Manifold method coupled velocity and pressure for Navier-Stokes equations and direct numerical solution of unsteady incompressible viscous flow

Numerical manifold method (NMM) application to direct numerical solution for unsteady incompressible viscous flow Navier-Stokes (N-S) equations was discussed in this paper, and numerical manifold schemes for N-S equations were derived based on Galerkin weighted residuals method as well. Mixed covers with linear polynomial function for velocity and constant function for pressure was employed in finite element cover system. The patch test demonstrated that mixed covers manifold elements meet the stability conditions and can be applied to solve N-S equations coupled velocity and pressure variables directly. The numerical schemes with mixed covers have also been proved to be unconditionally stable. As applications, mixed cover 4-node rectangular manifold element has been used to simulate the unsteady incompressible viscous flow in typical driven cavity and flow around a square cylinder in a horizontal channel. High accurate results obtained from much less calculational variables and very large time steps are in very good agreement with the compact finite difference solutions from very fine element meshes and very less time steps in references. Numerical tests illustrate that NMM is an effective and high order accurate numerical method for unsteady incompressible viscous flow N-S equations.     

A finite element based level-set method for multiphase flow applications

A numerical method for simulating incompressible two-dimensional multiphase flow is presented. The method is based on a level-set formulation discretized by a finite-element technique. The treatment of the specific features of this problem, such as surface tension forces acting at the interfaces separating two immiscible fluids, as well as the density and viscosity jumps that in general occur across such interfaces, have been integrated into the finite-element framework. Using a method based on the weak formulation of the Navier-Stokes equations has its advantages. In this formulation, the singular surface tension forces are included through line integrals along the interfaces, which are easily approximated quantities. In addition, differentiation of the discontinuous viscosity is avoided. The discontinuous density and viscosity are included in the finite element integrals. A strategy for the evaluation of integrals with discontinuous integrands has been developed based on a rigorous analysis of the errors associated with the evaluation of such integrals. Numerical tests have been performed. For the case of a rising buoyant bubble the results are in good agreement with results from a front-tracking method. The run presented here is a run including topology changes, where initially separated areas of one fluid merge in different stages due to buoyancy effects. Received: 1 March 1999 / Accepted: 17 June 1999     

Mixed analytical/numerical method for flow equations with a source term

A mixed analytical/numerical approach is studied for flow problems described by partial differential equations with source terms which are analytically integrable and which may involve a time scale (S-scale) much smaller than the mean flow time scale (M-scale). A rigorous error analysis based on the modified equation is conducted for a linear model equation and it is shown, both analytically and numerically, that the mixed scheme is more accurate than a conventional numerical method. Most interestingly, the mixed approach has a good accuracy for the M-scale structure even though the time step is larger than the S-scale, while a conventional scheme fails to work in this case by producing errors of order O(1) or larger.     

A numerical method for the transonic cascade flow problem

An implicit time-marching finite-difference method for solving the three-dimensional compressible Navier-Stokes equations for the relative flow of a turbomachine impeller in general curvilinear coordinates is presented. The fundamental equations of the method have the distinctive feature that the momentums of the contravariant velocities are employed as the dependent variables. The use of the momentum equations of the contravariant velocities makes possible correct and simple treatments of some boundary conditions. In order to obtain the stable solution for high Reynolds number turbulent flow, the Navier-Stokes equations and the k-ε turbulence model equations are solved simultaneously, and a high-resolution TVD upwind scheme is introduced. The calculated results of some two-dimensional turbulent flows agreed well with the experimental data. The calculated results of an axial-flow transonic compressor rotor flow showed that the leakage vortex from the tip clearance as well as the shock waves can be captured vividly, in spite of the relatively coarse grid.     

A numerical method for solving systems of nonlinear equations

An iterative solution method for systems of nonlinear equations is proposed, making use of a nonlinear technique for the construction of the approximations. The method is efficient for solving quadratic equations.Translated from Kibernetika, No. 3, pp. 60–64, 69, May–June 1990.     

A fourier series method for the numerical solution of hyperbolic equations

In this paper a Fourier series technique which reduces the given hyperbolic partial differential equation to a system of ordinary differential equations with one point boundary conditions is presented. The numerical results obtained indicate a considerable saving in time in addition to accuracy since no discretisation errors are incurred in the space variable.     

A numerical model for multiphase flow based on the GMPPS formulation. Part I: Kinematics

The CFD modeling of two-dimensional multiphase flows is a useful tool in industry, although accurate modeling itself remains a difficult task. One of the difficulties is to track the complicated topological deformations of the interfaces between different phases. This paper describes a marker-particle method designed to track fluid interfaces for fluid flows of at least three phases. The interface-tracking scheme presented in this paper is the first part of a series of papers presenting our complete model based on a one-field Godunov marker-particle projection scheme (GMPPS). In this part, we shall focus on the presentation of the interface-tracking scheme and the kinematic tests we conducted to examine the scheme’s ability to accurately track interfacial movements typified by vorticity-induced stretching and tearing of the interface. Our test results show that for a set of carefully designed and commonly used error measures, relative percentage errors never exceed 2% for all of the tests and grid sizes considered, provided a sufficient number of marker particles are used. We shall also demonstrate that the method is of second-order accuracy and the interface transition width remains constant never exceeding three cell widths.     

A numerical method for solving parabolic equations with opposite orientations

The solution of parabolic control problems is characterized by a system of two equations parabolic with respect to opposite orientations. In this paper a fast iterative method for solving such problems is proposed.     

A new numerical method for kinetic equations in several dimensions

We present first results of a numerical method solving inhomogeneous partial differential equation of first order with a conservation property. The method is based on the Finite Particle Schemes for homogeneous PDE's of the first order as the Vlasov-Poisson system in kinetic theory. The inhomogeneity is redefined as a flux. For the associated ‘velocity-field’ given by the Radon-Nikodym derivative of the flux, we give a numerical approximation. Together with the ‘velocity-field’ given, by the derivative terms of first order this gives the right hand side of the equations of motion of the particles. The computation can be done in a very efficient way and the results are in good agreement with the exact solution.     

A dynamic p-adaptive Discontinuous Galerkin method for viscous flow with shocks

A dynamic p-adaptive Discontinuous Galerkin method for compressible flows is proposed. The key element is a sensor, which measures the local regularity of the solution. This sensor is designed to preserve the compactness of the method and is easy to implement. In regions where the solution is quasi-uniform and in the vicinity of shocks the degree of the polynomial basis is decreased, while a high-degree basis is used in regions of smooth fluctuations of the flow. Numerical tests carried out in 1D and on 2D unstructured meshes prove that the convergence rate of the hybrid method is equal to the one of the highest degree polynomial basis, but with a significant CPU cost saving.     

A block-by-block method for the numerical solution of Volterra delay integro-differential equations

A block-by-block method based on interpolatory quadrature rules is applied to delay Volterra integrodifferential equations with variable delay. An error bound is obtained and a rate of convergence is found. Our numerical results are compared with those obtained by applying Neves's [19] algorithm and a cyclic linear multistep method of McKee [17]; they are also compared with those presented in Kemper [11].     

A two step method for the numerical integration of stiff differential equations

We examine a single-step implicit-integration algorithm which is obtained by a modification of the well-known Simpson rule. The accuracy and stability properties of these methods are investigated. The obtained new method is a fourth-order numerical process and preserves the property of A-stability of the Simpson rule. Numerical results for the solution of certain stiff equations.     

Barycentric interpolation collocation method for solving the coupled viscous Burgers' equations

The coupled viscous Burgers' equations have been an interesting and hot topic in mathematics and physics for a long time, and they have been solved by many methods. In order to make the numerical solutions more accurate, this paper introduces a new method to solve the equations. Compared to other methods, the present method can obtain higher accuracy with fewer nodes. Several numerical examples show the high accuracy of this method.     

Pseudospectral multi-domain method for incompressible viscous flow computation

We present a multi-domain pseudospectral method for the calculation of incompressible viscous flow. Governing equations are written in primitive variables formulation. Velocity components and pressure are discretized on the same grid of collocation points. A coupling algorithm for the Stokes problem is investigated and preliminary results are presented in the two-dimensional case.     

Solution domain decomposition method for viscous incompressible flow

A solution domain decomposition method is developed for steady state solution of the biharmonic-based Navier-Stokes equations. It consists of a domain decomposition in conjunction with Chebyshev collocation for spatial discretization. The interactions between subdomains are effectively decoupled by means of a superposition of auxiliary solutions to yield a set of independent elementary problems which can be solved concurrently on multiprocessor computers. Assessments are carried out to a number of test problems including the two-dimensional steady flow in a driven square cavity. Illustrative examples indicate a good performance of the proposed methodology which does not affect the convergence and stability of the discretization scheme. Spectral accuracy is retained with absolute error decaying in an exponential fashion. The numerical solutions for the driven cavity compare favorably against previously published numerical results except for a slight overprediction in the vertical velocity component at Reynolds number of 400. TheC ³ continuity is speculated to be its cause.     

An improved numerical solution of multiphase flow analysis in soil

The present paper is concerned with a numerical solution for the analysis of coupled water, a non-aqueous phase liquid, and gas and heat flow in unsaturated and saturated soil. First, it briefly presents all the problem governing equations and then concentrates on the development of a robust numerical solution algorithm. It improves the solution procedure by extending the primary variable switching scheme so that it can also be used for multiphase flow analysis and by developing a method for reducing the usual numerical oscillations. Finally, a sample numerical analysis of a spill of diesel fuel in a box filled with partially water-saturated soil is presented.     

A numerical method for the vorticity-velocity Navier-Stokes equations in two and three dimensions

This paper provides a numerical method for solving the steady-state vorticity-velocity Navier-Stokes equations in two and three dimensions. The vorticity transport equation is considered together with a Poisson equation for the velocity vector, the latter equation being parabolized in time according to the false transient approach. The two vector equations are discretized in time using the implicit Euler time stepping and the delta form of Beam and Warming. A staggered-grid spatial discretization is employed in conjunction with a deferred correction procedure. Second-order-accurate central differences are used to approximate the steady-state residuals, written in conservative form for accuracy reasons, whereas upwind differences are used for the advection terms in the implicit operator, to obtain diagonally-dominant tridiagonal systems. The discrete equations are solved sequentially by means of a robust alternating direction line-Gauss-Seidel iteration procedure combined with a simple multigrid strategy. For the model driven-cavity-flow problem in two and three dimensions, the method is found to be efficient and very accurate. For the first time, the three-dimensional discrete vorticity and velocity fields, computed using a Poisson equation for the velocity vector, are both solenoidal and satisfy their mutual relationship, exactly.