Spectral Entropy, Empirical Entropy and Empirical Exergy

A modified form of the Townsend equations for the fluctuating velocity wave vectors is applied to a laminar three-dimensional boundary-layer flow in a methane fired combustion channel flow environment. The objective of this study is to explore the applicability of a set of low dimensional, coupled, nonlinear differential equations for the prediction of possible deterministic ordered structures within a specific boundary-layer environment. Four increasing channel pressures are considered. The equations are cast into a Lorenz-type system of equations, which yields the low-dimensional set of equations. The solutions indicate the presence of several organized flow structures. Singular value decomposition of the nonlinear time series solutions indicate that nearly ninety-eight percent of the fluctuating directed kinetic energy is contained within the first four empirical modes of the decomposition. The empirical entropy computed from these results indicates that these four lowest modes are largely coherent structures with lower entropy rates. Four regions are observed: low-entropy structures over the first four modes; steep increase in entropy over three modes; steady, high entropy over seven modes; and an increase to maximum entropy over the last two modes. A measure, called the empirical exergy, characterizes the extent of directed kinetic energy produced in the nonlinear solution of the deterministic equations used to model the flow environment. The effect of increasing pressure is to produce more distinct ordered structures within the nonlinear time series solutions.