Low-diffusion preconditioning scheme with linear cut-off parameter

A low-diffusion preconditioning Roe scheme with an adjustable parameter to control the numerical dissipation is proposed. This scheme reflects the real physical dissipation in the extremely low-speed region. The preconditioning parameter in scheme is improved by linear cut-off and correction factor. The numerical results of low-Mach-number/low-Reynolds-number steady solutions of viscous flows past a circular cylinder and past a NACA0012 airfoil show the efficiency of the new scheme.     

Investigation of solution of Navier–Stokes equations using a variational formulation

A variational formulation for the solution of two dimensional, incompressible viscous flows has been developed by one of the authors.¹ The main objective of the present paper is to demonstrate the applicability of this approach for the solution of practical problems and in particular to investigate the introduction of boundary conditions to the Navier-Stokes equations through a variational formulation. The application of boundary conditions for typical internal and external flow problems is presented. Sample cases include flow around a cylinder and flow through a stepped channel. Quadrilateral, bilinear isoparametric elements are utilized in the formulation. A single-step, implicit, and fully coupled numerical integration scheme based on the variational principle is employed. Presented results include sample cases with different Reynolds numbers for laminar and turbulent flows. Turbulence is modelled using a simple mixing length model. Numerical results show good agreement with existing solutions.     

Research of influence of reduced-order boundary on accuracy and solution of interior points

The flow field with a high order scheme is usually calculated so as to solve complex flow problems and describe the flow structure accurately.However,there are two problems,i.e.,the reduced-order boundary is inevitable and the order of the scheme at the discontinuous shock wave contained in the flow field as the supersonic flow field is low.It is questionable whether the reduced-order boundary and the low-order scheme at the shock wave have an effect on the numerical solution and accuracy of the flow field inside.In this paper,according to the actual situation of the direct numerical simulation of the flow field,two model equations with the exact solutions are solved,which are steady and unsteady,respectively,to study the question with a high order scheme at the interior of the domain and the reduced-order method at the boundary and center of the domain.Comparing with the exact solutions,it is found that the effect of reduced-order exists and cannot be ignored.In addition,the other two model equations with the exact solutions,which are often used in fluid mechanics,are also studied with the same process for the reduced-order problem.     

Flux difference splitting for open-channel flows

A finite difference scheme based on flux difference splitting is presented for the solution of the one-dimensional shallow-water equations in open channels, together with an extension to two-dimensional flows. A linearized problem, analogous to that of Riemann for gas dynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearized problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second-order scheme which avoids non-physical, spurious oscillations. The scheme is applied to a one-dimensional dam-break problem, and to a problem of flow in a river whose geometry induces a region of supercritical flow. The scheme is also applied to a two-dimensional dam-break problem. The numerical results are compared with the exact solution, or other numerical results, where available.     

Laminar flow through a channel with uniformly porous walls of different permeability

Summary The steady laminar flow of a viscous incompressible fluid through a two-dimensional channel, having fluid sucked or injected with different velocities through its uniformly porous parallel walls is considered. A solution for small suction Reynolds number has been given by the authors in a previous paper. The purpose of this paper is to present a solution valid for large Reynolds numbers for the cases of (i) suction at both walls, and (ii) suction at one wall and injection at the other. A technique of matching outer and inner expansions is used to obtain an asymptotic solution for both of these cases. Further a perturbation solution for the case of suction at one wall and injection at the other is obtained by choosing the difference between two wall velocities as the perturbation parameter. Both asymptotic and perturbation solutions are confirmed by exact numerical solutions. As expected, the resulting solutions show the presence of the usual suction boundary layers in both types of flow considered in this paper.     

A Quasilinear Based Procedure for Saturated/Unsaturated Water Movement in Soils

The quasilinear form of Richards equation for one-dimensional unsaturated flow in soils can be readily solved for a wide variety of conditions. However, it cannot explain saturated/unsaturated flow and the constant diffusivity assumption, used to linearise the transient quasilinear equation, can introduce significant error. This paper presents a quasi-analytical solution to transient saturated/unsaturated flow based on the quasilinear equation, with saturated flow explained by a transformed Darcy's equation. The procedure presented is based on the modified finite analytic method. With this approach, the problem domain is divided into elements, with the element equations being solutions to a constant coefficient form of the governing partial differential equation. While the element equations are based on a constant diffusivity assumption, transient diffusivity behaviour is incorporated by time stepping. Profile heterogeneity can be incorporated into the procedure by allowing flow properties to vary from element to element. Two procedures are presented for the temporal solution; a Laplace transform procedure and a finite difference scheme. An advantage of the Laplace transform procedure is the ability to incorporate transient boundary condition behaviour directly into the analytical solutions. The scheme is shown to work well for two different flow problems, for three soil types. The technique presented can yield results of high accuracy if the spatial discretisation is sufficient, or alternatively can produce approximate solutions with low computational overheads by using large sized elements. Error was shown to be stable, linearly related to element size.     

Effective permeability of cracked unsaturated porous materials

This study focuses on a theoretical estimation of the effective permeability of unsaturated cracked porous media. The closed-form flow solution around and in a superconductive crack, embedded in an infinite porous matrix under a far-field condition, is recalled first. Then the solution of flow around a completely unsaturated (empty) crack that is considered as an obstruction against the flow is determined. The flow solution for partially saturated crack in special configurations is obtained by superposition of the two basic solutions for superconductive and empty cracks. The contribution of an unsaturated crack, with a given saturation degree, to the effective permeability is estimated by using dilute upscaling scheme. Numerical results obtained by Finite Elements Method, are in good agreement with the theoretical results for weak crack densities but show the additional effect of cracks interaction for higher densities.     

二维时-空守恒格式及其在激波传播计算中的应用

将改进后的一维时－空守恒格式推广到了二维情形,得到了一个新的一般形式的二维Ｅｕｌｅｒ方程时－空守恒格式,并用该格式对几个具有复杂波系的流场进行了数值模拟。结果表明,该格式保留了一维格式通用性好、结构简单的优点,其计算结果精度高,对激波等间断具有很强的分辨率。     

一般坐标系下的二维Euler方程时—空守恒格式

将文「１－３」中的一维时－空守恒格式推广到了二维情形,得到了一般坐标系下的二维Ｅｕｌｅｒ方程时－空守恒格式,并用几个典型算例进行了检验计算,结果表明：本文得到的二维时－空守恒格式保留了一维格式所有的优点,格式简单,通用性强,而且对激波等间断具有很高的分辨率。     

Theoretical solution of the boundary problem of subsonic aerodynamics

A general theoretical solution of the boundary problem of aerodynamics of high subsonic velocities is presented. The solution of the partial differential equation for the velocity potential is carried out in the physical plane in streamline co-ordinates. The principle of the solution is the representation flow of a compressible fluid around a given profile to a hypothetical flow of an incompressible fluid around a different associated profile. In other words, the problem of compressible flow is transformed to the problem of incompressible flow, which can easily be solved. The results of this solution show very good agreement with solutions of other authors and with experiments.     

Strain-hardening extrusion—I. Construction of slip-line fields from experimental flow patterns

A possible method of solving problems in strain-hardening flows is by perturbation of known perfectly-plastic solutions. It is shown that vertex singularities, which are possessed by most such solutions, are not admissible in steady hardening flows. The structure of regular local solutions, both perfectly-plastic and strain-hardening, is investigated, and it is shown how vertex singularities can be replaced by regular local corner solutions.A scheme for constructing strain-hardening slip-line fields based on experimental flow patterns is described, which uses the maximum shear strain-rate directions calculated from the digitized and smoothed flow pattern to perturb local Hencky-Prandtl nets, which are then patched together in conformity with the topology of the solution to form a complete slip-line field. This method has been implemented in a computer program to construct slip-line fields from flow patterns for extrusion through wedge-shaped dies, and some fields computed by the program are presented.     

A high‐order compact finite‐difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows

A high‐order compact finite‐difference lattice Boltzmann method (CFDLBM) is proposed and applied to accurately compute steady and unsteady incompressible flows. Herein, the spatial derivatives in the lattice Boltzmann equation are discretized by using the fourth‐order compact FD scheme, and the temporal term is discretized with the fourth‐order Runge–Kutta scheme to provide an accurate and efficient incompressible flow solver. A high‐order spectral‐type low‐pass compact filter is used to stabilize the numerical solution. An iterative initialization procedure is presented and applied to generate consistent initial conditions for the simulation of unsteady flows. A sensitivity study is also conducted to evaluate the effects of grid size, filtering, and procedure of boundary conditions implementation on accuracy and convergence rate of the solution. The accuracy and efficiency of the proposed solution procedure based on the CFDLBM method are also examined by comparison with the classical LBM for different flow conditions. Two test cases considered herein for validating the results of the incompressible steady flows are a two‐dimensional (2‐D) backward‐facing step and a 2‐D cavity at different Reynolds numbers. Results of these steady solutions computed by the CFDLBM are thoroughly compared with those of a compact FD Navier–Stokes flow solver. Three other test cases, namely, a 2‐D Couette flow, the Taylor's vortex problem, and the doubly periodic shear layers, are simulated to investigate the accuracy of the proposed scheme in solving unsteady incompressible flows. Results obtained for these test cases are in good agreement with the analytical solutions and also with the available numerical and experimental results. The study shows that the present solution methodology is robust, efficient, and accurate for solving steady and unsteady incompressible flow problems even at high Reynolds numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

A general numerical method for solution of gravity wave problems. Part 2: Steady nonlinear gravity waves

A type of numerical scheme for 2D and 3D steady non-linear water wave problems is described. It is based on the finite process method and is insensitive to initial solutions. The relationship between the finite process method and iterative techniques is discussed. As a numerical example the flow past a submerged vortex is solved and the results are compared with those of other authors.     

Nonlinear disturbance evolution in a two-dimensional boundary-layer along an elastic plate and induced radiated sound

The interaction between a boundary-layer flow and an elastic plate is addressed by direct numerical simulation, taking into account the full coupling between the fluid flow and the flexible wall. The convectively unstable flow state is harmonically forced and two-dimensional nonlinearly saturated wavelike disturbances are computed along archetype-plates with respect to stiffness and natural frequencies. In the aim of determining the low-Mach-number radiated sound for the system, the simulation data are used to solve the Lighthill's equation in terms of a Green's function in the wavenumber-frequency space. Different degrees of fluid-structure coupling are implemented in the radiated sound model and the resulting acoustic pressure levels are compared. The sound radiation levels are shown to be increased in the presence of flexible walls with however significant differences in the radiated pressure levels for different coupling assumptions.     

A high‐resolution scheme for low Mach number flows

A method for computing low Mach number flows using high‐resolution interpolation and difference formulas, within the framework of the Marker and Cell (MAC) scheme, is presented. This increases the range of wavenumbers that are properly resolved on a given grid so that a sufficiently accurate solution can be obtained without extensive grid refinement. Results using this scheme are presented for three problems. The first is the two‐dimensional Taylor–Green flow which has a closed form solution. The second is the evolution of perturbations to constant‐density, plane channel flow for which linear stability solutions are known. The third is the oscillatory instability of a variable density plane jet. In this case, unless the sharp density gradients are resolved, the calculations would breakdown. Under‐resolved calculations gave solutions containing vortices which grew in place rather than being convected out. With the present scheme, regular oscillations of this instability were obtained and vortices were convected out regularly. Stable computations were possible over a wider range of sensitive parameters such as density ratio and co‐flow velocity ratio. Copyright © 2004 John Wiley Sons, Ltd.     

A coupled solution of the vorticity–velocity formulation of the incompressible Navier–Stokes equations

A relatively novel formulation of the Navier-Stokes equations is used for obtaining solutions of two dimensional incompressible fluid flow and convective heat transfer problems. A vorticity transport equation along with two Poisson equations for the velocity components and the energy equation are solved by a finite difference scheme. A coupled solution procedure is used for solving simultaneously the dependent variables along a line, using a block tridiagonal matrix algorithm. The formulation is found to be stable and has features that may be desirable for solving a wide variety of flow and heat transfer problems.     

Parallel overlapping scheme for viscous incompressible flows

A 3D parallel overlapping scheme for viscous incompressible flow problems is presented that combines the finite element method, which is best suited for analysing flow in any arbitrarily shaped flow geometry, with the finite difference method, which is advantageous in terms of both computing time and computer storage. A modified ABMAC method is used as the solution algorithm, to which a sophisticated time integration scheme proposed by the present authors has been applied. Parallelization is based on the domain decomposition method. The RGB (recursive graph bisection) algorithm is used for the decomposition of the FEM mesh and simple slice decomposition is used for the FDM mesh. Some estimates of the parallel performance of FEM, FDM and overlapping computations are presented. © 1997 John Wiley & Sons, Ltd.     

Nonuniqueness of the solution of the problem of viscous interaction on an axisymmetric body

The article discusses solutions of the equations of the hypersonic boundary layer on an axisymmetric offset slender body (with a power exponent equal to 3/4), taking account of interactions with a nonviscous flow. It is shown that, in this case, the equations of the boundary layer have solutions differing from the self-similar solution corresponding to flow around a semi-infinite body. The solutions obtained are analogous to solutions for a strong interaction on a plate with slipping and triangular vanes [1–4], but are obtained over a wide range of values of the parameter of viscous interaction. An asymptotic solution is given to the problem with the approach to zero of the interaction parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 41–47, September–October, 1973.The authors thank V. V. Mikhailova for discussion of the work and useful advice.     

Internal three-dimensional viscous flow solutions using the vorticity-potential method

A numerical solution procedure for internal three-dimensional viscous flow is proposed in this paper. The formulation is based on the non-primitive variables, the vorticity and potentials, on a curvilinear grid. A new upwind difference scheme is introduced to overcome the convective instabilities arising in the central difference scheme for the vorticity transport equations, while keeping false diffusion to a minimum level. Developing flows in both straight and curved square ducts are simulated to validate the procedure. The results are compared with both experimental measurements and analytical solutions.