Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAmplitude equations for 3D double-diffusive convection interacted with a horizontal vortex
343
0
0.0
(
0
)
Added by
Sergey Kozitskiy Dr.
Publication date
2009
fields
Physics
and research's language is
English
Authors
S.B. Kozitskiy
Ask ChatGPT about the research
No Arabic abstract
Three dimensional roll-type double-diffusive convection in a horizontally infinite layer of an uncompressible liquid is considered in the neighborhood of Hopf bifurcation points. A system of amplitude equations for the variations of convective rolls amplitude is derived by multiple-scaled method. An attention is paid to an interaction of convection and horizontal vortex. Different cases of the derived equations are discussed.
rate research
Read More
We consider the problem of horizontal convection in which non-uniform buoyancy, $b_{rm s}(x,y)$, is imposed on the top surface of a container and all other surfaces are insulating. Horizontal convection produces a net horizontal flux of buoyancy, $mathbf{J}$, defined by vertically and temporally averaging the interior horizontal flux of buoyancy. We show that $overline{mathbf{J}cdotmathbf{ abla}b_{rm s}}=-kappalangle|boldsymbol{ abla}b|^2rangle$; overbar denotes a space-time average over the top surface, angle brackets denote a volume-time average and $kappa$ is the molecular diffusivity of buoyancy $b$. This connection between $mathbf{J}$ and $kappalangle|boldsymbol{ abla}b|^2rangle$ justifies the definition of the horizontal-convective Nusselt number, $Nu$, as the ratio of $kappa langle|boldsymbol{ abla}b|^2rangle$ to the corresponding quantity produced by molecular diffusion alone. We discuss the advantages of this definition of $Nu$ over other definitions of horizontal-convective Nusselt number currently in use. We investigate transient effects and show that $kappa langle|boldsymbol{ abla}b|^2rangle$ equilibrates more rapidly than other global averages, such as the domain averaged kinetic energy and bottom buoyancy. We show that $kappalangle|boldsymbol{ abla} b|^2rangle$ is essentially the volume-averaged rate of Boussinesq entropy production within the enclosure. In statistical steady state, the interior entropy production is balanced by a flux of entropy through the top surface. This leads to an equivalent surface Nusselt number, defined as the surface average of vertical buoyancy flux through the top surface times the imposed surface buoyancy $b_{rm s}(x,y)$. In experiments it is likely easier to evaluate the surface entropy flux, rather than the volume integral of $|mathbf{ abla}b|^2$ demanded by $kappalangle|mathbf{ abla}b|^2rangle$.
Vortices play an unique role in heat and momentum transports in astro- and geo-physics, and it is also the origin of the Earths dynamo. A question existing for a long time is whether the movement of vortices can be predicted or understood based on their historical data. Here we use both the experiments and numerical simulations to demonstrate some generic features of vortex motion and distribution. It can be found that the vortex movement can be described on the framework of Brownian particles where they move ballistically for the time shorter than some critical timescales, and then move diffusively. Traditionally, the inertia of vortex has often been neglected when one accounts for their motion, our results imply that vortices actually have inertial-induced memory such that their short term movement can be predicted. Extending to astro- and geo-physics, the critical timescales of transition are in the order of minutes for vortices in atmosphere and ocean, in which this inertial effect may often be neglected compared to other steering sources. However, the timescales for vortices are considerably larger which range from days to a year. It infers the new concept that not only the external sources alone, for example the solar wind, but also the internal source, which is the vortex inertia, can contribute to the short term Earths magnetic field variation.
In the preparation of Cafe Latte, spectacular layer formation can occur between the expresso shot in a glass of milk and the milk itself. Xue et al. (Nat. Commun., vol. 8, 2017, pp. 1-6) showed that the injection velocity of expresso determines the depth of coffee-milk mixture. After a while when a stable stratification forms in the mixture, the layering process can be modelled as a double diffusive convection system with a stably-stratified coffee-milk mixture cooled from the side. More specifically, we perform (two-dimensional) direct numerical simulations of laterally cooled double diffusive convection for a wide parameter range, where the convective flow is driven by a lateral temperature gradient while stabilized by a vertical concentration gradient. When the thermal driving force dominates over the stabilizing force, the flow behaves like vertical convection in which a large-scale circulation develops. However, with increasing strength of the stabilizing force, a meta-stable layered regime emerges. Initially, several vertically-stacked convection rolls develop, and these well-mixed layers are separated by sharp interfaces with large concentration gradients. The initial thickness of these emerging layers can be estimated by balancing the work exerted by thermal driving and the required potential energy to bring fluid out of its equilibrium position in the stably stratified fluid. In the layered regime, we further observe successive layer merging, and eventually only a single convection roll remains. We elucidate the following merging mechanism: As weakened circulation leads to accumulation of hot fluid adjacent to the hot sidewall, larger buoyancy forces associated with hotter fluid eventually break the layer interface. Then two layers merge into a larger layer, and circulation establishes again within the merged structure.
In a range of physical systems, the first instability in Rayleigh-Bernard convection between nearly thermally insulating horizontal plates is large scale. This holds for thermal convection of fluids saturating porous media. Large-scale thermal convection in a horizontal layer is governed by remarkably similar equations both in the presence of a porous matrix and without it, with only one additional term for the latter case, which, however, vanishes under certain conditions (e.g., two-dimensional flows or infinite Prandtl number). We provide a rigorous derivation of long-wavelength equations for a porous layer with inhomogeneous heating and possible pumping.
Convection over a wavy heated bottom wall in the air flow has been studied in experiments with the Rayleigh number $sim 10^8$. It is shown that the mean temperature gradient in the flow core inside a large-scale circulation is directed upward, that corresponds to the stably stratified flow. In the experiments with a wavy heated bottom wall, we detect large-scale standing internal gravity waves excited in the regions with the stably stratified flow. The wavelength and the period of these waves are much larger than the turbulent spatial and time scales, respectively. In particular, the frequencies of the observed large-scale waves vary from 0.006 Hz to 0.07 Hz, while the turbulent time in the integral scale is about 0.5 s. The measured spectra of these waves contains several localized maxima, that implies an existence of waveguide resonators for the large-scale standing internal gravity waves. For comparisons, experiments with convection over a smooth plane bottom wall at the same mean temperature difference between bottom and upper walls have been also conducted. In these experiments various locations with a stably stratified flow are also found and the large-scale standing internal gravity waves are observed in these regions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
