Showing papers on "Navier–Stokes equations published in 1985"
TL;DR: In this paper, a numerical method for computing three-dimensional, time-dependent incompressible flows is presented based on a fractional-step, or time-splitting, scheme in conjunction with the approximate-factorization technique.
2,997 citations
505 citations
TL;DR: In this article, the effect of dissipation models on the accuracy, stability, and convergence of transonic airfoils is investigated using an implicit approximate factorization code (ARC2D).
Abstract: Various artificial dissipation models that are used with central difference algorithms for the Euler equations are analyzed for their effect on accuracy, stability, and convergence rates. In particular, linear and nonlinear models are investigated using an implicit approximate factorization code (ARC2D) for transonic airfoils. Fully implicit application of the dissipation models is shown to improve robustness and convergence rates. The treatment of dissipation models at boundaries will be examined. It will be shown that accurate, error free solutions with sharp shocks can be obtained using a central difference algorithm coupled with an appropriate nonlinear artificial dissipation model. I. Introduction T HE solution of the Euler equations using numerical techniques requires the use of either a differencing method with inherent dissipation or the addition of dissipation terms to a nondissipative scheme. This is because the Euler equations do not provide any natural dissipation mechanism (such as viscosity in the Navier-Stokes equations) that would eliminate high frequencies which are caused by nonlinearitie s and especially shocks. A variety of numerical algorithms and computer codes for the Euler equations have been developed. Methods such as MacCormack's1 explicit
449 citations
14 Jan 1985
310 citations
01 Jan 1985
297 citations
222 citations
TL;DR: In this article, upwind relaxation algorithms for obtaining efficient steady-state solutions to the compressible Navier-Stokes equations are described, using third order flux splitting of the pressure and convective terms and second-order central differencing for shear and heat flux terms.
Abstract: The development of upwind relaxation algorithms for obtaining efficient steady-state solutions to the compressible Navier-Stokes equations is described. The method is second-order accurate spatially and naturally disipative, using third-order flux splitting of the pressure and convective terms and second-order central differencing for shear and heat flux terms. A line Gauss-Seidel relaxation approach, shown to be unconditionally stable for model convection and diffusion equations, is used. The algorithm is demonstrated for several flows using the thin-layer form of the equations, including the problem of shock-induced separation over a flat plate.
198 citations
TL;DR: The computation of steady incompressable flows by an Euler implicit algorithm is studied using both the incompressible equations and the low Mach number compressible equations, and a matrix preconditioning factor that accomplishes this is developed and demonstrated.
Abstract: The computation of steady incompressible flows by an Euler implicit algorithm is studied using both the incompressible equations and the low Mach number compressible equations. The incompressible equations are handled by adding an artificial time derivative to the continuity equation. This allows both the pressure and velocity to be obtained implicitly. In one-dimensional problems, both systems converge rapidly, even at low Mach numbers where the eigenvalues are very stiff. In two dimensions where approximate factorization is required, the presence of stiff eigenvalues is highly detrimental. Stiffness can be avoided in the incompressible equations by selecting an appropriate "pseudo"-Mach number. This insures reliable convergence and results in an efficient incompressible flow algorithm. In the case of compressible equations, the Mach number cannot be chosen arbitrarily and the contamination introduced by approximate factorization must be removed. A matrix preconditioning factor that accomplishes this is developed and demonstrated. With this modification, the convergence rate is the same as in the incompressible case and is independent of Mach number. Rapid convergence is observed at Mach numbers as low as 6.05.
191 citations
TL;DR: In this paper, a modified QUICK scheme, a higher-order upwind finite difference formulation, was proposed to simulate Taylor-Gortler-like vortices and other 3D effects.
Abstract: SUMMARY Previous three-dimensional simulations of the lid-driven cavity flow have reproduced only the most general features of the flow. Improvements to a finite difference code, REBUFFS, have made possible the first completely successful simulation of the three-dimensional lid-driven cavity flow. The principal improvement to the code was the incorporation of a modified QUICK scheme, a higher-order upwind finite difference formulation. Results for a cavity flow at a Reynolds number of 3200 have reproduced experimentally observed Taylor-Gortler-like vortices and other three-dimensional effects heretofore not simulated. Experimental results obtained from a unique experimental cavity facility validate the calculated results.
187 citations
TL;DR: In this paper, a control volume-based finite element method for solving the Navier-Stokes equations using equal-order velocity-pressure interpolation is presented, which calculates velocity and pressure at all the grid points in the domain.
Abstract: A control volume-based finite-element method for solving the Navier-Stokes equations using equal-order velocity-pressure interpolation is presented. Unlike the unequal-order-type methods that compute pressure at much fewer grid points than velocity, the proposed equal-order method calculates velocity and pressure at all the grid points in the domain. The validity of the method is established by applying it to solve some test problems
172 citations
14 Jan 1985
TL;DR: A class of explicit multistage time-stepping schemes is used to construct an algorithm for solving the compressible Navier-Stokes equations and Flexibility in treating arbitrary geometries is obtained with a finite-volume formulation.
Abstract: A class of explicit multistage time-stepping schemes is used to construct an algorithm for solving the compressible Navier-Stokes equations. Flexibility in treating arbitrary geometries is obtained with a finite-volume formulation. Numerical efficiency is achieved by employing techniques for accelerating convergence to steady state. Computer processing is enhanced through vectorization of the algorithm. The scheme is evaluated by solving laminar and turbulent flows over a flat plate and an NACA 0012 airfoil. Numerical results are compared with theoretical solutions or other numerical solutions and/or experimental data.
TL;DR: In this paper, the Navier-Stokes equations are used to calculate the flow of two fluids in random networks, where the fluids are assumed to be incompressible, immiscible, Newtonian, and of equal viscosity.
Abstract: To explore how the microscopic geometry of a pore space affects the macroscopic characteristics of fluid flow in porous media, the authors have used approximate solutions of the Navier-Stokes equations to calculate the flow of two fluids in random networks. The model pore space consists of an array of pores of variable radius connected by throats of variable length and radius to a random number of nearest neighbors. The various size and connectedness distributions may be arbitrarily assigned, as are the wetting characteristics of the two fluids in the pore space. The fluids are assumed to be incompressible, immiscible, Newtonian, and of equal viscosity. In the calculation, the authors use Stokes flow results for the motion of the individual fluids and incorporate microscopic capillary force via the Washburn approximation. At any time, the problem is mathematically identical to a random electrical network of resistors, batteries and diodes. From the numerical solution of the latter, the authors compute the fluid velocities and saturation rates of change, and use a discrete time-stepping procedure to follow the subsequent motion. The scale of the computation has so far restricted the authors to networks of modest size (100-400 pores) in two dimensions. Within these limitations,more » the authors discuss the dependence of residual oil saturations and interface shapes on network geometry and flow conditions.« less
TL;DR: In this article, a Fourier-Chebyshev spectral method for the incompressible Navier-Stokes equations is described, which is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time.
TL;DR: In this article, a self-adaptive-grid method is described for multidimensional steady and unsteady flow computations about airfoils in two dimensions, as well as a steady inviscid flow computation and a one-dimensional case.
Abstract: A self-adaptive-grid method is described that is suitable for multidimensional steady and unsteady computations. Based on variational principles, a spring analogy is used to redistribute grid points in an optimal sense to reduce the overall solution error. User-specified parameters, denoting both maximum and minimum permissible grid spacings, are used to define the all-important constants, thereby minimizing the empiricism and making the method self-adaptive. Operator splitting and one-sided controls for orthogonality and smoothness are used to make the method practical, robust, and efficient. Examples are included for both steady and unsteady viscous flow computations about airfoils in two dimensions, as well as for a steady inviscid flow computation and a one-dimensional case. These examples illustrate the precise control the user has with the self-adaptive method and demonstrate a significant improvement in accuracy and quality of the solutions.
TL;DR: In this article, a new combination of methods for solving nonlinear boundary value problems containing a parameter is discussed, combining methods of the continuation type with least squares formulations, preconditioned conjugate gradient algorithms and finite element approximations.
Abstract: We discuss in this paper a new combination of methods for solving nonlinear boundary value problems containing a parameter. Methods of the continuation type are combined with least squares formulations, preconditioned conjugate gradient algorithms and finite element approximations.We can compute branches of solutions with limit points, bifurcation points, etc.Several numerical tests illustrate the possibilities of the methods discussed in the present paper; these include the Bratu problem in one and two dimensions, one-dimensional bifurcation and perturbed bifurcation problems, the driven cavity problem for the Navier–Stokes equations.
14 Jan 1985
TL;DR: In this paper, the linear stability of plane Couette flow composed of two immiscible fluids in layers is considered, and it is shown that the arrangement with the heavier fluid on top can be linearly stable if the viscosity stratification, volume ratio, surface tension, Reynolds number, and Froude number are favorable.
Abstract: The linear stability of plane Couette flow composed of two immiscible fluids in layers is considered. The fluids have different viscosities and densities. For the case of equal densities, there is a critical Reynolds number above which the interfacial mode of the bounded problem is approximated by that of the unbounded problem for wavelengths that are not short enough to be in the asymptotic short‐wavelength range, as well as for short waves. The full linear analysis reveals unstable situations missed out by the long‐ and short‐wavelength asymptotic analyses, but which have comparable orders of magnitudes for the growth rates. For the case of unequal densities, it is found that the arrangement with the heavier fluid on top can be linearly stable if the viscosity stratification, volume ratio, surface tension, Reynolds number, and Froude number are favorable.
TL;DR: A new method is proposed that combines the Newton–Raphson scheme and Usawa's algorithm and converges to a divergence-free solution of the nonlinear Navier–Stokes equations.
Abstract: Usawa's algorithm provides an efficient method for solving the divergence-free Stokes problem. On the other hand, the Newton–Raphson scheme is very popular for the solution of the nonlinear Navier–Stokes equations. We propose here a new method that combines these two algorithms and converges to a divergence-free solution of the nonlinear Navier–Stokes equations.
TL;DR: In this paper, the Krook equation was used to simplify the discussion of the collisional dynamics of particle disks and to compute the effective collision rate in a disk of spherical particles with a power-law distribution of sizes.
TL;DR: The methods are based on the use of l ocal eigenvalues or wave speeds to control spat i al d ifferencing of inviscid terms and are aimed atcreasing the level of accuracy and stability achievable in computation.
Abstract: A class of implicit upwind differencing methods for the compressible Navier-Stokes equations is described and applied. The methods are based on the use of local eigenvalues or wave speeds to control spatial differencing of inviscid terms and are aimed at increasing the level of accuracy and stability achievable in computation. Techniques for accelerating the rate of convergence to a steady state solution are also used. Applications to inviscid and viscous transonic flows are discussed and compared with other methods and experimental measurements. It is shown that accurate and efficient transonic airfoil calculations can be made on the Cray-l computer in less than 2 min.
01 Jan 1985
TL;DR: In this paper, le comportement asymptotique des solutions des solutions for Navier Stokes equations is studied. But, le problem is not solvable by stationnaires.
Abstract: On considere le probleme aux valeurs initiales pour les equations de Navier Stokes non stationnaires. On etudie le comportement asymptotique des solutions
TL;DR: In this article, a spatial discretization of the Stokes problem in a domain of O(n 2 ) was studied, where the unknowns being the velocity and the pressure; optimal error was obtained by a finite element method.
Abstract: We study a spatial discretization of the Stokes problem in a domain of $\mathbb{R}^2 $ or $\mathbb{R}^3 $ by a finite element method, the unknowns being the velocity and the pressure; optimal error...
01 Jan 1985
TL;DR: In this paper, a new upwind scheme for computation of incompressible flow has been developed, which works well at high Reynolds number even using limited number of mesh points, and it was found that this scheme works well even with a small number of points.
Abstract: A new upwind scheme for computation of incompressible flow has been developed. It was found that this scheme works well at high Reynolds number even using limited number of mesh points.
TL;DR: Infinite pointwise stretching in a finite time for general initial conditions is found in a simulation of the Biot-Savart equation for a slender vortex tube in three dimensions.
Abstract: Infinite pointwise stretching in a finite time for general initial conditions is found in a simulation of the Biot-Savart equation for a slender vortex tube in three dimensions. Viscosity is ineffective in limiting the divergence in the vorticity as long as it remains concentrated in tubes. Stability has not been shown.
TL;DR: An exact solution of the unsteady Navier-Stokes equations is found in this article, which describes the stagnation region of an accelerating circular jet impinging obliquely on a flat plate.
Abstract: An exact solution of the unsteady Navier–Stokes equations is found. The solution describes the stagnation region of an accelerating circular jet impinging obliquely on a flat plate.
TL;DR: In this paper, numerical solutions for spatially developing axisymmetric and two-dimensional mixing layers are presented for forced spatially developed axisymmetric and 2D mixing layers.
Abstract: Numerical solutions are presented for forced spatially developing axisymmetric and two‐dimensional mixing layers. The numerical scheme employs quadratic upwind differencing for convection and a Leith type of temporal differencing in order to solve the incompressible Navier–Stokes and continuity equations. The applied forcing function is derived from linear inviscid stability theory. The resulting large‐scale vortex dynamics is visualized by means of streakline and isovorticity contour plots. It is seen that the vortex merging behavior in both types of mixing layers is determined by the subharmonics present in the forcing function. Manipulation of the vortex dynamics in a predictable fashion is possible through alterations in the frequency content of this applied forcing. Reynolds number is shown to be of only minor importance.
TL;DR: In this article, perfect gas exact solutions to the steady Navier-Stokes equations are given for laminar convective motion in open and closed vertical slots with large temperature differences using Sutherland law transport properties.
Abstract: Perfect gas exact solutions to the steady Navier–Stokes equations are given for laminar convective motion in open and closed vertical slots with large temperature differences using Sutherland law transport properties. The solutions are valid a few slot widths away from the ends in the asymptotic region where the opposite hot and cold wall boundary layers are fully merged. It is found that the static pressure (in the closed slot) and temperature and velocity distributions (in all cases) are very sensitive to property variations, even though the heat flux may not be. We observe the net horizontal and vertical heat fluxes to be the same as those obtained from the Boussinesq equations. Comparisons with constant property solutions and the well‐known Boussinesq limiting solution for small temperature differences are given for examples using air.
01 Jan 1985
TL;DR: In this paper, the Navier-Stokes equations were solved using Gauss-Seidel line relaxation and type dependent finite difference approximations, chosen according to the characteristic speeds of the flow.
Abstract: In the early 1970’s Murman and Cole in a landmark paper [1] set the groundwork for the development of computational fluid dynamics for years to come. Their paper demonstrated the use of type dependent finite difference approximations, chosen according to the characteristic speeds of the flow, and Gauss-Seidel line relaxation to solve the transonic small disturbance equation. Following their success, Jameson [2] introduced a “rotated” difference scheme and extended the Murman-Cole procedure to solve the full potential equation. These early contributions were responsible for the rapid growth and widespread application of computational fluid dynamics to aerodynamic design in the 1970’s. Last year Chakravarthy [3] applied flux-split type-dependent difference approximations and Gauss-Seidel line relaxation to solve the Euler equations. And this year Napolitano and Walters [4] and this author [5] used these procedures to solve the Navier-Stokes equations. On the one hand it’s amazing that the same key features of the Murman-Cole scheme can be applied to the more complete sets of governing equations, and on the other it’s amazing that it took a decade and a half to realize it. This paper outlines the use of these features for solving the Navier-Stokes equations and presents some computed results demonstrating high numerical efficiency.
TL;DR: In this paper, the authors modify the kinetic theory for suspensions of elastic dumbbells by introducing anisotropic hydrodynamic drag (expressed by a tensorial Stokes's law) and an isotropic Brownian motion, resulting from the introduction of a tensor into the Maxwellian velocity distribution.
Abstract: In this paper we modify the kinetic theory for suspensions of elastic dumbbells by introducing anisotropic hydrodynamic drag (expressed by a tensorial Stokes's law) and anisotropic Brownian motion (resulting from the introduction of a tensor into the Maxwellian velocity distribution). We show how our general formulation leads to Giesekus's 1982 constitutive equation, to the encapsulated dumbbell model of Bird and DeAguiar, and to several other constitutive equations. It is also shown how the Giesekus postulate for the form of the hydrodynamic drag tensor can be used to re‐interpret hydrodynamic interaction. Finally this development emphasizes the necessity for modifying the stress tensor expression when reptation is introduced into molecular theories.