# Current Research of Dr. Joseph Powers

Dr. Powers conducts his research at Fitzpatrick Hall of Engineering.

## Reduced Kinetic Schemes Coupled with Wavelet Collocation Analysis for Combustion Modelling

A key general problem in combustion science is that in which chemical reactions occur simultaneously with fluid flow. Some specific examples include candle flames, internal combustion engine chambers, jet propulsion engines, furnaces, incinerators, atmospheric chemistry, and global warming.

In many cases, the technical challenge to be met is not in writing suitable equations, but rather in actually solving them accurately in a reasonable computational time. The solution is made difficult because each problem is characterized by a variety of features which evolve on highly disparate space and time scales. For example, a candle flame is typically characterized by a global reaction zone structure which can be a few centimeters thick, while the underlying reactions which combine to form the global structure are typically on the order of microns. Similarly some reactions occur in a few nanoseconds, while others can take hundreds of seconds to complete.

Over the past several years, a study which has received support from the National Science Foundation, the Air Force Office of Scientific Research, and Los Alamos National Laboratories, has been conducted to develop improved algorithms to model general combustion processes. The goal of the research is to adapt for multi-dimensional unsteady combustion problems two recently developed methods: 1) a Wavelet Adaptive Multilevel Representation (WAMR) for more accurate spatial representation, and 2) an intrinsic low dimensional manifold (ILDM) method for efficient marching in time. The WAMR is similar to a standard Fourier analysis, except the basis functions, which in Fourier analysis are typically trigonometric functions, are instead functions which lead to a far more efficient representation of the underlying flow field. The ILDM method automatically selects processes occurring on very different time scales which can be approximated as in partial equilibrium, thereby reducing the number of equations which need to be solved using an expensive differential equation solver.

## Cavitation Suppression in High Performance Fuel Pumps

Jet engine performance is critically dependent on the reliable delivery of high mass flow rates of fuel to the combustor. A key factor which inhibits the rate of fuel delivery is cavitation in the fuel pump which becomes increasingly prominent at high mass flow rates. Cavitation, the process in which bubbles form in a low pressure region of the fuel, has long been known to have catastrophic consequences in a wide variety of devices which rely on fluid-structure interactions. Though a large number of studies have been conducted of cavitation in water, relatively few consider cavitation in liquid fuels.

In this study, supported by Honeywell, Inc., a joint experimental and theoretical study of this phenomena is just underway. Computational studies ranging from the molecular dynamics scale, to the scale of a single bubbles in fuel, to the scale of an actual fuel pump are being considered. Additionally an experimental apparatus is being constructed to study the nucleation of cavitation in common jet fuels.

## High Accuracy Computations for Detonations with Simple and Detailed Chemistry

Identification of the length scales over which a combustion event evolves is a critical step in determining the computational resources which must be marshaled in order to reliably predict the behavior. In this study, done in cooperation with personnel at Los Alamos National Laboratory, those scales are being determined and improved high order finite-difference-based techniques are being developed to address them. One component of this study employs simple single-step chemistry to study an unstable detonation. The instability can evolve over a broad range of length and time scales. Here an improved approach based on shock fitting coupled with an improved Weighted Essentially Non-Oscillatry (WENO) scheme has been employed to produce results with remarkably high accuracy. The high accuracy is mainly attributable to use of the shock fitting scheme, which enables the full rates of convergence of the high order differencing scheme to be realized. In contrast, typical shock capturing schemes can only achieve first order convergence rates, and for a comparable computational effort, produce a much less accurate result.

The other component of this study considers detonations in realistic systems with detailed chemistry. An eigenvalue analysis of a steady detonation in a hydrogen-air system has revealed that when the ambient state is at atmospheric conditions, the finest length scales are on the order of 100 nanometers, about three orders of magnitude less than generally realized. The implications of this for unsteady calculations is that far more computational resources are required for a scientifically verified solution to commonly used model equations. The full impact of these results are presently under consideration.

## Design Optimization for High Mach Number Applications

In a project which was supported by the USAF Palace Knight Program at Wright Laboratories, a novel Karhunen-Loeve (KL) Galerkin model for the supersonic, inviscid flow of a calorically perfect ideal gas about an axisymmetric, blunt body employing shock fitting is being developed. The motivation for developing the KL Galerkin model is the need for an accurate and computationally efficient model for use in the optimal design of hypersonic vehicles. In constructing a KL Galerkin model, a set of flow field solutions representative of the design space are required. For this, a global polynomial pseudospectral method for the generalized coordinate, nonconservative form of the Euler equations is implemented. The variables in the equations are collocated via Lagrange interpolating polynomials defined at the zeroes of the Chebyshev polynomials, i.e. Chebyshev-Gauss-Lobatto nodes. Code verification, validation and grid convergence results have been obtained from the pseudospectral code, and an optimal geometry has been identified form a single degree of freedom family of geometries. KL modes derived from pseudospectral solutions at Mach 3.5 from a uniform sampling of the design space are presently being used to develop a KL Galerkin model for the blunt body optimization.

## Transition to Detonation in Energetic Solids

The modeling of transition to detonation in solid energetic materials is the focus of an ongoing research project which has received support from Los Alamos National Laboratory. This research is motivated by concerns over the detonability of damaged solid propellants used in rocket motors, and over the accidental initiation of detonation in damaged solid explosives due to weak mechanical impact or thermal insult. In this study, we investigate the time-dependent development of self-propagating detonation in granular energetic material in response to slow heating. Continuum mixture equations are numerically solved using a high-resolution method. The numerical simulations predict many experimentally observed features including the initial formation of an inert compaction wave, followed by the onset of combustion, and the subsequent strengthening of the combustion wave leading to fully developed detonation. Furthermore, the model correctly predicts experimentally observed time scales, wave speeds, and pressure magnitudes.