Oscar P. Manley

Determining modes and fractal dimension of turbulent flows

Research on the abstract properties of the Navier–Stokes equations in three dimensions has cast a new light on the time-asymptotic approximate solutions of those equations. Here heuristic arguments, based on the rigorous results of that research, are used to show the intimate relationship between the sufficient number of degrees of freedom describing fluid flow and the bound on the fractal dimension of the Navier–Stokes attractor. In particular it is demonstrated how the conventional estimate of the number of degrees of freedom, based on purely physical and dimensional arguments, can be obtained from the properties of the Navier–Stokes equation. Also the Reynolds-number dependence of the sufficient number of degrees of freedom and of the dimension of the attractor in function space is elucidated.

Asymptotic analysis of the Navier-Stokes equations

Abstract New bounds are established on the number of modes which determine the solutions of the Navier-Stokes equations in two dimensions. The best bound available at present is nearly proportional to the generalized Grashof number (defined in the paper), and less than logarithmically dependent on the spatial structure, or the shape of the force driving the flow. To the extent than for the case of 2-dimensional Rayleigh-Benard convection, the generalized Grashof number may be identified with the usual Grashof number, the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds.

Empirical and Stokes eigenfunctions and the far-dissipative turbulent spectrum

It is shown that the Stokes eigenfunctions and their corresponding spectra, frequently used in mathematical investigations of the Navier–Stokes equations, provide estimates on the spectrum of the two‐point spatial covariance tensor. This, in turn, is used to estimate the far‐dissipative turbulent spectrum. An exponential falloff is predicted and evidence given which implies that this is a sharp estimate.

Estimates for the energy cascade in three-dimensional turbulent flows

Abstract The phenomenological theory of turbulence in three dimensions postulates that at large Reynolds numbers there exists an interval of wavenumbers within which the direct effects of the molecular viscosity are negligible. Within that interval, the so-called inertial range, an eddy characterized by a wavenumber given in that range decays principally by breaking down into smaller ones, with each of those smaller ones eventually breaking down into still smaller eddies, and so on, a process conventionally called a cascade in the wavenumber space. Such a cascade proceeds until the size of the descendant eddies is sufficiently small to enter the so-called dissipation range and disappear by the direct action of molecular viscosity. In this note, which is a continuation of [5], we prove the existence of the inertial range provided the Taylor wavenumber is sufficiently large. More precisely, we prove that the energy flux to higher modes is nearly equal to the energy dissipation rate throughout a certain range of wavenumbers much smaller than the Taylor wavenumber. These rigorous results show that the Taylor wavenumber is such that below it the conditions prevailing in the inertial range for the energy cascade are strictly satisfied. Moreover, we obtain several estimates concerning characteristic numbers and nondimensional numbers related to turbulent flows.

Exponential decay of the power spectrum of turbulence

The analyticity on a strip of the solutions of Navier-Stokes equations in 2D is shown to explain the observed fast decay of the frequency power spectrum of the turbulent velocity field. Some subtleties in the application of the Wiener-Khinchine method to turbulence are resolved by showing that the frequency power spectrum of turbulent velocities is in fact a measure exponentially decaying for frequency →±∞. Our approach also shows that the conventional procedures used in analyzing data in turbulence experiments are valid even in the absence of the ergodic property in the flow.

Approximate inertial manifolds and effective viscosity in turbulent flows

The recently formulated concept of approximate inertial manifolds is exploited as a means for eliminating systematically the fine structure of the velocity field in two‐dimensional flows. The resulting iterative procedure does not invoke any statistical properties of the solutions of Navier–Stokes equations. It leads to a modification of those equations, such that effective viscosity‐like terms arise in a natural way. The rigorous mathematical considerations can be related to the corresponding physical concepts and intuition. The result leading to a numerical algorithm, essentially a nonlinear Galerkin method, provides a basis for large eddy simulation in which the subgrid model is derived from the properties of the Navier–Stokes equations, rather than from more or less justifiable ad hoc arguments. Some limited speculations concerning the expected results in three dimensions are also offered.

Energy conserving Galerkin approximations for 2-D hydrodynamic and MHD Bénard convection

Abstract The upper bound on the minimum number of modes for a qualitatively correct Galerkin approximation to the two-dimensionl Benard convection is discussed. For that case and the ext case, selection rules are introduced insuring that the approximation satisfies the same energy balance conditions as the exact solution. These rules together with the proof of boundedness of the Galerkin approximation help establish bounds on the amplitudes of the modes. Finally, results of some mathematical (numerical) experiments carried out in this context are discussed.

Effects of the forcing function spectrum on the energy spectrum in 2-D turbulence

The response of the two‐dimensional velocity field to a single eigenmode driving a fluid flow is analyzed. It is shown that independent of its amplitude such a driving force cannot lead to Kraichnan’s inertial range spectrum. At least a pair of eigenmodes, one acting as a power source, and the other as a power sink, is necessary to obtain a separation of length scales and an accompanying statistically steady inertial spectral range.

Bounds for the mean dissipation of 2-D enstrophy and 3-D energy in turbulent flows

Abstract A rigorous statistical definition of the average enstrophy dissipation in two-dimensional flows, and of the average energy dissipation in three-dimensional flows leads to the establishment of rigorous lower bounds on the corresponding cut-off wave numbers. That in turn permits us to conjecture the corresponding lower bounds on the number of degrees of freedom in those flows. For a particular commonly considered strong body force driving a turbulent two-dimensional flow, an improved upper bound on the number of degrees of freedom is found.

Cascade of energy in turbulent flows

Abstract A starting point for the conventional theory of turbulence [12–14] is the notion that, on average, kinetic energy is transferred from low wave number modes to high wave number modes [19]. Such a transfer of energy occurs in a spectral range beyond that of injection of energy, and it underlies the so-called cascade of energy, a fundamental mechanism used to explain the Kolmogorov spectrum in three-dimensional turbulent flows. The aim of this Note is to prove this transfer of energy to higher modes in a mathematically rigorous manner, by working directly with the Navier–Stokes equations and stationary statistical solutions obtained through time averages. To the best of our knowledge, this result has not been proved previously; however, some discussions and partly intuitive proofs appear in the literature. See, e.g., [1,2,10,11,16,17,21], and [22]. It is noteworthy that a mathematical framework can be devised where this result can be completely proved, despite the well-known limitations of the mathematical theory of the three-dimensional Navier–Stokes equations. A similar result concerning the transfer of energy is valid in space dimension two. Here, however, due to vorticity constraints not present in the three-dimensional case, such energy transfer is accompanied by a similar transfer of enstrophy to higher modes. Moreover, at low wave numbers, in the spectral region below that of injection of energy, an inverse (from high to low modes) transfer of energy (as well as enstrophy) takes place. These results are directly related to the mechanisms of direct enstrophy cascade and inverse energy cascade which occur, respectively, in a certain spectral range above and below that of injection of energy [1,15]. In a forthcoming article [9] we will discuss conditions for the actual existence of the inertial range in dimension three.