摘要:

A survey of commonly used approximations to the Eulerian form of the equations of ideal fluid flow is given. Comparisons are made through amplitude and phase properties as determined by linear stability analysis. The unacceptable amplitude damping of first‐order approximations is reiterated. For second‐order approximations the discussion emphasizes numerical dispersion effects and shows that the familiar stable forms do not differ significantly in relative merit. Fourth‐order improvements are discussed with reference to further extensions which minimize dispersion. Conservative forms of the approximations are given along with experimentally determined properties regarding their nonlinear behavior in fluid‐dynamic calculations. Comments relating to fluid‐dynamic instability versus numerical instability are included.

ISSN:0031-9171    

DOI:10.1063/1.1692465

出版商:AIP    

年代:1969

数据来源: AIP

摘要:

Many of the existing computations of initial‐ and boundary‐value problems in fluid mechanics suffer from unrealistic treatment of boundary points. Three categories of boundaries are discussed briefly: rigid walls, arbitrary boundaries of a computational region in a subsonic flow, and shock waves. An attempt is made to show in what sense the numerical treatment of such boundaries may be physically wrong and what can be done instead. Examples from the blunt body problem, the transonic flow in a nozzle, the incompressible inviscid flow past a circle, and the quasi‐one‐dimensional flow in a Laval nozzle, are shown.

ISSN:0031-9171    

DOI:10.1063/1.1692426

出版商:AIP    

年代:1969

数据来源: AIP

摘要:

A finite‐difference method is presented for the solution of the elliptic differential equations for the steady transport of momentum, heat, and matter in two‐dimensional domains. Special features of the method include an unsymmetrical formulation for the convection terms, which promotes convergence at some cost in accuracy; obedience to the conservation equations for all subdomains; the use of Gauss‐Seidel iteration procedure; employment of grids having nonuniform mesh; and a novel treatment of the boundary condition for vorticity. Solutions are presented for the laminar flow and heat transfer inside a square cavity with a moving top, an impinging jet, and a Couette flow with mass transfer. The influence of the Reynolds and Prandtl numbers, and of the impinging jet “free” boundary conditions is studied, and the results of the computations are shown to agree with existing physical knowledge. The influence of mesh size, mesh nonuniformity, and the vorticity wall boundary condition on convergence and accuracy is studied. It is shown that convergence may be secured for a wide range of Reynolds numbers with coarse‐meshed grids. The convergence and computation speed appear to be satisfactory for many purposes; the accuracy of the solutions is discussed, and some improvements are suggested.

ISSN:0031-9171    

DOI:10.1063/1.1692439

出版商:AIP    

年代:1969

数据来源: AIP

摘要:

When hyperbolic systems are integrated, effects of so‐called approximation viscosity appear. To investigate the latter, there is a very effective method which allows the reduction of the problem of the stability of a difference scheme to the stability (correctness) of a system of differential equations (the so‐called first differential approximation). Such a reduction is possible in the classes of simple and majorant schemes. In a sense, the approximation viscosity of the scheme and its dissipative properties are determined by a first differential approximation. The stated method can be used in the investigation of difference schemes for the problems of compressible and viscous fluid.

ISSN:0031-9171    

DOI:10.1063/1.1692451

出版商:AIP    

年代:1969

数据来源: AIP

摘要:

An error analysis is formulated under the assumption that Lax's equivalence theorem is valid for quasilinear equations. Analysis is carried out for a class of difference approximations of the Burgers' equation. Explicit estimates of the errors due to various sources are obtained and verified by numerical integration. The boundary errors do not decay rapidly away from the boundary and often dominate the truncation errors. The boundary errors led to significant differences between apparently smooth and physically reasonable results of carefully executed computations of the same problem. The results based on difference formulations that are more nearly consistent with the differential formulation are likely to be more accurate. A conservative difference formulation is proposed to meet this consistency requirement as far as is practically possible. Calculations of multidimensional flow examples guided by these inferences are given. The results tend to support such inferences.

ISSN:0031-9171    

DOI:10.1063/1.1692466

出版商:AIP    

年代:1969

数据来源: AIP

摘要:

The occurrence of discontinuities in inviscid flows has led to the development of artificial viscosity techniques. In this paper, an investigation of the following two questions has been initiated. Is it possible to utilize a numerical scheme for the inviscid equations, such that the stability of the method does not depend on an artifical viscosity effect? Is it possible to have continuous nonisentropic, inviscid flows in which there are subsonic and supersonic regions? The results of two numerical studies presented here indicate that the finite difference representation that has been utilized to solve the Euler equations is stable, even in the presence of mathematical discontinuities, and that this stability is not due to what is commonly referred to as an artificial viscosity. Also, in the physical problem containing a region of constant entropy and a region of varying entropy it was found that the discontinuity formed only after the pressure wave had reached a region of constant entropy.

ISSN:0031-9171    

DOI:10.1063/1.1692467

出版商:AIP    

年代:1969

数据来源: AIP

摘要:

Fluid dynamics problems often involve boundary conditions at infinity. In the case of numerical computation the most serious difficulty arises from the practical impossibility of reaching such conditions. It is then necessary to transform the problem either into a Cauchy problem, or into a boundary‐value problem within a limited domain. Generally, this process entails the regrettable consequence of bringing up as parasites in the solution some of the eigenfunctions pertaining to the equation. A method is proposed to filter these parasite solutions; its principle is the introduction of a certain functional controlling the error due to the parasite solutions, in a given domain. The method is illustrated by some numerical results concerning the computation of a flow downstream of an obstacle within a gravitational field.

ISSN:0031-9171    

DOI:10.1063/1.1692468

出版商:AIP    

年代:1969

数据来源: AIP

摘要:

Numerical solutions of the steady Navier‐Stokes equations are presented for two‐dimensional flows past a circular cylinder in an infinite domain. The flow is assumed to be uniform at infinity upstream and the range of Reynolds numbers extends from 1 to 60. The Navier‐Stokes equations are replaced by a set of finite difference equations and the numerical solution is obtained by means of an iteration method. Conditions at “infinity” are applied by matching to Imai's asymptotic solution. The results are compared with those of other analytical and numerical computations as well as with experiments. In particular, the discussion is concerned with the drag, the base pressure, the shape of the standing vortices, and some formulas suggested for large Reynolds numbers. Excellent agreement with recent experiments of Acrivos, Leal, Snowden, and Pan is obtained.

ISSN:0031-9171    

DOI:10.1063/1.1692469

出版商:AIP    

年代:1969

数据来源: AIP

摘要:

Kawaguti and Jain have reported a systematic numerical investigation of unsteady viscous incompressible fluid flow past a circular cylinder for obtaining limiting steady‐state solutions at Reynolds numbers up to 50. After removing the symmetry conditions the flow pattern has been studied to investigate the existence of the limiting steady state or the Ka´rma´n vortex street numerically at Reynolds numberR = 40, 60, 100, and 200. AtR = 40and 60, steady‐state solutions have been obtained and found to be in agreement with the other results. Results atR = 100and 200 are also reported. Figures have been drawn to show the dependence of the flow pattern, the vorticity distribution, the pressure distribution, and the drag on the Reynolds number and the time.

ISSN:0031-9171    

DOI:10.1063/1.1692470

出版商:AIP    

年代:1969

数据来源: AIP

摘要:

Numerical solutions of the transient axisymmetric flow around a thin disk normal to the flow are obtained. The fluid is incompressible, homogeneous, and its flow is governed by the complete Navier‐Stokes equations. The Reynolds‐number range studied is 10‐600. The study investigates the effects that a very large curvature of the body has on the numerical procedure for the solution of the flow field, and the fundamental fluid‐dynamical phenomena of separation, of a recirculatory wake, and of vorticity shedding. A time‐dependent stream function‐vorticity formulation is adopted. The solutions are obtained on an oblate spheroidal grid system. The high curvature has profound adverse effects on the numerical stability and accuracy. This is due to the extreme gradients of vorticity that appear near the edge of the disk. The results show that no vorticity shedding occurs for axisymmetric flow in the Reynolds‐number range studied. In addition, some new interesting fluid‐dynamical features are revealed. These include a different behavior of the pressure distribution at low and high Reynolds numbers and a local maximum of vorticity inside the wake at the higher Reynolds numbers studied.

ISSN:0031-9171    

DOI:10.1063/1.1692471

出版商:AIP    

年代:1969

数据来源: AIP