Transition between Boundary-Limited Scaling and Mixing-Length Scaling of Turbulent Transport in Internally Heated Convection.

Heat transport in turbulent thermal convection increases with thermal forcing, but in almost all studies the rate of this increase is slower than it would be if transport became independent of the molecular diffusivities-the heat transport scaling exponent is smaller than the mixing-length (or "ultimate") value of 1/2. This is due to thermal boundary layers that throttle heat transport in configurations driven either by thermal boundary conditions or by internal heating, giving a scaling exponent close to the boundary-limited (or "classical") value of 1/3. With net-zero internal heating and cooling in different regions, the larger mixing-length exponent can be attained because heat need not cross a boundary. We report numerical simulations in which heating and cooling are unequal. As heating and cooling rates are made closer, the scaling exponent of heat transport varies from its boundary-limited value to its mixing-length value.

Global Stability of Fluid Flows Despite Transient Growth of Energy.

Verifying nonlinear stability of a laminar fluid flow against all perturbations is a central challenge in fluid dynamics. Past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. None show stability for the many flows that seem to be stable despite these energies growing transiently. Here a broadly applicable method to verify global stability of such flows is presented. It uses polynomial optimization computations to construct nonquadratic Lyapunov functions that decrease monotonically. The method is used to verify global stability of 2D plane Couette flow at Reynolds numbers above the the energy stability threshold found by Orr in 1907 [The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: A viscous liquid, Proc. R. Ir. Acad. Sect. A 27, 69 (1907)]. This is the first global stability result for any flow that surpasses the energy method.

Minimum wave speeds in monostable reaction-diffusion equations: sharp bounds by polynomial optimization.

Many monostable reaction-diffusion equations admit one-dimensional travelling waves if and only if the wave speed is sufficiently high. The values of these minimum wave speeds are not known exactly, except in a few simple cases. We present methods for finding upper and lower bounds on minimum wave speed. They rely on constructing trapping boundaries for dynamical systems whose heteroclinic connections correspond to the travelling waves. Simple versions of this approach can be carried out analytically but often give overly conservative bounds on minimum wave speed. When the reaction-diffusion equations being studied have polynomial nonlinearities, our approach can be implemented computationally using polynomial optimization. For scalar reaction-diffusion equations, we present a general method and then apply it to examples from the literature where minimum wave speeds were unknown. The extension of our approach to multi-component reaction-diffusion systems is then illustrated using a cubic autocatalysis model from the literature. In all three examples and with many different parameter values, polynomial optimization computations give upper and lower bounds that are within 0.1% of each other and thus nearly sharp. Upper bounds are derived analytically as well for the scalar reaction-diffusion equations.