Development of an artificial compressibility methodology using flux vector splitting

An implicit, upwind arithmetic scheme that is efficient for the solution of laminar, steady, incompressible, two-dimensional flow fields in a generalised co-ordinate system is presented in this paper. The developed algorithm is based on the extended flux-vector-splitting (FVS) method for solving incompressible flow fields. As in the case of compressible flows, the FVS method consists of the decomposition of the convective fluxes into positive and negative parts that transmit information from the upstream and downstream flow field respectively. The extension of this method to the solution of incompressible flows is achieved by the method of artificial compressibility, whereby an artificial time derivative of the pressure is added to the continuity equation. In this way the incompressible equations take on a hyperbolic character with pseudopressure waves propagating with finite speed. In such problems the ‘information’ inside the field is transmitted along its characteristic curves. In this sense, we can use upwind schemes to represent the finite volume scheme of the problem's governing equations. For the representation of the problem variables at the cell faces, upwind schemes up to third order of accuracy are used, while for the development of a time-iterative procedure a first-order-accurate Euler backward-time difference scheme is used and a second-order central differencing for the shear stresses is presented. The discretized Navier–Stokes equations are solved by an implicit unfactored method using Newton iterations and Gauss–Siedel relaxation. To validate the derived arithmetical results against experimental data and other numerical solutions, various laminar flows with known behaviour from the literature are examined. © 1997 John Wiley & Sons, Ltd.     

Improved artificial dissipation schemes for the Euler equations

The influence of artificial dissipation schemes on the accuracy and stability of the numerical solution of compressible flow is extensively examined. Using an implicit central difference factored scheme, an improved form of artificial dissipation is introduced which highly reduces the errors due to numerical viscosity. A function of the local Mach number is used to scale the amount of numerical damping added into the solution according to the character of the flow in several flow regimes. The resulting scheme is validated through several inviscid flow test cases.     

Prediction of incompressible flow separation with the approximate factorization technique

This paper describes one application of the approximate factorization technique to the solution of incompressible steady viscous flow problems in two dimensions. The velocity-pressure formulation of the Navier-Stokes equations written in curvilinear non-orthogonal co-ordinates is adopted. The continuity equation is replaced with one equation for the pressure by means of the artificial compressibility concept to obtain a system parabolic in time. The resulting equations are discretized in space with centred finite differences, and the steady state solution obtained by a time-marching ADI method requiring to solve 3 x 3 block tridiagonal linear systems. An optimized fourth-order artificial dissipation is introduced to damp the numerical instabilities of the artificial compressibility equation and ensure convergence. The resulting solver is applied to the prediction of a wide variety of internal flows, including both streamlined boundaries and sharp corners, and fast convergence and good results obtained for all the configurations investigated.     

A class of fully third-order accurate projection methods for solving the incompressible Navier-Stokes equations

In this paper, a fully third-order accurate projection method for solving the incompressible Navier-Stokes equations is proposed. To construct the scheme, a continuous projection procedure is firstly presented. We then derive a sufficient condition for the continuous projection equations to be temporally third-order accurate approximations of the original Navier-Stokes equations by means of the local- truncation-error-analysis technique. The continuous projection equations are discretized temporally and spatially to third-order accuracy on the staggered grids, resulting in a fully third-order discrete projection scheme. The possibility to design higher-order projection methods is thus demonstrated in the present paper. A heuristic stability analysis is performed on this projection method showing the probability of its being stable. The stability of the present scheme is further verified through numerical tests. The third-order accuracy of the present projection method is validated by several numerical test cases. The project supported by the China NKBRSF (2001CB409604) The English text was polished by Yunming Chen     

A Eulerian/Lagrangian model to calculate the evolution of a water droplet spray

We introduce a Eulerian/Lagrangian model to compute the evolution of a spray of water droplets inside a complex geometry. To take into account the complex geometry we define a rectangular mesh and we relate each mesh node to a node function which depends on the location of the node. The time-dependent incompressible and turbulent Navier-Stokes equations are solved using a projection method. The droplets are regarded as individual entities and we use a Lagrangian approach to compute the evolution of the spray. We establish the exchange laws related to mass and heat transfer for a droplet by introducing a mass transfer coefficient and a heat transfer coefficient. The numerical results from our model are compared with those from the literature in the case of a falling droplet in the atmosphere and from experimental investigation in a wind tunnel in the case of a polydisperse spray. The comparison is fairly good. We present the computation of a water droplet spray inside a complex and realistic geometry and determine the characteristics of the spray in the vicinity of obstacles.     

Multigrid pressure correction techniques for the computation of quasi-incompressible internal flows

A study is reported on the possibility of improving the speed of convergence of existing numerical programmes for the simulation of flow in combustion chambers by applying the multigrid method to the pressure correction phase only. A version of the multigrid algorithm is introduced for this purpose which achieves a 1:10 residual reduction in a single V(1, 1) cycle. The overall decrease in computation time with respect to an industry-standard SIMPLE algorithm with single-grid pressure correction ranges from four to five times for SIMPLE itself and several other well-known algorithms to six times for a newly developed pressure correction strategy we call difference operator triangularization (DOT).     

A problem of heat and mass transfer: Proof of the existence condition by a finite difference method

We solve by a finite difference method a system of simultaneous non-linear partial differential equations which modelizes the transfer of heat and mass when a fluid evaporates from the hot wall and condenses on the cold wall of an upright rectangular cavity. The need to verify a certain condition associating the physical parameters of the fluid for the existence of steady state solutions is proved.     

An analytic approximation of the drag coefficient for the viscous flow past a sphere

We give an analytic solution at the 10th order of approximation for the steady-state laminar viscous flows past a sphere in a uniform stream governed by the exact, fully non-linear Navier-Stokes equations. A new kind of analytic technique, namely the homotopy analysis method, is applied, by means of which Whitehead's paradox can be easily avoided and reasonably explained. Different from all previous perturbation approximations, our analytic approximations are valid in the whole field of flow, because we use the same approximations to express the flows near and far from the sphere. Our drag coefficient formula at the 10th order of approximation agrees better with experimental data in a region of Reynolds number R_d<30, which is considerably larger than that (R_d<5) of all previous theoretical ones.     

A new method for obtaining the exact solutions of the Navier-Stokes (NS) equations and its applications

In this paper we propose a new method for obtaining the exact solutions of the Mavier-Stokes (NS) equations for incompressible viscous fluid in the light of the theory of simplified Navier-Stokes (SNS) equations developed by the first author^[1,2], Using the present method we can find some new exact solutions as well as the well-known exact solutions of the NS equations. In illustration of its applications, we give a variety of exact solutions of incompressible viscous fluid flows for which NS equations of fluid motion are written in Cartesian coordinates, or in cylindrical polar coordinates, or in spherical coordinates. The project supported by National Natural Science Foundation of China.     

The numerical solution of the laminar flow in a constricted channel at moderately high Reynolds number using Newton iteration

The numerical solution of the flow in a stepped channel constricted to half its width has been obtained for Reynolds numbers up to 2000 using Newton's iteration to solve the ensuing algebraic system. In order to avoid high-frequency errors, a locally fine grid is effected near the corner by transformation of the independent variables. The results predict a downstream recirculation region, observed in experiments but not found in earlier numerical calculations. The inclusion of the Dennis–Hudson upwinding, added for stability in SOR methods, whilst giving the same characteristics of the flow, is less accurate by at least an order of magnitude.     

Numerical solution of the unsteady Euler equations for airfoils using approximate boundary conditions

This paper presents an efficient numerical method for solving the unsteady Euler equations on stationary rectilinear grids. Boundary conditions on the surface of an airfoil are implemented by using their first-order expansions on the mean chord line. The method is not restricted to flows with small disturbances since there are no restrictions on the mean angle of attack of the airfoil. The mathematical formulation and the numerical implementation of the wall boundary conditions in a fully implicit time-accurate finite-volume Euler scheme are described. Unsteady transonic flows about an oscillating NACA 0012 airfoil are calculated. Computational results compare well with Euler solutions by the full boundary conditions on a body-fitted curvilinear grid and published experimental data. This study establishes the feasibility for computing unsteady fluid-structure interaction problems, where the use of a stationary rectilinear grid offers substantial advantages in saving computer time and program design since it does not require the generation and implementation of time-dependent body-fitted grids.     

Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work

Buoyant flow is analysed for a vertical fluid saturated porous layer bounded by an isothermal plane and an isoflux plane in the case of a fully developed flow with a parallel velocity field. The effects of viscous dissipation and pressure work are taken into account in the framework of the Oberbeck–Boussinesq approximation scheme and of the Darcy flow model. Momentum and energy balances are combined in a dimensionless nonlinear ordinary differential equation solved numerically by a Runge–Kutta method. Both cases of upward pressure force (upward driven flows) and of downward pressure force (downward driven flows) are examined. The thermal behaviour for upward driven flows and downward driven flows is quite different. For upward driven flows, the combined effects of viscous dissipation and pressure work may produce a net cooling of the fluid even in the case of a positive heat input from the isoflux wall. For downward driven flows, viscous dissipation and pressure work yield a net heating of the fluid. A general reflection on the roles played by the effects of viscous dissipation and pressure work with respect to the Oberbeck–Boussinesq approximation is proposed.     

Simulation of 2D external viscous flows by means of a domain decomposition method using an influence matrix technique

Two-dimensional external viscous flows are numerically approximated by means of a domain decomposition method which combines a vortex method and a finite difference method. The vortex method is used in the flow region which is dominated by convective effects, whereas the finite difference method is used in the flow region where viscous diffusion effects are dominant. An influence matrix technique combined with the uniformity condition of the pressure is used to enforce the tangential velocity boundary condition. Comparisons between numerical and experimental data show that the method is well adapted for simulating two-dimensional flows.     

轻载径向滑动轴承中Taylor涡动的产生和影响研究

本文用原始变量法直接求解了三维的Ｎ－Ｓ方程，计算分析了高速旋转有限长圆柱轴承中油膜层流失稳产生涡动的临界Ｔａｙｌｏｒ数及流场、压力场和摩擦阻力。轴承端部泄油量等的变化。结果表明，在有限长同心圆柱轴承中，随着轴旋转速度的提高，轴承磨擦阻力线性增大，油膜层流失稳出现的涡动增加轴承摩擦阻力并减少轴承端部泄油量，油膜层流失稳后，轴承长度方向均匀地排列着一些流体涡，涡动的强度从轴承中间截面向轴承端部逐渐减弱     

Application of a Virtual-Boundary Method for the Numerical Study of Oscillations Developing Behind a Cylinder Near A Plane Wall

The conditions of onset and the character of the oscillations developing behind a circular cylinder located above a plane wall (screen) in a flow with a velocity profile of the boundary layer type are studied numerically. The dependence of the critical Reynolds number (at which a steady flow regime in the wake behind the cylinder is replaced by an oscillatory regime) on the cylinder-wall gap and the free-stream boundary layer thickness is found.     

Two-dimensional models of magma flows in a volcanic conduit taking the magma compressibility and thermal effects into account

Two steady-state models of magma flow in a conduit are considered, with and without allowance for magma compressibility. As distinct from studies [{xc1}–{xc6}], in which either simplified equations were solved or unrealistic values of the parameters were used, in the present study the complete systems of equations are solved and the values of the parameters correspond to magma flow in a volcanic conduit. The secondary flows obtained in [{xc5}] for model conditions are not formed when the magma is simulated by an incompressible fluid and all the terms of the equations are taken into account. When the magma compressibility is taken into account, in the isothermal case and for constant magma viscosity the entire flow is adequately described by the one-dimensional isothermalmodel, although this approach is not formally applicable.     

Development of the direct stress solution technique for three-dimensional hydrodynamic models using finite elements

Velocity varies rapidly near sheared boundaries. Therefore in many practical fluid problems it can be inefficient to solve discrete equations with velocity as the dependent variable. Conversely, shear stress varies slowly near sheared boundaries, suggesting that it may be well suited for use as the dependent variable in discrete equations. This paper describes a formulation of the internal mode equations for a three-dimensional hydrodynamic model using shear stress as the dependent variable. The resulting direct stress solution (DSS), coupled with a spatial discretization using linear finite elements, yields a system matrix that can be set up and solved with the efficiency of a banded matrix with bandwidth 8. If the eddy viscosity distribution is assumed to be piecewise linear over the depth (with an arbitrary number of time-varying segments), the recovery of velocity from stress can be easily accomplished in closed form, thereby avoiding any difficulty resulting from the logarithmic singularity in the velocity profile that occurs at a boundary. Results from tidal and wind-driven test cases with realistic boundary layers are used to demonstrate the accuracy and computational advantages of a DSS formulation versus a standard velocity-based formulation.     

Aerodynamic and aeroacoustic analysis of a pulsating internal flow by a multidomain weak collocation spectral method

A numerical methodology is presented for the coupled aerodynamic and aeroacoustic analysis of time dependent laminar viscous flows in general two-dimensional geometries. The overall procedure is constituted firstly by the solution of the incompressible Navier-Stokes equations and secondly by the solution of a linear evolution equation of second order in space and time which originates from Lighthill's acoustic analogy. A semi-implicit projection method is utilized for the time integration of the incompressible Navier-Stokes system, while a choice between a second order implicit Newmark scheme and a fourth order explicit Runge-Kutta-Nyström method is offered for the temporal discretization of the wave equation. The multidomain weak Legendre collocation spectral method on quadrilateral subdomain topologies is employed for the spatial approximation of both the fluid dynamic and the acoustic problems. A specific pulsating internal flow inside a plane constricted channel is selected as a representative application for the assessment of the capabilities of the proposed discrete algorithm. Numerous results are presented and discussed, so as to thoroughly demonstrate the behavior of the numerical procedure.     

Simulations of fibre orientation in dilute suspensions with front moving in the filling process of a rectangular channel using level-set method

The simulation of fibre orientation in dilute suspension with front moving is carried out using the projection and level-set methods. The motion of fibres is described using the Jeffery equation, and the contribution of fibres to the flow is accounted for by the configuration-field method. The dilute suspension of short fibres in Newtonian fluids is considered. The governing Navier–Stokes equation for the fluid flow is solved using the projection method with finite difference scheme, while the fibre-related equations are directly solved with the Runge–Kutta method. In the present study for fibres in dilute suspension flow for injection molding, the effects of various flow and material parameters on the fibre orientation, the velocity distributions and the shapes of the leading flow front are found and discussed. Our findings indicate that the presence of fibre motion has little influence on the front shape in the ranges of fibre parameters studied at the fixed Reynolds number. Influence of changing fibre parameters only causes variation of front shape in the region near the wall, and the front shape in the central core area does not vary much with the fibre parameters. On the other hand, the fibre motion has strong influence on the distributions of the streamwise and transverse velocities in the fountain flow. Fibre motion produces strong normal stress near the wall which leads to the reduction of transversal velocity as compared to the Newtonian flow without fibres, which in turn, leads to the increased streamwise velocity near the wall. Thus, the fibre addition to the flow weakens the strength of the fountain flow. The Reynolds number has also displayed significant influence on the distribution of the streamwise velocity behind the flow front for a given fibre concentration. It is also found that the fibre orientation is not always along the direction of the velocity vector in the process of mold filling. In the region of the fountain flow, the fibre near the centreline is more oriented across the streamwise direction compared to that in the region far behind the flow front. This leads to the fact that the fibre near the centreline in the region of fountain flow is more extended along the transverse direction. As the fibre orientation in the suspension flow and the shape of the flow front have great bearing on the quality of the product made from injection molding, this study has much implications for engineering applications. These results can also be useful in other fields dealing with fibre suspensions.     

On the implementation of symmetric and antisymmetric periodic boundary conditions for incompressible flow

In this paper we consider symmetric and antisymmetric periodic boundary conditions for flows governed by the incompressible Navier-Stokes equations. Classical periodic boundary conditions are studied as well as symmetric and antisymmetric periodic boundary conditions in which there is a pressure difference between inlet and outlet. The implementation of this type of boundary conditions in a finite element code using the penalty function formulation is treated and also the implementation in a finite volume code based on pressure correction. The methods are demonstrated by computation of a flow through a staggered tube bundle.