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The vertically averaged velocity of the 3D Rayleigh–Benard problem is analyzed and numerically simulated. This vertically averaged velocity satisfies a 2D incompressible Navier–Stokes system with a body force involving the 3D velocity. A time average of this force is estimated through time averages of the 3D velocity. Relations similar to those from 2D turbulence are then derived. Direct numerical simulation of the 3D Rayleigh–Be nard is carried out to test how prominent the features of 2D turbulence are for this Navier–Stokes system.

An algorithm for continuous data assimilation for the two-dimensional Benard convection problem is introduced and analyzed. It is inspired by the data assimilation algorithm developed for the Navier–Stokes equations, which allows for the implementation of variety of observables: low Fourier modes, nodal values, finite volume averages, and finite elements. The novelty here is that the observed data is obtained for the velocity field alone; i.e. no temperature measurements are needed for this algorithm. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate, the unique exact unknown solution of the Benard convection problem associated with the observed (finite dimensional projection of) velocity.

Rigorous upper and lower bounds are proved for the Taylor and the Kolmogorov wavenumbers for the three-dimensional space periodic Navier–Stokes equations. Under the assumption that Kolmogorov’s two-thirds power law holds, the bounds sharpen to κ T ∼ Gr 1 / 4 and κ ϵ ∼ Gr 3 / 8 respectively, where Gr is the Grashof number. This provides a rigorous proof that the power law implies (1) the energy cascade, (2) Kolmogorov dissipation law, and (3) a connection between κ T and κ ϵ . The portion of phase space where a key a priori estimate on the nonlinear term is sharp is shown to be significant by means of a lower bound on any probability measure associated with an infinite-time average.

We derive upper and lower bounds for ensemble averages of energy, enstrophy, and palinstrophy for the 2D periodic Navier–Stokes equations. This is carried out both in the general case, and in the case where the energy power law for fully developed turbulence holds. In the turbulent case, the bounds are sharp, up to a logarithm, and provide a new lower bound on the Landau–Lifschitz degrees of freedom. We also prove two properties of the inertial term under the turbulence assumption. One is that as the Grashof number is increased, the ensemble average of this term approaches the force. The other is that an estimate of it via the Ladyzhenskaya inequality is sharp on a considerable portion of the global attractor.

Several relations from the Kolmogorov theory of fully-developed three-dimensional turbulence are rigorously established for finite-time averages over Leray–Hopf weak solutions of the Navier–Stokes equations. The Navier–Stokes equations are considered with periodic boundary conditions and an external forcing term. The main parameter is the Grashof number associated with the forcing term. The relations rigorously proved in this article include estimates for the energy dissipation rate, the Kolmogorov wavenumber, the Taylor wavenumber, the Reynolds number, and the energy cascade process. For some estimates the averaging time depends on the macroscale wavenumber and the kinematic viscosity alone, while for others such as the Kolmogorov energy dissipation law and the energy cascade, the estimates depend also on the Grashof number. As compared with earlier works by some of the authors the more physical concept of finite-time average is replacing the concept of infinite-time average used before.

We compare the minimum number of modes needed to capture certain bifurcations of the 2D Navier-Stokes equations by a certain nonlinear Galerkin method and by the traditional Galerkin method. The boundary conditions are periodic and the forcing is in a single mode. Two different choices of the forcing mode are tested. We introduce and implement a homotopy which allows continuation between approximations of different size.