David Goluskin

Convection driven by internal heating

Abstract Two-dimensional direct numerical simulations are conducted for convection sustained by uniform internal heating in a horizontal fluid layer. Top and bottom boundary temperatures are fixed and equal. Prandtl numbers range from 0.01 to 100, and Rayleigh numbers ( R ) are up to 5 ⋅ 10 5 times the critical R at the onset of convection. The asymmetry between upward and downward heat fluxes is non-monotonic in R . In a broad high- R regime, dimensionless mean temperature scales as R − 1 / 5 . We discuss the scaling of mean temperature and heat-flux-asymmetry, which we argue are better diagnostic quantities than the conventionally used top and bottom Nusselt numbers.

Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization

We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit c...

Generation of Large-Scale Winds in Horizontally Anisotropic Convection.

We simulate three-dimensional, horizontally periodic Rayleigh-Bénard convection, confined between free-slip horizontal plates and rotating about a distant horizontal axis. When both the temperature difference between the plates and the rotation rate are sufficiently large, a strong horizontal wind is generated that is perpendicular to both the rotation vector and the gravity vector. The wind is turbulent, large-scale, and vertically sheared. Horizontal anisotropy, engendered here by rotation, appears necessary for such wind generation. Most of the kinetic energy of the flow resides in the wind, and the vertical turbulent heat flux is much lower on average than when there is no wind.

Internally heated convection beneath a poor conductor

We consider convection in an internally heated layer of fluid that is bounded below by a perfect insulator and above by a poor conductor. The poorly conducting boundary is modelled by a fixed heat flux. Using solely analytical methods, we find linear and energy stability thresholds for the static state, and we construct a lower bound on the mean temperature that applies to all flows. The linear stability analysis yields a Rayleigh number above which the static state is linearly unstable (

Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems

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Zonal flow driven by convection and convection driven by internal heating

), and the energy analysis yields a Rayleigh number below which it is globally stable (

Stabilities and Bounds

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A Family of Convective Models

). For various boundary conditions on the velocity, exact expressions for

Internally Heated Convection Experiments and Simulations

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Internally Heated Convection and Rayleigh-Bénard Convection

and