Roger Peyret

Roger Peyret
University of Nice Sophia Antipolis | UNS · Mathématics

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130
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Publications (130)
Preprint
Full-text available
In Memoriam of Professor Pierre Haldenwang Aix-Marseille iniversité expert Computational Fluids Dynamics, Spectral Methods, Fluids Mechanics, Convection ...
Chapter
One of the purposes of the present study was to show how the analysis of the equivalent system corresponding to the general class of schemes, ( µ = 1 + \tfracÖ5 2,b = \tfrac12)( \propto = 1 + \tfrac{{\sqrt 5 }}{2},\beta = \tfrac{1}{2}) minimizes the Maxh Î [ - 1,1]\mathop {Max}\limits_{\eta \in [ - 1,1]} E2. with the constraint E2O , the scheme...
Chapter
The paper is concerned with a hybrid finite element — spectral Chebyshev parallel solver of the incompressible Navier-Stokes equations. A domain decomposition, well adapted to the computation of wake or jet type flow, is assumed. Subdomains with complex geometry are handled with finite elements and the other ones with the highly accurate spectral m...
Chapter
The vorticity-streamfunction equations present some advantages over the velocity-pressure equations in the case of two-dimensional flows in simply connected domains. These advantages are well known: (1) the velocity field is automatically divergence-free, (2) the mathematical properties of the equations permit the construction of simple and robust...
Chapter
In this short chapter, the Navier-Stokes equations governing the motion of a viscous incompressible fluid are recalled. Two formulations are considered: the velocity-pressure formulation (also called “primitive variables” formulation) and the vorticity-streamfunction formulation. The formulation using vorticity and velocity as dependent variables w...
Chapter
The solution of the Navier-Stokes equations in velocity-pressure variables is the subject of this chapter. This set of equations is of broader application than the vorticity-streamfunction equations which are restricted to two-dimensional flows. First, the Fourier method for computing fully periodic flows is discussed. Then the major part of the ch...
Chapter
The more familiar spectral method is the Fourier method in which the basis functions are trigonometric functions. Such a basis is adapted to periodic problems. Spatial periodicity appears in a variety of flows. Beside the case of homogeneous turbulence, for which the assumption of periodicity in the three spatial directions is realistic, there also...
Chapter
The Fourier method is appropriate for periodic problems, but is not adapted to nonperiodic problems because of the existence of the Gibbs phenomenon at the boundaries. In the case of nonperiodic problems, it is advisable to have recourse to better-suited basis functions. Orthogonal polynomials, like Chebyshev polynomials, constitute a proper altern...
Chapter
In this chapter, we discuss the time-discretization of time-dependent equations. Although the methods apply to general nonlinear time-dependent equations, their analysis is developed in the linear case and, more especially, for the advection-diffusion equation. First, we address the stability of the spectral approximation, namely, the existence of...
Chapter
The aim of this introductory chapter is to present, in a general way, the spectral methods in their various formulations: Galerkin, tau, and collocation. By using the notion of residual, it will be shown how spectral approximation can be defined for the representation of a given function as well as for the solution of a differential problem. These...
Chapter
The domain decomposition method for the solution of differential problems consists of dividing the computational domain into a set of subdomains in which the solution is calculated by taking into account some transmission conditions at the interfaces between the subdomains. This chapter addresses some usual Chebyshev domain decomposition methods fo...
Chapter
This chapter discusses some techniques for handling stiff and singular problems, using Chebyshev methods. The solution of a stiff problem is regular but exhibits large variations in a region of small extent. For convergence reasons, the collocation points cannot be clustered arbitrarily in the rapid variation region, so that an appropriate distribu...
Book
Introduction * Fundamentals of Spectral Methods * Fourier Method * Chebyshev Method * Time-Dependent Equations * Navier-Stokes Equations for Incompressible Fluids * Vorticity-Streamfunction Equations * Velocity-Streamfunction Equations * Velocity-Pressure Equations * Stiff and Singular Problems * Domain Decomposition Method* Appendices * References...
Article
Two types of Chebyshev-collocation methods for the solution of the Navier-Stokes equations are considered according to the elliptic problem to be solved at each time-cycle : Stokes-type problem or Darcy problem. Typical features associated with the solution of these problems are discussed : spurious modes of pressure, multi-domain technique, outflo...
Article
The spectral calculations of the flow which develops near a heated vertical wall in a fluid stably stratified by a salinity gradient are carried out. The case of weak stratification and heating due to a constant heat flux is considered. The numerical results are compared to the experimental ones. Both on qualitative and quantitative arguments, a sa...
Article
The solution of fluid flow problems exhibits a singular behaviour when the conditions imposed on the boundary display some discontinuities or change in type. A treatment of these singularities has to be considered in order to preserve the accuracy of high-order methods, such as spectral methods. The present work concerns the computation of a singul...
Article
A spectral Chebyshev method for the computation of wakes in stratified fluids is presented. It is based on a divergence-free multi-domain solver of the generalized Oseen problem which results from the discretization in time of the Navier–Stokes equations, when taking into account implicitly the part of the convective term associated with the mean f...
Article
These lectures are devoted to a presentation and a discussion of high-order approximation methods currently used in computational fluid dynamics. The first part considers the spatial approximation: classical and Hermitian compact finite-difference, finite-volume and spectral methods. In particular, it is shown how approximation formulas can be deri...
Book
This book collects the lecture notes concerning the IUTAM School on Advanced Turbulent Flow Computations held at CISM in Udine September 7–11, 1998. The course was intended for scientists, engineers and post-graduate students interested in the application of advanced numerical techniques for simulating turbulent flows. The topic comprises two close...
Article
Highly-accurate solutions for the lid-driven cavity flow are computed by a Chebyshev collocation method. Accuracy of the solution is achieved by using a substraction method of the leading terms of the asymptotic expansion of the solution of the Navier–Stokes equations in the vicinity of the corners, where the velocity is discontinuous. Critical com...
Chapter
We present a new parallel hybrid method to solve numerically elliptic equations on a channel-like domain. The method combines the highly accurate Chebyshev — spectral method with a standard finite difference one, via the CGBI — domain decomposition procedure. By this approach the solution of linear elliptic boundary value problems is reduced to a m...
Article
This paper presents a spectral multidomain method for solving the Navier-Stokes equations in the vorticity-stream function formulation. The algorithm is based on an extensive use of the influence matrix technique and so leads to a direct method without any iterative process. Numerical results concerning the Czochralski melt configuration are report...
Chapter
We present and discuss some numerical experiments in double-diffusive convection where a layer of fluid, stably stratified by concentration, is subject to heating. By considering first the case where the fluid is heated from below we justify the need of highly accurate numerical approximations, like spectral methods. The spectral Chebyshev method f...
Article
A spectral Fourier-Chebyshev method for calculating unsteady two-dimensional free surface flows is presented and discussed. The vorticity-stream function equations are used in association with an influence matrix technique for prescribing the boundary and free surface conditions. The stability of the time-discretization scheme is analysed. Finally,...
Article
This paper presents a spectral multidomain method for solving Stokes equations in the vorticity-stream function formulation. Numerical results are reported and compared with spectral monodomain solutions to show the advantage of the domain decomposition for some problems with singular solution.
Article
The low Mach number approximation of the Navier—Stokes equations is of similar nature to the equations for incompressible flow. A major difference, however, is the appearance of a space- and time-varying density that introduces a supplementary non-linearity. In order to solve these equations with spectral space discretization, an iterative solution...
Article
Full-text available
The changes in the 2D Rayleigh-Benard flow that occur due to non-Boussinesq effects when density and/or diffusion coefficients vary strongly with temperature are investigated. The principal effect examined is a strong variation in the density considered primarily as a function of temperature as it is accounted for by the low Mach number equations....
Article
This paper reports some experiments on the use of adaptive Chebyshev pseudospectral methods for compressible mixing layer computations. Different functionals measuring the optimality of the polynomial approximation are discussed and compared. In particular, we address the problem of the practical computation of the various functionals. The utility...
Article
A spectral Fourier-Chebyshev method for the solution of the complete low Mach number equations of isobaric combustion is presented. It is used to calculate the evolution of a plane flame front that is unstable due to the hydrodynamical Darrieus-Landau instability. Simulations permit to determine numerically the growth rate of a perturbation when va...
Article
The Chebyshev multidomain technique for calculating solutions with steep gradients is first discussed regarding accuracy and stability in case of simple model equations. Then the method is described for the solution of the Navier-Stokes equations and applications to double-diffusive convection exhibiting thin inner layers are presented.
Article
This paper presents a two-dimensional pseudospectral method for low Mach number convection with one direction of periodicity. Preconditioned Uzawa-type iterations are used to solve the generalized Stokes problem resulting from the semi-implicit time scheme. Applications concern the Rayleigh-Bénard flow field with rigid boundaries.
Article
Full-text available
Pseudospectral methods are used for the computation of the time-dependent convective flows which arise in shallow cavities filled with low-Prandtí-number liquids when submitted to a horizontal temperature gradient. In similar situations several former numerical results have been shown to disagree about the determination of the threshold of oscillat...
Chapter
The computations are done using the Navier-Stokes equations within the vorticity-stream function formulation. These equations associated (in the case Pr=0.015) with the temperature equations are solved using the spectral method proposed in [1]. The time-differencing makes use of the semi-implicit Adams-Bashforth / second-order backward Euler scheme...
Article
A Chebyshev collocation method is proposed for the computation of laminar flame propagation in a two-dimensional gaseous medium. The method is based on a domain decomposition technique associated with co-ordinate transforms to map the infinite physical subdomains into finite computational ones. The influence matrix method is used to handle the patc...
Article
A Chebyshev collocation method for solving the unsteady two-dimensional Navier–Stokes equations in vorticity–streamfunction variables is presented and discussed. The discretization in time is obtained through a class of semi-implicit finite difference schemes. Thus at each time cycle the problem reduces to a Stokes-type problem which is solved by m...
Article
This paper presents a collection of experiments in which we investigate the question of grid adaption for flame propagation problems. Local refinement or mesh deformation procedures are applied to a sample of spatial approximations, including finite elements, finite differences and spectral methods. Comparisons are performed using the simplified th...
Article
This paper describes some aspects of the use of spectral methods for the numerical solution of systems of stiff partial differential equations. It is shown that despite the high spatial precision of these methods, a reasonable accuracy can only be attained with a large number of number and therefore, some kind of adaptive 'gridding' is necessary. A...
Article
The flow induced by a heat source in a fluid stratified horizontally by salinity gradients is investigated analytically. The influence-matrix technique is combined with a multidomain Fourier-Chebyshev method to solve the Navier-Stokes equations in a vorticity/stream-function formulation. Results are presented in extensive graphs and briefly charact...
Article
Recent advances in the numerical solution of the compressible Navier-Stokes equations are reviewed, with a focus on laminar flow problems. Topics addressed include the formulation of the equations, boundary conditions, approximations, spatial discretization, and mesh adaptation. Particular attention is given to time discretization, artificial visco...
Article
Full-text available
Spectral methods are proposed for the solution of the Navier-Stokes, energy and species equations. The closure of the governing system by using the ideal gas law and/or the Boussinesq approximation is discussed for binary mixture flows. Applications to crystal growth (by physical vapour transport) are presented for both microgravity and earth envir...
Article
Full-text available
Spectral methods (Fourier Galerkin, Fourier pseudospectral, Chebyshev Tau, Chebyshev collocation, spectral element) and standard finite differences are applied to solve the Burgers equation with small viscosity (). This equation admits a (nonsingular) thin internal layer that must be resolved if accurate numerical solutions are to be obtained. From...
Article
The bases of spectral methods (weighted residuals, collocation) using Fourier and Chebyshev approximation are introduced. Applications to the solution of simple partial differential equations are described. Methods for solving the Navier-Stokes equations are presented.
Article
Full-text available
A method for computing unsteady two dimensional incompressible flows, using the vorticity and the stream function as dependent variables approximated by Chebyshev polynomial expansions is presented. Boundary conditions for the vorticity are derived by the influence matrix technique. The theoretical and numerical difficulties associated with the two...
Article
A pseudo-spectral method for the calculation of steady binary gas mixture flows found in vapor crystal growth is presented. The method uses a pseudo-unsteady approach. The scheme is implicit for the diffusion terms and mainly explicit for the other terms. The spatial approximation uses Chebyshev polynomial expansions with a collocation technique. T...
Article
The stability of time-discretization schemes associated with the collocation-Chebyshev method is studied for the advection-diffusion equation. These schemes are applied to the Burgers equation and to the Navier-Stokes equations solved by psuedo-unsteady methods. Convergence and accuracy of finite difference, spectral, and hybrid methods are listed....
Article
The basic principles of finite difference and finite-volume methods for solving the Navier-Stokes equations, and approximations of them are presented. Each approach is illustrated by applying it to two and three-dimensional flow examples. Only the laminar case is considered.
Article
Numerical approaches are discussed, taking into account general equations, finite-difference methods, integral and spectral methods, the relationship between numerical approaches, and specialized methods. A description of incompressible flows is provided, giving attention to finite-difference solutions of the Navier-Stokes equations, finite-element...
Article
This paper presents a review of various pseudo-unsteady techniques based upon different principles used for the numerical calculation of inviscid or viscous flows. Two types of approaches to devise pseudo-unsteady methods are considered. The first, which relates to the mathematical formulation of the time-dependent problem, consists essentially in...
Article
A theoretical and numerical study is made of a multigrid method proposed by R. H. Ni (1982) for solving the Euler equations. Various parameters in the restriction and interpolation operators are introduced, and a class of schemes is defined. The stability analysis is based on a Fourier analysis well suited to the multigrid method. Then, it is point...
Article
The general Hermitian technique is described and applied to the solution of a linear ordinary differential equation. Questions related to boundary conditions are discussed, and the accuracy of the numerical solution obtained is evaluated. The Hermitian method is then applied to the solution of the steady Navier-Stokes equations. The solution is obt...
Chapter
The Navier-Stokes equations without external force can be written in dimensionless form $$\frac{{\partial V}}{{\partial t}} + A(V) + \nabla p = \frac{1}{{\operatorname{Re} }}{\nabla ^2}V$$ (6.1.1a) $$\nabla \cdot V = 0$$ (6.1.1b) where A(V) is expressed by one of the following equations according to the choice of conservative or nonconservative for...
Chapter
The previous discussions have presented a variety of numerical methods that can be employed to generate numerical solutions to flow problems. The apparent difference in these methods is not always easy to point out explicitly. However, we can demonstrate some explicit relationships between finite-difference, integral, and spectral methods for simpl...
Chapter
The field of spectral methods in fluid mechanics is still very much in the developmental phase. As a result, application of the method is still somewhat of an art. At this time, the spectral method has been applied to solve primitive-variable, stream-function vorticity and stream-function only formulations. In each of these formulations, the unstea...
Chapter
The Navier—Stokes equations apply equally to turbulent or laminar flows. Their completeness for turbulence computation have yet to be fully tested, however. For the present we will assume that the equations are adequate. The computation of turbulent flows with the above equations is a difficult task since the fully unsteady nature of turbulence mus...
Chapter
The introduction of the computer into engineering has resulted in the growth of a completely new field termed computational fluid dynamics. This field has led to the development of new mathematical methods for solving the equations of fluid mechanics. The improved methods have permitted advanced simulations of flow phenomena on the computer for wid...
Chapter
The solution of two-dimensional flow problems by finite-element methods can employ either the stream-function vorticity or the primitive-variable formulations along with least-squares or Galerkin techniques. The number of possible combinations including element geometry, function approximations, and formulation become extremely large. As a result,...
Chapter
Until recently, numerical methods for solving fluid-flow problems have been dominated by finite-difference approximations. These methods are powerful and play a major role in problem solutions. In this chapter we attempt to present the fundamental advances and insight into these methods. We begin by discussing the concept of discrete pointwise appr...
Chapter
In Chapter 2 the discussion centered on the use of finite-difference approximations to solve the differential equations of fluid flow. Various alternatives to the finite-difference approach are available, including integral approaches such as moment methods, least squares, Galerkin techniques, and Rayleigh—Ritz variational formulations. Each of the...
Chapter
In the area of inviscid compressible flows there are a variety of practical problems that arise in everyday engineering applications. These include rocket nozzle flows, aircraft and missile engine inlet flows, reentry vehicle and rocket aerodynamics, blast fields generated by different types of energy release, and aircraft flow fields. One can, of...
Chapter
In addition to the finite-difference, finite-element, and spectral techniques, there are computational techniques that do not fall directly into these categories. These techniques can be useful and, therefore, we have included them in this chapter.
Chapter
The calculations of steady viscous flows based on the Navier—Stokes equations are generally conducted with the unsteady equations by considering the limit of large time. Until recently most of the solutions were developed with explicit finite-difference schemes such as those of Thommen (1966) and MacCormack (1969). Such explicit schemes are easy to...
Article
In developing this book, we decided to emphasize applications and to provide methods for solving problems. As a result, we limited the mathematical devel­ opments and we tried as far as possible to get insight into the behavior of numerical methods by considering simple mathematical models. The text contains three sections. The first is intended to...
Article
A numerical model is developed for the effects of a jet introduced into a channel containing a stratified fluid at rest. The channel is regarded as semi-infinite, allowing the free fluid to escape downstream. A Boussinesq approximation is used for the Navier-Stokes equations, with disturbances of temperature and pressure linked to the temperature o...
Article
Two numerical approaches are presented for the computation of viscous compressible flows at high Reynolds' numbers. In the first approach, named global approach, the whole flow field, which includes viscous and inviscid regions, is determined as the solution of a single set of equations, which may be the full Navier-Stokes equations, or some approx...
Article
Numerical solutions of the initial value problems associated with linear or nonlinear hyperbolic systems, with application to the equations of one-dimensional unsteady gas dynamics, are studied. After recalling the bases of finite difference scheme theory for the solution of linear time-dependent equations, the concept of the equivalent system (the...
Article
Construction of various fourth-order Hermitian methods for a simple linear differential equation is studied and an error analysis is performed. A Hermitian method is then proposed for the solution of the steady Navier-Stokes equations within the framework of the velocity-pressure formulation.
Article
Fourth-order accurate Hermitian methods are proposed for the solution of the steady Navier-Stokes equations within the framework of the velocity-pressure formulation. Two types of Hermitian formulae for second derivatives (implicit and explicit formulae) are considered. The Navier-Stokes equations are solved by the method of artificial compressibil...
Article
A Hermitian-type finite-difference scheme for the numerical solution of Navier-Stokes equations is developed. The velocity-pressure formulation of the Navier-Stokes equations is treated by use of the scheme. The solution algorithm, based on an alternating-direction method, is tested for the case of an exact solution.
Article
The unsteady laminar flow due to the penetration of a horizontal jet of constant density into a stratified fluid is considered. A numerical solution of the Navier–Stokes equations under the Boussinesq approximation is obtained by means of an implicit finite-difference method. Results for different values of the Reynolds and internal Froude numbers...
Article
The paper considers a numerical model of two-dimensional convection by relatively heavy motile particles suspended in a viscous fluid. The motility consists of a steady motion upward, and may include in addition a random walk; moreover, the particles are advected by the fluid motion. The numerical method utilizes an implicit finite-difference schem...
Article
THE TWO-DIMENSIONAL SUPERSONIC FLOW ON A PARABOLIC, SEMI-INFINITE OBSTACLE WAS CALCULATED, AT MODERATE REYNOLDS NUMBERS, BY NUMERICAL SOLUTION OF THE UNSTEADY NAVIER-STOKES EQUATIONS. THE STEADY-STATE SOLUTION IS OBTAINED AS THE LIMIT, FOR INFINITE TIME, OF THE UNSTEADY-STATE SOLUTION. A NUMERICAL SEMI-IMPLICIT SCHEME IS USED, WITH ONE STEP IN TIME...
Article
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 presented. The following aspects of numerical methods are considered: limitation of the comp...
Article
Numerical methods for the solution of the Navier-Stokes equations for compressible fluids are discussed. A short review of the Navier-Stokes equations and of their qualitative mathematical properties, and a discussion of their interest in aerodynamics problems are presented. The following aspects of numerical methods are considered: limitation of t...
Article
The steady flow at low Reynolds number of an incompressible viscous flow in a channel with an abrupt change of section is considered. The problem is solved by an iterative matching of an expansion of the solution near the convex angle with a numerical solution of the Navier-Stokes equations computed outside the neighbourhood of the corner. The pres...
Article
The steady flow at low Reynolds number of an incompressible viscous flow in a channel with an abrupt change of section is considered. The problem is solved by an iterative matching of an expansion of the solution near the convex angle with a numerical solution of the Navier-Stokes equations computed outside the neighborhood of the corner. The prese...
Article
In this paper, the noncentered difference scheme given by Mac Cormack for the numerical solution of the gas dynamics equations is studied from a theoretical point of view, and its computational properties are tested for shock propagation problems. By means of a nonlinear analysis for a scalar equation, it is shown why there is no oscillations in th...

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