We study wetting front (traveling wave) solutions to the Richards equation that describe the vertical infiltration of water through one-dimensional periodically layered unsaturated soils. We prove the existence, uniqueness, and large time asymptotic stability of the traveling wave solutions under prescribed flux boundary conditions and certain constitutive conditions. The traveling waves are connections between two steady state solutions that form near the ground surface and towards the underground water table. We found a closed form expression of the wave speed. The speed of a traveling wave is equal to the ratio of the flux difference and the difference of the spatial averages of the two steady states. We give both analytical and numerical examples showing that the wave speeds in the periodic soils can be larger or smaller than those in the homogeneous soils which have the same mean diffusivity and conductivity

We analyze the power spectra and structure functions (SFs) of the temperature and radial velocity fields, calculated in the radial and azimuthal directions, in annular centrifugal Rayleigh–Bénard convection (ACRBC) for Rayleigh number Ra ∈[108,1011], Prandtl number Pr = 10.7, and inverse Rossby number Ro−1=16 using the spatial data obtained by quasi-two-dimensional direct numerical simulation. Bolgiano and Obukhov-like (BO59-like) scalings for the energy spectrum in both the azimuthal and radial directions and thermal spectrum in the azimuthal direction are observed. The range of BO59-like scaling becomes wider as Ra increases. At Ra=1011, it is found that BO59-like scaling Eu(kr)∼kr−11/5 spans nearly two decades for the energy spectrum calculated in the radial direction. Power-law fittings in the range larger than the Bolgiano scales, the scaling exponents of transverse and longitudinal velocity SFs vs the order coincide with the theoretical prediction of BO59 scaling ζ u p =3p/5 basically. The second-order temperature SFs exhibit a gradual transition from the Obukhov–Corrsin behavior at scales smaller than the Bolgiano scales to the BO59 behavior at scales larger than the Bolgiano scales. The slopes from the third to sixth-order temperature SFs are similar, which is similar to classical Rayleigh–Bénard convection and Rayleigh–Taylor turbulence. The probability density functions (p.d.f.) of temperature fluctuations δT/σT reveal the cold plumes are strong and the p.d.f. in different regions at high Ra are similar. The stronger turbulent-mixing and larger centrifugal buoyancy in ACRBC may result in the BO59-like scaling.

We perform a numerical study of the heat transfer and flow structure of Rayleigh-Benard (RB) convection in (in most cases regular) porous media, which are comprised of circular, solid obstacles located on a square lattice. This study is focused on the role of porosity in the flow properties during the transition process from the traditional RB convection with (so no obstacles included) to Darcy-type porous-media convection with approaching 0. Simulations are carried out in a cell with unity aspect ratio, for Rayleigh number from to and varying porosities , at a fixed Prandtl number , and we restrict ourselves to the two-dimensional case. For fixed , the Nusselt number is found to vary non-monotonically as a function of ; namely, with decreasing , it first increases, before it decreases for approaching 0. The non-monotonic behaviour of originates from two competing effects of the porous structure on the heat transfer. On the one hand, the flow coherence is enhanced in the porous media, which is beneficial for the heat transfer. On the other hand, the convection is slowed down by the enhanced resistance due to the porous structure, leading to heat transfer reduction. For fixed , depending on , two different heat transfer regimes are identified, with different effective power-law behaviours of versus , namely a steep one for low when viscosity dominates, and the standard classical one for large . The scaling crossover occurs when the thermal boundary layer thickness and the pore scale are comparable. The influences of the porous structure on the temperature and velocity fluctuations, convective heat flux and energy dissipation rates are analysed, further demonstrating the competing effects of the porous structure to enhance or reduce the heat transfer.

In the long-time scale, we consider the fluid dynamical limits for the kinetic equations when the fluctuation is decomposed into even and odd parts with respect to the microscopic velocity with different scalings. It is shown that when the background state is an absolute Maxwellian, the limit fluid dynamical equations are the incompressible Navier-Stokes equations with viscous heating. This is different from the case when the even and odd parts of the fluctuation have the same scaling where the standard incompressible Navier-Stokes equations without viscous heating are obtained. On the other hand, when the background is a local Maxwellian, it is shown that the above even-odd decomposition leads to a non-classical fluid dynamical system without viscous heating which has been used to describe the ghost effect in the kinetic theory. In addition, the above even-odd decomposition is justified rigorously for the Boltzmann equation for the former case when the background is an absolute Maxwellian.

Flow reversals in two-dimensional Rayleigh-Benard convection led by non-Oberbeck-Boussinesq (NOB) effects due to large temperature differences arc studied by direct numerical simulation. Perfect gas is chosen as the working fluid and the Prandtl number is 0.71 for the reference state. 11 NOB effects are included, the flow pattern P-11 with only one dominant roll often becomes unstable by the growth of the cold corner roll, which sometimes results in cession-led flow reversals. By exploiting the vorticity transport equation, it is found that the asymmetries of buoyancy and viscous forces are responsible for the growth of the cold corner roll because both such asymmetries cause an imbalance between the corner rolls and the large-scale circulation (I.SC). The buoyancy force near the cold wall increases and decreases near the hot wall originating from the temperature-dependent isobaric thermal expansion coefficient alpha = 1/T if NOB effects are included; Moreover, the decreased dissipation due to lower viscosity is favourable for the growth of the cold corner roll, while the increased viscosity further suppresses the growth of the hot corner roll. Finally, it is found that the boundary layer near the cold wall plays an important role in the mass transport from LSC to corner rolls subject to mass conservation.

The onset of thermal convection in a rapidly rotating spherical shell is studied by linear stability analysis based on the fully compressible Navier-Stokes equations. Compressibility is quantified by the number of density scale heights N-rho, which measures the intensity of density stratification of the motionless, polytropic base state. The nearly adiabatic flow with polytropic index n = 1.499 < na = 1.5 is considered, where na is the adiabatic polytropic index. By investigating the stability of the base state with respect to the disturbance of specified wavenumber, the instability process is found to be sensitive to the Prandtl number Pr and to N-rho. For large Pr and small N-rho, the quasi-geostrophic columnar mode loses stability first; while for relatively small Pr a new quasi-geostrophic compressible mode is identified, which becomes unstable first under strong density stratification. The inertial mode can also occur first for relatively small Pr and a certain intensity of density stratification in the parameter range considered. Although the Rayleigh numbers Ra for the onsets of the quasi-geostrophic compressible mode and columnar mode are different by several orders of magnitude, we find that they follow very similar scaling laws with the Taylor number. The critical Ra for convection onset is found to be always positive, in contrast with previous results based on the widely used anelastic model that convection can occur at negative Ra. By evaluating the relative magnitude of the time derivative of density perturbation in the continuity equation, we show that the anelastic approximation in the present system cannot be applied in the small-Ra and large-N-rho regime.

Based on the fully compressible Navier-Stokes equations, the linear stability of thermal convection in rapidly rotating spherical shells of various radius ratios eta is studied for a wide range of Taylor number Ta, Prandtl number Pr and the number of density scale height N-rho. Besides the classical inertial mode and columnar mode, which are widely studied by the Boussinesq approximation and anelastic approximation, the quasi-geostrophic compressible mode is also identified in a wide range of N-rho and Pr for all eta considered, and this mode mainly occurs in the convection with relatively small Pr and large N.. The instability processes are classified into five categories. In general, for the specified wavenumber m, the parameter space (Pr, N-rho) of the fifth category, in which the base state loses stability via the quasi-geostrophic compressible mode and remains unstable, shrinks as eta increases. The asymptotic scaling behaviours of the critical Rayleigh numbers Ra-c and corresponding wavenumbers m(c) to Ta are found at different eta for the same instability mode. As eta increases, the flow stability is strengthened. Furthermore, the linearized perturbation equations and Reynolds-Orr equation are employed to quantitatively analyse the mechanical mechanisms and flow instability mechanisms of different modes. In the quasi-geostrophic compressible mode, the time-derivative term of disturbance density in the continuity equation and the diffusion term of disturbance temperature in the energy equation are found to be critical, while in the columnar and inertial modes, they can generally be ignored. Because the time-derivative term of the disturbance density in the continuity equation cannot be ignored, the anelastic approximation fails to capture the instability mode in the small-Pr and large-N-rho system, where convection onset is dominated by the quasi-geostrophic compressible mode. However, all the modes are primarily governed by the balance between the Coriolis force and the pressure gradient, based on the momentum equation. Physically, the most important difference between the quasi-geostrophic compressible mode and the columnar mode is the role played by the disturbance pressure. The disturbance pressure performs negative work for the former mode, which appears to stabilize the flow, while it destabilizes the flow for the latter mode. As eta increases, in the former mode the relative work performed by the disturbance pressure increases and in the latter mode decreases.

We successfully perform the three-dimensional tracking in a turbulent fluid flow of small axisymmetrical particles that are neutrally-buoyant and bottom-heavy, i.e., they have a non-homogeneous mass distribution along their symmetry axis. We experimentally show how a tiny mass inhomogeneity can affect the particle orientation along the preferred vertical direction and modify its tumbling rate. The experiment is complemented by a series of simulations based on realistic Navier-Stokes turbulence and on a point-like particle model that is capable to explore the full range of parameter space characterized by the gravitational torque stability number and by the particle aspect ratio. We propose a theoretical perturbative prediction valid in the high bottom-heaviness regime that agrees well with the observed preferential orientation and tumbling rate of the particles. We also show that the heavy-tail shape of the probability distribution function of the tumbling rate is weakly affected by the bottom-heaviness of the particles.

The influences of non-Oberbeck-Boussinesq (NOB) effects on flow instabilities and bifurcation characteristics of Rayleigh-Benard convection are examined. The working fluid is air with reference Prandtl number Pr = 0.71 and contained in two-dimensional rigid cavities of finite aspect ratios. The fluid flow is governed by the low-Mach-number equations, accounting for the NOB effects due to large temperature difference involving flow compressibility and variations of fluid viscosity and thermal conductivity with temperature. The intensity of NOB effects is measured by the dimensionless temperature differential epsilon. Linear stability analysis of the thermal conduction state is performed. An epsilon(2) scaling of the leading-order corrections of critical Rayleigh number Ra-cr and disturbance growth rate sigma due to NOB effects is identified, which is a consequence of an intrinsic symmetry of the system. The influences of weak NOB effects on flow instabilities are further studied by perturbation expansion of linear stability equations with regard to epsilon, and then the influence of aspect ratio A is investigated in detail. NOB effects are found to enhance (weaken) flow stability in large (narrow) cavities. Detailed contributions of compressibility, viscosity and buoyancy actions on disturbance kinetic energy growth are identified quantitatively by energy analysis. Besides, a weakly nonlinear theory is developed based on centre-manifold reduction to investigate the NOB influences on bifurcation characteristics near convection onset, and amplitude equations are constructed for both codimension-one and -two cases. Rich bifurcation regimes are observed based on amplitude equations and also confirmed by direct numerical simulation. Weakly nonlinear analysis is useful for organizing and understanding these simulation results.

We derive macroscopic dynamics for self-propelled particles in a fluid. The starting point is a coupled Vicsek–Stokes system. The Vicsek model describes self-propelled agents interacting through alignment. It provides a phenomenological description of hydrodynamic interactions between agents at high density. Stokes equations describe a low Reynolds number fluid. These two dynamics are coupled by the interaction between the agents and the fluid. The fluid contributes to rotating the particles through Jeffery’s equation. Particle self-propulsion induces a force dipole on the fluid. After coarse-graining we obtain a coupled Self-Organised Hydrodynamics–Stokes system. We perform a linear stability analysis for this system which shows that both pullers and pushers have unstable modes. We conclude by providing extensions of the Vicsek–Stokes model including short-distance repulsion, finite particle inertia and finite Reynolds number fluid regime.