Alexandre J. Chorin

Numerical Solution of the Navier-Stokes Equations*

A finite-difference method for solving the time-dependent Navier-Stokes equations for an incompressible fluid is introduced. This method uses the primitive variables, i.e. the velocities and the pressure, and is equally applicable to problems in two and three space dimensions. Test problems are solved, and an application to a three-dimensional convection problem is presented.

Numerical solution of the Navier-Stokes equations

A finite-difference method for solving the time-dependent Navier- Stokes equations for an incompressible fluid is introduced. This method uses the primitive variables, i.e. the velocities and the pressure, and is equally applicable to problems in two and three space dimensions. Test problems are solved, and an ap- plication to a three-dimensional convection problem is presented.

NUMERICAL STUDY OF SLIGHTLY VISCOUS FLOW

A numerical method for solving the time-dependent Navier–Stokes equations in two space dimensions at high Reynolds number is presented. The crux of the method lies in the numerical simulation of the process of vorticity generation and dispersal, using computer-generated pseudo-random numbers. An application to flow past a circular cylinder is presented.

Random choice solution of hyperbolic systems

Abstract A random choice method for solving nonlinear hyperbolic systems of conservation laws is presented. The method is rooted in Glimms constructive proof that such systems have solutions. The solution is advanced in time by a sequence of operations which includes the solution of Riemann problems and a sampling procedure. The method can describe a complex pattern of shock wave and slip line interactions without introducing numerical viscosity and without a special handling of discontinuities. Examples are given of applications to one- and two-dimensional gas flow problems.

Vortex sheet approximation of boundary layers

Abstract A grid free method for approximating incompressible boundary layers is introduced. The computational elements are segments of vortex sheets. The method is related to the earlier vortex method; simplicity is achieved at the cost of replacing the Navier-Stokes equations by the Prandtl boundary layer equations. A new method for generating vorticity at boundaries is also presented; it can be used with the earlier vortex method. The applications presented include (i) flat plate problems, and (ii) a flow problem in a model cylinder-piston assembly, where the new method is used near walls and an improved version of the random choice method is used in the interior. One of the attractive features of the new method is the ease with which it can be incorporated into hybrid algorithms.

FLAME ADVECTION AND PROPAGATION ALGORITHMS

We present a simple algorithm for approximating the motion of a thin flame front of arbitrary shape and variable connectivity, which is advected by a fluid and which moves with respect to the fluid in the direction of its own normal. As an application, we examine the wrinkling of a flame front by a periodic array of vortex structures.

Numerical modelling of turbulent flow in a combustion tunnel

A numerical technique is presented for the analysis of turbulent flow associated with combustion. The technique uses Chorin’s random vortex method (r.v.m .), an algorithm capable of tracing the action of elementary turbulent eddies and their cumulative effects without imposing any restriction upon their motion. In the past, the r.v.m . has been used with success to treat non-reacting turbulent flows, revealing in particular the mechanics of large-scale flow patterns, the so-called coherent structures. Introduced here is a flame propagation algorithm , also developed by Chorin, in conjunction with volume sources modelling the mechanical effects of the exothermic process of combustion. As an illustration of its use, the technique is applied to flow in a combustion tunnel w here the flame is stabilized by a back-facing step. Solutions for both non-reacting and reacting flow fields are obtained. Although these solutions are restricted by a set of far-reaching idealizations, they nonetheless mimic quite satisfactorily the essential features of turbulent combustion in a lean propane—air mixture that were observed in the laboratory by means of high speed schlieren photography.

The evolution of a turbulent vortex

We examine numerically the evolution of a perturbed vortex in a periodic box. The fluid is inviscid. We find that the vorticity blows up. The support of theL2 norm of the vorticity converges to a set of Hausdorff dimension ∼2.5. The distribution of the vorticity seems to converge to a lognormal distribution. We do not observe a convergence of the higher statistics towards universal statistics, but do observe a strong temporal intermittency.

Discretization of a vortex sheet with an example of roll-up

Abstract The point vortex approximation of a vortex sheet in two space dimensions is examined and a remedy for some of its shortcomings is suggested. The approximation is then applied to the study of the roll-up of a vortex sheet induced by an elliptically loaded wing.

Optimal prediction with memory

Abstract Optimal prediction methods estimate the solution of nonlinear time-dependent problems when that solution is too complex to be fully resolved or when data are missing. The initial conditions for the unresolved components of the solution are drawn from a probability distribution, and their effect on a small set of variables that are actually computed is evaluated via statistical projection. The formalism resembles the projection methods of irreversible statistical mechanics, supplemented by the systematic use of conditional expectations and new methods of solution for an auxiliary equation, the orthogonal dynamics equation, needed to evaluate a non-Markovian memory term. The result of the computations is close to the best possible estimate that can be obtained given the partial data. We present the constructions in detail together with several useful variants, provide simple examples, and point out the relation to the fluctuation–dissipation formulas of statistical physics.