Bulletin of the American Physical Society
64th Annual Meeting of the APS Division of Fluid Dynamics
Volume 56, Number 18
Sunday–Tuesday, November 20–22, 2011; Baltimore, Maryland
Session D15: Rayleigh-Benard Convection II: Instability and Oscillations |
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Chair: Emmanuel Villermaux, Universit\'e Aix-Marseille Room: 318 |
Sunday, November 20, 2011 2:10PM - 2:23PM |
D15.00001: Forecasting Flows from Target Pattern Instability in Rayleigh-Benard Convection Balachandra Suri, Adam Perkins, Michael Schatz Using lab experiments combined with numerical simulations, we study systematically how the initial instability of an ordered pattern gradually evolves to a state of spatio-temporal complexity. The experiments begin from a reference pattern of axisymmetric convection rolls (a target pattern) that is reproducibly imposed using an optical technique for actuating fluid flow. For sufficiently large Rayleigh numbers, the axisymmetric pattern loses stability to patterns where the target's bull's-eye shifts off- center. We analyze an experimental ensemble of unstable patterns with nearby initial conditions to extract the spatial structure of the dominant modes and corresponding growth rates. We then test the extent to which a Boussinesq numerical model, in combination with a state estimation algorithm (Local Ensemble Transform Kalman Filter (LETKF)), can be used to predict the subsequent evolution of the experimentally observed patterns. [Preview Abstract] |
Sunday, November 20, 2011 2:23PM - 2:36PM |
D15.00002: Dynamic stabilization of the Rayleigh-B\'{e}nard instability in rectangular enclosures: A computational model Randy Carbo, Robert Smith, Matt Poese If an acoustic field is imposed on a fluid within a container, the critical Rayleigh number is a strong function of the frequency and amplitude of that acoustic field as noted by G. Swift. and S. Backhaus, [\textit{J. Acoust. Soc. Am.} \textbf{126}(5), 2009]. Results will be reported for nonlinear and linear models constructed to predict the modified critical Rayleigh number, based on a full field solution of the hydrodynamic equations using the approach of A. Yu. Gelfgat, [\textit{J. Comp. Phys.} \textbf{156}, 1999]. The spatial portion of the differential equations were solved using the Galerkin method and the dynamic stability for the linear model was determined using Floquet analysis. One of the benefits of the approach compared to the averaging methods used by G. Gershuni and D. Lyubimov, \textit{Thermal Vibration Convection}, (Wiley, New York, 1998) is that the parametric stability boundary can be recovered. This study includes a variety of container aspect ratios, boundary conditions, and Rayleigh numbers ranging from $10^3$ to $10^8$. [Preview Abstract] |
Sunday, November 20, 2011 2:36PM - 2:49PM |
D15.00003: Cellular Convection in a Chamber with a Warm Surface Raft John Whitehead, Erin Shea, Mark Behn We calculate velocity and temperature fields for Rayleigh-Benard convection in a chamber with a warm raft that can float along the top surface for Rayleigh number up to Ra=20,000. Two-dimensional, infinite Prandtl number, Boussinesq approximation equations are numerically advanced in time from a motionless state in a chamber of length L' and depth D'. We consider cases with an insulated raft and a raft of fixed temperature. Either oscillatory or stationary flow exists. The case of an insulated raft has three governing parameters: Ra, scaled chamber length L=L'/D', and scaled raft width W. For W=0 and L=1, the marginal state is at Ra=779.3. For smallest W (determined by numerical grid size) and Ra$<$790 the raft approaches the center monotonically in time. For 790$<$Ra$<$811 the raft has a decaying oscillation consisting of raft movement back and forth (and convection cell reversal). For 811$>$Ra$>$871 amplitude is steady, starting small and increasing with larger Ra and for Ra$>$871 raft movement ceases. For larger W, a range of W and Ra has raft oscillation up to Ra=20,000. Rafts in longer cavities (L=2 and 4) have almost no oscillatory behavior. With a raft of temperature Tr rather than insulating, Ra=20,000, and with internal heating, there are wider ranges of oscillating flow. Thus the presence or absence of motion is very sensitive to W, L, raft thermal properties and Ra. Reasons why are discussed. [Preview Abstract] |
Sunday, November 20, 2011 2:49PM - 3:02PM |
D15.00004: Spiral Defect Chaos in Generalized Swift-Hohenberg Models with Mean Flow Alireza Karimi, Zhi-Feng Huang, Mark Paul We present a numerical study of spiral defect chaos in two generalized models of the Swift-Hohenberg equation with mean flow. We use large-sized square domains with periodic boundaries and integrate the equations for very long times (up to one million time units) to study the effect of mean flow on pattern dynamics as the strength of mean flow is varied. The magnitude of mean flow is adjusted via a continuous parameter characterizing the type of the velocity boundary conditions on the horizontal surfaces in a convective flow. We show that the typical parameters used in the literature for spiral defect chaos studies yield a weak mean flow magnitude and leads to a state dominated by large and slowly moving target defects. When the magnitude of mean flow is sufficiently large, it is demonstrated that spatiotemporal chaos is feasible as indicated by a positive Lyapunov exponent. In addition, we compare the spatial distribution of the mean flow for these models with that of large aspect ratio Rayleigh-B\'{e}nard convection undergoing spiral defect chaos and discuss their differences near spiral defects. [Preview Abstract] |
Sunday, November 20, 2011 3:02PM - 3:15PM |
D15.00005: 1:2 Resonance and Pattern Formation in Thermal Convections Kaoru Fujimura Resonant interaction of steady modes having wavenumbers in the ratio 1:2 was re-examined on a hexagonal lattice in two-layered Rayleigh-Benard convection, where the exact resonance may take place between critical modes. Amplitude equations were derived by means of the center manifold reduction. Numerical examination of the equations reveals that the heteroclinic orbit of AGH-type exists, but loses its stability with respect to disturbances in four-dimensional complex space. Instead, a new type of heteroclinic cycle arises in rather narrow range of parameters. The cycle is a connection between rolls and hexagons. [Preview Abstract] |
Sunday, November 20, 2011 3:15PM - 3:28PM |
D15.00006: Natural Convection Due to a Long Wavelength Heating Ali Asgarian, Mohammed Z. Hossain, Jerzy M. Floryan Natural convection in an infinite horizontal layer subject to a spatially periodic heating is considered. The analysis is focused on the heating wave number $\alpha \to $0. It has been shown that convection has a simple topology consisting of one pair of counter-rotating rolls per heating period when the heating intensity does not exceed the critical value of the Rayleigh number Ra = 427. Secondary motions in the form of rolls aligned in the direction of the primary rolls and concentrated around the hot spots occur for more intense heating. When 427$<$Ra$<$470 the secondary motions are described by supercritical pitchfork bifurcations and can occur only if $\alpha $ is reduced below 0.14. One of the branches of such bifurcations is associated with an odd number of secondary rolls per half period, with rolls at the hot spots rotating in the direction opposite to the primary rolls. The other branch is associated with an even number of secondary rolls per half period, with the rolls at the hot spots co-rotating with the primary rolls. The number of rolls increases without limit as $\alpha $ decreases with new rolls being pinched off in pairs. Increase of heating intensity to Ra$>$470 results in secondary motions occurring at larger values of $\alpha $, i.e., $\alpha >$0.14, and bifurcation changing character into ``bifurcations from infinity.'' It is shown that the observed phenomena are strictly associated with the small wave number limit of the external heating. [Preview Abstract] |
Sunday, November 20, 2011 3:28PM - 3:41PM |
D15.00007: Natural Convection in a Fluid Layer Subject to Periodic Heating Mohammed Z. Hossain, Jerzy M. Floryan Natural convection in a horizontal layer exposed to periodic heating is considered. The primary response leads to stationary convection in the form of rolls orthogonal to the heating wave vector. For large $\alpha $ a uniform conductive layer emerges at the upper section of the fluid layer. Secondary convection gives rise either to the longitudinal rolls, or to the transverse rolls or to the oblique rolls depending on $\alpha $. Three mechanisms of instability have been identified. In the case of small and moderate $\alpha $ the parametric resonance leads to the pattern of instability that is locked-in with the pattern of the heating as $\delta _{cr}=\alpha $/2, where $\delta _{cr}$ denotes component of the disturbance wave vector. The second mechanism is associated with the formation of patterns of vertical temperature gradients and the primary convection currents, operates approximately in the same range of $\alpha $ and provides direct modulation with structure dictated by $\alpha $. The third mechanism operates for large $\alpha $ where the instability is driven by the uniform mean vertical temperature gradient created by the primary convection with the critical disturbance wave vector $\delta _{cr} \quad \to $ 1.56. Competition between the first and second effects gives rise to the appearance of soliton lattices. [Preview Abstract] |
Sunday, November 20, 2011 3:41PM - 3:54PM |
D15.00008: Flow simulation for a high Prandtl number liquid heated from above Erdem Uguz, Franck Pigeonneau, Gerard Labrosse, Ranga Narayanan Natural convection in a glass furnace has crucial importance for glass manufacturing as it affects mixing and bubble dynamics. To understand the convective flows in a glass furnace where the Prandtl numbers typically are greater than 1000, a numerical study is performed using the spectral collocation Chebyshev method. Calculations were done for a 2D rectangular geometry for various aspect ratios (i.e. height divided by length) and Rayleigh (Ra) numbers for a constant viscosity fluid. Depending upon the value of Ra, and for a fixed aspect ratio, secondary and even tertiary transient cell formations were observed. In the process of heating (i.e. in the process of increasing Ra) thermal and velocity boundary layers are formed. Also with increasing Ra the solution becomes unstable and the critical Ra number that defines this transition has a strong dependence on the aspect ratio. [Preview Abstract] |
Sunday, November 20, 2011 3:54PM - 4:07PM |
D15.00009: Relaxation Dynamics of Spatiotemporal Chaos in the Nematic Liquid Crystal Fahrudin Nugroho, Tatsuhiro Ueki, Yoshiki Hidaka, Shoichi Kai We are working on the electroconvection of nematic liquid crystals, in which a kind of spatiotemporal chaos called as a soft-mode turbulence (SMT) is observed. The SMT is caused by the nonlinear interaction between the convective modes and the Nambu--Goldstone (NG) modes. By applying an external magnetic field \textbf{H}, the NG mode is suppressed and an ordered pattern can be observed. By removing the suppression effect the ordered state relax to its original SMT pattern. We revealed two types of instability govern the relaxation process: the zigzag instability and the free rotation of wavevector \textbf{q}(\textbf{r}). [Preview Abstract] |
Sunday, November 20, 2011 4:07PM - 4:20PM |
D15.00010: Rayleigh-B\'{e}nard-Poiseuille flow: Optimal growth of streamwise-uniform disturbances J John Soundar Jerome, Jean-Marc Chomaz, Patrick Huerre An investigation of the dominant transient growth mechanisms in plane Poiseuille flow subjected to a destabilizing cross-stream temperature gradient is presented. It was pointed out by the same authors in DFD meeting 2009 that only the streamwise-uniform and nearly-streamwise-uniform disturbances are highly influenced by the Rayleigh number \textbf{\textit{Ra}} and Prandtl number \textbf{\textit{Pr}}. Here, it is demonstrated that the \textit{short-time} behavior is governed by the classical inviscid lift-up mechanism and the optimal input for the largest \textit{long-time} response is given by the adjoint of the dominant eigenmode with respect to the energy scalar product: the Rayleigh-B\'{e}nard eigenmode without its streamwise velocity component. These short and long-time responses are then shown to depict, up to leading order, the optimal transient growth \textbf{\textit{G(t)}}. It is thereby brought out that, at moderately large \textbf{\textit{Ra}} (or small \textbf{\textit{Pr}} at a fixed \textbf{\textit{Ra}}), the dominant adjoint mode is a good approximation to the optimal initial condition for all time. The results remain qualitatively similar over a general class of norms that can be considered as growth functions. For instance, the dominant adjoint eigenmode still approximates the maximum optimal response. [Preview Abstract] |
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