Jay P. Boris

Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works

Abstract This paper describes a class of explicit, Eulerian finite-difference algorithms for solving the continuity equation which are built around a technique called “flux correction.” These flux-corrected transport algorithms are of indeterminate order but yield realistic, accurate results. In addition to the mass-conserving property of most conventional algorithms, the FCT algorithms strictly maintain the positivity of actual mass densities so steep gradients and inviscid shocks are handled particularly well. This first paper concentrates on a simple one-dimensional version of FCT utilizing SHASTA, a new transport algorithm for the continuity equation, which is described in detail.

New insights into large eddy simulation

Fluid dynamic turbulence is one of the most challenging computational physics problems because of the extremely wide range of time and space scales involved, the strong nonlinearity of the governing equations, and the many practical and important applications. While most linear fluid instabilities are well understood, the nonlinear interactions among them makes even the relatively simple limit of homogeneous isotropic turbulence difficult to treat physically, mathematically, and computationally. Turbulence is modeled computationally by a two-stage bootstrap process. The first stage, direct numerical simulation, attempts to resolve the relevant physical time and space scales but its application is limited to diffusive flows with a relatively small Reynolds number (Re). Using direct numerical simulation to provide a database, in turn, allows calibration of phenomenological turbulence models for engineering applications. Large eddy simulation incorporates a form of turbulence modeling applicable when the large-scale flows of interest are intrinsically time dependent, thus throwing common statistical models into question. A promising approach to large eddy simulation involves the use of high-resolution monotone computational fluid dynamics algorithms such as flux-corrected transport or the piecewise parabolic method which have intrinsic subgrid turbulence models coupled naturally to the resolved scales in the computed flow. The physical considerations underlying and evidence supporting this monotone integrated large eddy simulation approach are discussed.

Flux-corrected transport II: Generalizations of the method

Abstract The recently developed method of Flux-Corrected Transport (FCT) can be applied to many of the finite-difference transport schemes presently in use. The result is a class of improved algorithms which add to the usual desirable properties of such schemes—conservation, stability, second-order (in some cases) accuracy, etc.—the property of maintaining the intrinsic positivity of quantities like density, energy density, and pressure. Illustrations are given for algorithms of the Lax-Wendroff, leapfrog, and upstreaming types. The errors introduced by the flux-correction process which lies at the heart of the method are cataloged and their effect described. Phoenical FCT, a refinement which minimizes residual diffusive errors, is analyzed. Applications of FCT to general fluid systems, multidimensions, and curvilinear geometry are described. The results of computer tests are shown in which the various types of FCT are compared with one another and with some conventional algorithms.

Flux-corrected transport. III. Minimal-error FCT algorithms

Abstract This paper presents an error analysis of numerical algorithms for solving the convective continuity equation using flux-corrected transport (FCT) techniques. The nature of numerical errors in Eulerian finite-difference solutions to the continuity equation is analyzed. The properties and intrinsic errors of an “optimal” algorithm are discussed and a flux-corrected form of such an algorithm is demonstrated for a restricted class of problems. This optimal FCT algorithm is applied to a model test problem and the error is monitored for comparison with more generally applicable algorithms. Several improved FCT algorithms are developed and judged against both standard flux-uncorrected transport algorithms and the optimal algorithm. These improved FCT algorithms are found to be four to eight times more accurate than standard non-FCT algorithms, nearly twice as accurate as the original SHASTA FCT algorithm, and approach the accuracy of the optimal algorithm.

Theory and Simulation of Turbulent Heating by the Modified Two-Stream Instability.

Results of an analytical and numerical study of the nonresonant, modified plasma two‐stream instability, which is driven by relative streaming of electrons and ions across a magnetic field B0 are presented. The instability has characteristic frequency and growth rate comparable to the lower‐hybrid frequency. The linear theory is discussed both in the electrostatic and fully electromagnetic cases, and a detailed numerical study of the dependence of the unstable roots of the dispersion relation for a wide range of plasma parameters is presented. The nonlinear theory includes discussions of (1) quasilinear theory, (2) trapping, which is responsible for nonlinear stabilization, (3) a derivation of a fully nonlinear scaling law which shows how results scale with electron‐ion mass ratio, and (4) the effect of cross‐field vortex‐like motion caused by turbulence induced E × B drifts. One‐and two‐dimensional computer simulations with dense k‐space spectra are presented in support of this theory. The simulations sh...

Detailed Modelling of Combustion Systems

Abstract : The purpose of this paper is to acquaint the reader with some of the basic principles of detailed modelling as applied to combustion systems. Detailed modelling is also known as numerical simulation. It can be used to describe the chemical and physical evolution of a complex reactive flow system by solving numerically the governing time-dependent conservation equations for mass, momentum and energy. Solving these equations requires input data such as the species present, the chemical reactions that can occur, transport coefficients for viscosity, thermal conductivity, molecular diffusion, and thermal diffusion, the equation of state for the various materials present, and a set of boundary, source and initial conditions. Given this information, the equations contain in principle all the information we might want from the largest macroscopic space scales down to the point where the fluid approximation itself breaks down. Flame, detonation, turbulence phenomena, and all multidimensional effects are included in the solutions of these equations. An important goal of detailed modelling is to develop a computational model with a well-understood range of validity. This model can then be used in a predictive role to evaluate the feasibility and validity of new concepts. It can also be used to interpret experimental measurements, to extend our knowledge to new parameter regimes, and perhaps as an engineering design tool. Throughout these various applications, the model may serve as an excellent way to test our understanding of the interactions of the individual physical processes which control the behavior of a reactive flow system.

Determination of detonation cell size and the role of transverse waves in two-dimensional detonations☆

Abstract Two-dimensional time-dependent numerical simulations have been performed to study the structure and propagation of self-sustained detonations. The simulations are first used to develop a systematic approach for determining the detonation cell size. This approach involves simulating systems with channel widths both larger and smaller than the transverse cell spacing. The cell size estimated using this approach is compared with experimental data. The simulations also provide insight into some aspects of the mechanism by which a two-dimensional, self-sustained detonation propagates. The evolution of the curvature of the transverse wave appears to be a crucial feature. It is shown that depending on the curvature of the transverse wave at the time of its reflection from either a neighboring transverse wave or a wall, flattened cells or pockets of unreacted gas can be formed.

Weak and strong ignition. I. Numerical simulations of shock tube experiments

Abstract Detailed one-dimensional calculations have been performed to simulate reflected shock tube experiments in the weak and strong ignition regime in hydrogenoxygenargon mixtures. It is found that the experiments and simulations agree well in the strong ignition case studied. In the weak ignition case, the simulations show the same qualitative behavior as the experiments. Here ignition starts at a distance away from the reflecting wall at a time much earlier than the calculated chemical induction time. This latter effect is shown to arise because of the sensitivity of the chemical induction time to fluctuations in the calculation. In the calculations these fluctuations arise because of numerical inaccuracies. In experiments, they can arise from a number of sources including nonuniformities in the incident shock wave leading to nonuniform reflection, thermal conduction to the walls, and interactions with boundary layers.

A barely implicit correction for flux-corrected transport

Abstract The barely implicit correction (BIC) removes the stringent limit on the timestep imposed; by the sound speed in explicit methods. This is done by adding one elliptic equation which has to be solved implicitly. BIC is combined with the flux-corrected transport algorithm in order to represent sharp gradients in subsonic flows accurately. The resultant conservative algorithm costs about the same per timestep as a single explicit timestep calculated using an optimized FCT module. Several examples show the techniques ability to solve nearly incompressible flows very economically.

Numerical simulations of detonations in hydrogen-air and methane-air mixtures

Detailed numerical simulations of supersonic reactive flow and gas phase detonation problems are very expensive due to their computer time and memory requirements. The bulk of this cost is in integrating the ordinary differential equations describing chemical reactions. A global induction parameter model has thus been developed which describes the chemical induction time of a mixture and allows for release of energy over a finite time period. The specific gases for which it has been calibrated are stoichiometric mixtures of hydrogen and methane in air. The relatively inexpensive induction parameter model is then used in time-dependent one- and two-dimensional simulations of supersonic reactive flows.