Samuel Paolucci

On slow manifolds of chemically reactive systems

This work addresses the construction of slow manifolds for chemically reactive flows. This construction relies on the same decomposition of a local eigensystem that is used in formation of what are known as Intrinsic Low Dimensional Manifolds (ILDMs). We first clarify the accuracy of the standard ILDM approximation to the set of ordinary differential equations which model spatially homogeneous reactive systems. It is shown that the ILDM is actually only an approximation of the more fundamental Slow Invariant Manifold (SIM) for the same system. Subsequently, we give an improved extension of the standard ILDM method to systems where reaction couples with convection and diffusion. Reduced model equations are obtained by equilibrating the fast dynamics of a closely coupled reaction/convection/diffusion system and resolving only the slow dynamics of the same system in order to reduce computational costs, while maintaining a desired level of accuracy. The improvement is realized through formulation of an ellipt...

Accurate Spatial Resolution Estimates for Reactive Supersonic Flow with Detailed Chemistry

Ar obust method is developed and used to provide rational estimates of reaction zone thicknesses in onedimensional steady gas-phase detonations in mixtures of inviscid ideal reacting gases whose chemistry is described by detailed kinetics of the interactions of N molecular species constituted from L atomic elements. The conservation principles are cast as a set of algebraic relations giving pressure, temperature, density, velocity, and L species mass fractions as functions of the remaining N‐L species mass fractions. These are used to recast the N‐L species evolution equations as a self-contained system of nonlinear ordinary differential equations of the form dYi/dx = fi (Y1 ,..., YN‐L). These equations are numerically integrated from a shock to an equilibrium end state. The eigenvalues of the Jacobian of fi are calculated at every point in space, and their reciprocals give local estimates of all length scales. Application of the method to the standard problem of a stoichiometric Chapman‐Jouguet hydrogen‐air detonation in a mixture with ambient pressure of 1 atm and temperature of 298 K reveals that the finest length scale is on the order of 10 −5 cm; this is orders of magnitude smaller than both the induction zone length, 10 −2 cm, and the overall reaction zone length, 10 0 cm. To achieve numerical stability and convergence of the solution at a rate consistent with the order of accuracy of the numerical method as the spatial grid is refined, it is shown that one must employ a grid with a finer spatial discretization than the smallest physical length scale. It is shown that published results of detonation structures predicted by models with detailed kinetics are typically underresolved by one to five orders of magnitude.

Numerical simulation of filling and solidification of permanent mold castings

Filling of a mold is an essential part of the permanent mold casting process and affects significantly the heat transfer and solidification of the melt. For this reason, accurate prediction of the temperature field in permanent mold castings can be achieved only by including simulation of filling in the analysis. In this work we model filling and solidification of a casting of an automotive piston produced from an aluminum alloy. Filling of the three-dimensional mold is modeled by using the volume-of-fluid method. Fluid mechanics and heat transfer equations are solved by a finite element method. Comparisons of numerical results to available experimental data show that the formulated model provides a solution of acceptable accuracy despite some uncertainty in material properties and boundary and initial conditions. This implies that the model can be a viable tool to design permanent molds.

Viscous detonation in H2-O2-Ar using intrinsic low-dimensional manifolds and wavelet adaptive multilevel representation

A standard ignition delay problem for a mixture of hydrogen-oxygen-argon in a shock tube is extended to the viscous regime and solved using the method of intrinsic low-dimensional manifolds (ILDM) coupled with a wavelet adaptive multilevel representation (WAMR) spatial discretization technique. An operator-splitting method is used to describe the reactions as a system of ordinary differential equations at each spatial point. The ILDM method is used to eliminate the stiffness associated with the chemistry by decoupling processes which evolve on fast and slow time scales. The fast time scale processes are systematically equilibrated, thereby reducing the dimension of the phase space required to describe the reactive system. The WAMR technique captures the detailed spatial structures automatically with a small number of basis functions thereby further reducing the number of variables required to describe the system. A maximum of only 300 collocation points and 15 scale levels yields results with striking resolution of fine-scale viscous and induction zones. Additionally, the resolution of physical diffusion processes minimizes the effects of potentially reaction-inducing artificial entropy layers associated with numerical diffusion.

Stability of natural convection flow in a tall vertical enclosure under non-Boussinesq conditions

Abstract We have examined the linear stability of the fully developed natural convection flow in a differentially heated tall vertical enclosure under non-Boussinesq conditions. The three-dimensional analysis of the stability problem was reduced to a two-dimensional one by the use of Squires theorem. The resulting eigenvalue problem was solved using an integral Chebyshev collocation method. The influence of non-Boussinesq effects on the stability was studied. We have investigated the dependence of the critical Rayleigh number on the temperature difference. The results show that two different modes of instability are possible, one of which is new and due entirely to non-Boussinesq effects. Both types of instability are oscillatory, and the critical disturbance wave speed is zero only in the Boussinesq limit.

The G-Scheme: A framework for multi-scale adaptive model reduction

The numerical solution of mathematical models for reaction systems in general, and reacting flows in particular, is a challenging task because of the simultaneous contribution of a wide range of time scales to the system dynamics. However, the dynamics can develop very-slow and very-fast time scales separated by a range of active time scales. An opportunity to reduce the complexity of the problem arises when the fast/active and slow/active time scales gaps becomes large. We propose a numerical technique consisting of an algorithmic framework, named the G-Scheme, to achieve multi-scale adaptive model reduction along-with the integration of the differential equations (DEs). The method is directly applicable to initial-value ODEs and (by using the method of lines) PDEs. We assume that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. The G-Scheme introduces locally a curvilinear frame of reference, defined by a set of orthonormal basis vectors with corresponding coordinates, attached to this decomposition. The evolution of the curvilinear coordinates associated with the active subspace is described by non-stiff DEs, whereas that associated with the slow and fast subspaces is accounted for by applying algebraic corrections derived from asymptotics of the original problem. Adjusting the active DEs dynamically during the time integration is the most significant feature of the G-Scheme, since the numerical integration is accomplished by solving a number of DEs typically much smaller than the dimension of the original problem, with corresponding saving in computational work. To demonstrate the effectiveness of the G-Scheme, we present results from illustrative as well as from relevant problems.

Stability of mixed-convection flow in a tall vertical channel under non-boussinesq conditions

We have examined the linear stability of the fully developed mixed-convection flow in a differentially heated tall vertical channel under non-Boussinesq conditions. The three-dimensional analysis of the stability problem was reduced to an equivalent twodimensional one by the use of Squire’s transformation. The resulting eigenvalue problem was solved using an integral Chebyshev pseudo-spectral method. Although Squire’s theorem cannot be proved analytically, two-dimensional disturbances are found to be the most unstable in all cases. The influence of the non-Boussinesq effects on the stability was studied. We have investigated the dependence of the critical Grashof and Reynolds numbers on the temperature difference. The results show that four different modes of instability are possible, two of which are new and due entirely to non-Boussinesq effects.

Nonlinear analysis of convection flow in a tall vertical enclosure under non-Boussinesq conditions

A weakly nonlinear theory, based on the combined amplitude–multiple timescale expansion, is developed for the flow of an arbitrary fluid governed by the low-Mach-number equations. The approach is shown to be different from the one conventionally used for Boussinesq flows. The range of validity of the applied analysis is discussed and shown to be sufficiently large. Results are presented for the natural convection flow of air inside a closed differentially heated tall vertical cavity for a range of temperature differences far beyond the region of validity of the Boussinesq approximation. The issue of possible resonances of two different types is noted. The character of bifurcations for the shear- and buoyancy-driven instabilities and their interaction is investigated in detail. Lastly, the energy transfer mechanisms are analysed in supercritical regimes.

WAMR: An adaptive wavelet method for the simulation of compressible reacting flow. Part I. Accuracy and efficiency of algorithm

Abstract The Wavelet Adaptive Multiresolution Representation (WAMR) method provides a robust method for controlling spatial grid adaptation — fine grid spacing in regions where a solution varies greatly (i.e., near steep gradients, or near-singularities) and a much coarser grid where the solution varies slowly. Subsequently, a wide range of spatial scales, often demanded in challenging continuum physics problems, can be efficiently captured. Furthermore, the wavelet transform provides a direct measure of local error at each collocation point, effectively producing automatically verified solutions. The method is applied to the solution of unsteady, compressible, reactive flow equations, and includes detailed diffusive transport and chemical kinetics models. Accuracy and performance of the method are examined on several test problems. The sparse grids produced by the WAMR method exhibit an impressive compression of the solution, reducing the number of collocation points used by factors of many orders of magnitude when compared to uniform grids of equivalent resolution.

Implementation of a Level Set Interface Tracking Method in the FIDAP and CFX-4 Codes

We present a streamline-upwind-Petrov-Galerkin (SUPG) finite element level set method that may be implemented into commercial computational fluid dynamics (CFD) software, both finite element (FE) and finite volume (FV) based, to solve problems involving incompressible, two-phase flows with moving interfaces. The method can be used on both structured and unstructured grids. Two formulations are given. The first considers the coupled motion of the two phases and is implemented within the framework of the commercial CFD code CFX-4. The second can be applied for those gas-liquid flows for which effects of the gaseous phase on the motion of the liquid phase are negligible; consequently, the gaseous phase is removed from consideration. This level set formulation is implemented in the commercial CFD code FIDAP. The resulting level set formulations are tested and validated on sample problems involving two-phase flows with density ratios of the order of 10 3 and viscosity ratios as high as 1.6X 10 5 .