# Current Research of Dr. Samuel Paolucci

Dr. Paolucci conducts his research in the Computer Graphics Laboratory and other laboratories at Fitzpatrick Hall of Engineering.

## Transition and Chaos in Differentially Heated Vertical Cavities

Thermal convection in differentially heated cavities has been studied extensively because of its relevance in a number of diverse fields. The majority of prior work has been concerned with steady-state laminar flows. Yet in many of the fields of application, the flow is unsteady and possibly turbulent. Since many variables of engineering interest depend strongly on the flow regime, it is essential to understand the different physical processes responsible for the conversion of an initially laminar flow to a turbulent one. Numerical and analytical methods are being used in this study. Recent results have shown that an oscillatory approach to steady-state, oscillatory instabilities, quasi-periodic flow, chaotic flow, and turbulent flow exists for the flow regimes investigated. We are also investigating the effect of variable properties on the flow.

## Stability of Mixed Convection Flow

Fabrication of semiconductors using chemical vapor deposition (CVD) depends critically upon the uniformity of the deposition processes which in turn is strongly affected by convective and diffusive transport. A typical reactor configuration comprises of a flowing gas in a channel with deposition occurring on a heated substrate via surface reactions. The flow and temperature fields, and subsequently the uniformity of deposition, are completely affected by any possible instabilities of the flow. These effects are important for the design and operation of such CVD reactors as demands for deposition uniformity become more stringent. The objective of this research is to identify the choices of design parameters that lead to such instabilities, and characterize the resulting flows.

## Chemically Reactive Flows

A wide variety of combustion processes involve a large number of elementary reactions occurring simultaneously within a complex flow field. These processes are modeled by a large number of partial differential equations representing the evolution of numerous reactive chemical species, coupled with the full Navier-Stokes equations. Fully resolved solution of these model equations, which incorporate detailed finite rate chemical kinetics, often requires a prohibitive amount of computational resources. Hence, there is a need to develop methods which rationally reduce the model equations such that numerical simulations can be accomplished in a reasonable amount of computational time. Elementary chemical reactions occur over a wide range of time scales which is manifested as stiffness in the model equations, and subsequently high computational costs. For stable systems, this stiffness can be reduced by systematically equilibrating the fast time scale chemical processes and resolving only the relevant slow time scale chemical processes. The reduced model equations describe the slow dynamics under the assumption that the fast dynamics can be neglected. Most chemical time scales are faster than time scales associated with fluid dynamical phenomena such as convection and diffusion. Nevertheless, it is important that the reduced model equations maintain the coupling of the flow processes with those chemical processes which occur at similar time scales. In this work we illustrate how this coupling of fluid and chemical processes can be maintained such that an approximate and less expensive numerical solution of the reduced model equations is consistent with the more accurate and expensive numerical solution of the full model equations.

## Heat Transfer During Solidification of Metals

The purpose of this study is to look at the significant role that heat transfer and fluid mechanics play in the manufacturing of castings from metal dies, with the primary intent of modeling such processes. The solidification is a phase transformation that is accompanied by release of thermal energy. The essential feature of this phase change transformation is the presence of a moving, generally multidimensional, solid-liquid interface separating regions having different thermophysical properties and dynamics. One of the major manufacturing problems is to extract the heat in a controlled manner. Both the structural perfection and the microstructure of the casting greatly depend on solidification parameters, and there is a close interaction between these parameters and heat transfer in the solid and melt during solidification of metals and alloys. We are presently investigating the important issues which affect the manufacturing process.

## Wavelet Collocation Methods for Solving Partial Differential Equations in Multiple Dimensions

Wavelet analysis is a new numerical concept which allows one to represent a function in terms of a set of basis functions, called wavelets, which are localized both in location and scale. Because of these properties, numerical methods based on wavelet bases are able to attain good spatial and spectral resolution. A fast, dynamically adaptive, multilevel wavelet collocation method is being developed for the solution of partial differential equations in multiple dimensions. The multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution. High resolution computations are performed only in regions where singularities or sharp transitions occur. Preliminary numerical results demonstrate the ability of the method to resolve localized structures such as shocks, which change their location and steepness in space and time. The method can handle general boundary conditions. The computational cost of the algorithm is independent on the dimensionality of the problem and is of the order of the total number of collocation points. Current results indicate that the method is very competitive with other well established numerical algorithms. Future progress will be primarily focused in solving very challenging problems involving transient multidimensional compressible and incompressible fluid flows.