Showing papers on "Navier–Stokes equations published in 1986"
TL;DR: It is shown that a class of deterministic lattice gases with discrete Boolean elements simulates the Navier-Stokes equation, anc, and can be used to design simple, massively parallel computing machines.
Abstract: We show that a class of deterministic lattice gases with discrete Boolean elements simulates the Navier-Stokes equation, anc can be used to design simple, massively parallel computing machines.
2,288 citations
TL;DR: In this paper, a method for solving the Navier-Stokes equations in primitive variables using a coupled block-implicit multigrid procedure is presented, which is applicable to finite-difference formulations using staggered locations of the flow variables.
637 citations
TL;DR: In this article, a model of cellular automata is presented, in which particles with discrete velocities populate the links of a fixed array of sites, and equations for microscopic particle distributions are derived.
Abstract: Continuum equations are derived for the large-scale behavior of a class of cellular automaton models for fluids. The cellular automata are discrete analogues of molecular dynamics, in which particles with discrete velocities populate the links of a fixed array of sites. Kinetic equations for microscopic particle distributions are constructed. Hydrodynamic equations are then derived using the Chapman-Enskog expansion. Slightly modified Navier-Stokes equations are obtained in two and three dimensions with certain lattices. Viscosities and other transport coefficients are calculated using the Boltzmann transport equation approximation. Some corrections to the equations of motion for cellular automaton fluids beyond the Navier-Stokes order are given.
621 citations
TL;DR: In this article, the results of Harten and Tadmor are generalized to the compressible Navier-Stokes equations including heat conduction effects and a symmetric form of the equations is derived in terms of entropy variables.
Abstract: Results of Harten and Tadmor are generalized to the compressible Navier-Stokes equations including heat conduction effects. A symmetric form of the equations is derived in terms of entropy variables. It is shown that finite element methods based upon this form automatically satisfy the second law of thermodynamics and that stability of the discrete solution is thereby guaranteed ab initio.
507 citations
TL;DR: In this paper, large time behavior of solutions to the navier-stokes equations is studied. But the authors focus on the Navier-Stokes equations and do not consider the large time behaviour of solutions of the non-navier Stokes equations.
Abstract: (1986). Large time behaviour of solutions to the navier-stokes equations. Communications in Partial Differential Equations: Vol. 11, No. 7, pp. 733-763.
375 citations
TL;DR: In this paper, the authors consider the equations which describe the motion of a viscous compressible fluid, taking into consideration the case of inflow and/or outflow through the boundary.
Abstract: We consider the equations which describe the motion of a viscous compressible fluid, taking into consideration the case of inflow and/or outflow through the boundary. By means of some a priori estimates we prove the existence of a global (in time) solution. Moreover, as a consequence of a stability result, we show that there exist a periodic solution and a stationary solution.
280 citations
TL;DR: In this article, the Navier-Stokes equations are solved with exact solutions for two-and three-dimensional shear flows of unbounded extent, including two-dimensional stagnation point flows and 2-dimensional flows with uniform vorticity.
Abstract: New classes of exact solutions of the incompressible Navier-Stokes equations are presented. The method of solution has its origins in that first used by Kelvin (Phil. Mag. 24 (5), 188-196 (1887)) to solve the linearized equations governing small disturbances in unbounded plane Couette flow. The new solutions found describe arbitrarily large, spatially periodic disturbances within certain two- and three-dimensional 'basic' shear flows of unbounded extent. The admissible classes of basic flow possess spatially uniform strain rates; they include two- and three-dimensional stagnation point flows and two-dimensional flows with uniform vorticity. The disturbances, though spatially periodic, have time-dependent wavenumber and velocity components. It is found that solutions for the disturbance do not always decay to zero; but in some instances grow continuously in spite of viscous dissipation. This behaviour is explained in terms of vorticity dynamics.
280 citations
TL;DR: An implicit, finite difference computer code has been developed to solve the incompressible Navier-Stokes equations in a three-dimensional curvilinear coordinate system based on the pseudocompressibility approach.
Abstract: An implicit, finite difference computer code has been developed to solve the incompressible Navier-Stokes equations in a three-dimensional curvilinear coordinate system. The pressure field solution is based on the pseudocompressibility approach in which a time derivative pressure term is introduced into the mass conservation equation. The solution procedure employs an implicit, approximate factorization scheme. The Reynolds Stresses, which are uncoupled from the implicit scheme, are lagged by one time step to facilitate implementing various levels of the turbulence model. Test problems for external and internal flows are computer and the results are compared with existing experimental data. The application of this technique for general three-dimensional problems is then demonstrated.
275 citations
TL;DR: In this paper, assumptions about the stability of a solution are introduced in the numerical analysis of the nonstationary Navier-Stokes problem, for the purpose of extending local a priori error estimates, and local a posteriori error estimate, globally in time.
Abstract: In this paper, assumptions about the stability of a solution are introduced in the numerical analysis of the nonstationary Navier–Stokes problem, for the purpose of extending local a priori error estimates, and local a posteriori error estimates, globally in time.
232 citations
TL;DR: In this paper, a redefinition of the Leonard term, the subgrid scale cross term, and the sub-grid scale Reynolds stress is proposed, consistent with a general statement concerning turbulent stresses.
Abstract: A redefinition of the Leonard term, the subgrid scale cross term, and the subgrid scale Reynolds stress is proposed, consistent with a general statement concerning turbulent stresses. The main advantage of the new redefined stresses consists of their term by term Galilean invariance.
TL;DR: In this article, it was shown that the breakdown of smooth solutions to the 3-D incompressible slightly viscous Navier-Stokes equations cannot occur without breakdown in the corresponding solution of the Euler (ideal fluid) equation.
Abstract: One of the central problems in the mathematical theory of turbulence is that of breakdown of smooth (indefinitely differentiable) solutions to the equations of motion. In 1934 J. Leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3—D incompressible Navier-Stokes equations. The problem is still open. We show in this report that breakdown of smooth solutions to the 3—D incompressible slightly viscous (i.e. corresponding to high Reynolds numbers, or “highly turbulent”) Navier-Stokes equations cannot occur without breakdown in the corresponding solution of the incompressible Euler (ideal fluid) equation. We prove then that solutions of distorted Euler equations, which are equations closely related to the Euler equations for short term intervals, do breakdown.
TL;DR: In this article, the behavior of numerical ODE methods for the solution of systems of differential equations coupled with algebraic constraints is investigated, and it is shown that backward differentiation formulas converge with the expected order of accuracy for these systems.
Abstract: In this paper we investigate the behavior of numerical ODE methods for the solution of systems of differential equations coupled with algebraic constraints. Systems of this form arise frequently in the modelling of problems from physics and engineering; we study some particular examples from electrical networks, fluid dynamics and constrained mechanical systems. We show that backward differentiation formulas converge with the expected order of accuracy for these systems.
TL;DR: In this article, an isoparametric spectral element discretization for the solution of flow problems in fully general, curvy geometries is proposed, and a time-splitting scheme for the Navier-Stokes equations is presented.
01 Jan 1986
TL;DR: In this article, implicit finite difference schemes for solving two-dimensional and three-dimensional Euler and thin layer Navier-Stokes equations are presented in fully vectorized codes for a Cray type architecture The Beam and Warming implicit approximate factorization algorithm in generalized coordinates is used The methods are either time accurate or accelerated non-time accurate steady state schemes Acceleration and efficiency modifications such as matrix reduction, diagonalization, and flux split schemes are presented
Abstract: Implicit finite difference schemes for solving two-dimensional and three-dimensional Euler and thin layer Navier-Stokes equations are addressed The methods are demonstrated in fully vectorized codes for a Cray type architecture The Beam and Warming implicit approximate factorization algorithm in generalized coordinates is used The methods are either time accurate or accelerated non-time accurate steady state schemes Acceleration and efficiency modifications such as matrix reduction, diagonalization, and flux split schemes are presented Two dimensional inviscid and viscous calculations (eg, airfoils with a deflected spoiler, circulation control airfoils, and unsteady buffeting) and of three dimensional viscous elliptical bodies, exhausting boattails, and generic oblique wing computations are discussed
TL;DR: In this paper, the Euler equation for incompressible fluids in R2 is well posed in the (vector-valued) Lebesgue spaces, where Lsp = (1 -?)-s/2 Lp(R2) with s > 1 + 2/p, 1 < p < 8.
Abstract: In this paper we show that the Euler equation for incompressible fluids in R2 is well posed in the (vector-valued) Lebesgue spaces
Lsp = (1 - ?)-s/2 Lp(R2) with s > 1 + 2/p, 1 < p < 8
and that the same is true of the Navier-Stokes equation uniformly in the viscosity ?.
25 Jun 1986
TL;DR: In this article, the authors propose a method to solve the problem of "uniformity": 4.123 4.13 4.12 4.23... 4.
Abstract: 123 4.
TL;DR: In this article, an algorithm for local Taylor-series expansion solutions of Navier-Stokes and continuity equations to arbitrary order can be found, where the truncated series solution satisfies the full equations of motion to within a specified accuracy.
Abstract: An algorithm has been developed which enables local Taylor-series-expansion solutions of the Navier-Stokes and continuity equations to be generated to arbitrary order. Much of the necessary algebra for generating these solutions can be done on a computer. Various properties of the algorithm are investigated and checked by making comparisons with known solutions of the equations of motion. A method of synthesising nonlinear viscous-flow patterns with certain required properties is developed and applied to the construction of a number of two- and three-dimensional flow-separation patterns. These patterns are asymptotically exact solutions of the equations of motion close to the origin of the expansion. The region where the truncated series solution satisfies the full equations of motion to within a specified accuracy can be found.
06 Jan 1986
TL;DR: A multigrid method for solving the compressible Navier Stokes equations is presented and results are presented for laminar and turbulent two dimensional flows over airfoils.
Abstract: A multigrid method for solving the compressible Navier Stokes equations is presented. The dimensionless conservation equations are discretized by a finite volume technique and time integration is performed by using a mltistage explicit algorithm. Convergence to a steady state. is enhanced by local time stepping, implicit smoothing of the residuals and the use of mltiple grids. The raethod has been implemented in two different ways: firstly a cell centered and secondly a corner point formulation ( i . e . the unknown variables are defined either at the center of a computational cell or at its vertices). laminar and turbulent two dimensional flows over airfoils. Computed results are presented for
TL;DR: Linearized alternating direction implicit (ADI) forms of a class of total variation diminishing (TVD) schemes for the Euler and Navier-Stokes equations have been developed as mentioned in this paper.
Abstract: Linearized alternating direction implicit (ADI) forms of a class of total variation diminishing (TVD) schemes for the Euler and Navier-Stokes equations have been developed. These schemes are based on the second-order-accurate TVD schemes for hyperbolic conservation laws developed by Harten (1983, 1984). They have the property of not generating spurious oscillations across shocks and contact discontinuities. In general, shocks can be captured within 1-2 grid points. These schemes are relatively simple to understand and easy to implement into a new or existing computer code. One can modify a standard three-point central-difference code by simply changing the conventional numerical dissipation term into the one designed for the TVD scheme. For steady-state applications, the only difference in computation is that the current schemes require a more elaborate dissipation term for the explicit operator; no extra computation is required for the implicit operator. Numerical experiments with the proposed algorithms on a variety of steady-state airfoil problems illustrate the versatility of the schemes.
12 May 1986
TL;DR: In this article, the Navier-Stokes equations for an incompressible fluid with a viscosity that is allowed to depend on the pressure were discussed and the Dirichlet initial-boundary valu problem is well posed as long as the equations do not change type.
Abstract: We discuss the Navier-Stokes equations for an incompressible fluid wit a viscosity that is allowed to depend on the pressure. Ellipticity and the complementing condition of Agmon, Douglis and Nirenberg [l] are discussec It is found that the pressure dependence of viscosity leads to the possibilit of a change of type. It is shown that the Dirichlet initial-boundary valu problem is well posed as long as the equations do not change type
TL;DR: In this paper, the existence of weak solutions for the Cauchy problem for the Navier-Stokes equations for one-dimensional, isentropic flow when the initial velocity is in L 2 and the initial density is in BV is proved.
Abstract: We prove the global existence of weak solutions for the Cauchy problem for the Navier-Stokes equations for one-dimensional, isentropic flow when the initial velocity is in L2 and the initial density is in L2 ∩ BV. Solutions are obtained as limits of approximations obtained by building heuristic jump conditions into a semi-discrete difference scheme. This allows for a rather simple analysis in which pointwise control is achieved through piecewise H1 and total variation estimates.
Journal Article•
TL;DR: In this paper, a finite element model of the vocal tract was made in which the time-dependent, compressible, Navier Stokes equations were solved, and the model was used to study the various effects that may be observed which are due to the nonlinear and stress-related terms.
01 May 1986
TL;DR: In this article, a new upwind algorithm based on Roe's scheme has been developed to solve the two-dimensional parabolized Navier-Stokes (PNS) equations, which does not require the addition of user specified smoothing terms for the capture of discontinuities such as shock waves.
Abstract: A new upwind algorithm based on Roe's scheme has been developed to solve the two-dimensional parabolized Navier-Stokes (PNS) equations. This method does not require the addition of user specified smoothing terms for the capture of discontinuities such as shock waves. Thus, the method is easy to use and can be applied without modification to a wide variety of supersonic flowfields. The advantages and disadvantages of this adaptation are discussed in relation to those of the conventional Beam-Warming scheme in terms of accuracy, stability, computer time and storage, and programming effort. The new algorithm has been validated by applying it to three laminar test cases including flat plate boundary-layer flow, hypersonic flow past a 15 deg compression corner, and hypersonic flow into a converging inlet. The computed results compare well with experiment and show a dramatic improvement in the resolution of flowfield details when compared with the results obtained using the conventional Beam-Warming algorithm.
TL;DR: A new splitting scheme is proposed for the numerical solution of the time-dependent incompressible Navier-Stokes equations by spectral methods using staggered grid, improved intermediate boundary conditions, and spectral multigrid techniques for the solution ofThe implicit equations.