Buoyancy Boundary Layer Research Articles

Finite-amplitude instability of the buoyancy boundary layer in a thermally stratified medium

The finite-amplitude instability of the buoyancy-driven boundary layer is considered on a vertical plate immersed in a thermally stratified ambient medium, where the wall and surrounding fluid have different temperature gradients. Although the linear stability in this configuration has been investigated, the finite-amplitude solution arising from the critical instability has been studied only for specific parameter values. We extend this by using the amplitude expansion method. The primary bifurcations to the two-dimensional least unstable mode for different temperature gradient ratios ( $0 \leqslant \lambda \leqslant 10$ ) and Prandtl numbers ( $10^{-1} \leqslant Pr \leqslant 10^{4}$ ) are investigated. Only supercritical bifurcations are found to occur when $0 \leqslant \lambda < 2$ and $Pr \leqslant 2800$ , while subcritical bifurcations are also found for larger values of temperature gradient ratio and Prandtl number. Analysis of the contribution of the nonlinear terms in the Landau coefficient reveals that the interaction of the modification of the mean flow and second harmonic for velocity with the fundamental mode for temperature plays an important role in subcritical bifurcation. Based on the Landau equation, the threshold amplitude of the nonlinear equilibrium solution is discussed as well. These encouraging results should be helpful for understanding such a buoyancy-driven flow system.

Analysis of Turbulent Characteristics of Sheared Convective Boundary Layer under Different Shear and Potential Temperature Gradients

Effects of shear and potential temperature gradient on the formation mechanism and turbulence characteristics of the sheared convective boundary layer are studied through Large-Eddy Simulation (LES) experiments. Four cases with different velocity difference ΔU of 1.5ms−1 and 3.0ms−1 and different potential temperature gradient Γθ of 0.006Km−1 and 0.015Km−1 were conducted in the horizontal homogeneous atmospheric convective boundary layer of 6000m × 6000m × 2000m. The results show that the formation of the entrainment layer is affected integratedly by the horizontal shear effect and the vertical buoyancy effect, in which turbulence shows strong anisotropy and intermittent characteristics, wherein the mixed layer is dominated by stable small eddies, and the entrainment layer by strip-shaped large eddies. Compared with the pure buoyancy boundary layer, the entrainment flux ratio Af is 0.176∼0.385, and the height of the entrainment layer zi is shifted to the side with bigger velocity (U2). ΔU promotes entrainment and Γθ inhibits, especially for the upper limit of the entrainment layer h2. Obviously, the shear-shelter entrainment phenomenon appears under the weak shear and strong potential temperature gradient.

Lower bound for transient growth of inclined buoyancy layer

The relationship between stabilities of the buoyancy boundary layers along an inclined plate and a vertical plate immersed in a stratified medium is studied theoretically and numerically. The eigenvalue problem of energy stability is solved with the method of descending exponentials. The disturbance energy is found to be able to grow to 11.62 times as large as the initial disturbance energy for Pr = 0.72 when the Grashof number is between the critical Grashof numbers of the energy stability and the linear stability. We prove that, with a weighted energy method, the basic flow of the vertical buoyancy boundary layer is stable to finite-amplitude streamwise-independent disturbances.

Constant-flux discrete heating in a unit aspect-ratio annulus

Natural convection in an annulus with a discrete heat source on the inner cylinder is studied numerically. The outer cylinder is isothermally cooled at a fixed low temperature, and the top wall, the bottom wall and unheated portions of the inner cylinder are thermally insulated. For low applied heat flux through the heater, as measured non-dimensionally by a Grashof number, Gr, the flow in the annular gap consists of a single-cell overturning meridional flow driven by the radial temperature gradient between the heater on the inner cylinder and the cold outer cylinder. In this regime, the flow is very weak and heat is transported primarily via conduction. The flow structure does not change until Gr ∼ 104, although the flow strength steadily increases with Gr. As the nonlinear convection terms become more important, the meridional circulation sweeps the isotherms from being almost vertical near the outer cylinder to almost horizontal near the bottom wall. By the end of the transition from the conduction-dominated regime (Gr < 104) to the convection-dominated regime (Gr ∼ 106), the flow becomes segregated into three distinct regions: (i) for vertical levels below that of the bottom of the heater, an essentially cold stagnant pool develops, with the heat flux through the outer cylinder dropping to zero. (ii) At vertical levels between the bottom and the top of the heater, most of the region in between the two cylinders is stably stratified with a relatively weak radial flow from the cold to the heated cylinder. The horizontal isotherms adjust to the temperatures on the cylinders in thin buoyancy boundary layers which drive fluid down the cold cylinder and up the heated cylinder segment. The boundary layer on the heater is about half as thick as that on the cold cylinder, but about twice as intense. (iii) The third region is above the heater top. The boundary layer flow from the heater continues upward where it meets the top endwall and bounces off of it. A wavy jet bounces on its way radially outward between the top insulated wall and the stably stratified region below. Flow separation on the top wall leads to the formation of a recirculation zone there. The vast majority of the heat flux through the outer cylinder occurs at this upper level, and is heavily concentrated near the very top. Geometric factors, such as the radius ratio of the cylinders and the heater placement, have quantitative effects, which are described, but the overall qualitative picture remains unchanged.

Multi-layered diffusive convection. Part 1. Spontaneous layer formation

Diffusive convection in an infinite two-dimensional fluid with linear vertical gradients of temperature and salinity is studied numerically and analytically. When the density gradient ratio exceeds a critical value above which diffusive convection grows according to the linear stability analysis, spontaneous layer formation is found to occur. At the first stage nearly steady oscillating motions, the horizontal scale of which is of the order of the buoyancy boundary layer scale δ, arise. After several tens of the oscillation cycle, a transition to the second stage occurs in which overturning convective motions develop and well-mixed regions are formed. The convective motions resemble Rayleigh–Bénard convection at a high Rayleigh number. The well-mixed regions are gradually organized into horizontal layers, a typical thickness of which is of the order of δ. A detailed analysis of the nonlinear process during the layer formation reveals that four modes are responsible for the layer formation: The first mode is the linear fastest-growing mode with wavenumber vector (k0, 0). The second mode with (k0, m0) is weakly growing. The third mode with (0, m0) is dissipating, and the fourth mode is its higher harmonic having (0, 2m0). It is shown that a truncated spectral model with the four modes well reproduces the results of the full numerical simulation.

Supercritical Solutions for the Buoyancy Boundary Layer

Two-dimensional, finite amplitude solutions for the Prandtl buoyancy boundary layer are obtained for Reynolds numbers close to the critical values by an expansion in terms of disturbance amplitude. Supercritical nonlinear wave solutions are found to occur, rather than subcritical instabilities. The dependence of the wave amplitude and Nusselt number upon wall inclination, relative wall temperature, and Prandtl number is discussed.

Instabilities in buoyancy-driven boundary-layer flows in a stably stratified medium

An analysis is made of the stability of a buoyancy boundary layer existing on an inclined wall which is either heated or cooled relative to ambient, stably stratified fluid. A Boussinesq fluid, with various Prandtl numbers, is considered. Detailed calculations of the linear stability boundaries are made for both streamwise periodic, travelling disturbances and spanwise periodic, stationary disturbances. The former type is found to become unstable first for all angles of tilt, but calculations at a particular angle indicate that the latter can have a higher growth rate once the Reynolds number is sufficiently above the critical value. Energy integrals are evaluated at the critical Reynolds number for various angles of tilt in order to clarify the mechanisms for energy transfer to the disturbance.

Energy stability of the buoyancy boundary layer

The critical value RE of the Reynolds number R is predicted by the application of the energy theory. When R &lt; RE, the buoyancy boundary layer is the unique steady solution of the Boussinesq equations and the same boundary conditions, and is, further, stable in a slightly weaker sense than asymptotically stable in the mean. The critical value RE is determined by numerically integrating the relevant Euler–Lagrange equations. Analytic lower bounds to RE are obtained. Comparisons are made between RE and RL, the critical value of R according to linear theory, in order to demark the region of parameter space, RE &lt; R &lt; RL, in which subcritical instabilities are allowable.

Analogous behavior of homogeneous, rotating fluids and stratified, non-rotating

It has been pointed out in the past that analyses of homogeneous, rotating fluids (system S) and stratified, non-rotating fluids (system S) lead to the same mathematical problem in certain circumstances. The analogy is explored here for the case of steady, linear two-dimensional flows where the temperature variations imposed along side boundaries drive the stably stratified flow and swirl or zonal velocity imposed at the top and bottom boundaries drive the rotating flow. In the fluid S buoyancy boundary layers exist near the side boundaries and these serve the same function in adjusting the temperature between boundary and interior values that the Ekman layer serves to adjust the zonal velocity in the rotating case. The mathematical properties of the two layers are identical. In system S any changes which take place in the interior and tend to alter the stratification are resisted just as changes in the interior which tend to alter the vertical vorticity are resisted in system ?. Processes which give rise to changes in the interior are concentrated in the boundary layers. Simple problems are worked out which exhibit the analogous behavior for two-dimensional systems driven by symmetric and by anti-symmetric conditions imposed at the boundaries. The temperature in the interior of the stratified fluid is the average of the imposed boudary values. This means that with symmetric driving temperatures the fluid is simply restratified by the imposed conditions and with anti-symmetric boundary values the temperature of the fluid interior is unchanged. Buoyancy boundary layers are necessary to adjust interior to boundary temperatures in the latter case. The rotating system has an analogous behavior. In three dimensions the analogy breaks down. The reason is that the third dimension in system S does not involve the constraint and therefore is literally an added degree of freedom whereas in system ? the third dimension also involves the constraint. The difference between the two systems is exhibited by an analysis of the three-dimensional stratified problem with symmetric and anti-symmetric boundary conditions. The symmetric problem in system S differs qualitatively from both the two-dimensional case and the rotating fluid problem. The three-dimensional problem in system ? is qualitatively similar to the two-dimensional one. DOI: 10.1111/j.2153-3490.1967.tb01485.x

Analogous behavior of homogeneous, rotating fluids and stratified, non-rotating fluids

It has been pointed out in the past that analyses of homogeneous, rotating fluids (system S) and stratified, non-rotating fluids (system S) lead to the same mathematical problem in certain circumstances. The analogy is explored here for the case of steady, linear two-dimensional flows where the temperature variations imposed along side boundaries drive the stably stratified flow and swirl or zonal velocity imposed at the top and bottom boundaries drive the rotating flow. In the fluid S buoyancy boundary layers exist near the side boundaries and these serve the same function in adjusting the temperature between boundary and interior values that the Ekman layer serves to adjust the zonal velocity in the rotating case. The mathematical properties of the two layers are identical. In system S any changes which take place in the interior and tend to alter the stratification are resisted just as changes in the interior which tend to alter the vertical vorticity are resisted in system ?. Processes which give rise to changes in the interior are concentrated in the boundary layers. Simple problems are worked out which exhibit the analogous behavior for two-dimensional systems driven by symmetric and by anti-symmetric conditions imposed at the boundaries. The temperature in the interior of the stratified fluid is the average of the imposed boudary values. This means that with symmetric driving temperatures the fluid is simply restratified by the imposed conditions and with anti-symmetric boundary values the temperature of the fluid interior is unchanged. Buoyancy boundary layers are necessary to adjust interior to boundary temperatures in the latter case. The rotating system has an analogous behavior. In three dimensions the analogy breaks down. The reason is that the third dimension in system S does not involve the constraint and therefore is literally an added degree of freedom whereas in system ? the third dimension also involves the constraint. The difference between the two systems is exhibited by an analysis of the three-dimensional stratified problem with symmetric and anti-symmetric boundary conditions. The symmetric problem in system S differs qualitatively from both the two-dimensional case and the rotating fluid problem. The three-dimensional problem in system ? is qualitatively similar to the two-dimensional one. DOI: 10.1111/j.2153-3490.1967.tb01485.x