ALBERT

Description: A stability analysis is undertaken to theoretically study the effects of gravity modulation and cross-diffusion on the onset of convection in horizontally unbounded doubly diffusive fluid layers. We investigate the stability of doubly stratified incompressible Boussinesq fluid layers with stress-free and rigid boundaries when the stratification is either imposed or induced by Soret separation. The stability criteria are established by way of Floquet multipliers of the amplitude equations. The topology of neutral curves and stability boundaries exhibits features not found in modulated singly diffusive or unmodulated multiply diffusive fluid layers. A striking feature in gravity-modulated doubly cross-diffusive layers is the existence of bifurcating neutral curves with double minima, one of which corresponds to a quasi-periodic asymptotically stable branch and the other to a subharmonic neutral solution. As a consequence, a temporally and spatially quasi-periodic bifurcation from the basic state is possible, in which case there are two incommensurate critical wavenumbers at two incommensurate onset frequencies at the same Rayleigh number. In some instances, the minimum of the subharmonic branch is more sensitive to small parameter variations than that of the quasi-periodic branch, thus affecting the stability criteria in a way that differs substantially from that of unmodulated layers. © 1993, Cambridge University Press. All rights reserved.

Description: An experimental and numerical investigation has been carried out into the instability characteristics of natural convection of an ethanol-water solution in a vertical tank with aspect ratio (height/width) of 15. The solution contains 39 wt% ethanol with Prandtl number Pr=26. The density anomaly due to the Soret effect may be safely ignored in the present test configuration. Onset of instability, in the form of multicellular convection located in the mid-height of the tank, occurs at Grashof number Gr ≅ 13 500. These convection cells are unsteady even at low supercritical states, similar to earlier observations for higher Pr fluids. The cause of such unsteadiness of the flow has been determined by studying the streak images constructed by superposing individual frames of a digital movie sequence. New cells are generated in the upper and lower portions of the tank and then migrate toward the centre, causing the convection cells in the mid-section to merge. At higher Gr, even the tertiary cells, which rotate in the opposite direction of the secondary cells, participate in the merging process. Numerical simulations of the two-dimensional natural convection of a Boussinesq fluid with constant thermophysical properties, carried out at low supercritical Gr equivalent to the experimental value, show the same process of cell generation and merging as that observed in the experiments. By analysing the substantial time rate of change of the kinetic energy of the fluid using the mechanical energy equation, it is determined that the energy needed for the cell generation process is supplied by the work of the dynamic pressure. The subsequent migration of the cells toward the middle is caused by the pressure gradient in the tank. The total kinetic energy of the fluid attains a relative maximum right after a merging process due to the reduction of dissipation associated with the region of strong shear between the cells. © 2004 Cambridge University Press.

Description: The second-order problem of Helmholtz flow past lifting hydrofoils and symmetric struts has been formulated and solved. The solution involves elementary operations on the known solutions of the first-order problem. The second-order lift and drag coefficients are given in integral form. Results obtained for a flat plate at incidence and a symmetric wedge agree with the exact solutions up to the second order. In terms of quantitative improvements, the present second-order theory predicts a lift coefficient for a flat plate at 45° incidence with an error of 8%, and a drag coefficient for a symmetric wedge of 50° included angle with an error of 5%; the corresponding angles at which the linear theory would predict force coefficients incurring the same errors are 5° and 15° respectively. © 1962, Cambridge University Press. All rights reserved.

Description: The stability of the flow induced by an impulsively started inner cylinder in a Couette flow apparatus is investigated by using a linear stability analysis. Two approaches are taken; one is the treatment as an initial-value problem in which the time evolution of the initially distributed small random perturbations of given wavelength is monitored by numerically integrating the unsteady perturbation equations. The other is the quasi-steady approach, in which the stability of the instantaneous velocity profile of the basic flow is analyzed. With the quasi-steady approach, two stability criteria are investigated; one is the standard zero perturbation growth rate definition of stability, and the other is the momentary stability criterion in which the evolution of the basic flow velocity field is partially taken into account. In the initial-value problem approach, the predicted critical wavelengths agree remarkably well with those found experimentally. The kinetic energy of the perturbations decreases initially, reaches a minimum, then grows exponentially. By comparing with the experimental results, it may be concluded that when the perturbation kinetic energy has grown a thousand-fold, the secondary flow pattern is clearly visible. The time of intrinsic instability (the time at which perturbations first tend to grow) is about ¼ of the time required for a thousandfold increase, when the instability disks are clearly observable. With the quasi-steady approach, the critical times for marginal stability are comparable to those found using the initial-value problem approach. The predicted critical wavelengths, however, are about 1½ to 2 times larger than those observed. Both of these points are in agreement with the findings of Mahler, Schechter & Wissler (1968) treating the stability of a fluid layer with time-dependent density gradients. The zero growth rate and the momentary stability criteria give approximately the same results. © 1971, Cambridge University Press. All rights reserved.

Description: We examine two-dimensional motion of a stably stratified fluid containing two solutes with different molecular diffusivities in an inclined slot. The two solutes have continuous opposing gradients with the slower-diffusing one more dense at the bottom. It is found that, in the steady state, there exists a slow upward flow along the slope driven by the slight buoyancy difference near the wall, not unlike the solution found by Phillips (1970) for a single solute. The magnitude of the flow is less than that in Phillips’ solution by a factor of approximately (1-λ)/(1-λτ), where λ is the ratio of the density gradient and τ−1 is the ratio of the diffusivity of the faster-diffusing solute to that of the slower-diffusing one. For the time-dependent flow resulting from switching on the diffusivities at t = 0, there may be a dramatic reversal of the flow near the walls depending on the relative magnitude of λ and τ. If λ is somewhat greater than τ, the initial flow is downward, along the slope, reaching a maximum magnitude about one order of magnitude greater than the steady-state value. Then the ‘reverse’ flow gradually diminishes and approaches the steady state rather slowly. For λ ≳ τ, the approach to the steady state is monotonic; there is no ‘reverse’ flow near the wall. The existence of the downward flow, which was observed by Turner & Chen (1974), may lead to double-diffusive instabilities which eventually result in horizontal convecting layers. © 1975, Cambridge University Press. All rights reserved.

Description: The two-dimensional motion of a stably stratified fluid containing two solutes with different molecular diffusivities in an inclined slot has recently been examined by Chen (1975, hereafter referred to as I). The two solutes have continuous opposing gradients with the slower-diffusing one more dense at the bottom. It is found that, in the steady state, there exists a slow upward flow along the slope driven by the slight buoyancy difference near the wall, not unlike the solution found by Wunsch (1970) and Phillips (1970) for a single solute. For the time-dependent flow resulting from switching on the diffusivities at t = 0, there may be a flow reversal near the wall depending on the relative magnitude of λ and τ (where λ is the ratio of the density gradient and τ−1 is the ratio of the diffusivity of the faster-diffusing solute T to that of the slower-diffusing one S). By examining the distributions of S and T across the slot, it becomes apparent that in cases with flow reversal double-diffusive instability is likely to occur. In this paper, we examine the stability of time-dependent double-diffusive convection in an inclined slot both analytically and experimentally. The time-dependent perturbation equations are numerically integrated starting with an initial distribution of small random disturbances in the vorticity. The growth or decay of the kinetic energy of the perturbations serves to indicate whether the flow is unstable or stable. The results show that the flow becomes more unstable (a) with increasing λ at a given angle of inclination with respect to the vertical and (b) with increasing angle of inclination at a given value of λ. Experiments were carried out in a 2.54 cm wide slot using sugar and salt solutions at angles of inclination of 30°, 45° and 60°. Results obtained confirm the trends predicted by the analysis. Good agreement was obtained between the predicted and the experimental values of the critical wavelength for the case λ = 0.7. © 1977, Cambridge University Press. All rights reserved.

Description: Linear stability analysis is applied to the problem of a density-stratified fluid contained in an inclined slot being subjected to a lateral temperature gradient. Stability equations are solved using the Galerkin technique with 12 terms in the truncated expansion series. Within the range of θ considered, |θ| 〈 75°, critical instability was found to be of the stationary type. Results of critical thermal Rayleigh numbers and wavenumbers at all inclination angles are in good agreement with the experimental results obtained earlier (Paliwal & Chen 1980). Contrary to intuition, these results show that the system is more stable when the lower wall is heated. This is shown to be the result of the increased vertical solute gradient in the steady state prior to the onset of instabilities when the heating is from below. © 1980, Cambridge University Press. All rights reserved.

Description: The nonlinear double-diffusive convection in a Boussinesq fluid with stable constant vertical solute gradient, and bound by two differentially heated rigid inclined parallel plates is considered. The analysis was carried out by a Galerkin method for the cases when the angle of inclination was 0°, - 45° and + 45° (positive angle denotes heating from below, and negative angle denotes heating from above). The counter-rotating cells predicted by the linear theory merge into single cells with the same sense of rotation within a very short period of time even under slightly supercritical conditions. This is consistent with the experimental observations. Furthermore, as observed in the experiments, the evolution of instability is more rapid when heating is from above than when heating is from below. Our results for a salt-heat system are in excellent agreement with those based on the limiting case of Lewis number 0 and Schmidt number →. © 1982, Cambridge University Press. All rights reserved.

Description: The effect of surface tension on the onset of convection in a horizontal double-diffusive layer was studied both experimentally and by linear stability analysis. The experiments were conducted in a rectangular tank with base dimension of 25 x 13 cm and 5 cm in height. A stable solute (NaCl) stratification was first established in the tank, and then a vertical temperature gradient was imposed. Vertical temperature and concentration profiles were measured using a thermocouple and a conductivity probe and the flow patterns were visualized by a schlieren system. Two types of experiments were carried out which illustrate the effect of surface tension on the onset of convection. In the rigid-rigid experiments, when the critical thermal Rayleigh number, RT, is reached, large double-diffusive plumes were seen simultaneously to rise from the heated bottom and descend from the cooled top. In the rigid-free experiments, owing to surface-tension effects, the first instability onset was of the Marangoni type. Well-organized small plumes were seen to emerge and persist close to the top free surface at a relatively small RMT (where subscript M denotes ‘Marangoni’). At larger RbT 〉 RtT (where subscript t denotes ‘top’) these plumes evolved into larger double-diffusive plumes. The onset of double-diffusive instability at the bottom region occurred at a still higher RbT 〉 RtT (where subscript b denotes ‘bottom’), A series of stability experiments was conducted for a layer with an initial top concentration of 2wt% and different concentration gradients. The stability map shows that in the rigid-free case the early Marangoni instability in the top region reduces significantly the critical RT for the onset of double-diffusive convection. Compared with the rigid-rigid case, the critical RT in the top region is reduced by about 60 % and in the bottom region by about 30%. The results of the linear stability analysis, which takes into account both surface-tension and double-diffusive effects, are in general agreement with the experiments. The analysis is then applied to study the stability characteristics of such a layer as gravity is reduced to microgravity level. Results show that even at 10-4go, where g0is the gravity at sea level, the double-diffusive effect is of equal importance to the Marangoni effect. © 1995, Cambridge University Press. All rights reserved.

Description: Experiments are conducted to study the longitudinal vortices that develop in the boundary layer on the upper surface of an inclined, heated plate. An isothermal plate in water is inclined at angles ranging from 20 to 60 degrees (from the vertical) while the temperature difference is varied from 2 to 23°C. A double-pass Schlieren system is used to visualize the vortices and particle image velocimetry (PIV) is used to measure velocities. In addition, a unique method is developed such that both the Schlieren visualization and PIV can be performed simultaneously. The wavelengths of the vortices and the critical modified Reynolds numbers (R̃) for the onset, merging, and breakup of the vortices are determined from Schlieren images for Pr = 5.8. The critical values for R̃ and the critical wavelengths are compared to results of previous experiments and stability analyses. The spatial growth rates of vortices are determined by using the PIV measurements to determine how the circulation in the vortices grows with distance from the leading edge. This is the first time that the growth rate of the vortices have been found using velocity measurements. These spatial growth rates are compared to the results of Iyer & Kelly (1974) and found to be in general agreement. By defining a suitable circulation threshold, the critical R̃ for the onset of the vortices can be found from the growth curves.