Large time behaviour of the 2D thermally non-diffusive Boussinesq equations with Navier-slip boundary conditions

The goal of this paper is to study the large-time bahaviour of a buoyancy driven fluid without thermal diffusion and Navier-slip boundary conditions in a bounded domain with Lipschitz-continuous second derivatives. After showing global well-posedness and regularity of classical solutions, we study their large-time asymptotics. Specifically we prove that, in suitable norms, the solutions converge to the hydrostatic equilibrium. Moreover, we prove linear stability for the hydrostatic equilibrium when the temperature is an increasing affine function of the height, i.e. the temperature is vertically stably stratified. This work is inspired by results in [Doe+18] for free-slip boundary conditions.Comment: 37 page

Lattice Boltzmann Methods for thermal flows: continuum limit and applications to compressible Rayleigh-Taylor systems

We compute the continuum thermo-hydrodynamical limit of a new formulation of lattice kinetic equations for thermal compressible flows, recently proposed in [Sbragaglia et al., J. Fluid Mech. 628 299 (2009)]. We show that the hydrodynamical manifold is given by the correct compressible Fourier- Navier-Stokes equations for a perfect fluid. We validate the numerical algorithm by means of exact results for transition to convection in Rayleigh-B\'enard compressible systems and against direct comparison with finite-difference schemes. The method is stable and reliable up to temperature jumps between top and bottom walls of the order of 50% the averaged bulk temperature. We use this method to study Rayleigh-Taylor instability for compressible stratified flows and we determine the growth of the mixing layer at changing Atwood numbers up to At ~ 0.4. We highlight the role played by the adiabatic gradient in stopping the mixing layer growth in presence of high stratification and we quantify the asymmetric growth rate for spikes and bubbles for two dimensional Rayleigh- Taylor systems with resolution up to Lx \times Lz = 1664 \times 4400 and with Rayleigh numbers up to Ra ~ 2 \times 10^10.Comment: 26 pages, 13 figure

On the Boussinesq equations with non-monotone temperature profiles

In this article we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles $T(y)$. As a first main result we show that if $T'$ is of size at most $\nu^{1/3}$ in a suitable norm, then the linearized Boussinesq equations with only vertical dissipation of the velocity but not of the temperature are stable. Thus, mixing enhanced dissipation can suppress Rayleigh-B\'enard instability in this linearized case. We further show that these results extend to the (forced) nonlinear equations with vertical dissipation in both temperature and velocity.Comment: 30 pages; updated and added reference

On the Boussinesq Equations with Non-monotone Temperature Profiles

In this article, we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles T(y). As a first main result, we show that if ′ is of size at most $^{1/3}$ in a suitable norm, then the linearized Boussinesq equations with only vertical dissipation of the velocity but not of the temperature are stable. Thus, mixing enhanced dissipation can suppress Rayleigh–Bénard instability in this linearized case. We further show that these results extend to the (forced) nonlinear equations with vertical dissipation in both temperature and velocity

On the Boussinesq equations with non-monotone temperature profiles

Abstract. In this article we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles $T(y)$. As a first main result we show that if $T′$ is of size at most $\nu^{1/3}$ in a suitable norm, then the linearized Boussinesq equations with only vertical dissipation of the velocity but not of the temperature are stable. Thus, mixing enhanced dissipation can suppress Rayleigh-Bénard instability in this linearized case. We further show that these results extend to the (forced) nonlinear equations with vertical dissipation in both temperature and velocity

Fluid flow dynamics under location uncertainty

We present a derivation of a stochastic model of Navier Stokes equations that relies on a decomposition of the velocity fields into a differentiable drift component and a time uncorrelated uncertainty random term. This type of decomposition is reminiscent in spirit to the classical Reynolds decomposition. However, the random velocity fluctuations considered here are not differentiable with respect to time, and they must be handled through stochastic calculus. The dynamics associated with the differentiable drift component is derived from a stochastic version of the Reynolds transport theorem. It includes in its general form an uncertainty dependent "subgrid" bulk formula that cannot be immediately related to the usual Boussinesq eddy viscosity assumption constructed from thermal molecular agitation analogy. This formulation, emerging from uncertainties on the fluid parcels location, explains with another viewpoint some subgrid eddy diffusion models currently used in computational fluid dynamics or in geophysical sciences and paves the way for new large-scales flow modelling. We finally describe an applications of our formalism to the derivation of stochastic versions of the Shallow water equations or to the definition of reduced order dynamical systems

Stability of 2D partially dissipative Boussinesq equations and 3D rotating Boussinesq equations

Examining the stability of nonlinear partial differential systems near a physically relevant equilibrium with suitable perturbation is a fundamental problem in fluid dynamics. This dissertation solves the stability and large-time behavior of solutions to the nonlinear Boussinesq equations.We study the two-dimensional Boussinesq equations for buoyancy-driven fluids with degenerate dissipations due to their application in a specific physical scenario. Furthermore, these degenerate dissipations help reveal the inner structure of the system when we perform various interactions between the velocity and temperature. We perturb the solutions of two different two-dimensional Boussinesq systems near the hydrostatic equilibrium in a different domain. We prove that the temperature stabilizes the buoyancy-driven fluids for the first system, which has only vertical dissipation and horizontal thermal diffusion. For the second system containing only horizontal dissipation and vertical thermal diffusion, we establish the stability of the solutions and stratifying patterns of the buoyancy-driven fluids as mathematically rigorous facts.Along with this, we study the stability of the three-dimensional rotating Boussinesq equations with only horizontal dissipation, which have a special two-dimensional solution that is dynamic and independent of depth. On large scales, this unique solution provides the bulk averaged properties of the fluid motion. To achieve the global existence, uniqueness, and stability result, we perturb the three-dimensional rotating Boussinesq equations near this dynamic solution