles. Expressions are presented for these scalar constants for a number of solids of revolution, including spheroids, dumbbells of arbitrary aspect ratio and long slender bodies. These are employed to calculate rheological properties for a variety of different shear flows, including uniaxial and biaxial extensional flows, simple shear flows, and general two-dimensional shear flows. It is demonstrated that the rheological properties appropriate to a general two-dimensional shear flow can be deduced immediately from those for a simple shear flow. This observation greatly extends the utility of much of the prior Couette flow literature, especially the extensive numerical calculations of Scheraga et al. (1951, 1955).The commonality of many disparate results dispersed and diffused in earlier publications is emphasized, and presented from a unified hydrodynamic viewpoint.     

Calculation of the Flow around a High-Pressure Turbine Blade with Cooling-Jet Injection from Slots at the Leading Edge

A fully vectorised, basically three-dimensional finite-volume multi-block method for flows with complex boundaries is applied in a stripped-down two-dimensional version for the investigation of the flow field in the leading-edge region of a high-pressure turbine blade with slot cooling-jet injection. The calculation results are compared with experiments and the numerical results of other investigators. The present method yields excellent agreement with the experiments for the isentropic Mach number distributions on the blade surface. All numerical results for the velocity field were found to be in very good agreement with each other and with experiments on the suction side, while the agreement is not as good on the pressure side.     

Three dimensional finite element analysis of the flow of polymer melts

The finite element simulation of a selection of two- and three-dimensional flow problems is presented, based upon the use of four different constitutive models for polymer melts (Oldroyd-B, Rolie-Poly, Pom-Pom and XPP). The mathematical and computational models are first introduced, before their application to a range of visco-elastic flows is described. Results demonstrate that the finite element models used here are able to re-produce predictions made by other published numerical simulations and, significantly, by carefully conducted physical experiments using a commercial-grade polystyrene melt in a three-dimensional contraction geometry. The paper also presents a systematic comparison and evaluation of the differences between two- and three-dimensional simulations of two different flow regimes: flow of an Oldroyd-B fluid around a cylinder and flow of a Rolie-Poly fluid into the contraction geometry. This comparison allows new observations to be made concerning the relatively poor quality of two-dimensional simulations for flows in even quite deep channels.     

Natural convection in a bottom-heated top-cooled cubic cavity with a baffle at the median height: experiment and model validation

This paper presents an experimental and numerical investigation on the natural convection flow and heat transfer in an enclosure with a single-hole baffle at the median height. The temperature in the fluid is quantified by means of temperature sensitive thermo-chromic liquid crystal (TLC) particles. The fluid flow velocity is measured non-intrusively with a full field particle tracking technique. The three-dimensional numerical model, developed and validated with experimental data, provides a computational tool for further investigation of mass and energy transport through the baffle openings in these types of enclosures. The experimentally visualized and numerically simulated flow structures show a pair of streams across the baffle-hole. The two chambers communicate through this pair of streams which carry the fluid exchange and heat transfer between the two chambers. At the baffle opening, the two streams are aligned in a diagonal direction across of the enclosure. The streams are accelerated and form jet-like flows that drive the whole circulation in the chambers. The jet-like flows leave the baffle opening, approach the vertical centerline of the cavity, and finally impinge on the top/bottom walls.     

Spiral thermocapillary flows in two-layer systems

Parallel thermocapillary flows in infinite layers can be calculated easily. However, in reality thermocapillary flows occur in channels or closed cavities, and they are three-dimensional. In the present paper, a two-layer fluid system filling a channel with a rectangular cross section is considered. A numerical investigation of three-dimensional spiral thermocapillary flows generated by a temperature gradient imposed along the channel has been performed. Both the case of a zero longitudinal pressure gradient (a through flow in the channel) and that of a zero longitudinal fluid flux (a flow in the closed cavity) are investigated. Steady and oscillatory thermocapillary motions, and transitions between them are studied.     

Introduction of turbulent model in a mixed finite volume/finite element method

The aim of this work is to present a new numerical method to compute turbulent flows in complex configurations. With this in view, a k-? model with wall functions has been introduced in a mixed finite volume/finite element method. The numerical method has been developed to deal with compressible flows but is also able to compute nearly incompressible flows. The physical model and the numerical method are first described, then validation results for an incompressible flow over a backward-facing step and for a supersonic flow over a compression ramp are presented. Comparisons are performed with experimental data and with other numerical results. These simulations show the ability of the present method to predict turbulent flows, and this method will be applied to simulate complex industrial flows (flow inside the combustion chamber of gas turbine engines). The main goal of this paper is not to test turbulence models, but to show that this numerical method is a solid base to introduce more sophisticated turbulence model.     

Triadic resonant modes: dynamical model and truncation criterion

In the present paper we propose a three-dimensional theoretical model describing the local evolution of any two- or three-dimensional disturbance superimposed on a basic shear flow. A numerical investigation, using a truncated form of the evolution model, was performed to select resonant three-dimensional disturbance in a transitional cylinder wake. A truncation criterion of non-linear triadic resonance, which is a generalisation of the standard linear-resonance condition, allowed us to locate some triadic sets of linear eigenmodes. Numerical integration of the corresponding low-dimensional dynamical systems showed interesting results that are in qualitative agreement with those performed by Williamson and Prasad [J. Fluid Mech. 256 (1993) 269.] This seems to validate the truncation criterion which, together with the theoretical model, proved to be an appropriate tool for describing the local evolution of large-scale structures in shear flows.     

On the theory of supersonic inviscid flow separation in gasdynamic problems

A general schematic flow representation that explains the mechanism of inviscid gas separation in time-dependent and three-dimensional gas flows is presented. The scenario of gas flow separation from a body surface or a mixing layer is described as a vortex which induces in the flowfield a velocity opposing to that of the main flow, thus decelerating it. Within the framework of this scenario the analytical conditions of separation are obtained for conical and self-similar gas flows which coincide with the results of experimental and numerical simulations.     

Calculation of the permeability and apparent permeability of three-dimensional porous media

In this study, creeping and inertial incompressible fluid flows through three-dimensional porous media are considered, and an analytical–numerical approach is employed to calculate the associated permeability and apparent permeability. The multiscale homogenization method for periodic structures is applied to the Stokes and Navier–Stokes equations (aided by a control-volume type argument in the latter case), to derive the appropriate cell problems and effective properties. Numerical solutions are then obtained through Galerkin finite-element formulations. The implementations are validated, and results are presented for flows through cubic lattices of cylinders, and through the dendritic zone found at the solid–liquid interface during solidification of metals. For the interdendritic flow problem, a geometric configuration for the periodic cell is built by the approximate matching of experimental and numerical results for the creeping-flow problem; inertial effects are then quantified upon solution of the inertial-flow problem. Finally, the functional behavior of the apparent permeability results is analyzed in the light of existing macroscopic seepage laws. The findings contribute to the (numerical) verification of the validity of such laws.     

Computation of flow of viscoelastic fluids by parameter differentiation

A technique combining the features of parameter differentiation and finite differences is presented to compute the flow of viscoelastic fluids. Two flow problems are considered: (i) three-dimensional flow near a stagnation point and (ii) axisymmetric flow due to stretching of a sheet. Both flows are characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. The exact numerical solutions are obtained using the technique described in the paper. Also, the first-order perturbation solutions (in terms of the viscoelastic fluid parameter) are derived. A comparison of the results shows that the perturbation method is inadequate in predicting some of the vital characteristic features of the flows, which can possibly be revealed only by the exact numerical solution.