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Showing papers on "Navier–Stokes equations published in 1975"


Journal ArticleDOI
TL;DR: In this article, the equations of fluid motion have been formulated in a generalized noncartesian, nonorthogonal coordinate system and a particular coordinate transformation which transforms a domain with an irregular lower boundary into a cube has been constructed.

645 citations


ReportDOI
01 Jan 1975
TL;DR: In this article, a finite difference technique is presented for solving the Navier-Stokes equations for an incompressible fluid, based on the Marker-and-Cell method, to facilitate its use by persons with little or no experience in numerical fluid dynamics.
Abstract: : A finite difference technique is presented for solving the Navier-Stokes equations for an incompressible fluid. The technique, based on the Marker-and-Cell method, is simplified to facilitate its use by persons with little or no experience in numerical fluid dynamics. Section I of the report describes the basic algorithm, SOLA, for confined flows; Sec. II describes modifications necessary for free or curved rigid surface boundaries. Each includes a flow chart and a FORTRAN listing. Sample problems show how to incorporate simple modifications into the basic code to adapt it to a variety of problems.

541 citations


Proceedings ArticleDOI
01 Jan 1975
TL;DR: In this article, a numerical method for solving the compressible form of the unsteady Navier-Stokes equations is described, with emphasis on the choice of a computational mesh for high Reynolds number flows, finitedifference approximations for mixed partial derivatives, extension of the Courant-Friedrichs-Lewy stability condition for viscous flows, mesh boundary conditions, and numerical smoothing for strong shock-wave calculations.
Abstract: A numerical method for solving the compressible form of the unsteady Navier-Stokes equations is described. This method was originally presented in 1970 and has since been modified during the development of computer programs at Ames for implementing models that account for the effects of turbulence in shock-induced separated flows. Although this paper does not describe the turbulence models themselves, a complete description of the basic numerical method is given with emphasis on the choice of a computational mesh for high Reynolds number flows, finite-difference approximations for mixed partial derivatives, extension of the Courant-Friedrichs-Lewy stability condition for viscous flows, mesh boundary conditions, and numerical smoothing for strong shock-wave calculations.

252 citations



Book ChapterDOI
01 Jan 1975

134 citations


Journal ArticleDOI
TL;DR: The analytical structure of the hydrodynamic frequencies for the sound, heat, and shear modes and of hydrodynamics equations for a monatomic fluid are discussed in this article.
Abstract: The analytic structure of the hydrodynamic frequenciesz(k) for the sound, heat, and shear modes and of the hydrodynamic equations for a monatomic fluid are discussed on the basis of the mode-mode coupling theory. It is shown that the hydrodynamic frequencies depend on the wave numberk, for smallk, asz(k) = ak + bk 2 + $$z(k) = ak + bk^2 + \sum olimits_{n = 1}^\infty {c_n k^{3 - 2^{ - n} } } $$ and that some of the correlation functions that appear in the Fourier-Laplace transforms of the hydrodynamic equations contain branch point singularities. The implications of these results for the derivation of linear hydrodynamic equations, such as the Burnett equations, and for the long-time behavior of time correlation functions are discussed.

107 citations


01 Sep 1975
TL;DR: In this paper, problems relating to the computation of viscous compressible flows based on numerical solutions of the Navier-Stokes equations are reviewed and a discussion of their interest in aerodynamic problems are presented.
Abstract: : Problems relating to the computation of viscous compressible flows based on numerical solutions of the Navier-Stokes equations are reviewed. A general introduction to the Navier-Stokes equations and a discussion of their interest in aerodynamic problems are first presented. Then the following aspects of numerical methods are considered: limitation of the computational domain and boundary conditions on the outer boundary; various approaches in finite difference methods and description of some representative schemes; treatment of boundary conditions at a solid wall; treatment of shock waves, and general considerations on accuracy and computing times. Finally reported computations of two-dimensional or three-dimensional flows are presented in table form with summary indications on the problems treated and the methods used.

106 citations


Book ChapterDOI
TL;DR: In this article, the authors describe the simulations of turbulent shear flows by direct numerical solution of the three-dimensional Navier-Stokes equations, the dynamical equations, and boundary conditions employed with particular emphasis on the momentum less wake model.
Abstract: Publisher Summary This chapter describes the simulations of turbulent shear flows by direct numerical solution of the three-dimensional Navier-Stokes equations, the dynamical equations, and boundary conditions employed with particular emphasis on the momentum less wake model The numerical computation approach provides several advantages over more conventional approaches—the complete flow field is obtained at all times so that detailed flow characteristics may be obtained that would be difficult to measure in the laboratory, and the initial conditions can be accurately controlled so that their effect may be determined The technique of imposing the initial conditions allows arbitrary mean velocity profile, turbulence intensity profile, and local turbulence energy spectrum The present simulations of turbulent shear flows are a first step toward proper understanding of the basic mechanisms and dynamics of these flows Detailed comparisons and tests are presently underway between various turbulence modeling hypotheses, laboratory experiments, and the present simulations As time and the art of numerical simulation progress, simulations like the present ones should be expected to fulfill more and more need of a laboratory workhorse

98 citations


Journal ArticleDOI
TL;DR: In this paper, a modified eddy viscosity model is incorporated into the compressible Navier-Stokes equations to reproduce the response of turbulence to a severe pressure gradient in the flowfield.
Abstract: A modified eddy viscosity model is incorporated into the compressible Navier-Stokes equations. The modification attempts to reproduce the response of turbulence to a severe pressure gradient in the flowfield. This relaxation phenomenon is described by an exponential decay of the unperturbed eddy viscosity coefficient downstream of the perturbation in terms of a prescribed length scale. The system of equations is solved by MacCormack's time-splitting explicit numerical scheme for a series of compression corner configurations. Computations are performed for ramp angles varying from 15 to 25° at a Mach number of 2.96 and a Reynolds number of 10 7. Calculations utilizing the modified eddy viscosity for the interacting turbulent flow compare very well with experimental measurements, particularly in the prediction of the upstream pressure propagation and location of the separation and the reattachment points. Good agreement is also attained between the measured and calculated density profiles in the viscous-inviscid interaction region.

81 citations


Journal ArticleDOI
TL;DR: In this paper, a finite difference scheme for solving the equations of fluid motion in a generalized coordinate system has been constructed, which conserves mass and all the first integral moments of the motion.

68 citations


Journal ArticleDOI
TL;DR: In this article, a Cartesian orthogonal frame is introduced for viscous incompressible liquid to occupy a volume of three-dimensional Euclidean space E3, where the scale may be chosen in such a way that the density is equal to unity.
Abstract: Let viscous incompressible liquid occupy a volume ~ of three-dimensional Euclidean space E3. Introducing a Cartesian orthogonal frame xl, x2, x3 in E3, we denote a point (xl, x2, x3) by x. Let us denote the velocity and pressure of the liquid at the point x and at the instant t by v(x, t) and p(x, t), respectively, v~(x, t), i-1, 2, 3, being the projections of the velocity on axes x~. The scale may be chosen in such a way that the density is equal to unity. Then according to the Navier-Stokes theory the motion of the liquid caused by external forces f(x, t) will be described by the following system of four equations : 3 v~-vAv+ ~ vkv~,~ = -gradp+f, (1) k=l


Journal ArticleDOI
TL;DR: In this paper, the influence of wall slip and catalytic atom-recombination on the flow field and wall heat flux are calculated for high-altitude flight and arcjet-flow conditions.
Abstract: The influence of wall slip and catalytic atom-recombination on the flowfield and wall heat flux are calculated for high-altitude flight and arcjet-flow conditions. Boundary equations, which include velocity slip, temperature jump, and wall catalytic atom-recombination, are coupled to the viscous reacting multicomponent NavierStokes equation. These equations are solved using a time-dependent finite difference technique applied to spheres in an arcjet flow (Reynolds number of 550) and a high-altitude flight case representative of the space shuttle orbiter (Reynolds number of 450). The results indicate that catalysis strongly influences the temperature jump, but not the velocity slip. Slip increases the atom fraction and temperature at both the wall and in the flowfield. Likewise, the shock stand-off distance, the wall heat flux, and friction coefficient are increased over the nonslip cases. The reacting gas calculations confirm the chemically frozen nature of the shock layer in arcjet flows. design the reusable space shuttle orbiter for highaltitude, low Reynolds number, atmospheric entry necessitates a more comprehensive treatment and understanding of the interaction between a high enthalpy gas flow and a relatively cold surface. At low densities the continuum model of the gas breaks down in regions of large gradients such as those near a cold body. Corrections to the equations for the boundary conditions are then required for the flow. These influences are reflected in the calculations of surface heat-transfer rate and chemical composition of the flow near the wall. The aim of this paper is to quantify how these low density phenomena interact, how they influence interpretation of test data obtained on thermal protection systems, and how they alter the predictions of heating rates and performance of the space shuttle orbiter thermal protection system (TPS) during its long high-altitude entry. The approach taken here is to obtain finite difference solutions to the reacting Navier-Stokes equations for the flow around spheres in both space shuttle flight and arcjet environments. The wall boundary conditions for these solutions are obtained from slip and jump relations for a nonequilibrium multicomponent gas mixture. They include the effects of catalytic atom recombination. From the solutions one can assess the effects. of the boundary conditions on the flow properties, the heat flux, etc. At low densities, the continuum-flow equations are no longer adequate close to the wall because the mean free path becomes long compared to characteristi c lengths associated with significant changes in macroscopic-flow parameters. The flow in a region next to the wall having a thickness on the order of a mean free path (the Knudsen layer) cannot be described by the Navier-Stokes description (Kogan1). To determine the flow properties within the Knudsen layer requires the direct solution of the Boltzmann equation matched to the solutions for the outer flow (Navier-Stokes equation) and the wall boundary condition. This is most conveniently done through the use of a slip model in which slip and jump properties are used for the boundary conditions for the Navier-Stokes equations.

Journal ArticleDOI
TL;DR: In this paper, three iterative methods for numerically solving the steady-state Navier-Stokes equations are presented, i.e., the Laplacian Driver (LAD), the Numerical Oseen (NOS), and the Split NOS method.

Journal ArticleDOI
TL;DR: In this article, a method for the solution of transient, incompressible viscous flow in two dimensions was described, where dependent variables, stream function and vorticity, were approximated over each triangular element using linear interpolation functions.

01 Jan 1975
TL;DR: In this article, the authors proposed a method for numerical solution of the Navier-Stokes equations that can treat the unsteady laminar flow about bodies of arbitrary shape, such as two-dimensional airfoils, multiple air foils, and submerged hydrofoils.
Abstract: A procedure for numerical solution of the time-dependent, two-dimensional incompressible Navier-Stokes equations that can treat the unsteady laminar flow about bodies of arbitrary shape, such as two-dimensional airfoils, multiple airfoils, and submerged hydrofoils, as naturally as it can deal with the flow about simple bodies. The solution is based on a method of automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multiconnected region containing any number of arbitrarily shaped bodies. The curvilinear coordinates are generated as the solution of two elliptical partial differential equations with Dirichlet boundary conditions, one coordinate being specified to be constant on each of the boundaries, and a distribution of the other being specified along the boundaries. The solution compares excellently with the Blasius boundary layer solution for the flow past a semiinfinite flat plate.

Journal ArticleDOI
TL;DR: In this article, the results of aerodynamic studies of highly turbulent flows produced in the downstream region of an abrupt change in flow area were derived from the autocorrelation function of light scattered in laser velocimeter experiments, and compared with numerical solutions of the Navier Stokes equations incorporating a two-equation model.
Abstract: The paper gives the results of aerodynamic studies of highly turbulent flows produced in the downstream region of an abrupt change in flow area. Information has been deduced from the autocorrelation function of light scattered in laser velocimeter experiments, and is compared with numerical solutions of the Navier Stokes equations incorporating a two-equation model for turbulence.

Journal ArticleDOI
TL;DR: A mathematical model for the flow development upstream of the region of air entrainment on a spillway is presented in this paper, which is based on numerical solutions of the time-averaged two-dimensional form of the Navier-Stokes equations.
Abstract: A mathematical model for the flow development upstream of the region of air entrainment on a spillway is presented. The model is based on numerical solutions of the time-averaged two-dimensional form of the Navier-Stokes equations. Predictions from the model of mean velocity distributions, boundary layer growth, and water surface profiles are compared with experimental data, both model and prototype. Excellent agreement is obtained. Furthermore, it is shown conclusively that self-aeration in a spillway flow commences at the point where the bed induced turbulence reaches the free surface.

Journal ArticleDOI
TL;DR: The Biharmonic Driver (BID) method as mentioned in this paper uses direct (non-iterative) linear biharmonic solvers to obtain solutions to the nonlinear two-dimensional steady-state incompressible Navier-Stokes equations.




Journal ArticleDOI
TL;DR: In this article, a forward-marching procedure for separated boundary-layer flows which permits the rapid and accurate solution of flows of limited extent is presented, where the streamwise convection of vorticity in the reversed flow region is neglected, and this approximation is incorporated into a previously developed (Carter, 1974) inverse boundary-layered procedure.
Abstract: A forward-marching procedure for separated boundary-layer flows which permits the rapid and accurate solution of flows of limited extent is presented. The streamwise convection of vorticity in the reversed flow region is neglected, and this approximation is incorporated into a previously developed (Carter, 1974) inverse boundary-layer procedure. The equations are solved by the Crank-Nicolson finite-difference scheme in which column iteration is carried out at each streamwise station. Instabilities encountered in the column iterations are removed by introducing timelike terms in the finite-difference equations. This provides both unconditional diagonal dominance and a column iterative scheme, found to be stable using the von Neumann stability analysis.

Book ChapterDOI
Sin-I Cheng1
01 Jan 1975
TL;DR: In this paper, the mathematical foundation and various practical aspects of the numerical solution of gas dynamic equations are critically reviewed with emphasis on obtaining quantitatively accurate solutions for application in various engineering and sciences.
Abstract: Abstract : The mathematical foundation and the various practical aspects of the numerical solution of gas dynamic equations are critically reviewed with emphasis on obtaining quantitatively accurate solutions for application in various engineering and sciences. Computational stability rate of convergence and accuracy (or error estimate) are discussed. The promises and problems of the 4th generation computers are outlined within this perspective. Computational stability shoud not be obtained at the sacrifice of the convergence rate to and the accuracy of the final solution. With accuracy in mind, the explicit algorithms are likely preferrable to the implicit ones. Strict conservation of the difference formulation is recommended and exemplified to avoid the accumulation of local truncation errors and to facilitate the estimate of the errors in a steady state solution. Illustrative examples are given including supersonic flows with shocks. (Author)



01 Jan 1975
TL;DR: In this paper, a numerical solution of the Navier-Stokes equations for the flow about arbitrary airfoils or other bodies is presented using a numerically generated curvilinear coordinate system.
Abstract: A method for numerical solution of the Navier-Stokes equations for the flow about arbitrary airfoils or other bodies is presented. This method utilizes a numerically generated curvilinear coordinate system having a coordinate line coincident with the body contour. Streamlines, velocity profiles, and pressure and force coefficients for several airfoils and an arbitrary rock are given. Potential flow solutions are also presented. The procedure capable of treating multiple-element airfoils, and potential flow results are presented.

Book ChapterDOI
01 Jan 1975
TL;DR: An interesting oscillatory flow field over a concave shape, which has not been observed experimentally, has been calculated by numerically solving the unsteady Navier-Stokes equations as mentioned in this paper.
Abstract: An interesting oscillatory flow field over a concave shape, which has not been observed experimentally, has been calculated by numerically solving the unsteady Navier-Stokes equations.The unique characteristic of this flow is that it is not separated and yet is oscillatory in nature.All previously observed oscillatory flow fields contain separated regions. The general characteristics of the flow pattern have been described, together with the hypothesized flow mechanism which produces the oscillation. The flow field appears to be hydrodynamically unstable which results in an oscillatory variation of the flow field downstream of the forebody pressure minimum. Various changes in the afterbody shape do not alter the period of the oscillation and results in minor changes in the magnitude of the pressure oscillation. A number of numerical experiments have been performed and described, the results of which indicate that the oscillation is of physical origin. Further work is continuing to understand these flow fields and the underlying governing flow mechanisms. Experimental verification of this flow phenomenon is presently being pursued by Holden [1974] and a more detailed description of the flow will be forthcoming pending the results of this experimental study.

Journal ArticleDOI
TL;DR: In this article, the authors used the computer to integrate various solutions and solution properties of the sub-flowfields which made up the entire flowfield without resorting to a finite difference solution to the complete Navier-Stokes equations.
Abstract: A method is developed to determine the flowfield of a body of revolution in separated flow. The technique employed is the use of the computer to integrate various solutions and solution properties of the sub-flowfields which make up the entire flowfield without resorting to a finite difference solution to the complete Navier-Stokes equations. The technique entails the use of the unsteady cross flow analogy and a new solution to the required two-dimensional unsteady separated flow problem based upon an unsteady discrete-vorticity wake. Data for the forces and moments on bodies of revolution at high angle of attack (outside the range of linear inviscid theories) such that the flow is substantially separated are produced which compare well with experimental data at low speeds. In addition, three-dimensional steady separation regions and wake vortex patterns are determined.

Journal ArticleDOI
TL;DR: In this article, an asymptotic method is developed for the linearized Navier-Stokes equations governing the time-dependent motion of a viscous fluid flow down an inclined plane.
Abstract: An asymptotic method is developed for the linearized Navier-Stokes equations governing the time-dependent motion of a viscous fluid flow down an inclined plane. A diffusion equation for the first order approximation of the fluid surface elevation in a perturbation scheme is derived and a critical Reynolds number is defined based upon the well-posedness of the equation. Under a set of sufficient conditions it is shown that the solution of the diffusion equation is a uniform asymptotic approximation to the generalized solution of the full equations for all time by means of various $L_2 $ and pointwise estimates.