Showing papers on "Navier–Stokes equations published in 1987"

Navier-Stokes Equations and Nonlinear Functional Analysis

Abstract: Preface to the second edition Introduction Part I. Questions Related to the Existence, Uniqueness and Regularity of Solutions: 1. Representation of a Flow: the Navier-Stokes Equations 2. Functional Setting of the Equations 3. Existence and Uniqueness Theorems (Mostly Classical Results) 4. New a priori Estimates and Applications 5. Regularity and Fractional Dimension 6. Successive Regularity and Compatibility Conditions at t=0 (Bounded Case) 7. Analyticity in Time 8. Lagrangian Representation of the Flow Part II. Questions Related to Stationary Solutions and Functional Invariant Sets (Attractors): 9. The Couette-Taylor Experiment 10. Stationary Solutions of the Navier-Stokes Equations 11. The Squeezing Property 12. Hausdorff Dimension of an Attractor Part III. Questions Related to the Numerical Approximation: 13. Finite Time Approximation 14. Long Time Approximation of the Navier-Stokes Equations Appendix. Inertial Manifolds and Navier-Stokes Equations Comments and Bibliography Comments and Bibliography Update for the Second Edition References.

Lattice gas hydrodynamics in two and three dimensions

Abstract: Hydrodynamical phenomena can be simulated by discrete lattice gas models obeing cellular automata rules (U. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett. 56, 1505, (1986); D. d'Humieres, P. Lallemand, and U. Frisch, Europhys. Lett. 2, 291, (1986)). It is here shown for a class of D-dimensional lattice gas models how the macrodynamical (large-scale) equations for the densities of microscopically conserved quantities can be systematically derived from the underlying exact ''microdynamical'' Boolean equations. With suitable restrictions on the crystallographic symmetries of the lattice and after proper limits are taken, various standard fluid dynamical equations are obtained, including the incompressible Navier-Stokes equations in two and three dimensions. The transport coefficients appearing in the macrodynamical equations are obtained using variants of fluctuation-dissipation and Boltzmann formalisms adapted to fully discrete situations.

On pressure boundary conditions for the incompressible Navier‐Stokes equations

Abstract: The pressure is a somewhat mysterious quantity in incompressible flows. It is not a thermodynamic variable as there is no ‘equation of state’ for an incompressible fluid. It is in one sense a mathematical artefact—a Lagrange multiplier that constrains the velocity field to remain divergence-free; i.e., incompressible—yet its gradient is a relevant physical quantity: a force per unit volume. It propagates at infinite speed in order to keep the flow always and everywhere incompressible; i.e., it is always in equilibrium with a time-varying divergence-free velocity field. It is also often difficult and/or expensive to compute. While the pressure is perfectly well-defined (at least up to an arbitrary additive constant) by the governing equations describing the conservation of mass and momentum, it is (ironically) less so when more directly expressed in terms of a Poisson equation that is both derivable from the original conservation equations and used (or misused) to replace the mass conservation equation. This is because in this latter form it is also necessary to address directly the subject of pressure boundary conditions, whose proper specification is crucial (in many ways) and forms the basis of this work. Herein we show that the same principles of mass and momentum conservation, combined with a continuity argument, lead to the correct boundary conditions for the pressure Poisson equation: viz., a Neumann condition that is derived simply by applying the normal component of the momentum equation at the boundary. It usually follows, but is not so crucial, that the tangential momentum equation is also satisfied at the boundary.

Finite Element Flux-Corrected Transport (FEM-FCT) for the Euler and Navier-Stokes equations

Abstract: A high resolution finite element method for the solution of problems involving high speed compressible flows is presented. The method uses the concepts of flux-corrected transport and is presented in a form which is suitable for implementation on completely unstructured triangular or tetrahedral meshes. Transient and steady state examples are solved to illustrate the performance of the algorithm.

Decay Results for Weak Solutions of the Navier–Stokes Equations on Rn

Abstract: On considere les solutions faibles du probleme de Cauchy pour l'equation de Navier-Stokes sur R n , n≥2. On etablit des resultats concernant la decroissance de la L 2 -norme d'une solution faible sous des hypotheses appropriees relatives au second membre et a la valeur initiale

An LU-SSOR scheme for the Euler and Navier-Stokes equations

Abstract: A new multigrid relaxation scheme, lower-upper symmetric successive overrelaxation (LU-SSOR) is developed for the steady-state solution of the Euler and Navier-Stokes equations. The scheme, which is based on central differences, does not require flux splitting for approximate Newton iteration. Application to transonic flow shows that the new method is efficient and robust. The vectorizable LU-SSOR scheme needs only scalar diagonal inversions.

Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations†‡

Abstract: The current status of streamline-upwind/Petrov-Galerkin (SUPG) methods for the analysis of flow problems is surveyed in an analytical review. Problem areas addressed include classical Galerkin, upwind, artificial-diffusion, SUPG, discontinuous Galerkin, space-time FEM, and discontinuity-capturing approaches to the scalar advection-diffusion equation; incompressible flows; advective-diffusive systems; and the compressible Euler and Navier-Stokes equations. Graphs and diagrams are provided, and the good stability properties of state-of-the-art SUPG methods are pointed out.

A comparison of numerical flux formulas for the Euler and Navier-Stokes equations

Abstract: Numerical flux formulas for the convection terms in the Euler or Navier-Stokes equations are analyzed with regard to their accuracy in representing steady nonlinear and linear waves (shocks and entropy/shear waves, respectively) Numerical results are obtained for a one-dimensional conical Navier-Stokes flow including both a shock and a boundary layer Analysis and experiments indicate that for an accurate representation of both layers the flux formula must include information about all different waves by which neighboring cells interact, as in Roe's flux-difference splitting In comparison, Van Leer's flux-vector splitting, which ignores the linear waves, badly diffuses the boundary layer The results of MacCormack's scheme, if properly tuned, are significantly better The use of a sufficiently detailed flux formula appears to reduce the number of cells required to resolve a boundary layer by a factor 1/2 to 1/4 and thus pays off

Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data

Abstract: We prove the global existence of weak solutions of the Cauchy problem for the Navier-Stokes equations of compressible, isentropic flow of a polytropic gas in one space dimension. The initial velocity and density are assumed to be in L2 and L2 n BV respectively, modulo additive constants. In particular, no smallness assumptions are made about the intial data. In addition, we prove a result concerning the asymptotic decay of discontinuities in the solution when the adiabatic constant exceeds 3/2.

Forced Convection Cooling Across Rectangular Blocks

Abstract: Conjugate heat transfer for two-dimensional, developing flow over an array of rectangular blocks, representing finite heat sources on parallel plates, is considered. Incompressible flow over multiple blocks is modeled using the fully elliptic form of the Navier-Stokes equations. A control-volume-based finite difference procedure with appropriate averaging for the diffusion coefficients is used to solve the coupling between the solid and fluid regions. The heat transfer characteristics resulting from recirculating zones around the blocks are presented. The analysis is extended to study the optimum spacing between heat sources for a fixed heat input and a desired maximum temperature at the heat source.

The three‐dimensional instability of strained vortices in a viscous fluid

Abstract: The recent theory describing 3‐D exact solutions of the Navier–Stokes equations is applied to the problem of stability of 2‐D viscous flow with elliptical streamlines. An intrinsically inviscid instability mechanism persists in all such flows provided the length scale of the disturbance is sufficiently large. Evidence is presented that this mechanism may be responsible for 3‐D instabilities in high Reynolds number flows whose vortex structures can be locally described by elliptical streamlines.

Artificial dissipation and central difference schemes for the Euler and Navier-Stokes equations

Abstract: An artificial dissipation model, including boundary treatment, that is employed in many central difference schemes for solving the Euler and Navier-Stokes equations is discussed. Modifications of this model such as the eigenvalue scaling suggested by upwind differencing are examined. Multistage time stepping schemes with and without a multigrid method are used to investigate the effects of changes in the dissipation model on accuracy and convergence. Improved accuracy for inviscid and viscous airfoil flow is obtained with the modified eigenvalue scaling. Slower convergence rates are experienced with the multigrid method using such scaling. The rate of convergence is improved by applying a dissipation scaling function that depends on mesh cell aspect ratio.

Finite element approximation of incompressible Navier-Stokes equations with slip boundary condition

Abstract: We consider mixed finite element approximations of the stationary, incompressible Navier-Stokes equations with slip boundary condition simultaneously approximating the velocity, pressure, and normal stress component. The stability of the schemes is achieved by adding suitable, consistent penalty terms corresponding to the normal stress component and to the pressure. A new method of proving the stability of the discretizations allows, us to obtain optimal error estimates for the velocity, pressure, and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition. The schemes can easily be implemented into existing finite element codes for the Navier-Stokes equations with standard Dirichlet boundary conditions.

On the accuracy of upwind schemes for the solution of the navier-stokes equations

Abstract: This paper is concerned with an investigation of the influence of the upwind Euler discretization on the solution of the Navier-Stokes equations. A number of elements of the upwind discretization were investigated, like the type of upwind discretization, the limiter and boundary formulation of the Euler fluxes, and the influence of flux splitting on the total enthalpy. Their influence on the accuracy of the viscous solution was demonstrated in test problems.

Lagrangian finite element analysis applied to viscous free surface fluid flow

Abstract: This paper presents the development of a new scheme to simulate numerically the 2-D flow of an incompressible fluid with a free surface. Comparisons of results are presented with previous theoretical, numerical and experimental results.

An L(p)-Theory for the n-Dimensional, Stationary, Compressible, Navier-Stokes Equations, and the Incompressible Limit for Compressible Fluids. The Equilibrium Solutions.

Abstract: In this paper we study the system (1.1), (1.3), which describes the stationary motion of a given amount of a compressible heat conducting, viscous fluid in a bounded domain Ω ofR n, n≧2. Hereu(x) is the velocity field, σ(x) is the density of the fluid, ς(x) is the absolute temperature,f(x) andh(x) are the assigned external force field and heat sources per unit mass, andp(σ, ς) is the pressure. In the physically significant case one hasg=0. We prove that for small data (f, g, h) there exists a unique solution (u, σ, ς) of problem (1.1), (1.3)1, in a neighborhood of (0,m, ς0); for arbitrarily large data the stationary solution does not exist in general (see Sect. 5). Moreover, we prove that (for barotropic flows) the stationary solution of the Navier-Stokes equations (1.8) is the incompressible limit of the stationary solutions of the compressible Navier-Stokes equations (1.7), as the Mach number becomes small. Finally, in Sect. 5 we will study the equilibrium solutions for system (4.1). For a more detailed explanation see the introduction.

Explicit multigrid algorithm for quasi-three-dimensional viscous flows in turbomachinery

Abstract: A rapid quasi-three-dimensional analysis has been developed for blade-to-blade flows in turbomachinery. The analysis solves the unsteady Euler or thin-layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stream surface thickness. The BaldwinLomax eddy viscosity model is used for turbulent flows. The equations are solved using a two-stage RungeKutta scheme made efficient by the use of vectorization, a variable time step, and a flux-based multigrid scheme, all of which are described. Results of a stability analysis are presented for the two-stage scheme. Results for a flat-plate model problem show the applicability of the method to axial, radial, and rotating geometries. Results for a centrifugal impeller and a radial diffuser show that the quasi-three-dimensional viscous analysis can be a practical design tool.

The PARC Code: Theory and Usage

Abstract: : The PARC code is a flow-field simulation computer program based on the strong conservation law form of the Navier Stokes equations. Several variants of these equations are treated, including 2-D, axisymmetric, and 3-D versions and inviscid, laminar, and turbulence specializations. Use is made of perfect has equations of state and Sutherland's viscosity law. The equations are expressed in curvilinear coordinates while retaining strong conservation law forms. The Beam and Warming approximate factorization algorithm is used for forming the implicit central-difference algorithm. Pulliam's scalar pentadiagonal transformation is applied to produce an efficient steady-state solver. Artificial viscosity is added both implicity and explicitly and follows the switched fourth-order and second-order form of Jameson. The metrics are calculated in a finite volume manner so as to maintain free steam. Variable spatial and temporal time-steps are used for improved algorithm efficiency and robustness. Generalized boundary conditions are selected through code inputs and may be located anywhere within the computational grid. Use of the PARC code is straight-forward, requiring only a restart file and file of NAMELIST inputs. These inputs allow the selection of flow equation specialization, thermodynamic properties, grid size, artificial viscosity parameters, iteration controls, and output parameters are provided through NAMELIST input.

Vortex dynamics and the existence of solutions to the Navier–Stokes equations

Abstract: The Biot–Savart model for a vortex filament predicts a finite time singularity in which the maximum velocity diverges as (t*−t)−1/2 for the time t tending to t*. The filament pairs with itself, yet remains locally smooth even though the characteristic length scales as (t*−t)1/2. A multiscale perturbative treatment of the Euler equations is developed for solutions that are locally a two‐dimensional vortex dipole centered on a slowly varying three‐dimensional space curve. For short periods of time the Euler and Biot–Savart solutions agree. Provided this correspondence persists, a sufficiently small viscosity ν will not control the divergence in the maximum velocity until it is of order exp(cst/ν), where cst is a constant of order the filament circulation. Singularities in the Navier–Stokes equations cannot be easily dismissed. The most questionable step in the arguments presented occurs for ν=0, namely whether the Euler vortex dipole solutions break down when they self‐stretch.

Application of an upwind algorithm to the three-dimensional parabolized Navier-Stokes equations

Abstract: A new computer code for the solution of the three-dimensional parabolized Navier-Stokes equations has been developed. The code employs a state-of-the-art upwind algorithm to capture strong shock waves. The algorithm is implicit, uses finite volumes, and is second-order accurate in the crossflow directions. The new code is validated through application to laminar hypersonic flows past two simple body shapes: a circular cone of 10 deg half-angle, and a generic all-body hypersonic vehicle. Cone flow solutions were computed at angles of attack of 12, 20, and 24 deg and results are in agreement with experimental data. Results are also presented for the flow past the all-body vehicle at angles of incidence of 0 and 10 deg.

Small-time existence for the Navier-Stokes equations with a free surface

Abstract: We prove the small-time existence of a solution of the Navier-Stokes equations, for any initial data, for a free boundary fluid with surface tension taken into account. A fixed point method is used. The linearized problem is hyperbolic and dissipative. The classical methods to solve it seem to fail and the method used here could perhaps be applied for equations of the same kind.

The Connection Between the Navier-Stokes Equations, Dynamical Systems, and Turbulence Theory

Abstract: Publisher Summary There are three major distinct but not mutually contradictory theoretical views on turbulence—(a) the statistical view, in which the turbulence is considered as the observed behavior of the evolution of statistical distributions of flows instead of the evolution of one individual flow; (b) the viewpoint of regularity breakdown, in which the turbulence is considered to result from the blow-up of the vorticity in finite time, albeit necessarily on a set of small Hausdorff dimension; and (c) the dynamical systems view, in which the turbulence is a phenomenologic perception of the long time complicated behavior of the individual flows. The chapter discusses that it is reasonable to test them by trying to establish rigorous mathematical facts based on the Navier–Stokes equations, the interpretation of which are consistent with any of the three viewpoints mentioned above. The chapter presents the contribution by Ciprian Foias and Roger Temam to this program from the dynamical systems point of view. The chapter presents a proof of a regularity and backward uniqueness theorem on an open dense subset of the universal attractor of the 3D Navier–Stokes equations.

Cascade viscous flow analysis using the Navier-Stokes equations

Abstract: A previously developed explicit, multiple-grid, time-marching Navier-Stokes solution procedure has been modified and extended for the calculation of steady-state high Reynolds number turbulent flows in cascades. Particular attention has been given to the solution accuracy of this procedure as compared with boundary-layer theory and experimental data. A new compact discretization scheme has been implemented for the viscous terms which has the same finite-difference molecule as the inviscid terms of the Navier-Stokes equations. This compact operator has been found to yield accurate and stable solutions in regions of the flow where the gradients are large and the computational mesh is relatively sparse. A modified C grid generation procedure has been developed for cascades that greatly reduces grid skewing in the midgap region. As a result, numerical errors associated with the use of numerical smoothing on skewed grids are reduced considerably. In addition, a body normal grid system has also been generated for accurately determining the eddy viscosity distribution based on an algebraic turbulence model and for comparing the results directly with boundary-layer theory. A combined second and fourth difference numerical smoothing operation has been carefully constructed to prevent oscillations in the solution for the flow over complicated geometries without contaminating the velocity profiles near the wall. Results from turbine and compressor applications are presented to demonstrate the accuracy of the present scheme through comparisons with experimental data and attached boundary-layer theory.

Diagonal implicit multigrid algorithm for the Euler equations

Abstract: A multigrid implementation of the Alternating Direction Implicit algorithm has been developed to solve the Euler equations of inviscid, compressible flow. The equations are approximated using a finite-volume spatial approximation with added dissipation provided by an adaptive blend of second and fourth differences. For computational efficiency, the equations are diagonalized by a local similariity transformation so that only a decoupled system of scalar pentadiagonal systems need be solved along each line. Results are computed for transonic flows past airfoils and include pressure distributions to verify the accuracy of the basic scheme and convergence histories to demonstrate the efficiency of the method.

Some numerical methods for the computation of capillary free boundaries governed by the Navier-Stokes equations

Abstract: In §, the trial method, Newton’s method, and the total linearization method are applied to a simple one-dimensional free boundary (FB) problem for which the solution is known. Attention will be focused on convergence speed and computer implementation facilities of these methods. Section 2, discusses static FB problems in two- and three-dimensional containers. These FBs are governed by the Laplace–Young equation. In §3, two stationary FB problems governed by the Navier–Stokes equations are formulated and these problems are solved using the three methods introduced in §1. A thermocapillary FB problem governed by the Navier–Stokes equations coupled with the heat equation is studied in §4. In §5, a moving FB problem governed by the Navier–Stokes equations will be considered within the context of perturbation theory. Finally, in §6, some remarks are made on present and future research in the field of capillary FB problems governed by the Navier–Stokes equations.

Calculating Rotordynamic Coefficients of Seals by Finite-Difference Techniques

Abstract: For modelling the turbulent flow in a seal the Navier-Stokes equations in connection with a turbulence (kappa-epsilon) model are solved by a finite-difference method. A motion of the shaft round the centered position is assumed. After calculating the corresponding flow field and the pressure distribution, the rotor-dynamic coefficients of the seal can be determined. These coefficients are compared with results obtained by using the bulk flow theory of Childs and with experimental results.

On the existence of stationary solutions to compressible Navier-Stokes equations

Abstract: We prove the existence of a stationary solution to the Navier–Stokes equations for compressible fluids, under the assumption that the external force field is small in a suitable sense. The proof is based over an existence result for a linearized problem, followed by a fixed point argument.