Influence of acoustic streaming on the stability of a laterally heated three-dimensional cavity.

The flows induced by acoustic streaming in a three-dimensional side-heated parallelepiped cavity of length Ax representative of crystal growth configurations are numerically studied. Both the structure of the flows and their stability properties are determined. The flows have different symmetries, belonging to the group D4 for pure streaming, Z2xZ2 for pure buoyancy, and Z2 for the mixed case, but these symmetries are generally broken at the first bifurcation points. Bifurcation diagrams are obtained which show that the flows become oscillatory periodic at a Hopf bifurcation, either directly on the primary steady solution branch, or on a secondary branch which bifurcates from the primary branch at a steady bifurcation point. The critical Grashof numbers for these bifurcation points are calculated as a function of the cavity length Ax, the Prandtl number Pr and the acoustic streaming parameter A. The thresholds are generally found to increase when the acoustic streaming contribution is enhanced, which indicates a stabilizing effect induced by acoustic streaming and may explain the observed improvement of the crystal quality when ultrasound waves are applied during the growth process. Destabilization effects are, however, found in some parameter range.

Linear temporal and spatiotemporal stability analysis of two-layer falling films with density stratification.

A detailed temporal and spatiotemporal stability analysis of two-layer falling films with density and viscosity stratification is performed by using the Chebyshev collocation method to solve the full system of linear stability equations. From the neutral curves Re(k) for the surface mode and the interface mode of instability, obtained for different density ratios gamma of the upper layer to the lower layer, it is found that smaller density ratios make the surface mode and the short-wave interface mode much more stable, and can even make the short-wave interfacial instability disappear. Moreover, through the study of the local growth rates of the spatiotemporal instability as a function of the ray velocity V , it is found that for not too small incline angles like theta=0.2, the two-layer flow is always convectively unstable, and there is a transition between long- and short-wave instabilities which is determined by the Briggs-Bers collision criterion. Due to the existence of the absolute Rayleigh-Taylor instability for gamma>0 and theta=0, a transition from convective to absolute instability can be detected at small incline angles, and the corresponding boundary curves are plotted for different Reynolds numbers, viscosity ratios, and incline angles. It is found that there exists a limit Reynolds number above which the two-layer film flow can only be convectively unstable for a fixed small incline angle. The spatial amplification properties of the convective waves are finally presented for both surface and interface modes.

Stability of a flow down an incline with respect to two-dimensional and three-dimensional disturbances for Newtonian and non-Newtonian fluids.

Squire's theorem, which states that the two-dimensional instabilities are more dangerous than the three-dimensional instabilities, is revisited here for a flow down an incline, making use of numerical stability analysis and Squire relationships when available. For flows down inclined planes, one of these Squire relationships involves the slopes of the inclines. This means that the Reynolds number associated with a two-dimensional wave can be shown to be smaller than that for an oblique wave, but this oblique wave being obtained for a larger slope. Physically speaking, this prevents the possibility to directly compare the thresholds at a given slope. The goal of the paper is then to reach a conclusion about the predominance or not of two-dimensional instabilities at a given slope, which is of practical interest for industrial or environmental applications. For a Newtonian fluid, it is shown that, for a given slope, oblique wave instabilities are never the dominant instabilities. Both the Squire relationships and the particular variations of the two-dimensional wave critical curve with regard to the inclination angle are involved in the proof of this result. For a generalized Newtonian fluid, a similar result can only be obtained for a reduced stability problem where some term connected to the perturbation of viscosity is neglected. For the general stability problem, however, no Squire relationships can be derived and the numerical stability results show that the thresholds for oblique waves can be smaller than the thresholds for two-dimensional waves at a given slope, particularly for large obliquity angles and strong shear-thinning behaviors. The conclusion is then completely different in that case: the dominant instability for a generalized Newtonian fluid flowing down an inclined plane with a given slope can be three dimensional.

Near-field acoustic streaming jet.

A numerical and experimental investigation of the acoustic streaming flow in the near field of a circular plane ultrasonic transducer in water is performed. The experimental domain is a parallelepipedic cavity delimited by absorbing walls to avoid acoustic reflection, with a top free surface. The flow velocities are measured by particle image velocimetry, leading to well-resolved velocity profiles. The theoretical model is based on a linear acoustic propagation model, which correctly reproduces the acoustic field mapped experimentally using a hydrophone, and an acoustic force term introduced in the Navier-Stokes equations under the plane-wave assumption. Despite the complexity of the acoustic field in the near field, in particular in the vicinity of the acoustic source, a good agreement between the experimental measurements and the numerical results for the velocity field is obtained, validating our numerical approach and justifying the planar wave assumption in conditions where it is a priori far from obvious. The flow structure is found to be correlated with the acoustic field shape. Indeed, the longitudinal profiles of the velocity present a wavering linked to the variations in acoustic intensity along the beam axis and transverse profiles exhibit a complex shape strongly influenced by the transverse variations of the acoustic intensity in the beam. Finally, the velocity in the jet is found to increase as the square root of the acoustic force times the distance from the origin of the jet over a major part of the cavity, after a strong short initial increase, where the velocity scales with the square of the distance from the upstream wall.

Stability of two-layer shear-thinning film flows.

The stability of a two-layer film flow of non-Newtonian fluids is studied with a linear temporal approach. Shear-thinning fluids are considered, which follow the four-parameter inelastic Carreau model. A modified Orr-Sommerfeld equation system is obtained, which is solved by using a spectral Tau collocation method based on Chebyshev polynomials. The effects of density and viscosity stratification are considered, as well as the influence of the shear-thinning properties of the fluid. It is found that, when the viscosity is stronger in the upper layer, the base flow and the stability properties are almost not influenced by the change of the shear-thinning properties in this upper layer. In the other situations, the shear-thinning properties have an influence on the different instabilities, the long-wave surface instability and the short- and long-wave interface instabilities.

Three-dimensional continuation study of convection in a tilted rectangular enclosure.

A continuation method developed from a three-dimensional spectral finite element code is used to study natural convection in a tilted rectangular cavity. The cavity has its length equal to two times the side of its square cross section and it contains a fluid with a Prandtl number Pr = 1. A detailed bifurcation diagram is first obtained in the case without inclination in order to get the sequence of the different branches of solutions and determine the stable solutions. The focus is then put on the stable solutions in the inclined cavity, when the tilt occurs around its longest axis. The subtle changes induced by the tilt on the convective system are clarified. Three different stable solutions are obtained: the longitudinal roll L- solution (with the same sense of rotation as the inclination angle), which develops smoothly from zero Rayleigh number on the leading branch; the longitudinal roll L+ solution (with a sense of rotation opposite to the inclination angle), which is on a disconnected branch and is stabilized beyond a secondary bifurcation point; the oblique roll O ± solutions (corresponding to transverse roll solutions perturbed by the longitudinal flow induced by the tilt), which quickly appear beyond saddle-node points on new disconnected branches. The domain of existence of these stable solutions is eventually obtained and described in the Rayleigh number-inclination parameter space. Finally, the Nusselt number is determined as a function of the inclination at a constant Rayleigh number for the different stable solutions. The Nusselt number is maximum at an inclination of 49.55° for the leading longitudinal roll L- solution.

Instabilities in the Rayleigh-Bénard-Eckart problem.

This study is a linear stability analysis of the flows induced by ultrasound acoustic waves (Eckart streaming) within an infinite horizontal fluid layer heated from below. We first investigate the dependence of the instability threshold on the normalized acoustic beam width H(b) for an isothermal fluid layer. The critical curve, given by the critical values of the acoustic streaming parameter, A(c), has a minimum for a beam width H(b) ≈ 0.32. This curve, which corresponds to the onset of oscillatory instabilities, compares well with that obtained for a two-dimensional cavity of large aspect ratio [A(x) = (length/height) = 10]. For a fluid layer heated from below subject to acoustic waves (the Rayleigh-Bénard-Eckart problem), the influence of the acoustic streaming parameter A on the stability threshold is investigated for various values of the beam width H(b) and different Prandtl numbers Pr. It is shown that, for not too small values of the Prandtl number (Pr > Pr(l)), the acoustic streaming delays the appearance of the instabilities in some range of the acoustic streaming parameter A. The critical curves display two behaviors. For small or moderate values of A, the critical Rayleigh number Ra(c) increases with A up to a maximum. Then, when A is further increased, Ra(c) undergoes a decrease and eventually goes to 0 at A = A(c), i.e., at the critical value of the isothermal case. Large beam widths and large Prandtl numbers give a better stabilizing effect. In contrast, for Prandtl numbers below the limiting value Pr(l) (which depends on H(b)), stabilization cannot be obtained. The instabilities in the Rayleigh-Bénard-Eckart problem are oscillatory and correspond to right- or left-traveling waves, depending on the parameter values. Finally, energy analyses of the instabilities at threshold have indicated that the change of the thresholds can be connected to the modifications induced by the streaming flow on the critical perturbations.

Instabilities in a cylindrical cavity heated from below with a free surface. I. Effect of Biot and Marangoni numbers.

Convective instabilities in a cylindrical cavity heated from below, with a free surface at the top, are numerically investigated using a spectral-element code. Both buoyancy and surface tension forces are taken into account, and heat exchange is considered at the upper surface. This configuration corresponds to the Bénard-Marangoni situation. The primary thresholds associated with azimuthal eigenmodes and corresponding to the onset of convection are first given as a function of the aspect ratio of the cavity A (radius/height), the Biot number Bi, and the Marangoni number Ma. Particular attention is paid to the influence of the Biot and Marangoni numbers: a stabilizing surface tension effect (Ma>0) induces an increase of the primary thresholds, which is magnified for small values of Bi, but may also change the flow structure by creating counter-rotating rolls near the free surface. The nonlinear evolution of the convection beyond its onset is given through bifurcation diagrams for A=1.5. Two different branches of axisymmetric solutions, either with upflow or downflow at the center, emerge at the onset. The destabilization of these solutions and the further dynamical evolution of the flow has been highlighted for widely varying Biot numbers.

Instabilities in a cylindrical cavity heated from below with a free surface. II. Effect of a horizontal magnetic field.

The effect of a constant and uniform horizontal magnetic field on the flow in a cylindrical cavity heated from below, with a free surface at the top, is numerically investigated. The azimuthal modes, which usually trigger convection in a cylinder, are changed by the horizontal magnetic field to oriented modes, either parallel or perpendicular to the magnetic field direction. The corresponding primary thresholds increase with the Hartmann number Ha. This increase, however, depends on the structure of the modes and is the weakest for the parallel modes and the strongest for the perpendicular modes. The changes that affect the evolution of the primary thresholds with the aspect ratio for nonzero Ha are also emphasized. The nonlinear evolution of the convection with a horizontal magnetic field is presented through bifurcation diagrams for different values of the Prandtl number Pr. For Pr=1 and small values of Ha, the structuring effect of the horizontal magnetic field, which involves modifications of the flow structures and bifurcation points, is put into light. Results are finally shown for smaller Pr values corresponding to liquid metals.

Stability of buoyant convection in a layer submitted to acoustic streaming.

The linear stability of the flows induced in a fluid layer by buoyant convection (due to an applied horizontal temperature gradient) and by acoustic streaming (due to an applied horizontal ultrasound beam) is studied. The vertical profiles of the basic flows are determined analytically, and the eigenvalue problem resulting from the temporal stability analysis is solved by a spectral Tau Chebyshev method. Pure acoustic streaming flows are found to be sensitive to a shear instability developing in the plane of the flow (two-dimensional instability), and the thresholds for this oscillatory instability depend on the normalized width Hb of the ultrasound beam with a minimum for Hb=0.32 . Acoustic streaming also affects the stability of the buoyant convection. For a centered beam, effects of stabilization are obtained at small Prandtl number Pr for large beam widths Hb (two-dimensional shear instability) and for moderate Pr (three-dimensional oscillatory instability), but destabilization is also effective at small Pr for small beam widths Hb and at large Pr with a spectacular decrease of the thresholds of the three-dimensional steady instability. An adequate decentring of the ultrasound beam can enhance the stabilization. Insight into the stabilizing and destabilizing mechanisms is gained from the analysis of the fluctuating energy budget associated with the disturbances at threshold. The modifications affecting the two-dimensional shear instability thresholds are strongly connected to modifications of the velocity fluctuations when acoustic streaming is applied. Concerning the three-dimensional steady instability, the spectacular decrease of the thresholds is explained by the extension of the zone with inverse stratification in the lower half of the layer.