Transitional Intermittency Exponents Through Deterministic Boundary-Layer Structures and Empirical Entropic Indices

Lavar Isaacson

doi:10.3390/e16052729

Abstract

A computational procedure is developed to determine initial instabilities within a three-dimensional laminar boundary layer and to follow these instabilities in the streamwise direction through to the resulting intermittency exponents within a fully developed turbulent flow. The fluctuating velocity wave vector component equations are arranged into a Lorenz-type system of equations. The nonlinear time series solution of these equations at the fifth station downstream of the initial instabilities indicates a sequential outward burst process, while the results for the eleventh station predict a strong sequential inward sweep process. The results for the thirteenth station indicate a return to the original instability autogeneration process. The nonlinear time series solutions indicate regions of order and disorder within the solutions. Empirical entropies are defined from decomposition modes obtained from singular value decomposition techniques applied to the nonlinear time series solutions. Empirical entropic indices are obtained from the empirical entropies for two streamwise stations. The intermittency exponents are then obtained from the entropic indices for these streamwise stations that indicate the burst and autogeneration processes.

Highlights

Considerable progress has been made in understanding the basic physical processes underlying the transition of a laminar boundary layer into the turbulent state
These results indicate an oscillatory motion in the normal-spanwise plane, again, demonstrating that the normalspanwise flow configuration plays a significant role in the “burst” and “sweep” phases of the transition of the laminar boundary layer state to the turbulent state
This article has presented the results of a computational scenario for the fluctuating velocity field embedded in a streamwise laminar boundary layer that yields the prediction of deterministic structures developing within the boundary layer flow

Summary

Introduction

Considerable progress has been made in understanding the basic physical processes underlying the transition of a laminar boundary layer into the turbulent state. A new approach has been introduced which uses dynamical systems theory in which the system’s trajectory traverses among mutually repelling flow states. Some of these flow states exist on the boundaries between laminar and turbulent states within the flow environment. Et al [1] and Blau [2], have recently extended the concept of the edge of turbulence to a spatially developing flat plate boundary-layer flow. Their results indicate the presence of ordered structures, such as hairpin vortices and streamwise streaks. A “minimal seed,” is introduced that is considered to be the smallest flow structure for which the energy growth is maximized over short times

Results

Discussion

Conclusion