Session A15: Rayleigh-Benard Convection I: Instability and Oscillations

Sunday, November 20, 2011

The onset of three-dimensional thermal convection of a fluid in a rectangular cavity is numerically studied, under the assumption that its all walls are rigid and of perfect thermal conductance with vertically linear temperature profile. The dependence of the critical Rayleigh number and the symmetry of the most unstable mode on the aspect ratios of the cavity is obtained.   

Sunday, November 20, 2011

We report on high resolution numerical studies of infinite Prandtl number convection using a simplified model with relevance to the motion of the Earth's mantle. Our model uses Navier-Stokes equations with the Boussinesq approximation and free slip velocity boundary conditions that is driven soley by internal heating. The 2D simulations are calculated using pseudospectral Fourier and Chebyshev methods. We examine the transition from conduction to steady convection, to unsteady laminar convection and lastly to chaotic convection. As the heating rate is increased, we report on the relationship between the non- dimensional heat Rayleigh number (proportional to the heating rate) and the averaged temperature (spatially and temporally). We also compare different aspect ratios (width to height) to see the impact this change has on our system.   

Sunday, November 20, 2011

Convective flow in porous media undergoes a transition around Rayleigh number $Ra = 1300$, from predominantly large-scale (quasi-) periodic rolls to vigorous columnar exchange flow driven by unsteady plume formation in boundary layers. The dynamics of these structures determine the flux of heat or solute through the system, as described by the Nusselt number $Nu$. This is of particular interest for understanding how convection affects the rate of dissolution of sequestered CO$_2$ in a saline aquifer. High resolution 2D numerical simulations of porous media Rayleigh-Benard convection are presented, which show that, for $2000 [Preview Abstract]

Sunday, November 20, 2011

One of the typical yet important type of flows in porous media is the case in which two miscible fluids begin to interact with each other due to the gravitational instability that exists due to the density differences between the underlying fluids. This problem has been extensively studied in literature. However many of these studies, have not considered the role the viscosity difference can play during the onset of convection. The onset of convection cannot be captured by most typical linear stability analysis, due to the continuous spectrum of the concerned operator. We convert our governing equations such that they effectively involve a discrete spectrum which in turn provides more accurate and reliable platform for conducting linear stability analysis. In order to verify our analysis, we compare the obtained growth rates with those obtained by fully resolved non-linear 2D simulations.   

Sunday, November 20, 2011

Density or viscosity mismatch among different solutions in porous media may result in interfacial instability. Many underground reservoir flow processes such as in-situ upgrading of heavy oil, in-situ water remediation and geological storage of carbon dioxide involve chemical reactions and natural transverse flows. As a result, further complicated hydrodynamical instabilities may be encountered. In this study, a $2D$ vertical porous medium saturated with a reactant $A$ placed at the top of another reactant $B$ is considered where a chemical product $C$ is formed. All chemical components are assumed to have different physical properties namely the density and the viscosity. Moreover, a transverse flow is introduced parallel to the initial reactant interface. In the limit of small times the stability characteristics of the flow are examined. Thereafter, the evolutions of unstable modes are analyzed by conducting full nonlinear simulations. Chemical production concentration iso-surfaces are tracked and analyzed in time. A quantitative analysis is performed in terms of the total amount of chemical product. A physical discussion on how fluid reactivity and presence of transverse flows can affect the fate of hydrodynamical instabilities is presented.   

Sunday, November 20, 2011

Dissolution by convective mixing is one of the main trapping mechanisms during CO$_{2}$ sequestration in saline aquifers. Initially, the buoyant CO$_2$ dissolves into the underlying brine by diffusion. The CO$_{2}$-brine mixture is denser than the two initial fluids, leading to a Rayleigh-B{\'e}nard-type instability known as convective mixing, which greatly accelerates CO$_{2}$ dissolution. Although this is a well-known process, it remains unclear how convective mixing scales with the governing parameters of the system and its impact on the actual mixing of CO$_{2}$ and brine. We explore the dependence of the CO$_{2}$ dissolution flux on the nonlinearity of the density and viscosity of the fluid mixture by means of high-resolution numerical simulations and laboratory experiments with an analogue fluid system (water and propylene glycol). We find that the value of the concentration for which the density of the mixture is maximum, and the viscosity contrast between the fluids, both exert a powerful control on the convective flux. From the experimental and simulation results, we obtain the scaling behavior of convective mixing, and clarify the role of nonlinear density and viscosity feedbacks.   

Sunday, November 20, 2011

In an arrangement of brine over water across a horizontal permeable membrane, where the unstable density difference across the membrane initiates convection, we discover a new convection regime where the Sherwood number scales approximately as the Rayleigh number. Inferring from the planforms of plume pattern and the estimates of velocity through the membrane, we show that such a regime occurs when advection balances diffusion in the membrane pore. Utilizing mass balance and symmetry assumptions in the top and bottom fluids, convection-diffusion equation for the membrane pore is solved to obtain concentration drops across and in the boundary layers above and below the membrane. With the observation that the normalized net flux is constant in the new regime, an expression for the flux scaling in the new regime is derived. The scaling matches with the experiments and has correct asymptotes in the advection and diffusion regimes (Puthenveettil {\&} Arakeri, JFM, V542, 2005; V609, 2008). The plume spacings in the new regime are distributed lognormally, and their mean follow the trend in the advection regime.   

Sunday, November 20, 2011

Convection in multicomponent mixtures can show a variety of flow patterns due to several heat and mass transfer mechanisms: convection, heat conduction, main and cross diffusion, and the Soret effect. Convective stability of multicomponent fluids has not been widely investigated so far. The use of simplifying assumptions (e.g. the absence of cross-diffusion) may lead to the disagreement between theory and experiment. We study the stability of a plane multicomponent fluid layer heated from above/below in gravity field. In the basic state, the fluid is at rest and temperature gradient induces concentration gradients due to the Soret effect. The problem is reduced to that without cross-diffusion and Soret effect by a special transformation. Several types of boundary conditions are considered: 1) free, permeable 2) rigid, permeable 3) rigid, impermeable. The theorems, which generalize the exchange of stability principle to multicomponent fluids, are proved for boundary conditions 1 and 2. An explicit formula for critical Rayleigh numbers is obtained for boundary conditions 1. The stability problem for boundary conditions 3 was solved numerically for a ternary mixture. The stability maps are constructed in a wide range of parameters.   

Sunday, November 20, 2011

Convective filtration near a highly conductive cylindrical solid inclusion in porous medium saturated by a fluid is studied for the case of heating from above. Convective flow is absent far from inclusion and near inclusion conditions for the conductive state are not satisfied and convection arises. The study is performed in the framework of Darcy-Boussinesq approximation. Two-dimensional flow uniform along the cylinder axis is considered. We use Oseen-like approach where, however, quasi-linearization is applied to the nonlinear terms in the energy equation is not in the momentum. In the first approximation we obtain a system of linear equations with the Rayleigh number as a parameter. An asymptotic representation of solutions uniformly valid at all distances from the inclusion is obtained for small values of Rayleigh number. It is shown that convective flow at the distances from the inclusion larger than the inclusion size has the form of horizontal jets directed away from the body and slowly expanding with the increase of distance from the body. The compensational flow is not of the jet type. The case of finite thermal conductivity of the inclusion is studied in the framework of similar approach.