Showing papers on "Navier–Stokes equations published in 1980"

Implicit Finite-Difference Simulations of Three-Dimensional Compressible Flow

Abstract: An implicit finite-difference procedure for unsteady three-dimensional flow capable of handling arbitrary geometry through the use of general coordinate transformations is described. Viscous effects are optionally incorporated with a "thin-layer" approximation of the Navier-Stokes equations. An implicit approximate factorization technique is employed so that the small grid sizes required for spatial accuracy and viscous resolution do not impose stringent stability limitations. Results obtained from the program include transonic inviscid or viscous solutions about simple body configurations. Comparisons with existing theories and experiments are made. Numerical accuracy and the effect of three-dimensional coordinate singularities are also discussed.

Transition to turbulence in plane Poiseuille and plane Couette flow

Abstract: Direct numerical solutions of the three-dimensional time-dependent Navier-Stokes equations are presented for the evolution of three-dimensional finite-amplitude disturbances of plane Poiseuille and plane Couette flows. Spectral methods using Fourier series and Chebyshev polynomial series are used. It is found that plane Poiseuille flow can sustain neutrally stable two-dimensional finite-amplitude disturbances at Reynolds numbers larger than about 2800. No neutrally stable two-dimensional finite-amplitude disturbances of plane Couette flow were found.Three-dimensional disturbances are shown to have a strongly destabilizing effect. It is shown that finite-amplitude disturbances can drive transition to turbulence in both plane Poiseuille flow and plane Couette flow at Reynolds numbers of order 1000. Details of the resulting flow fields are presented. It is also shown that plane Poiseuille flow cannot sustain turbulence at Reynolds numbers below about 500.

Numerical Simulation of Steady Supersonic Viscous Flow

Abstract: A noniterative, implicit, space-marching, finite-difference algorithm is developed for the steady thin-layer Navier-Stokes equations in conservation-law-form. The numerical algorithm is applicable to steady supersonic viscous flow over bodies of arbitrary shape. In addition, the same code can be used to compute supersonic inviscid flow or three-dimensional boundary layers. Computed results from two-dimensional and three-dimensional versions of the numerical algorithm are in good agreement with those obtained from more costly time-marching techniques.

Shock-wave structure via nonequilibrium molecular dynamics and Navier-Stokes continuum mechanics

Abstract: A strong steady dense-fluid shock wave is simulated with 4800-atom nonequilibrium molecular dynamics. The resulting density, stress, energy, and temperature profiles are compared with corresponding macroscopic profiles we derive from Navier-Stokes continuum mechanics. The differences found are relatively small.

The Navier-Stokes equations on a bounded domain

Abstract: SupposeU is an open bounded subset of 3-space such that the boundary ofU has Lebesgue measure zero. Then for any initial condition with finite kinetic energy we can find a global (i.e. for all time) weak solutionu to the time dependent Navier-Stokes equations of incompressible fluid flow inU such that the curl ofu is continuous outside a locally closed set whose 5/3 dimensional Hausdorff measure is finite.

Analyses of Pressure Oscillations in an Open Cavity

Abstract: Open cavities an aircraft exposed to high-speed flow, such as weapon bays, can give rise to intense selfinduced pressure oscillations. The amplitude of these oscillations, under certain flight conditions, can cause structural damage. Substantial experimental and analytical efforts have investigated these pressure fluctuations, resulting in some understanding of the complex interaction of the external shear layer and cavity acoustical disturbances. However, no numerical computations have been obtained for the complete governing fluid mechanical equations. The purpose of this study is to obtain numerical solutions of the Navier-Stokes equations for an open cavity in order to provide a new tool for the analysis of this phenomenon. '

Darcy's Law for Flow in Porous Media and the Two-Space Method,

Abstract: A common problem in engineering and science is to derive simple equations governing complicated phenomena. Often complicated governing equations are known, but they are too difficult to analyze. An example of a simplified equation is Darcy’s law, which describes flow of a viscous fluid through a porous medium. The more complicated equation for the same phenomenon is the Navier-Stokes equation. As an example of a general method for simplifying equations, This chapter shows how to derive Darcy’s law from the Navier-Stokes equation. Simplified equations are often called “homogenized equations,” and the procedure of replacing the original equations by them is often called “homogenization.” The chapter discusses the two-space method for deriving simplified equations by using an example of the flow of a compressible viscous fluid through a rigid porous medium.

Navier-Stokes Solutions for Spin-Up in a Filled Cylinder

Abstract: A predictor-corr ector multiple-iteration scheme is adapted and used to solve the unsteady Navier-Stokes equations. Numerical solutions for Reynolds numbers up to 50,000 are obtained for the transient spin-up flow in a cylindrical container. The grid point distribution is optimized using coordinate transformations to resolve simultaneously details of both the interior and endwall/sidewall boundary-layer flows formed during spin-up; Calculations for five test problems show very good agreement with previous computations and experimental measurements. Transient phenomena occurring at early time near the sidewall, including reversed flow regions and inertial oscillations, are discussed as well as certain aspects of the endwall Ekman layer flow.

Nonlinear stability of parallel flows by high-order amplitude expansions

Abstract: * * The Landau-Stuart theory and its subsequent modifications suffer from some restrictions and from the nonuniqueness in determining higher-order terms of the amplitude expansions, which limit the range of applicability as well as the validity of the results. In the present paper, a well-defined amplitude is introduced a priori. In this way, uniqueness for terms of any order is achieved. Moreover, Watson's method is no longer restricted to almost neutral disturbances. This offers not only more accurate approximations and numerical studies on convergence but, as a consequence, a whole series of new applications. As a first example, the nonlinear equilibrium states of the plane Poiseuille flow are investigated. The numerical results are discussed in context with the author's solutions of the nonlinear equations and with special emphasis on the convergence of Landau's series. f

A semi-implicit calculation procedure for flows described in boundary-fitted coordinate systems

Abstract: A semi-implicit numerical procedure is described for solving the Navier-Stokes equations in boundary-fitted coordinate systems. The procedure solves the steady-state equations directly without marching in time, and in the current study it is applied to flows with a predominant flow direction. Two illustrative flow situations have been analyzed and are reported. The required computing times are modest, and are less than those for fully explicit schemes, when the steady-state behavior is of prime concern.

Steady solutions of the Navier-Stokes equations representing plane flow in channels of various types

Abstract: 5. l~ d e c a y a t in f in i ty for channe l s of t y p e I ' 5.1. Channels of t y p e I ' . . . . . . . . . . . . . . . . . . . . 109 5.2. E s t i m a t e s in a b o u n d a r y n e i g h b o u r h o o d . . . . . . . . . . . 111 5.3. E s t i m a t e s of t he vo r t i c i t y . . . . . . . . . . . . . . . . . 114 5.4. Averages over arcs o f p a n d ]p] . . . . . . . . . . . . . . 117 5.5. Po in twise decay of t he pressure a n d ve loc i ty . . . . . . . . . 120

Numerical solutions to the partially parabolized navier-stokes equations for developing flow in a channel

Abstract: The partially parabolized Navier-Stokes equations are a set of approximate governing equations that are applicable to flows possessing a predominant flow direction and are obtained when terms representing viscous diffusion of momentum in the main flow direction are dropped from the full Navier-Stokes equations. These reduced equations differ from Prandtl's boundary-layer equations in that no further simplifying assumptions have been made regarding the transverse pressure gradients and transverse momentum equations. Hence, transmission of influences through pressure can be significant even against the main flow direction. The present study was motivated by an interest in evaluating the merits of this reduced flow model for conditions where the boundary-layer equations are known to be inadequate. The principle features of a numerical scheme used to solve the partially parabolized Navier-Stokes equations are described. Results for the flow in the inlet region of a channel are compared with solutions to the f...

Computations of the Magnus effect for slender bodies in supersonic flow

Abstract: A recently reported Parabolized Navier-Stokes code has been employed to compute the supersonic flow field about spinning cone, ogive-cylinder, and boattailed bodies of revolution at moderate incidence. The computations were performed for flow conditions where extensive measurements for wall pressure, boundary layer velocity profiles and Magnus force had been obtained. Comparisons between the computational results and experiment indicate excellent agreement for angles of attack up to six degrees. The comparisons for Magnus effects show that the code accurately predicts the effects of body shape and Mach number for the selected models for Mach numbers in the range of 2-4.

The steady state Stefan problem with convection

Abstract: For steady-state Stefan problems with convection in the fluid phase governed by either the Stokes equations or the Navier Stokes equations, and with adherence of the fluid on all boundaries, the existence of a weak solution is obtained via the introduction of a temperature dependent penalty term in the fluid flow equation together with application of various compactness arguments.

Flow in Streamwise Corners of Arbitrary Angle

Abstract: A formulation of the problem posed by the flow along stream wise corners is presented. The corners are imagined to be formed by the abutment of two quarter-infinite planes along a side edge parallel to a uniform stream, and the treatment is general to any angle between the planes except the limiting angles 0 and 360 deg. The secondary shear layer, which exists in streamwise corner flows, is shown to exhibit a simple antisymmetric behavior for angles equidistant from 180 deg. This is explained in terms of the matching requirement between the potential flow and the outflow from the quasi-two-dimensional boundary layers adjacent to the corner layer. Numerical results are given in the form of velocity and wall shear stress distributions for corners of different angle. Agreement between these new results and available solutions for rectangular corners is excellent. I. Introduction T HE flow in a rectangular streamwise corner formed by two semi-infinite rigid planes with coplanar leading edges has been treated in detail by Rubin et al. 1-3 More recently, Desai and Mangier4 have published a method for dealing with corners of arbitrary angle, and Ghia5 has reconsidered the rectangular corner. The solutions by Rubin and Ghia are in close agreement in most respects, but differ significantly from Desai and Mangier's results for the rectangular corner, particularly with regard to the secondary flow or crossflow. A new formulation of the problem posed by the corner of arbitrary angle is presented herein, and the result is a set of equations that, when specialized to meet the rectangular corner situation, coincide with those obtained by Rubin. 1 Numerical results are given for the velocity and wall shear stress distributions for corner included angles ranging from 90 to 270 deg. The numerical method of solution adopted is rather similar to that used by Ghia, and the results for a rectangular corner are in excellent agreement with Ghia's solution for that case. II. Analysis A. Equations of Motion The corner is formed by the abutment of two quarterinfinite planes joined along a side edge which is parallel to an infinite uniform stream of velocity U. A coordinate system suitable for the present purpose is the x*,x2,x3 system indicated in Fig. 1. The origin is in the symmetry plane at the leading edge. The x1 axis is directed downstream. The x2 and x3 axes are normal to x1 and lie, respectively, in the symmetry plane and in one of the joining quarter-infinite planes in order to form a right-handed oblique reference frame. The disturbance caused by the corner to the otherwise uniform stream will result in a flow velocity vector which is a function of position and whose physical components in the x*,x2,x3 directions we will denote by t;(l), v(2)9 v(3), respectively. We will omit the routing but laborious procedure which shows that the governing equations, namely the Navier-Stokes and mass continuity equations, for steady incompressible flow can be written as

Thin-layer approximation for three-dimensional supersonic corner flows

Abstract: The thin-layer approximation is extended to an axial corner that is formed by the intersection of two perpendicular plates, one of which has an inclination angle with respect to the free stream. A computer code developed by Hung and MacCormack (1978) is modified for the thin-layer approximation, and a case with Mach 5.9 and a wedge angle of 6 deg is computed. In addition, it is shown that it is not necessary to solve the complete Navier-Stokes equations for a three-dimensional high-Reynolds-number corner flow.

Classical solutions of the Navier-Stokes equations

Abstract: The simplest, most elementary proofs of the existence of solutions of the Navier-Stokes equations are given via Galerkin approximation. The core of such proofs lies in obtaining estimates for the approximations from which one can infer their convergence (or at least the convergence of a subsequence of the approximations) as well as some degree of regularity of the resulting solution. The first to use this approach was Hopf [ 5 ], who based an existence theorem for the initial boundary value problem on an energy estimate for Galerkin approximations. However, based on this single estimate, Hopf's theorem provides very little regularity of the solution, in fact, insufficient regularity to prove the solution's uniqueness if the domain is three-dimensional. To remedy this situation, Kiselev and Ladyzhenskaya [ 7 ] introduced a second estimate for the approximations which yields enough further regularity for a uniqueness theorem. As is well known, this second estimate holds only locally in time unless the data are small or the domain is two-dimensional, a circumstance which has stimulated much speculation over the question of "unique solvability in the large". On the other hand, even during the time interval for which it holds, the estimate of Kiselev and Ladyzhenskaya provides far less than the full classical regularity of the solution.

The asymptotic behaviour of solutions of the Navier-Stokes equations near sharp corners

Abstract: When the boundary of a fluid domain has any sharp corners, an understanding of the asymptotic structure of the flow near the corners is invariably helpful as a preliminary to determining the overall flow structure, and as a means of checking subsequent numerical calculations. The simplest example is that of Poiseuille flow along a sharp corner; a less simple example is the low Reynolds number flow between two hinged plates in relative angular motion. These problems are discussed here in §§1 and 2 (full details are given in Moffatt & Duffy 1979) and it is shown that the local similarity solution is not invariably asymptotically valid near the corner, but that it may (for certain ranges of corner angle) be dominated by ‘eigenfunction’ contributions which are sensitive to conditions remote from the corner. For critical angles, a transitional behaviour occurs.

A finite element approximation of Navier-Stokes equations for incompressible viscous fluids. Iterative methods of solution

Abstract: We present in this paper a method for the numerical solution of the steady and unsteady Navier-Stokes equations for incompressible viscous fluids. This method is based on the following techniques: • A mixed finite element approximation acting on a pressure-velocity formulation of the problem, • A time discretization by finite differences for the unsteady problem, • An iterative method using — via a convenient nonlinear least square formulation — a conjugate gradient algorithm with scaling; the scaling makes a fundamental use of an efficient Stokes solver associated to the above mixed finite element approximation.

A mixed finite element approximation of the Navier-Stokes equations

Abstract: We study in this paper the convergence of a new mixed finite element approximation of the Navier-Stokes equations. This approximation uses low order Lagrange elements, leads to optimal order of convergence for the velocity and the pressure, and induces an efficient numerical algorithm for the solution of this problem.

Navier-Stokes Solution of the Turbulent Flowfieid about an Isolated Airfoil

Abstract: A COMPRESSIBLE time-dependent solution of the Navier-Stokes equations, including a transitionturbulence model, is obtained for the isolated airfoil flowfield problem. The equations are solved by a consistently split linearized block implicit scheme due to Briley and McDonald. A nonorthogonal body-fitted coordinate system is used, which has maximum resolution near the airfoil surface and in the region of the airfoil leading edge. The transitionturbulence model is based upon the turbulence kinetic energy equation and predicts regions of laminar, transitional, and turbulent flow. Mean flowfield and turbulence field results are presented for an NACA 0012 airfoil at zero and nonzero incidence angles at Reynolds number up to one million and low subsonic Mach numbers.