Showing papers on "Navier–Stokes equations published in 1978"

An Implicit Factored Scheme for the Compressible Navier-Stokes Equations

Abstract: An implicit finite-difference scheme is developed for the numerical solution of the compressible Navier-Stokes equations in conservation- law form. The algorithm is second-order- time accurate, noniterative, and spatially factored. In order to obtain an efficient factored algorithm, the spatial cross derivatives are evaluated explicitly. However, the algorithm is unconditional ly stable and, although a three-time-lev el scheme, requires only two time levels of data storage. The algorithm is constructed in a "delta" form (i.e., increments of the conserved variables and fluxes) that provides a direct derivation of the scheme and leads to an efficient computational algorithm. In addition, the delta form has the advantageous property of a steady state (if one exists) independent of the size of the time step. Numerical results are presented for a two-dimensiona l shock boundary-layer interaction problem.

Implicit Finite-Difference Simulation of Flow about Arbitrary Two-Dimensional Geometries

Abstract: Finite-difference procedures are used to solve either the Euler equations or the "thin-layer" Navier-Stokes equations subject to arbitrary boundary conditions. An automatic grid generation program is employed, and because an implicit finite-difference algorithm for the flow equations is used, time steps are not severely limited when grid points are finely distributed. Computational efficiency and compatibility to vectorized computer processors is maintained by use of approximate factorization techniques. Computed results for both inviscid and viscous flow about airfoils are described and compared to viscous known solutions.

Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier-Stokes equations

Abstract: We consider the problem of finding the restrictions on the domain Ω⊂R n,n=2,3, under which the space of the solenoidal vector fields from coincides with the space , the closure in W 2 1 (Ω) of the set of all solenoidal vectors from . We give domains Ω⊂Rn, for which the factor space has a finite nonzero dimension. A similar problem is considered for the spaces of solenoidal vectors with a finite Dirichlet integral. Based on this, one compares two generalized formulations of boundary-value problems for the Stokes and Navier-Stokes systems. The following auxiliary problems are studied: , .

Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral

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Qualitative Properties of Navier-Stokes Equations

Abstract: Publisher Summary This chapter discusses the qualitative properties of Navier–Stokes equations. It is known that for small Reynolds numbers, if a steady excitation is applied to the fluid then there is a unique stable steady state that actually appears for large t ( t →∞). The chapter presents some new properties of the set of steady-state solutions to the Navier–Stokes equations of a viscous incompressible fluid. The chapter provides the description of Navier–Stokes equation and their functional setting. The results of Navier–Stokes equation on generic bifurcations are discussed in the chapter.

Qualitative Properties of Navier-Stokes Equations

Abstract: Publisher Summary This chapter discusses the qualitative properties of Navier–Stokes equations. It is known that for small Reynolds numbers, if a steady excitation is applied to the fluid then there is a unique stable steady state that actually appears for large t ( t →∞). The chapter presents some new properties of the set of steady-state solutions to the Navier–Stokes equations of a viscous incompressible fluid. The chapter provides the description of Navier–Stokes equation and their functional setting. The results of Navier–Stokes equation on generic bifurcations are discussed in the chapter.

Spatial Decay Estimates for the Navier–Stokes Equations with Application to the Problem of Entry Flow

Abstract: The development of velocity profiles in the inlet region of channels or pipes is a classic problem of laminar flow theory which has given rise to an extensive literature. Most previous work on this entry flow problem has involved some degree of simplification either in flow geometry or in the governing equations. Here we treat the flow development within the general framework of the Navier–Stokes equations governing the steady laminar flow of an incompressible viscous fluid in a cylindrical pipe of arbitrary cross-section.The problem to be treated is that of an “end effect” involving comparison between two distinct solutions of the Navier–Stokes equations. Thus we consider the spatial evolution of the difference between the base flow and the fully developed solution. The corresponding velocity difference clearly satisfies a condition of zero net inflow. In this way, we draw an analogy between the issue of concern here and the celebrated “Saint-Venant’s Principle” of elasticity theory involving the effect ...

The Navier-Stokes equations in space dimension four

Abstract: Solutions to the Navier-Stokes equations in four space dimensions are continuous except for a closed set whose three dimensional Hausdorff measure is finite.

Hypersonic Three-Dimensional Viscous Shock-Layer Flows over Blunt Bodies

Abstract: The viscous shock-layer equations have been extended to treat blunt three-dimensional bodies at angle of attack. Numerical solutions have been obtained on sphere-cones at angles of attack up to 38 deg. Comparisons are made with available experimental data, inviscid solutions, and solutions of the parabolized Navier-Stokes equations. The experimental data consisted of heat-transfer distributions, pressure distributions, and drag coefficients in a Mach number range from 10-18, Reynolds numbers of the order 1.3 x 10 4/ft, and a from 0-40 deg. Two cases were compared with the parabolized Navier-Stokes solutions at Mach numbers of 22.77 and 25.81 and altitudes of 180 and 240 kft at angles of attack of 23 and 38 deg, respectively. In general, the shocklayer predictions were in good agreement with the available experimental and numerical data, but the parabolic treatment of the crossflow viscous shock-layer equations prevented solutions on the leeward side of long bodies at large angles of attack. cp C'

Theory of turbulence in a homogeneous fluid induced by an oscillating grid

Abstract: A theoretical model of steady turbulence caused by an oscillating grid in a box of homogeneous fluid agrees fairly well with observations of root‐mean‐square velocity and integral length scale. In the unsteady case the depth of the turbulent layer increases as the square root of time.

Development of a predictive tool for in-cylinder gas motion in engines

Abstract: A method is described of calculating the flow, temperature, and turbulence fields in cylinder configurations typical of a direct-injection diesel engine. The method operates by solving numerically the Navier Stokes equations that govern the flow, together with additional equations representing the effects of turbulence. A general curvilinear-orthogonal grid that translates with the piston motion is used for the calculations in the complex-shaped piston bowl, while an expanding/contracting grid is used elsewhere. Predictions are presented showing the evolution of the velocity and turbulence fields during the compression and expansion phases of a motored engine cycle, for various shapes of axisymmetric piston bowl and various initial swirl levels. The method is capable of solving the conservation equations governing in-cylinder flow and heat transfer in typical diesel and stratified-charge configurations with swirl under monitoring conditions, but the accuracy of the solutions remains to be assessed. The presence of a bowl provokes, even in the absence of swirl, a complex flow pattern containing at least two strong vortices, one being formed during the approach to top dead center by the squish phenomenon and the other in the clearance gap by reverse squish during the initial descent of the piston. The evolution of the turbulence intensity distribution can be explained largely in terms of the flow behavior. Changes in piston bowl shape and swirl level may profoundly alter the flow structures.

On the stream function‐vorticity finite element solutions of Navier‐Stokes equations

Abstract: The stream function-vorticity method for solving Navier-Stokes equations with finite elements is often disregarded in view of the necessity of an iterative proceudre for satisfying the boundary conditions. The present work shows briefly that such iterations are unnecessary. Sample solutions for flows in a square cavity and in a channel with a step show good agreement with existing solutions.

Finite-element solution procedures for solving the incompressible, Navier-Stokes equations using equal order variable interpolation

Abstract: The conventional finite-element formulation of the equations of motion (written in pressure-velocity variables) requires that the order of interpolation for pressure be one less than that used for the velocity components. This constraint is inconvenient and can be argued to be physically inconsistent when inertial effects are dominant. The origins of the constraint are discussed and three new finite-element formulations are advanced that permit equal order representation of pressure and velocity. Of these, the velocity correction scheme, similar to that commonly used in finite-difference procedures, offers superior performance for the examples examined in this paper.

Plane entry flows and energy estimates for the navier-stokes equations

Abstract: One of the classic problems of laminar flow theory is the development of velocity profiles in the inlet regions of channels or pipes. Such entry flow problems have been investigated extensively, usually by approximate techniques. In a recent paper [4], Horgan & Wheeler have provided an alternative approach, based on an energy method for the stationary Navier-Stokes equations. In [4], concerned with laminar flow in a cylindrical pipe of arbitrary cross-section, an analogy is drawn between the end effect issue of concern here, called the “end effect”, and the celebrated “Saint-Venant's Principle” of the theory of elasticity.

On Some Problems Connected with Navier Stokes Equations

Abstract: Publisher Summary This chapter presents some results on the flows of nonhomogeneous fluids and on the flows of fluids in media with many obstacles in a periodic structure. This chapter presents a simple model, which, although an oversimplication of real problems, is of some physical interest and leads to many interesting mathematical and numerical questions. It highlights a semi-Galerkin's approximation (i.e. a Galerkin's approximation on the velocity, an infinite dimensional approximation on the density), and a priori estimates of Kajikov; these a priori estimates give results on some sort of fractional derivative in time of the velocity (they are weaker than those obtained by Fourier transform in t but the Fourier transform method seems to fail in the present situation). Similar estimates have been used in the context of variational inequalities of evolution by H. Brezis.

Simultaneous Solution of the Boundary Layer and Freestream with Separated Flow

Abstract: A general procedure for calculating the boundary layer simuifaneousry with the outer, inviscid flow is described Integra! equations for ihe boundary layer and finite-difference equations for the inviscid flow at a given longitudinal position form a linear set with a tridiagonai coefficient matrix The set is solved simultaneously ai each position, beginning at the upstream boundary and iterating over Ihe flowfield (successive line relaxation) Convergence of ihe procedure with separated flow is demonstrated by a numerical example

The Mathematics of Finite Elements and Applications II

Abstract: : This paper describes theory and methods for developing a posteriori error estimates and an adaptive strategy for hp-finite element approximations of the incompressible Navier-Stokes equations. For an error estimation, use is made of a new approach which is based on the work of Ainsworth, Wu and the author. That theory has been shown to produce good results for general elliptic systems and general hp-finite element method% Recently, these techniques have been extended to the Navier-Stokes equations A three-step adaptive algorithm is also described which produces reasonable hp meshes very efficiently. These techniques are applied to representative transient and steady state problems of incompressible viscous flow.

Error estimates for MAC-like approximations to the linear Navier-Stokes equations

Abstract: In this paper a priori error estimates are derived for the discretization error which results when the linear Navier-Stokes equations are solved by a method which closely resembles the MAC-method of Harlow and Welch. General boundary conditions are permitted and the estimates are in terms of the discreteL 2 norm. A solvability result is given which also applies to a generalization of the method to the nonlinear case. This generalization is used in the last section to produce a numerical solution to the problem of flow around an obstacle.

Viscous Compressible Flow in the Boundary Region of an Axial Corner

Abstract: Laminar flow along a 90-deg corner, formed by the intersection of two semi-infinite flat plates, is analyzed, under the assumption of shock-free supersonic flow. The analysis is based on the method of matched asymptotic expansions. Numerical solutions are obtained for the streamwise self-similar corner flow, using the ADI method. Appropriate mapping functions are employed, so that the far-field boundary conditions are imposed at true infinity. Results are presented for subsonic and supersonic flows, with adiabatic as well as heat-transfer wall conditions. The effects of location of the far-field boundary and of the model-fluid assumption are assessed accurately. Effects of Prandtl number, viscosity law, and level of freestream temperature on the solutions are also studied. The numerical method developed is efficient and the optimization techniques implemented have wider applications. A numerical "compressibility-transformation" is also suggested for obtaining a suitable finite-difference grid for highly supersonic flows.

Stability analysis of some finite difference schemes for the Navier-Stokes equations

Abstract: The stability analysis of the hyperbolic and parabolic explicit difference schemes presented in our previous paper 1 is improved on. Analysis of the Fourier components gives rise to stability limits that are very near the actual numerical stability limits.

The numerical solution of the Navier-Stokes equations for a three-dimensional laminar flow in curved pipes using finite- difference methods

Abstract: The numerical method presented treats the primitive-variable form of the Navier-Stokes equations. It is shown how to treat the generalised orthogonal coordinate form of the equations in order to retain the numerical stability of the linearised equations when these are approximated by finite differences. A property analogous to diagonal dominance in more simple systems is shown to exist for the complete set of difference approximations to the flow equations so that the matrix of the finite-difference equations has all of its eigen values in the left-hand half-plane. It follows that the linearized equations are unconditionally stable. An entirely new difference scheme for the continuity equation is derived and shown to be superior to the more commonly used “central-difference” approximations for the high-Reynolds-number flow considered. The total “package” is tested against experiment on a shear flow through a 90° rectangular bend. The experimental measurements are of total-pressure distributions, and these indicate the presence of a strong secondary flow. The computed results give a close agreement to the experimental results.