One-Dimensional Turbulence: From Extreme Reynolds Number Turbulence to Hypersonic Turbulent Boundary Layers

Abstract

Turbulence remains one of the most important unsolved problems in classical physics. The difficulty of arriving at a complete understanding of turbulence is rooted in the non-linear and multi-scale nature of the phenomenon. Although fluid mechanical turbulence is governed by the deterministic Navier-Stokes equations, there is no unified theory to obtain a generalized closed form solution for arbitrary turbulent flow configurations. Direct Numerical Simulations (DNS), Large Eddy Simulations (LES) and Reynolds-Averaged Navier-Stokes (RANS) solvers utilize different approaches to arrive at a solution, but each method has complications and limitations of its own.

The description of turbulence as a superposition of coherent structures has provided unique insights into the phenomenon. The possibility of representing turbulence from the statistical and structural viewpoints provides an opportunity to explore models that can provide alternative frameworks combining the features of these two viewpoints.

This work explores the use of dimensionality reduction which couples with a structural and statistical approach to model turbulence. Even though dimensionality reduction is a simplified representation of the complex physical reality, the model retains its usefulness if it captures the essence of the non-linear and multi-scale nature of turbulence. Establishing this simplified model, while retaining the main features of the complex fluid flow, is the primary objective of this work. The one-dimensional turbulence (ODT) method originally proposed by Kerstein (1999) is the foundation of this work, which utilizes geometric mapping of properties at multiple spatio-temporal scales in the domain, to represent three-dimensional turbulent eddies. The use of the one-dimensional approach is particularly favourable when the chosen flow configuration exhibits a characteristic direction of variation, which is often the case with the self-similar canonical flows that form the core of turbulence theory.

The developed model builds on the foundational ODT model to address two building block flow configurations of enormous importance in turbulence literature: the homogeneous isotropic turbulence (HIT) and the turbulent boundary layer (TBL). Both of these flows have been the subject of extensive DNS studies, but the extreme computational cost of the simulations is prohibitive at extreme Reynolds numbers. Extreme Reynolds numbers flows are of paramount importance in many scientific and engineering applications. In addition, the high-Reynolds number simulations are also needed to propose and establish turbulence theory, since idealized turbulence should approach the vanishing viscosity limit.

In this work, we develop a fully-compressible Eulerian ODT solver to study different aspects of canonical high-Reynolds number turbulent flows. This thesis comprises several journal articles, published or under revision for publication at the time of submission of this thesis, that address the following aspects: (i) the study of homogeneous isotropic turbulence at high Taylor-scale Reynolds numbers (up to 5428), with descriptions of high-order statistics and phenomenology of extreme events, (ii) an analysis of the key features of one-dimensional turbulence model with the Townsend's Attached-Eddy Model, and (iii) the study of compressible turbulent boundary layers at hypersonic conditions with Mach numbers of about 5.86, with cold walls at wall-to-recovery temperature ratio as low as 0.26.

The developed methodology shows significant promise in extending the state-of-the-art turbulence simulations to high Reynolds numbers, which is made possible by the computational cost reduction of ODT. The multi-scale nature of ODT captures classical features of homogeneous isotropic turbulence in terms of turbulent energy spectrum, energy flux spectra, normalized dissipation rate and intermittency characteristics. Identifying the salient aspects of Attached-Eddy hypothesis in the ODT model, we establish the importance of probability distribution for eddies alongside the formulations for eddy-induced velocities and displacements. The developed compressible ODT model is also used for prediction of compressible turbulent boundary layer. The model produces the mean statistics of velocity and temperature profiles at hypersonic conditions, agreeing with DNS results, along with the characterization of temperature fluctuations.