Multiple modes of instability in a box heated from the side in low-Prandtl-number fluids (original) (raw)
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Journal of Fluid Mechanics, 2003
The multiplicity, stability and bifurcations of low-Prandtl-number steady natural convection in a two-dimensional rectangular cavity with partially and symmetrically heated vertical walls are studied numerically. The problem represents a simple model of a set-up in which the height of the heating element is less than the height of the molten zone. The calculations are carried out by the global spectral Galerkin method. Linear stability analysis with respect to two-dimensional perturbations, a weakly nonlinear approximation of slightly supercritical states and the arclength path-continuation technique are implemented. The symmetry-breaking and Hopf bifurcations of the flow are studied for aspect ratio (height/length) varying from 1 to 6. It is found that, with increasing Grashof number, the flow undergoes a series of turning-point bifurcations. Folding of the solution branches leads to a multiplicity of steady (and, possibly, oscillatory) states that sometimes reaches more than a dozen distinct steady solutions. The stability of each branch is studied separately. Stability and bifurcation diagrams, patterns of steady and oscillatory flows, and patterns of the most dangerous perturbations are reported. Separated stable steady-state branches are found at certain values of the governing parameters. The appearance of the complicated multiplicity is explained by the development of the stably and unstably stratified regions, where the damping and the Rayleigh-Bénard instability mechanisms compete with the primary buoyancy force localized near the heated parts of the vertical boundaries. The study is carried out for a low-Prandtl-number fluid with Pr = 0.021. It is shown that the observed phenomena also occur at larger Prandtl numbers, which is illustrated for Pr = 10. Similar three-dimensional instabilities that occur in a cylinder with a partially heated sidewall are discussed.
Multiple flow transitions in a box heated from the side in low-Prandtl-number fluids
Physical Review E, 2007
The determination of the flow transitions in a cavity heated from the side in low-Prandtl-number fluids has been a challenge for many years. Contrarily to the Rayleigh-Bénard situation, these transitions occur in already very intense convective flows, and the problem has been up to now mainly treated in two-dimensional situations. Thanks to a performing numerical method, the thresholds corresponding to the first flow transition in a three-dimensional ͑3D͒ parallelepipedic cavity have been determined for a wide range of aspect ratios and Prandtl number values. We obtain a kind of map of the transitions involved. Such a map of transitions is quite usual for Rayleigh-Bénard or Marangoni-Bénard situations, but completely new for 3D cavities heated from the side. The most striking result is the very frequent change of stability branches when the aspect ratios or Prandtl number are changed, which indicates different flow structures triggered at the thresholds, either steady or oscillatory, and breaking some of the symmetries of the problem.
Proceeding of International Heat Transfer Conference 12, 2002
Applications to nuclear reactors have revived interest in natural convection. A rectangular closed cavity with internal heat generation and wall-cooling roughly simulating a channel of an internally-cooled homogeneous reactor core has been studied theoretically and experimentally. The basic equations of continuity, Navier-Stokes, and a modified energy relation including a volumetric heat source are normalized to show the dependence on the following nondimensional parameters: i) Nusselt number based on width; ii) Prandtl number, and iii) product of Rayleigh number based on width and aspect ratio, a/b y of the cavity. The complexity of these equations allows only numerical solutions, which are obtained following a modified Squire's method consisting in assuming temperature and velocity profiles. These are substituted into the nondimensional equations, and integrated across the cavity, resulting in a still complex system of differential equations in which the dependent variables and unknown functions are the thickness, velocity, and temperature of the rising core of fluid. The coefficients in the equations are functions of the core thickness, more or less complicated according to the velocity and temperature profiles assumed. Two cases are considered: a simplified temperature profile, as used by Lighthill; and a more sophisticated profile with a positive maximum. Both velocity profiles are Lighthill's. Digital computer calculations using a fourth-order Kunge-Kutta method yielded solutions that follow the typical one-fourth power law: Nu = C(m,cr)[(a/6)Ra] 1 / 4 , where 1/2m is the slope of the wall temperature distribution, assumed linear. To include liquid metals, C was computed for 0.01 < cr ^ 10. The parallel experimental study confirms the existence of a positive maximum in the temperature profile, previously not reported. Introduction of this innovation in the theoretical treatment leads to excellent agreement with experimental results, and has the general effect of lowering the theoretical curves Nu = /[<x b (a/&JRa]. Semiquantitative experimental data on the velocity field also indicate the existence of a positive maximum in the velocity profile until now not reported.
Physics of Fluids, 1999
Three-dimensional steady flows are simulated in a circular cylindrical cavity of aspect ratio A ϭH/D, where H is the height and D the diameter of the cavity. The cavity is heated from below and its sidewalls are considered to be adiabatic. The effect of the geometry of the cavity on the onset of convection and on the structure and symmetries of the flow is analyzed. The nonlinear evolution of the convection beyond its onset is presented through bifurcation diagrams for two typical aspect ratios Aϭ0.5 and Aϭ1. Axisymmetric (mϭ0) and asymmetric ͑mϭ1 and mϭ2͒ azimuthal modes ͓exp (im)͔ are observed. For Aϭ0.5, the axisymmetric solution loses its stability to a three-dimensional solution at a secondary bifurcation point. Better understanding of the mechanisms leading to this instability is obtained by analyzing the energy transfer between the basic state and the critical mode. To study the influence of the Prandtl number on the flow pattern and on the secondary bifurcation, three values of the Prandtl number are investigated: Prϭ0.02 ͑liquid metal͒, Prϭ1 ͑transparent liquids͒, and Prϭ6.7 ͑water͒.
Onset of double-diffusive convection in a rectangular cavity with stress-free upper boundary
Physics of Fluids, 2010
Double-diffusive buoyancy convection in an open-top rectangular cavity with horizontal temperature and concentration gradients is considered. Attention is restricted to the case where the opposing thermal and solutal buoyancy effects are of equal magnitude ͑buoyancy ratio R =−1͒. In this case, a quiescent equilibrium solution exists and can remain stable up to a critical thermal Grashof number Gr c. Linear stability analysis and direct numerical simulation show that depending on the cavity aspect ratio A, the first primary instability can be oscillatory, while that in a closed cavity is always steady. Near a codimension-two point, the two leading real eigenvalues merge into a complex coalescence that later produces a supercritical Hopf bifurcation. As Gr further increases, this complex coalescence splits into two real eigenvalues again. The oscillatory flow consists of counter-rotating vortices traveling from right to left and there exists a critical aspect ratio below which the onset of convection is always oscillatory. Neutral stability curves showing the influences of A, Lewis number Le, and Prandtl number Pr are obtained. While the number of vortices increases as A decreases, the flow structure of the eigenfunction does not change qualitatively when Le or Pr is varied. The supercritical oscillatory flow later undergoes a period-doubling bifurcation and the new oscillatory flow soon becomes unstable at larger Gr. Random initial fields are used to start simulations and many different subcritical steady states are found. These steady states correspond to much stronger flows when compared to the oscillatory regime. The influence of Le on the onset of steady flows and the corresponding heat and mass transfer properties are also investigated.
Convection in a vertical cavity submitted to crossed uniform heat fluxes
International Journal of Heat and Mass Transfer, 2003
In this study, free convection in a vertical cavity heated from the four walls by uniform heat fluxes is considered. Analytical solutions are derived for a fully developed base flow, for which linear stability analysis predicts the growth of oblique, three-dimensional disturbances in general. A Hopf type bifurcation occurs at the critical Rayleigh number, over the entire range of Prandtl numbers and heat flux ratios considered, characterized by oscillating instabilities. Depending mostly on the value of the Prandtl number, either thermal, for Pr > 1, or hydrodynamic, for Pr < 1, instability modes are predicted. For small Prandtl numbers, both modes can occur at the codimension two intersection points of the critical branches.
Linear stability analysis of cylindrical Rayleigh Benard convection
The instabilities and transitions of flow in a vertical cylindrical cavity with heated bottom, cooled top and insulated sidewall are investigated by linear stability analysis. The stability boundaries for the axisymmetric flow are derived for Prandtl numbers from 0.02 to 1, for aspect ratio A (A = H/R = height/radius) equal to 1, 0.9, 0.8, 0.7, respectively. We found that there still exists stable non-trivial axisymmetric flow beyond the second bifurcation in certain ranges of Prandtl number for A = 1, 0.9 and 0.8, excluding the A = 0.7 case. The finding for A = 0.7 is that very frequent changes of critical mode (azimuthal Fourier mode) of the second bifurcation occur when the Prandtl number is changed, where five kinds of steady modes m = 1, 2, 8, 9, 10 and three kinds of oscillatory modes m = 3, 4, 6 are presented. These multiple modes indicate different flow structures triggered at the transitions. The instability mechanism of the flow is explained by kinetic energy transfer analysis, which shows that the radial or axial shear of base flow combined with buoyancy mechanism leads to the instability results.
Experiments in Fluids, 2001
Rayleigh-Bénard convection in a cubical cavity with adiabatic or conductive sidewalls is experimentally analyzed at moderate Rayleigh numbers (Ra ≤ 8 × 104) using silicone oil (Pr=130) as the convecting fluid. Under these conditions the flow is steady and laminar. Three single-roll-type structures and an unstable toroidal roll have been observed inside the cavity with nearly adiabatic sidewalls. The sequence from the conductive state consists of a toroidal roll that evolves to a diagonally oriented single roll with increasing Rayleigh number. This diagonal roll, which is stabilized by the effect of the small but finite conductivity of the walls, shifts its axis of rotation towards to two opposite walls, and back to the diagonal orientation to allow for the increase in circulation that occurs as the Rayleigh number is further increased. Conduction at the sidewalls modifies this sequence in the sense that the two initial single rolls finally evolve into a four-roll structure. Once formed, this four-roll structure remains stable when decreasing the Rayleigh number until the initial single diagonally oriented roll is again recovered. The topology and the velocity fields of all structures, characterized with visualization and particle image velocimetry, respectively, are in good agreement with numerical results reported previously for the cavity with adiabatic walls, as well as with the numerical predictions obtained in the present study for perfectly conducting lateral walls.
Flow transitions of a low-Prandtl-number fluid in an inclined 3D cavity
European Journal of Mechanics - B/Fluids, 2001
We present a numerical and theoretical investigation on the natural convection of a low Prandtl number fluid (Pr = 0.025) in 2D and 3D side-heated enclosures tilted α = 80 • with respect to the vertical position. The choice of this inclination angle comes from a previous linear stability analysis of the basic (plane-parallel) flow that predicts the same critical Ra for longitudinal oscillatory and stationary transversal modes. In both the 2D and 3D enclosures the first transition gradually leads to a transversal stationary centered shear roll. In the 2D geometry the flow becomes time-dependent and multicellular (3 rolls) at the onset of a Hopf bifurcation, followed by subsequent period-doubling. On the other hand, in the 3D enclosure, the onset of oscillations is due to a fully three-dimensional standing wave composed of three counter-rotating longitudinal rolls. The further evolution of the 3D flow qualitatively agrees with previous experiments (J. Crystal Growth, 102 (1990) pp. 54-68): a quasiperiodic flow followed by a frequency locked state. The main contribution of this work is the analysis of the flow structure underlying the secondary frequency: a transversal wave composed of two shear rolls that coexist with the three longitudinal cells. This is the first numerical work that explicitly illustrates this scenario which was suggested at the onset of the biperiodic regime in many of the previous experiments. 2001 Éditions scientifiques et médicales Elsevier SAS
A numerical study of natural convection in cavity filled with air has been carried out under large temperature gradient. The flows 10 under study are generated by a heated solid body located close to the bottom wall in a rectangular cavity with cold vertical walls 11 and insulated horizontal walls. They have been investigated by direct simulations using a two-dimensional finite volume numerical code 12 solving the time-dependent Navier-Stokes equations under the low Mach number approximation. This model permits to take into 13 account large temperature variations unlike the classical Boussinesq model which is valid only for small temperature differences. We were 14 particularly interested in the first transitions which occur when the Rayleigh number is increased for flows in cavities of aspect ratio 15 A = 1, 2, 4. Starting from a steady state, the results obtained for A = 1 and A = 4 show that the first transition occurs through a super-16 critical Hopf bifurcation. The induced disturbances determined for weakly supercritical regimes indicate the existence of two instability 17 types driven by different physical mechanisms: shear and buoyancy-driven instabilities, according to whether the flow develops in a 18 square or in a tall cavity. For A = 2, the flow undergoes a pitchfork bifurcation leading to an asymmetric steady state which in turn 19 becomes periodic via a supercritical Hopf bifurcation point. In both the cases, the flow is found to be strongly deflected towards one 20 vertical wall and instabilities are found to be of shear layers type. 21 55 interferometer to observe the disturbances as they were 56 convected downstream. The experimental results show that 57 sufficiently high frequency disturbances are stable as they 58 are convected downstream. Later, calculations of the same 59 problem have been carried out by Haaland and Sparrow [5] 60 taking into account non-parallel and higher-order effects of 61 the base flow in the linear stability analysis. The authors 62 obtained a lower branch of neutral curve and then a critical 63 Grashof number. Their results show that the unstable 64 region is smaller than that obtained from the quasi-parallel 65 theory . The stability problem of non-parallel flows were 66 also investigated by Wakitani [7] using the method of mul-67 tiple scales. This method considers that the various distur-68 bance quantities have different amplification rates. Their 69 results relating to the amplification rate of disturbances 70 within unstable regions show a substantial deviation from 71 that predicted by the quasi-parallel theory. 72