Limit Of Large Rayleigh Number Research Articles

The energy flux spectrum of internal waves generated by turbulent convection

We present three-dimensional direct numerical simulations of internal waves excited by turbulent convection in a self-consistent, Boussinesq and Cartesian model of mixed convective and stably stratified fluids. We demonstrate that in the limit of large Rayleigh number ($Ra\in [4\times 10^{7},10^{9}]$) and large stratification (Brunt–Väisälä frequencies$f_{N}\gg f_{c}$, where$f_{c}$is the convective frequency), simulations are in good agreement with a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region as described in Lecoanet & Quataert (Mon. Not. R. Astron. Soc., vol. 430 (3), 2013, pp. 2363–2376). Specifically, we demonstrate that the wave energy flux spectrum scales like$k_{\bot }^{4}\,f^{-13/2}$for weakly damped waves (with$k_{\bot }$and$f$the waves’ horizontal wavenumbers and frequencies, respectively), and that the total wave energy flux decays with$z$, the distance from the convective region, like$z^{-13/8}$.

Axisymmetric natural convection-driven evaporation of water: Analysis and numerical solution

Coupled steady laminar heat and mass transfer from an evaporating circular pool of water into quiescent dry air is considered in the limit of large Rayleigh number (Ra). The resulting boundary-layer equations are reformulated in similarity-like variables that are necessary for numerical implementation; different variables are necessary, depending on whether the water is hotter than the ambient air, or vice versa. The governing partial differential equations are solved using a finite-element method and the dependences of the dimensionless Nusselt (Nu) and Sherwood (Sh) numbers on the water temperature and the ambient air temperature are determined.

On the boundary-layer structure of high-Prandtl-number horizontal convection

This paper describes the boundary-layer structure of the steady flow of an infinite Prandtl number fluid in a two-dimensional rectangular cavity driven by differential heating of the upper surface. The lower surface and sidewalls of the cavity are thermally insulated and the upper surface is assumed to be either shear-free or rigid. In the limit of large Rayleigh number (R → ∞), the solution involves a horizontal boundary layer at the upper surface of depth of order R−1/5 where the main variation in the temperature field occurs. For a monotonic temperature distribution at the upper surface, fluid is driven to the colder end of the cavity where it descends within a narrow convection-dominated vertical layer before returning to the horizontal layer. A numerical solution of the horizontal boundary-layer problem is found for the case of a linear temperature distribution at the upper surface. At greater depths, of order R−2/15 for a shear-free surface and order R−9/65 for a rigid upper surface, a descending plume near the cold sidewall, together with a vertically stratified interior flow, allow the temperature to attain an approximately constant value throughout the remainder of the cavity. For a shear-free upper surface, this constant temperature is predicted to be of order R−1/15 higher than the minimum temperature of the upper surface, whereas for a rigid upper surface it is predicted to be of order R−2/65 higher.

Large Rayleigh number thermal convection: Heat flux predictions and strongly nonlinear solutions

We investigate the structure of strongly nonlinear Rayleigh–Bénard convection cells in the asymptotic limit of large Rayleigh number and fixed, moderate Prandtl number. Unlike the flows analyzed in prior theoretical studies of infinite Prandtl number convection, our cellular solutions exhibit dynamically inviscid constant-vorticity cores. By solving an integral equation for the cell-edge temperature distribution, we are able to predict, as a function of cell aspect ratio, the value of the core vorticity, details of the flow within the thin boundary layers and rising/falling plumes adjacent to the edges of the convection cell, and, in particular, the bulk heat flux through the layer. The results of our asymptotic analysis are corroborated using full pseudospectral numerical simulations and confirm that the heat flux is maximized for convection cells that are roughly square in cross section.

Relaxation in Reactive Flows

We consider convective systems in a bounded domain, in which viscous fluids described by the Stokes system are coupled using the Boussinesq approximation to a reaction-advection-diffusion equation for the temperature. We show that the resulting flows possess relaxation-enhancing properties in the sense of [CoKRZ]. In particular, we show that solutions of the nonlinear problems become small when gravity is sufficiently strong due to the improved interaction with the cold boundary. As an application, we deduce that the explosion threshold for power-like nonlinearities tends to infinity in the large Rayleigh number limit. We also discuss the behavior of the principal eigenvalues of the corresponding advection-diffusion problem and the quenching phenomenon for reaction-diffusion equations.

Effect of anisotropy on the development of convective boundary layer flow in porous media

The effects of anisotropy on the development of thermal boundary layer flow in a rectangular porous cavity is studied. The side walls of the cavity are respectively heated and cooled isothermally. Top and bottom walls are insulated. The porous medium is anisotropic both in permeability and thermal conductivity with its principal axes oriented in a direction that is oblique to the gravity vector. Scale analysis is applied to predict the orders of magnitude involved in the boundary layer regime. In the large Rayleigh number limit, the governing boundary layer equations are solved in closed form, using an intergral approach. A finite difference method is used to obtain numerical solutions of the full governing equations. The effects of the anisotropy in permeability and thermal conductivity on the development of free convective boundary layer flow are found to be significant.

Free convection along a vertical heated plate in a porous medium with anisotropic permeability

Natural convection from a semi‐infinite vertical plate embedded in a fluid saturated porous medium is studied both analytically and numerically. The plate is assumed to be heated isothermally or by a constant heat flux. The porous medium, modeled according to Darcy’s law, is anisotropic in permeability with its principal axes oriented in a direction that is oblique to the gravity vector. In the large Rayleigh number limit, the governing boundary‐layer equations are solved in closed form, using a similarity transformation. Comparisons between the numerical solution of the full equations and analytical solutions are presented for a wide range of the governing parameters. The effects of the anisotropic permeability ratio K*, of the orientation angle of the principal axes θ, and of the Rayleigh number RH on the flow and heat transfer are investigated. Results indicate that the anisotropic properties of the porous medium considerably modify the heat transfer, velocity and temperature profiles from that expected under isotropic conditions.

Directional solidification into static stability

Consider the directional solidification of a binary alloy rejecting a heavy solute as it solidifies upward. If the solidification front is planar, the fluid melt ahead of the front is stably stratified and convection is not expected. In this paper we analyse the linear stability of planar solidification asymptotically in the limit of large solutal Rayleigh number, R. Three distinct linear modes are found which correspond to internal waves, buoyancy edge waves, or morphological modes. Of these three modes, only the morphological modes are subject to an instability. We find that for large Rayleigh number this instability first occurs at long wavelengths with wavenumbers that scale on R−1/14. The scalings derived from the linear analysis are used to construct a nonlinear theory for the morphological instability in the large Rayleigh number limit. Similarity solutions are found which describe steadily convecting, non-planar growth reminiscent of an observed phenomenon known as steepling.

Boundary-layer analysis for natural convection in a vertical porous layer filled with a non-Newtonian fluid

An analytical and numerical study is reported of steady-state natural convection in a two-dimensional rectangular porous cavity saturated by a non-Newtonian fluid. The enclosure is heated and cooled isothermally from the vertical sides, while the horizontal walls are adiabatic. The modified Darcy power-law model proposed by Pascal (1983) is used to characterize the non-Newtonian fluid behavior. In the large Rayleigh number limit, the boundary-layer equations are solved analytically upon introducing a similarity transformation. The core structure is determined using an integral form of the energy equation. Numerical integrations are carried out using the Runge-Kutta method. Solutions for the flow and temperature fields and Nusselt numbers are obtained in terms of a modified Rayleigh number R, the aspect ratio of the cavity A, and the power-law index n. A numerical study of the same phenomenon, obtained by solving the complete system of governing equations, is also conducted, and results are reported in the range 10 2 ≤ R ≤ 10 3, 4 ≤ 8, and 0.6 ≤ n ≤ 1.4. The numerical experiments confirm the flow features and scales anticipated by the approximate boundary-layer solution.

Convection in a fluid-saturated porous medium due to internal heat generation

Two-dimensional convective motions driven by uniformly distributed internal heat sources in a fluid-saturated porous medium are analyzed in the large Rayleigh number limit. An integral method is used to obtain a simple analytical representation of the flow field within a rectangular cavity whose vertical side walls are isothermal and whose horizontal boundaries are adiabatic. The results are in excellent agreement with numerical computations based on a finite-difference technique.

Thermal convection in a cavity filled with a porous medium: A classification of limiting behaviours

Thermally driven flows in a 2-dim. rectangular cavity filled with a fluid-saturated porous medium are considered in the large Rayleigh number limit when the applied temperature gradient is perpendicular to the gravity vector. The standard boundary layer description of these flows is shown to be valid only for R ⪢ 10 4 L 2, where R is the Rayleigh number based on the cavity height and L is the cavity aspect ratio (length/depth). For flows in which R = 0( L 2), L → ∞, the horizontal layers merge and a different solution regime exists. A discussion of the relationship between the merged layer limit ( R/ L 2 fixed, L → ∞) and the classical Hadley cell limit (R fixed, L → ∞)is given. It is found that the intermediate regime R/ L fixed, L → ∞, provides the necessary bridge between the merged layer and the Hadley solutions. Bounds on the extent of the various regimes in the ( R, L) parameter space are deduced. The scaling suggested by the merged layer regime gives rise to an alternative correlation of heat transfer data. Existing theories for the Nusselt number are reviewed and shown to be consistent with the new scaling law.

Convection in a porous layer

Steady convective motions inside a rectangular cavity filled with a porous material are examined in the large Rayleigh number limit for flows driven by a horizontal temperature gradient. The boundary-layer structure on the side walls is determined using an integral relations approach. This method leads to results for the core mass flux, for the core-temperature gradient and for the heat-transfer characteristics which are in excellent agreement with numerical solutions of the boundary-layer equations.

Boundary Layer Regime for Laminar Free Convection between Horizontal Circular Cylinders

The steady, buoyancy-driven, laminar motion induced in the annulus of two horizontal, concentric, circular cylinders by a difference in the boundary temperatures is studied analytically in the large Rayleigh number limit. The flowfield is divided into five physically distinct regions: (1) an inner free convection boundary layer near the inner cylinder, (2) an outer free convection boundary layer near the outer cylinder, (3) a vertical plume above the inner cylinder, (4) a stagnant region below the inner cylinder, and (5) a core region surrounded by the other four regions. Zeroth-order solutions which account for the coupling of those five regions are obtained in the high Prandtl number limit using a boundary-layer approximation and integral methods. Comparisons of the calculated heat transfer and temperature fields with experiment and numerical finite-difference results are favorable.

Fast viscous bénard convection

Abstract We have previously described a method for the precise study of steady two-dimensional convection problems in the asymptotic limit of large Rayleigh number R, assuming a very viscous fluid with the Prandtl number effectively infinite. The method here is applied to cellular convection between isothermal horizontal planes (Bénard convection). The result for the dimensionless heat flux, or Nusselt number, is where c is ⅓ for free-surface convection and ⅕ for fixed surface convection. The constant b is found numerically, as a function of the cell aspect ratio. Previous approximate analyses have obtained c values of ⅓ and ¼ for the two cases; b values were not determined.