The modeling of transient effects in burners is becoming increasingly important. The problem of ensuring the safe performance of an industrial burner, for example, is much more difficult during the startup or shutdown phases of operation. The peak formation of pollutants is also much more dependent on transient behavior, in particular, on peak temperatures, than on average operating conditions. In this paper we present a new methodology for the modeling of unsteady, nonpremixed, reacting flow in industrial burners. The algorithm uses a second-order projection method for unsteady, low-Mach number reacting flow and accounts for species diffusion, convective and radiative heat transfer, viscous transport, turbulence, and chemical kinetics. The time step used by the method is restricted solely by an advective CFL condition. The methodology is applicable only in the low-Mach number regime (M < .3), typically met in industrial burners. The projection method for low-Mach number reacting flow is an extension of a higher-order projection method for incompressible flow [9, 5, 3,4] to the low-Mach number equations of reacting flow. Our method is based on an approximate projection formulation. Radiative transport is modeled using the discrete ordinates method. The main goal of this work is to introduce and investigate themore » simulation of burners using a higher-order projection method for low-Mach number combustion. As such, the methodology is applied here only to axisymmetric flow in gas-fired burners for which the boundaries can be aligned with a rectangular grid. The perfect gas law is also assumed. In addition, we use a one-step reduced kinetics mechanism, a {kappa} {minus} {epsilon} model for turbulent transport, and a simple turbulent combustion model.« less

CASTRO is a new, multi-dimensional, Eulerian AMR radiation-hydrodynamics code designed for astrophysical simulations. The code includes routines for various equations of state and nuclear reaction networks, and can be used with Cartesian, cylindrical or spherical coordinates. Time integration of the hydrodynamics equations uses unsplit PPM with new limiters. Self-gravity can be calculated on the adaptive hierarchy using a simple monopole approximation or a full Poisson solve for the gravitational potential. CASTRO includes gray and multigroup radiation diffusion. Multi-species neutrino diffusion for supernovae is nearing completion. The adaptive framework of CASTRO is based on an time-evolving hierarchy of nested rectangular grids with refinement in both space and time; the entire implementation is designed to run on thousands of processors. Our initial applications of CASTRO include Type Ia and Type II supernovae. 1Lawrence Berkeley National Lab; 2Lawrence Livermore National Lab; 3UC Santa Cruz; 4University of Michigan; 5Princeton University; 6Stony Brook University

We present a new code, CASTRO, that solves the multicomponent compressible hydrodynamic equations for astrophysical flows including self-gravity, nuclear reactions, and radiation. CASTRO uses an Eulerian grid and incorporates adaptive mesh refinement (AMR). Our approach to AMR uses a nested hierarchy of logically rectangular grids with simultaneous refinement in both space and time. The radiation component of CASTRO will be described in detail in the next paper, Part II, of this series.

We develop a scheme for the system coupling the radiation diffusion and matter energy balance equations. The method is based on fully implicit, first-order, backward Euler differencing; Picard–Newton iterations solve the nonlinear system. We show that iterating on the radiation energy density and the emission source is more robust. Since the Picard–Newton scheme may not converge for all initial conditions and time steps, pseudo-transient continuation (Ψtc) is introduced. The combined Ψtc–Picard–Newton scheme is analyzed. We derive conditions on the Ψtc parameter that guarantee physically meaningful iterates, e.g., positive energies. Successive Ψtc iterates are bounded and the radiation energy density and emission source tend to equilibrate. The scheme is incorporated into a multiply dimensioned, massively parallel, Eulerian, radiation–hydrodynamic computer program with automatic mesh refinement (AMR). Three examples are presented that exemplify the scheme's performance. (1) The Pomraning test problem that models radiation flow into cold matter. (2) A similar, but more realistic problem simulating the propagation of an ionization front into tenuous hydrogen gas with a Saha model for the equation-of-state. (3) A 2D axisymmetric (R,Z) simulation with real materials featuring jetting, radiatively driven, interacting shocks.

Block-structured meshes provide the ability to concentrate grid points and computational effort in interesting regions of a flow field, without sacrificing the efficiency and low memory requirements of a regular grid. We describe an algorithm for simulating radiation diffusion on such a mesh, coupled to multi-fluid gasdynamics. Conservation laws are enforced by using locally conservative difference schemes along with explicit synchronization operations between different levels of refinement. In unsteady calculations each refinement level is advanced at its own optimal timestep. Particular attention is given to the appropriate coupling between the fluid energy and the radiation field, the behavior of the discretization at sharp interfaces, and the form of synchronization between levels required for energy conservation in the diffusion process. Two- and three-dimensional examples are presented, including parallel calculations performed on an IBM SP-2.

We present a numerical method using the level set approach for solving incompressible two-phase flow with surface tension. In the level set approach, the free surface is represented as the zero level set of a smooth function; this has the effect of replacing the advection of density, which has steep gradients at the free surface, with the advection of the level set function, which is smooth. In addition, the free surface can merge or break up with no special treatment. We maintain the level set function as the signed distance from the free surface in order to accurately compute flows with high density ratios and stiff surface tension effects. In this work, we couple the level set scheme to an adaptive projection method for the incompressible Navier–Stokes equations, in order to achieve higher resolution of the free surface with a minimum of additional expense. We present two-dimensional axisymmetric and fully three-dimensional results of air bubble and water drop computations.

In this paper we present an adaptive projection method for modeling unsteady, lowMachreacting flow in an unconfined region. The equations we solve are based on amodel for low-Mach number combustion that consists of the evolution equations fordensity, species concentrations, enthalpy, and momentum coupled with a constraint onthe divergence of the flow. The algorithm is based on a projection methodology in whichwe first advance the evolution equations and then solve an elliptic equation to...

We describe a new code for numerical solution of three-dimensional self-gravitational hydrodynamics problems. This code utilizes the technique of local adaptive mesh refinement (AMR), employing multiple grids at multiple levels of resolution and automatically and dynamically adding and removing these grids as necessary to maintain adequate resolution. This technology allows solution of problems that would be prohibitively expensive with a code using fixed resolution, and it is more versatile and efficient than competing methods of achieving variable resolution. In particular, we apply this technique to simulate the collapse and fragmentation of a molecular cloud, a key step in star formation. The simulation involves many orders of magnitude of variation in length scale as fragments form at positions that are not a priori discernible from general initial conditions. In this paper, we describe the methodology behind this new code and present several illustrative applications. The criterion that guides the degree of adaptive mesh refinement is critical to the success of the scheme, and, for the isothermal problems considered here, we employ the Jeans condition for this purpose. By maintaining resolution finer than the local Jeans length, we set new benchmarks of accuracy by which to measure other codes on each problem we consider, including the uniform collapse of a finite pressured cloud. We find that the uniformly rotating, spherical clouds treated here first collapse to disks in the equatorial plane and then, in the presence of applied perturbations, form filamentary singularities that do not fragment while isothermal. Our results provide numerical confirmation of recent work by Inutsuka & Miyama on this scenario of isothermal filament formation.

The discrete ordinates form of the radiative transport equation (RTE) is spatially discretized and solved using an adaptive mesh refinement (AMR) algorithm. This technique permits local grid refinement to minimize spatial discretization error of the RTE. An error estimator is applied to define regions for local grid refinement; overlapping refined grids are recursively placed in these regions; and the RTE is then solved over the entire domain. The procedure continues until the spatial discretization error has been reduced to a sufficient level. The following aspects of the algorithm are discussed: error estimation, grid generation, communication between refined levels, and solution sequencing. This initial formulation employs the step scheme and is valid for absorbing and isotropically scattering media in two-dimensional enclosures. The utility of the algorithm is tested by comparing the convergence characteristics and accuracy to those of the standard single-grid algorithm. For two simple benchmark problems, the AMR algorithm maintains the convergence characteristics of the standard single-grid algorithm, but it does not provide any efficiency gains due to a lack of disparate spatial scales. In a third, more localized problem, however, the AMR algorithm demonstrates significant memory and CPU time reductions.

In Sussman, Smereka and Osher (1994), a numerical method using the level set approach was formulated for solving incompressible two-phase ow with surface tension. In the level set approach, the interface is represented as the zero level set of a smooth function; this has the e ect of replacing the advection of density, which has steep gradients at the interface, with the advection of the level set function, which is smooth. In addition, the interface can merge or break up with no special treatment. We maintain the level set function as the signed distance from the interface in order to robustly compute ows with high density ratios and sti surface tension e ects. In this work, we couple the level set scheme to an adaptive projection method for the incompressible Navier-Stokes equations, in order to achieve higher resolution of the interface with a minimum of additional expense. We present two-dimensional axisymmetric and fully three-dimensional results of air bubble and water drop computations.

We demonstrate with a new three-dimensional adaptive mesh refinement code that perturbations arising from discretization of the equations of self-gravitational hydrodynamics can grow into fragments in multiple-grid simulations, a process we term “artificial fragmentation.” We present star formation calculations of isothermal collapse of dense molecular cloud cores. In simulation of a Gaussian-profile cloud free of applied perturbations, we find artificial fragmentation can be avoided across the isothermal density regime by ensuring the ratio of cell size to Jeans length, which we call the Jeans number, J ≡ Δx/λJ, is kept below 0.25. We refer to the constraint that λJ be resolved as the Jeans condition. When an m=2 perturbation is included, we again find it necessary to keep J≤0.25 to achieve a converged morphology. Collapse to a filamentary singularity occurs without fragmentation of the filament, in agreement with the predictions of Inutsuka & Miyama. Simulation beyond the time of this singularity requires an arresting agent to slow the runaway density growth. Physically, the breakdown of isothermality due to the buildup of opacity acts as this agent, but many published calculations have instead used artificial viscosity for this purpose. Because artificial viscosity is resolution dependent, such calculations produce resolution-dependent results. In the context of the perturbed Gaussian cloud, we show that use of artificial viscosity to arrest collapse results in significant violation of the Jeans condition. We also show that if the applied perturbation is removed from such a calculation, numerical fluctuations grow to produce substantial fragments not unlike those found when the perturbation is included. These findings indicate that calculations that employ artificial viscosity to halt collapse are susceptible to contamination by artificial fragmentation. The Jeans condition has important implications for numerical studies of isothermal self-gravitational hydrodynamics problems insofar as it is a necessary but not, in general, sufficient condition for convergence.

Embedded boundary methods model e uid e ows in complex geometries by treating boundaries as tracked interfaces in a regular mesh. Though often referred to as Cartesian grid methods, they are equally well-suited to axisymmetric problems. This paper describes a formulation of the discrete ordinates method for radiative transfer calculations with embedded boundaries. The method uses diamond-difference stencils in the interior with a conservative extension to boundary cells based on a volume-of-e uid approach. Numerical examples are presented in both two-dimensional Cartesian and axisymmetric geometries, including a model of the BERL 300-kW natural gas burner.

A higher-order, embedded boundary projection method for axisymmetric, unsteady, low-Mach number combustion is used to model a natural gas flame from a 300 kW IFRF burner in the Burner Engineering Research Laboratory (BERL) at Sandia National Laboratory under hot wall conditions. The numerical predictions presented are the late simulated-time results of a computation of unsteady flow in the furnace. The predictions are compared both with measurements completed in the BERL as part of the GRI SCALING 400 Project and with results from a steady-state axisymmetric reacting flow code in order to evaluate the combustion model and the numerical method. The results compare favorably with the experimental data.

A new methodology for the modeling of unsteady, nonpremixed, axisymmetric reacting flow in industrial furnaces is presented. The method is an extension of previous work by the authors to complex geometries, multistep kinetics mechanisms, and realistic properties, especially thermochemical data. The walls of the furnace are represented as an embedded boundary in a uniform, rectangular grid. The grid then consists of uniform rectangular cells except at the furnace wall where irregular (mixed) cells may be present. We use finite volume differencing techniques for the convective, viscous, and radiative heat transport terms in the mixed cells, while a finite element-based technique is used to solve the elliptic equation arising from the low-Mach number formulation. Results from the simulation of an experimental natural gas-fired furnace are shown.