On the nature of NavierStokes turbulence
Abstract
Several turbulent and nonturbulent solutions of the NavierStokes equations are obtained. The unaveraged equations are used numerically in conjunction with tools and concepts from nonlinear dynamics, including time series, phase portraits, Poincare sections, largest Liapunov exponents, power spectra, and strange attractors. Initially neighboring solutions for a lowReynoldsnumber fully developed turbulence are compared. The solutions, separate exponentially with time, having a positive Liapunov exponent. Thus the turbulence is characterized as chaotic. In a search for solutions which contrast with the turbulent ones, the Reynolds number is reduced. Several qualitatively different flows are noted. These are, fully chaotic, complex period, weakly chaotic, simple periodic, and fixedpoint. Of these, only the fully chaotic flows are classified as turbulent. Those flows have both a positive Liapunov exponent and Poincare sections without pattern. By contrast, the weakly chaotic flows have some pattern in their Poincare sections. The fixedpoint and periodic flows are nonturbulent, since turbulence, is both timedependent and aperiodic. Turbulent solutions are obtained in which energy cascades from large to smallscale motions. In general, the spectral energy transfer takes place between wavenumber bands that are considerably separated. The special transfer can occur either as a result of nonlinear turbulence selfinteraction or by interaction of turbulence with mean gradients. Turbulent systems are compared with those studied in kinetic theory. The two types of systems are fundamentally different (continuous and dissipative as opposed to discrete and conservative), but there are similarities. For instance, both are nonlinear and show sensitive dependence on initial conditions. <Also, the turbulent and molecular stress tensors are identical if the macroscopic velocities for the turbulent stress are replaced by molecular velocities.
 Publication:

Ph.D. Thesis
 Pub Date:
 May 1989
 Bibcode:
 1989PhDT........20D
 Keywords:

 Energy Transfer;
 Flow Characteristics;
 NavierStokes Equation;
 Turbulence;
 Kinetic Theory;
 Liapunov Functions;
 Low Reynolds Number;
 Poincare Problem;
 Power Spectra;
 Strange Attractors;
 Stress Tensors;
 Time Series Analysis;
 Turbulent Mixing;
 Fluid Mechanics and Heat Transfer