From theory to simulation: delving into the dynamics of Burgers turbulence

Abstract

This thesis presents a comprehensive exploration of the dynamics of linear and non-linear Burgers equations in the context of one-dimensional Burgers turbulence. The initial phase of the research examines how viscosity influences the solutions of these equations, emphasizing the interaction between inertial and viscous effects and exploring the variation of dissipation across different wavenumbers and scales. The study then shifts its focus to Burgers’ turbulence, beginning with an in-depth statistical characterization of Decaying Burgers’ turbulence. This includes an analysis of energy spectra, inverse cascade of energy, probability density, and various central moments. The research further extends to the investigation of Forced Burgers’ turbulence under varying initial conditions and forcing terms. A key aspect of this investigation is examining the development of Turbulent Kinetic Energy in forced Burgers turbulence and its invariance to changes in Reynolds number. Our findings confirm that, upon reaching equilibrium, velocity profiles demonstrate remarkable consistency across different Reynolds numbers, aligning with the literature’s observations. Furthermore, the thesis highlights self-similarity in decaying Burgers turbulence, evident up to the fourth-order correlation of the velocity field. Furthermore, this thesis establishes the unique behavior of the Burgers equation in contrast to Navier-Stokes turbulence, highlighting its diminished sensitivity to initial conditions and the lack of chaotic dynamics, alongside its adaptability to external influences. The study concludes in an extensive examination of the statistical characteristics of forced Burgers turbulence, including energy spectrum, probability density functions (PDFs), correlation functions, and various statistical moments, providing significant insights into the complexities of turbulence. The findings of this study contribute to the broader field of fluid dynamics, enhancing our comprehension of turbulence phenomena.

Date	27 Mar 2024
Original language	American English
Awarding Institution	École de technologie supérieure
Supervisor	Louis Dufresne (Supervisor)