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In a commonly implemented version of the vorticity-velocity formulation, the governing equations for the fluid dynamics are expressed as two Poisson-like velocity equations together with the vorticity transport equation. However, for some flows with large vorticity gradients, spurious mass loss or gain can be observed. In order to conserve mass, a modification to the vorticity-velocity formulation is proposed, involving the substitution of the kinematic definition of vorticity in certain terms of the fluid-dynamic equations. This modified formulation results in a broader computational stencil when the equations are in a second-order-accurate discretized form, and a stronger coupling between the predicted vorticity and the curl of the predicted velocity field. The resulting system of elliptic equations - which includes the energy and species transport equations for the reacting flow case - is discretized with finite differences on a nonstaggered grid and is then solved using Newton's method. Both the unmodified and modified vorticity-velocity formulations are applied to two problems with high vorticity gradients: (1) incompressible, axisymmetric fluid flow through a suddenly expanding pipe and (2) a confined, axisymmetric laminar flame with detailed chemistry and multicomponent transport, generated on a burner whose inner tube extends above the burner surface. The modified formulation effectively eliminates the spurious mass loss in the two test cases to within an acceptable tolerance. The two cases demonstrate the broader range of applicability of the modified formulation, as compared with the unmodified formulation.

A novel, stable, implicit compact scheme solver that is higher order in space, suitable for modeling steady-state and time-dependent phenomena on nonuniform grids for one-dimensional configurations, is presented. Several properties of compact scheme discretizations are introduced to develop efficient algorithms for Jacobian matrix generation and Jacobian-vector multiplication using a new component form for Jacobian operations. Composite nonuniform grids are introduced that enable the implicit compact scheme solver to achieve sixth order accuracy. A robust Newton's method is employed with explicit generation of Jacobian matrices. Superior resolution characteristics of the implicit compact scheme solver are demonstrated with several steady-state and time-dependent problems for the Burgers equation. The example of the solution of stiff flame problem is given. An analysis of spectral properties of Jacobian matrices is presented, which shows that the condition number and the eigenvalue distributions behave similarly to those found in Jacobians associated with low-order discretizations. Two sparsification strategies are developed for the systematic approximation of a dense Jacobian aimed at the practical implementation of linear system preconditioning through partial Jacobians.

A new solution-adaptive gridding method has been developed for the solution of discretized systems of coupled nonlinear elliptic partial differential equations on rectangular domains. Such a method is required for the numerical solution of realistic combustion problems, in which physical quantities may vary by orders of magnitude over one-tenth of a millimeter at atmospheric pressure, or over micrometers at higher pressures. The local rectangular refinement (LRR) method maintains orthogonality at grid-line intersections but lifts the tensor product restriction common to traditional grids, producing unstructured grids. Governing equations are discretized throughout the domain using newly derived forms, and Newton's method is used to solve the resulting system. On a simple test case with a known solution, the LRR method and its new discretizations are found to be more accurate than gridding methods representative of those appearing previously in the literature. For the more realistic problem of nonreacting driven square cavity flow, the LRR solution agrees very well with previously published data. When the LRR method is applied to a practical reacting flow (a rich axisymmetric laminar Bunsen flame with complex chemistry, multicomponent transport, and an optically thin radiation submodel), grid spacing highly influences the inner flame's position, which stabilizes only with adequate refinement. The vorticity?velocity formulation of the governing equations is shown to produce valid results when used in conjunction with the LRR gridding technique. Furthermore, each LRR grid is used to form a nonuniform equivalent tensor product (ETP) grid and also, in most cases, an equispaced fully refined (FR) grid; these additional grids are supersets of the LRR grids and thus contain refinement in exactly the same regions. Performance comparisons between the LRR, ETP, and FR grids indicate that the LRR method provides substantial savings in execution time and computer memory requirements, without compromising solution accuracy.

A numerical model of a three-dimensional, horizontal channel, chemical vapor deposition reactor is presented in order to study gallium arsenide growth from trimethylgallium and arsine source reactants. Fluid flow and temperature predictions inside the reactor are obtained using the vorticity-velocity form of the three-dimensional, steady-state Navier?Stokes equations coupled with a detailed energy balance equation inside the reactor and on its walls. Detailed gas phase and surface chemistry mechanisms are used to predict the chemical species profiles inside the reactor, the growth rate distribution on the substrate, and the level of carbon incorporation into the grown layer. The species diffusion velocities are written using the recent theory of iterative transport algorithms and account for both thermal diffusion and multicomponent diffusion processes. The influence of susceptor temperature and inlet composition on growth rate and carbon incorporation is found to agree well with previous numerical and experimental work.

The vorticity-velocity formulation of the Navier-Stokes equations is extended to the solution of three-dimensional compressible fluid flow and heat transfer problems. The basic governing equations are expressed in terms of three Poisson-like equations for the velocity components together with a vorticity transport equation and an energy equation. The resulting seven coupled partial differential equations are solved by a finite difference method on a single grid and a discrete solution is obtained by combining a steady-state and a time-dependent Newton's method. Once a converged solution is obtained, one of the velocity equations can be removed from the system and replaced by the continuity equation and a "conservative" solution is obtained by using the previous solution as a starting estimate for Newton's method with only a few additional iterations. The numerical procedure is evaluated by applying it to natural and mixed convection problems. The formulation is found to be stable at high Rayleigh numbers and it may be applied to a wide variety of flow and heat transfer problems.

In this paper we present a primitive variable Newton-based solution method with a block-line linear equation solver for the calculation of reacting flows. The present approach is compared with the stream function-vorticity Newton's method and the SIMPLER algorithm on the calculation of a system of fully elliptic equations governing an axisymmetric methane-air laminar diffusion flame. The chemical reaction is modeled by the flame sheet approximation. The numerical solution agrees well with experimental data in the major chemical species. The comparison of three sets of numerical results indicates that the stream function-vorticity solution using the approximate boundary conditions reported in the previous calculations predicts a longer flame length and a broader flame shape. With a new set of modified vorticity boundary conditions, we obtain agreement between the primitive variable and stream function-vorticity solutions. The primitive variable Newton's method converges much faster than the other two methods. Because of much less computer memory required for the block-line tridiagonal solver compared to a direct solver, the present approach makes it possible to calculate multidimensional flames with detailed reaction mechanisms. The SIMPLER algorithm shows a slow convergence rate compared to the other two methods in the present calculation.

Conclusions derived from the solution of premixed laminar flames in a stagnation point flow are important in the study of pollutant formation, the determination of chemically controlled extinction limits and in the ability to characterize the combustion processes occurring in turbulent flames. In the neighborhood of the stagnation point produced in these flames, a chemically reacting boundary layer is established. For a given equivalence ratio, the input flow velocity can be varied and solutions can be determined for increasing values of the strain rate. As the strain rate increases, the flame nears extinction. In the vicinity of the extinction point, however, the Jacobian of the system becomes singular. To avoid computational difficulties, we employ numerical bifurcation techniques to generate the approriate steady-state profiles (both physical and nonphysical). The method is applied to study the extinction behavior of a one-step kinetics model of a premixed hydrogen-air and methane-air flame in a counterflow geometry.

A numerical technique has been developed for integrating the one-dimensional steady state premixed laminar flame equations. A global finite difference approach is used in which the nonlinear difference equations are solved by a damped-modified Newton method. An assumed temperature profile helps to generate a converged numerical solution on an initial coarse grid. Mesh points are inserted in regions where the solution profiles exhibit high gradient and high curvature activity. These features are discussed and illustrated in the paper, and the method is used to calculate the temperature and species profiles of several laboratory flames.

A direct and very efficient approach for obtaining sensitivities of two-point boundary value problems solved by Newton's method is studied. The link between the solution method and the sensitivity equations is investigated together with matters of numerical accuracy and efficiency. This approach is employed in the analysis of a model three species, unimolecular, steady-state, premixed laminar flame. The numerical accuracy of the sensitivities is verified and their values are utilized for interpretation of the model results. It is found that parameters associated directly with the temperature play a dominant role. The system's Green's functions relating dependent variables are also controlled strongly by the temperature. In addition, flame speed sensitivities are calculated and shown to be a special class of derived sensitivity coefficients. Finally, some suggestions for the physical interpretation of sensitivities in model analysis are given.

Solid-solid alloying reactions occur in a variety of pyrotechnical applications. They arise when a mixture of powders composed of appropriate oxidizing and reducing agents is heated. The large quantity of heat evolved produces a self-propagating reaction front that is often very narrow with sharp changes in both the temperature and the concentrations of the reacting species. Solution of problems of this type with an equispaced or mildly nonuniform grid can be extremely inefficient. In this paper we develop a two-dimensional fully adaptive method for solving problems of this class. The method adaptively adjusts the number of grid points needed to equidistribute a positive weight function over a given mesh interval in each direction at each time level. We monitor the solution from one time level to another to ensure that the local error per unit step associated with the time differencing method is below some specified tolerance. The method is applied to several examples involving exothermic, diffusion-controlled, self-propagating reactions in packed bed reactors.