Showing papers on "Navier–Stokes equations published in 1984"

The three‐dimensional flow due to a stretching flat surface

Abstract: An exact similarity solution of the Navier–Stokes equations is found. The solution represents the three‐dimensional fluid motion caused by the stretching of a flat boundary.

A modified finite element method for solving the time‐dependent, incompressible Navier‐Stokes equations. Part 1: Theory

Abstract: SUMMARY Beginning with the Galerkin finite element method and the simplest appropriate isoparametric element for modelling the Navier-Stokes equations, the spatial approximation is modified in two ways in the interest of cost-effectiveness: the mass matrix is ‘lumped’ and all coefficient matrices are generated via l-point quadrature. After appending an hour-glass correction term to the diffusion matrices, the modified semi-discretized equations are integrated in time using the forward (explicit) Euler method in a special way to compensate for that portion of the time truncation error which is intolerable for advection-dominated flows. The scheme is completed by the introduction of a subcycling strategy that permits less frequent updates of the pressure field with little loss of accuracy. These techniques are described and analysed in some detail, and in Part 2 (Applications), the resulting code is demonstrated on three sample problems: steady flow in a lid-driven cavity at Res10,000, flow past a circular cylinder at Re 5400, and the simulation of a heavy gas release over complex topography.

Determination of the solutions of the Navier-Stokes equations by a set of nodal values

Abstract: We consider the Navier-Stokes equations of a viscous incompresible fluid, and we want to see to what extent these solutions can be determined by a discrete set of nodal values of these solutions. The results presented here are exact results and not approximate ones: we show, in several cases, that the solutions are entirely determined by their values on a discrete set, provided this set contains enough points and these points are sufficiently densely distributed (in a sense described in the article). Two typical results are the following ones; two stationary solutions coincide if they coincide on a set sufficiently dense but finite; similarly if the large time behavior of the solutions to the Navier-Stokes equations is known on an appropriate discrete set, then the large time behavior of the solution itself is totally determined.

Conforming finite element methods for incompressible and nearly incompressible continua

Abstract: : Interest here is in finite element discretizations of problems involving an incompressibility condition. As model problems we consider the Stokes equations for the flow of a viscous, incompressible fluid and the equations of linear plane-strain elasticity for the deformation of an isotropic, nearly incompressible solid. In both cases the incompressibility condition takes the form of a divergence constraint. Although this is the most simple formulation, the proper understanding of how an approximate method satisfies the constraint represents an important step towards the understanding of more complicated situations, involving e.g. the Navier-Stokes equations or the equations of nonlinear elasticity. The finite element methods we study have the property that the approximations to the velocities, respectively to the displacements, are continuous; such methods are generally referred to as conforming.

The Baldwin-Lomax Turbulence Model for Two-Dimensional Shock-Wave/Boundary-Layer Interactions

Abstract: The algebraic turbulent eddy viscosity model of Baldwin and Lomax has been critically examined for the case of two-dimension al (2-D) supersonic compression corner interactions. The flowfields are computed using the Navier-Stokes equations together with three different versions of the Baldwin-Lomax model, including the incorporation of a relaxation technique. The turbulence models are evaluated by a detailed comparison with available experimental data for compression ramp flows over a range of corner angle and Reynolds number. The Baldwin-Lomax outer formulation is found to be unsuitable for separated 2-D supersonic interactions due to the unphysical streamwise variation of the computed length scale in the vicinity of separation. Minor modifications are proposed to partially remedy this difficulty. The use of relaxation provides significant improvement in the flowfield prediction upstream of the corner. However, the relaxation length required is one-tenth of that employed in a previous computational study. AH of the turbulence models tested here fail to simulate the rapid recovery of the boundary layer downstream of reattachment.

Finite Difference Methods for the Stokes and Navier–Stokes Equations

Abstract: This paper presents a new finite difference scheme for the Stokes equations and incompressible Navier–Stokes equations for low Reynolds number. The scheme uses the primitive variable formulation of the equations and is applicable with nonuniform grids and nonrectangular geometries. Several other methods of solving the Navier–Stokes equations are also examined in this paper and some of their strengths and weaknesses are described. Computational results using the new scheme are presented for the Stokes equations for a region with curved boundaries and for a disk with polar coordinates. The results show the method to be second-order accurate.

The spatial stability of a class of similarity solutions

Abstract: The spatial stability of a class of exact similarity solutions of the Navier–Stokes equations whose longitudinal velocity is of the form xf′(y), where x is the streamwise coordinate and f′(y) is a function of the transverse, cross‐streamwise, coordinate y only, is determined. These similarity solutions correspond to the flow in an infinitely long channel or tube whose surface is either uniformly porous or moves with a velocity linear in x. Small perturbations to the streamwise velocity of the form xλg′(y) are assumed, resulting in an eigenvalue problem for λ which is solved numerically. For the porous wall problem, it is shown that similarity solutions in which f′(y) is a monotonic function of y are spatially stable, while those that are not monotonic are spatially unstable. For the accelerating‐wall problem, the interpretation of the stability results is not unambiguous and two interpretations are offered. In one interpretation the conclusions are the same as for the porous problem—monotonic solutions ar...

On the smoothness of the nonlinear spectral manifolds associated to the Navier-Stokes equations

Abstract: On considere les equations de Navier-Stokes dans Ω×]0,∞[ ou Ω est un ouvert borne lisse de R n , n=2, 3 ou le cube ]O,L[ n . On montre que la variete spectrale associee est une variete analytique lisse

Existence of solutions to the nonhomogeneous steady Navier-Stokes equations

Abstract: On considere les equations de Navier-Stokes dans un domaine borne ΩCR 2 :−ν 0 ⊇u+(u.⊇)u=f−⊇p, ⊇.u=0 dans Ω, u=g~ sur ∂Ω. On suppose que ∂Ω est infiniment differentiable, f∈C ∞ (Ω→R 2 ), g~∈C ∞ (∂Ω→R 2 ) et ν 0 >0

The circular cylinder in subsonic and transonic flow

Abstract: This paper provides new results describing compressible fluid flow around a cylinder. The investigation was restricted to subsonic and transonic flow at Reynolds numbers of about 10s. The experiments showed that a strong coupling exists between the flow over a cylinder and the vortex street formed in the near wake. The phenomenon was investigated using high-speed visualization synchronized with unsteady pressure measurements. Various coupling regimes were classified and instantaneous pressure distributions were obtained at different times during the vortex street period. From these elements it was possible to deduce the unsteady force.

On the Large Time Galerkin Approximation of the Navier–Stokes Equations

Abstract: In this article we are interested in the difficult problem of relating the large time behaviour of the Navier–Stokes equations to the large time behaviour of their Galerkin approximation. We restrict ourselves to the simplest situation dealing with stationary solutions and the question is the following one; if the computed approximation of the time dependent equations “seems to converge” to some limit as $t \to \infty $, is the same true for the exact problem and are the limits related? We give here some sufficient conditions (reasonably easy to verify) which guarantee a positive answer to this question.

Stable viscosities and shock profiles for systems of conservation laws

Abstract: : Many equations of mathematical physics take the form of nonlinear hyperbolic systems of conservation laws. With small dissipative effects neglected, typically smooth solutions must develop discontinuities (shocks) in finite time. Re-incorporating dissipation helps select those discontinuities which are physically relevant. For this purpose, many different sorts of dissipation will do; in particular, the physical viscosity is typically degenerate and not convenient. In this paper the author provide an understanding of what high order viscosity terms smooth the physical discontinuities. A natural class of admissible viscosity terms is determined based on a simple linearized stability criterion. In addition, they determine a class of degenerate second order viscosity terms of physical type which are admissible. These results are applied to the equations of compressible fluid dynamics, to determine what conditions ensure the existence of the shock layer with viscosity and heat conduction. This should be of interest to others interested in general equations of state for compressible fluids, such as those investigating phase transitions.

Numerical investigation of unsteady inlet flowfields

Abstract: The flow field within an unsteady, two-dimensional inlet is studied numerically, using a two dimensional Navier Stokes and a one-dimensional inviscid model. Unsteadiness is introduced by varying the outflow pressure boundary condition. The cases considered include outflow pressure variations which were a single pressure pulse, a rapid increase and a sine function. The amplitude of the imposed exit plane pressure disturbance varied between 1 percent and 20 percent of the mean exit pressure. At the higher levels of pressure fluctuation, the viscous flow field results bore little resemblance to the inviscid ones. The viscous solution included such phenomena as shock trains and bifurcating separation pockets. The induced velocity at the outflow plane predicted by the viscous model differs significantly from accoustical theory or small perturbation results.

Influence of surface tension on jet-stripped continuous coating of sheet materials†

Abstract: : Air jets are used as a wiping device to reduce the final thickness in some industrial coating applications. Existing theories allow only a given internal normal stress, i.e. impose a pressure distribution in the coating liquid, due to the jet. In this note, surface tension and tangential stresses are also included. In particular, numerical results are obtained which quantify the extent to which surface tension reduces the wiping effect of the jet. Additional keywords: Navier Stokes equations, Differential equations, Reprints. (Author).

A finite element method for the three-dimensional non-steady Navier-Stokes equations

Abstract: The algorithm for solving the three-dimensional non-steady Navier-Stokes equations by the explicit forward Euler method is shown and the Galerkin finite element formulation is presented. As a numerical example, an entrace flow in a square duct is illustrated.

A time-split finite-volume algorithm for three-dimensional flowfield simulation

Abstract: A general finite-volume algorithm is developed for solving three-dimensional, time-dependent, compressible Navier-Stokes equations for high Reynolds number flows over an arbitrary geometry. This algorithm adapts MacCormack's (1982) explicit-implicit scheme to a time-split, three-dimensional finite-volume concept in a general coordinate system. It is shown that the thin-layer approximation in all three spatial directions significantly reduces the evaluation of viscous terms and allows the algorithm to solve more complicated geometries with all boundaries in two or all three directions. The calculated results using this method are found to be in good agreement with the experimental measurements of a blunt-fin induced shock wave and boundary-layer interaction problems. Observations of the existence of peak pressure, primary horseshoe and secondary vortices, and reversed supersonic zones show that computational fluid dynamics can effectively supplement the wind tunnel tests for aerodynamic design as well as for understanding basic fluid dynamics.

Prediction of cascade flow fields using the averaged Navier-Stokes equations

Abstract: A numerical solution procedure for the ensemble-averaged, compressible, time-dependent Navier-Stokes equations is applied to predict the flow about a cascade of airfoils operating in the transonic flow regime. The equations are solved by the consistently split, linearized block implicit (LBI) method of Briley and McDonald. Boundary conditions are set so as to specify upstream total pressure and downstream static pressure. Turbulence is modeled by a mixing length model. Predictions are made for flow through a compressor cascade configuration. The method yields converged solutions within a relatively small number of time steps ( ≈ 150), which give good comparisons with experimental data.

Prediction of Transonic Separated Flows

Abstract: Johnson et al. (1982) have provided a detailed comparison between a thoroughly documented transonic flow with shock-induced separations and solutions of the flow using the Navier-Stokes equations. According to this comparison, there were several deficiencies in the computations. The present investigation takes into account new experimental data which have been obtained in a larger wind tunnel with the same test model for a wider range of freestream Mach numbers. The results of new Navier-Stokes computations using more compatible boundary conditions are shown, and the effects of the turbulence model choice on predicting Mach number trends are assessed.

Fixed point limit behavior of N-mode truncated Navier-Stokes equations as N increases

Abstract: The fixed point behavior ofN-mode truncations of the Navier-Stokes equations on a two-dimensional torus is investigated asN increases. FromN=44 on the behavior does not undergo any qualitative change. Furthermore, the bifurcations occur at critical parameter values which clearly tend to stabilize asN approaches 100.

A strongly implicit procedure for the compressible Navier-Stokes equations

Abstract: The strongly implicit procedure of Stone is used to obtain numerical solutions of the compressible Navier-Stokes equations in conservative form. In contrast to the spatially split Douglas-Gunn type methods, the method is shown to be numerically stable for the three-dimensional wave equation. The method is applied to a variety of external and internal two-dimensional flow problems involving shock wave boundary-layer interaction for both laminar and turbulent flows. The results are in good agreement with other methods and/or experiments. The storage penalty associated with the method is discussed and a simple, yet effective, means of minimizing the problem is presented.

Analysis of cylindrical Couette flows by use of the direction simulation method

Abstract: Cylindrical Couette flows of a rarefied gas are analyzed by use of Bird’s direct‐simulation method An extremely large sample size is employed to reduce the statistical fluctuations of the obtained solutions to a negligible magnitude The solutions should, though numerical, be regarded as the exact solutions of the full nonlinear Boltzmann equation The results are compared with the existing analyses and experiment The solution of the Navier–Stokes equations for slip boundary conditions is accurate up to Kn∼01 The solution for Kn∼1 of the ellipsoidal model equation shows a qualitative agreement with the present solution; it certainly gives the density minimum found here The experimental data for Kn∼004 agree with the present solution The data for Kn≳008 show only a qualtitative agreement Much more accurate measurements are necessary to ascertain the present solutions