Oscillating Onset of the Rayleigh-Bénard Convection with Viscoelastic Fluids in a Slightly Tilted Cavity (original) (raw)
Related papers
Journal of Non-Newtonian Fluid Mechanics, 2010
A study of heat transport in Rayleigh-Bénard convection in viscoelastic liquids with/without gravity modulation is made using a most minimal representation of Fourier series and a representation with higher modes. The Oldroyd-B constitutive relation is considered. The resulting non-autonomous Lorenz model (generalized Khayat-Lorenz model of four modes and seven modes) is solved numerically using the adaptive-grid Runge-Kutta-Fehlberg45 method to quantify the heat transport. The effect of gravity modulation is shown to be stabilizing there by leading to a situation of reduced heat transfer. The Deborah number is shown to have an antagonistic influence on convection compared to the stabilizing effect of modulation amplitude and elastic ratio. The results in respect of Maxwell, Rivlin-Ericksen and Newtonian liquids are obtained as particular cases of the present study. A transformation of the momentum equations illustrates the equivalence of present approach and the one due to Khayat that uses normal stresses explicitly.
Physics of Fluids, 2023
The effect of viscoelasticity on the flow and heat transport in the Rayleigh-Bénard convection (RBC) in a rectangular with horizontal periodic boundary is investigated via direct numerical simulation. The working fluid is described by the finitely extensible nonlinear elastic-Peterlin (FENE-P) constitutive model that is able to capture some of the most important polymeric flow behaviors. Numerical simulations are conducted at a low concentration β = 0.9, where β = μs/μ0, μs is the solvent viscosity, μ0 = μs + μp is sum of μs and the polymer viscosity μp. A parametric analysis is performed to understand the influence of the Weissenberg number Wi, the viscosity ratio β and the extension length L on the oscillating mode of the viscoelastic RBC. It is found that both Wi and β weakly inhibit the convection onset and the transition from steady to oscillatory convection. The amplitude and frequency of the oscillations in the oscillatory flow regime are both suppressed. However, the elastic nonlinearity may make the flow transition irregular and even may bring about the relaminarization or lead to the convection cells traveling in the horizontal direction. The extension length L may induce multiple pairs of roll flow patterns at a specific setting of (Ra, Wi). Heat transport is reduced by elasticity but still obeys the power law with Ra if the flow pattern has one pair of rolls. However, heat transfer enhancement occurs if multiple pairs of rolls are induced.
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.
Physical Review Fluids, 2023
The time-dependent oscillating convection leading to the formation of reverse flowing cells is a special phenomenon induced by viscoelasticity in the Rayleigh-Bénard convection (RBC). The causes and the evolution of this overstability problem have not yet been investigated in-depth. Numerical simulations of the viscoelastic Rayleigh-Bénard convection (VRBC) have been conducted in this work with viscoelastic working fluids abiding by the nonlinear Phan-Thien-Tanner (PTT) constitutive structure in two-dimensional cavities. To understand the impact of the nonlinearity and the rheological parameters on the mechanism of the regular reverse flow numerical simulations have been performed over the range of β = (0.1, 0.2) (where β = μs/μ0, μs is the solvent viscosity, μ0 = μs + μp is sum of solvent viscosity μs and polymer viscosity μp) and Weissenberg number (We ∈ [0.075, 0.25]), using an in-house finite-difference code. The remaining constitutive parameters of the (PTT) fluid representing elongational and slippage characteristics of the fluid were kept fixed at = 0.1 and ξ = 0.05, respectively. A viscoelastic kinetic-energy budget method was used to analyze the energy transport in this time-dependent reverse flow process. An original parametric analysis is developed to gain an insight into the dynamics of the reversal flow observed recently in our work, Zheng et al. [Phys. Rev. Fluids 7, 023301 (2022)], as well as observed by Park and Ryu [Rheol. Acta 41, 427 (2002)] and Lappa and Boaro [J. Fluid Mech. 904, A2 (2020)]. The emergence of the reversal convection can be explained by the transfer of potential energy between flow and fluid elasticity during the reversal process. The existence of time phase differences of different potentials in the evolution drive this potential-energy transfers.
Pattern Selection in Rayleigh Bénard Convection with Non-Linear Viscoelastic Fluids
Physical Review Fluids, 2022
Rayleigh-Bénard convection in a rectangular enclosure of aspect ratio 2:1 filled by a class of non-linear viscoelastic fluids represented by the Phan Thien-Tanner (PTT) constitutive equation is investigated numerically. Governing equations are discretized by finite difference methods in space and time. The momentum and PTT constitutive equations are written in a quasi-linear formulation. Quasi-linear terms are treated with the High-Order Upwind Central (HOUC) method and velocity-pressure coupling is handled through the projection method. The developed model is validated for Oldroyd-B type of working fluids. The onset of time-dependent convection is observed and the critical Rayleigh number is determined for PTT type of fluids. Time-dependent flow pattern transition is investigated and explained. Transition from time-dependent flow to steady-state flow is observed at a higher Rayleigh number and the corresponding critical Rayleigh number is computed, for the first time in the literature. This is a new original finding. The effect of the rheological parameters on heat transfer is investigated.
Physics of Fluids, 2013
We present the results of direct numerical simulations of flow patterns in a low-Prandtl-number (P r = 0.1) fluid above the onset of oscillatory convection in a Rayleigh-Bénard system rotating uniformly about a vertical axis. Simulations were carried out in a periodic box with thermally conducting and stress-free top and bottom surfaces. We considered a rectangular box (L x ×L y ×1) and a wide range of Taylor numbers (750 ≤ T a ≤ 5000) for the purpose. The horizontal aspect ratio η = L y /L x of the box was varied from 0.5 to 10. The primary instability appeared in the form of two-dimensional standing waves for shorter boxes (0.5 ≤ η < 1 and 1 < η < 2). The flow patterns observed in boxes with η = 1 and η = 2 were different from those with η < 1 and 1 < η < 2. We observed a competition between two sets of mutually perpendicular rolls at the primary instability in a square cell (η = 1) for T a < 2700, but observed a set of parallel rolls in the form of standing waves for T a ≥ 2700. The three-dimensional convection was quasiperiodic or chaotic for 750 ≤ T a < 2700, and then bifurcated into a two-dimensional periodic flow for T a ≥ 2700. The convective structures consisted of the appearance and disappearance of straight rolls, rhombic patterns, and wavy rolls inclined at an angle φ = π 2 − arctan (η −1) with the straight rolls.
Onset of oscillatory binary fluid convection in three-dimensional cells
Theoretical and Computational Fluid Dynamics, 2004
The purpose of this work is to investigate the influence of the transverse walls on the onset of convection in a horizontal rectangular cavity of infinite length filled with a binary mixture when heated from below. For the first time we take into account the effect of the third dimension without making any approximation and considering realistic boundary conditions. In previous numerical works the width of the cell was either taken to be infinity (bulk mixtures) or different approximations usually valid in the narrow cell limit were assumed (i.e. Hele-Shaw and non-ideal Hele-Shaw approximations). The results we find show that the presence of the walls has a considerable effect on the onset of convection even for intermediate transverse aspect ratio cells. They also show that the approximations generally assumed fail to reproduce the correct behaviour of the critical parameters in an important range of aspect ratio values when the primary bifurcation is oscillatory. We have compared the critical values of the Rayleigh number and the frequency that we obtain with those reported in the literature (Ohlsen et al., 1990) and we find a quantitative agreement within the experimental error.
A New Approach to the Numerical Modeling of the Viscoelastic Rayleigh-Bénard Convection
Conference: ASME 2019 International Mechanical Engineering Congress and Exposition, Salt Lake City, Utah USA November 11-14, 2019; IMECE 2019 Proceedings, Paper No. IMECE2019-11675; published online January 21, 2020, ASME Digital Collection, 2019
A new approach to the numerical simulation of incompressible viscoelastic Rayleigh-Bénard convection in a cavity is presented. Due to the fact that the governing equations are of elliptic-hyperbolic type, a quasi-linear treatment of the hyperbolic part of the equations is proposed to overcome the strong instabilities that can be induced and is handled explicitly in time. The elliptic part related to the mass conservation and the diffusion is treated implicitly in time. The time scheme used is semi-implicit and of second order. Second-order central differencing is used throughout except for the quasi-linear part treated by third order space scheme HOUC. Incompressibility is handled by a projection method. The numerical approach is validated first through comparison with a Newtonian benchmark of Rayleigh-Bénard convection and then by comparing the results related to the convection setup in a 2 : 1 cavity filled with an Oldroyd-B fluid. A preliminary study is also conducted for a PTT fluid and shows that PTT fluid is slightly more unstable than Oldroyd-B fluid in the configuration of Rayleigh-Bénard convection.
European Journal of Mechanics - B/Fluids, 2007
In this paper we analyze transitional regimes and mean flow structures for the thermally driven convective flow in a cylindrical cell of aspect-ratio (diameter over cell height) Γ = 1/2. The investigation is carried out through the numerical integration of the three-dimensional unsteady Navier-Stokes equations with the Boussinesq approximation. In particular the critical Rayleigh numbers for the onset of convection, for the unsteady, chaotic and turbulent regimes are computed for two values of the Prandtl number and comparisons with cylindrical cells of larger aspect-ratio are performed. The effect of the background rotation on the flow dynamics is also described showing that the heat transfer increase, already evidenced in the literature, is only obtained for a range of rotation rates. The rotation can enhance or inhibit the heat transfer and, at low Rayleigh numbers, it is a very effective way to inhibit vertical motions and to prevent horizontal thermal gradients. This is highly desirable in solidification and crystal growth processes where thermally induced motions cause material defects and crystal inhomogeneities.
Journal of Fluid Mechanics, 2015
The convective and absolute nature of instabilities in Rayleigh-Bénard-Poiseuille (RBP) mixed convection for viscoelastic fluids is examined numerically with a shooting method as well as analytically with a one-mode Galerkin expansion. The viscoelastic fluid is modelled by means of a general constitutive equation that encompasses the Maxwell model and the Oldroyd-B model. In comparison to Newtonian fluids, two more dimensionless parameters are introduced, namely the elasticity number λ 1 and the ratio Γ between retardation and relaxation times. Temporal stability analysis of the basic state showed that the three-dimensional thermoconvective problem can be Squire-transformed. Therefore, one must distinguish mainly between two principal roll orientations: transverse rolls TRs (rolls with axes perpendicular to the Poiseuille flow direction) and longitudinal rolls LRs (rolls with axes parallel to the Poiseuille flow direction). The critical Rayleigh number for the appearance of LRs is found to be independent of the Reynolds number (Re). Depending on λ 1 and Γ , two different regimes can be distinguished. In the weakly elastic regime, the emerging LRs are stationary, while they are oscillatory in the strongly elastic regime. For TRs, it is found that in the weakly elastic regime, the stabilization effect of Re is more important than in Newtonian fluids. Moreover, for sufficiently elastic fluids a jump is observed in the oscillation frequencies and wavenumbers for moderate Re. In the strongly elastic regime, the effect of the imposed throughflow is to promote the appearance of the upstream moving TRs for low values of Re, which are replaced by downstream moving TRs for higher values of Re. Moreover, the results proved that, contrary to the case where Re = 0, the elasticity number λ 1 (the ratio Γ ) has a strongly stabilizing (destabilizing) effect when the throughflow is added. The influence of the rheological parameters on the transition curves from convective to absolute instability in the Reynolds-Rayleigh number plane is also determined. We show that the viscoelastic character of the fluid hastens the transition to absolute instability and even may suppress the convective/absolute transition. Throughout this paper, similarities and differences with the corresponding problem for Newtonian fluids are highlighted.