Bulletin of the American Physical Society
68th Annual Meeting of the APS Division of Fluid Dynamics
Volume 60, Number 21
Sunday–Tuesday, November 22–24, 2015; Boston, Massachusetts
Session D11: Convection and Buoyancy-Driven Flows: Analytic Techniques |
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Chair: Gunnar Peng, DAMTP, University of Cambridge Room: 111 |
Sunday, November 22, 2015 2:10PM - 2:23PM |
D11.00001: How does the diffusion fish swim? Gunnar Peng, Neil Balmforth, William Young An asymmetric object (such as a wedge) placed in a stably stratified fluid moves with a steady horizontal speed. We explain how this spontaneous motion is caused by the diffusion-driven buoyancy layers that form on the sloping surfaces of the object, and calculate the speed for a variety of two-dimensional configurations using the method of matched asymptotic expansions. Surprisingly, in many cases, the leading-order speed depends on neither the viscosity nor the stratification strength. [Preview Abstract] |
Sunday, November 22, 2015 2:23PM - 2:36PM |
D11.00002: Optimizing exit times Jean-Luc Thiffeault, Florence Marcotte, Charles Doering, William Young A heat exchanger can be modeled as a closed domain containing an incompressible fluid. The fluid has some temperature distribution obeying the advection-diffusion equation, with zero temperature boundary conditions at the walls. The goal is then to start from some initial positive heat distribution, and to flux it through the walls as fast as possible. For a steady flow, this is a time-dependent problem, which can be hard to optimize. Instead, we consider the mean exit time of Brownian particles starting from inside the domain. A flow favorable to heat exchange should lower the exit time, and so we minimize some norm of the exit time. This is a simpler, time-independent optimization problem, which we then proceed to solve analytically in some limits, and numerically otherwise. [Preview Abstract] |
Sunday, November 22, 2015 2:36PM - 2:49PM |
D11.00003: ABSTRACT WITHDRAWN |
Sunday, November 22, 2015 2:49PM - 3:02PM |
D11.00004: Non-Boussinesq Rayleigh-Benard linear stability Thierry Alboussiere, Yanick Ricard The simplest Rayleigh-Benard configuration consists in a horizontal fluid layer maintained at a higher temperature on the under side, with no shear stress on its boundary. In the Boussinesq approximation, Rayleigh obtained an analytic value, $27 \pi ^4 / 4$, for the critical stability threshold of a dimensionless parameter which now bears his name. There are two ways to go away from the Boussinesq approximation: when there is a significant temperature difference across the layer compared to the average thermodynamic temperature, or when gravity creates a significant compression of the fluid near the bottom. We have determined an approximate analytical expression for the critical Rayleigh number depending on the those two non-Boussinesq causes. We have also determined the critical threshold for the intermediate model called the 'anelastic liquid approximation' in which the adiabatic temperature gradient is taken into account, while density flucuations are assumed to be solely due to temperature fluctuations. It is found that a small product $\alpha T$ (thermal expansion coefficient times temperature) does not make the anelastic liquid approximation any better, for a Gr\"uneisen parameter close to unity. [Preview Abstract] |
(Author Not Attending)
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D11.00005: Buoyancy effects in a wall jet over a heated horizontal plate Ramon Fernandez-Feria, Francisco Castillo-Carrasco A similarity solution of the boundary layer equations for a wall jet on a heated horizontal surface taking into account the coupling of the temperature and velocity fields by buoyancy is described. It exists for any positive value of $\Lambda =Gr/Re^2$, characterizing this coupling between natural and forced convection over the horizontal plate; i.e., only when the plate temperature is larger than the ambient one. The flow structure is qualitatively very different from the well known Glauert's similarity solution for a wall jet without buoyancy effects ($\Lambda=0$): basically coincides for both a radially spreading jet and a two-dimensional jet, and the maximum of the horizontal velocity increases as the jet spreads over the surface, with the power $1/5$. The similarity solution is checked by solving numerically the boundary layer equations for a jet with uniform velocity and temperature emerging from a slot of height $\delta$ and radius $r_0$ (in the radial case). An approximate, analytical similarity solution near the jet exit is also found that helps to start the numerical integration. The similarity solution is reached for any set of the non-dimensional parameters governing the problem provided that the plate is heated ($\Lambda >0$). [Preview Abstract] |
Sunday, November 22, 2015 3:15PM - 3:28PM |
D11.00006: Convective dissolution in partially miscible systems: classification of the effect of reactions V. Loodts, C. Thomas, L. Rongy, A. De Wit Dissolution-driven convection in partially miscible systems has regained much interest in the context of CO$_2$ sequestration. A buoyantly unstable density stratification can build up upon dissolution of a species into the host fluid phase, thereby developing convection. Chemical reactions can impact such convection as they affect concentrations and thus the density of the host phase. We theoretically classify the effects of reactions on the convective instability as a function of the contributions to density and diffusion coefficients of the chemical species involved. To do so, we compute the reaction-diffusion density profiles in the host phase and assess their stability with regard to buoyancy-driven convection by a linear stability analysis. The buoyancy-driven instability grows faster when the product of the reaction contributes sufficiently more to density than the initially dissolved reactant. We illustrate this by experimental results showing that reactions accelerate the development of buoyancy-driven fingering during the convective dissolution of CO$_2$ into aqueous solutions of alkali hydroxides. [Preview Abstract] |
Sunday, November 22, 2015 3:28PM - 3:41PM |
D11.00007: Nonlinear convection in unbounded vertical channels Rishad Shahmurov, Layachi Hadji We investigate the linear and weakly nonlinear solutions to a convection problem that was first studied by Ostroumov in 1947. The problem pertains to the stability of the equations governing convective motion in an infinite vertical fluid layer that is heated from below. Ostroumov's linear stability analysis yields instability threshold conditions that are characterized by zero wavenumber for the Fourier mode in the vertical direction and by eigenfunctions that are independent of the vertical coordinate. Thus, any undertaking at determining the supercritical nonlinear solutions and their stability through a small amplitude expansion fails. This failure is attributed to the fact that the nonlinear interaction of the linear modes vanish identically. In this paper, we put forth exact and stable similarity type solutions to the Ostroumov problem. These solutions are characterized by the same linear threshold conditions as Ostroumov's solutions. Moreover, we are able to extend the analysis to the supercritical regime through a small amplitude analysis to obtain steady two-dimensional solutions for a small range of Prandtl numbers. These solutions are found to be stable to general two-dimensional, time-dependent disturbances. [Preview Abstract] |
Sunday, November 22, 2015 3:41PM - 3:54PM |
D11.00008: Pattern selection in ternary mushy layers Peter Guba, Daniel Anderson We consider finite-amplitude convection in a mushy layer during the primary solidification of a ternary alloy. A previous linear theory identified, for the case of vanishing latent heat, solute rejection and background solidification, a direct mode of convective instability when all the individual stratifying agencies (thermal and two solutal) were statically stabilizing. The physical mechanism behind this instability was attributed to the local-phase-change effect on the net solute balance through the liquid-phase solutal diffusivity. A weakly nonlinear development of this instability is investigated in detail. We examine the stability of two-dimensional roll, and three-dimensional square and hexagonal convection patterns. The amplitude evolution equations governing roll/square and roll/hexagon competition are derived. We find that any of rolls, squares or hexagons can be nonlinearly stable, depending on the relative importance of a number of physical effects as reflected in the coefficients of the amplitude equations. The results for a special case are found to isolate a purely double-diffusive phase-change mechanism of pattern selection. Subcritical behaviour is identified inside the domain of individual static stability. [Preview Abstract] |
Sunday, November 22, 2015 3:54PM - 4:07PM |
D11.00009: Using Covariant Lyapunov Vectors to Build a Physical Understanding of Spatiotemporal Chaos in Rayleigh-B\'enard Convection Mu Xu, Mark Paul We explore the high-dimensional spatiotemporal chaos of Rayleigh-B\'enard convection using covariant Lyapunov vectors. We integrate the three-dimensional and time dependent Boussinesq equations for a convection layer for very long-times and for a range of Rayleigh numbers. We simultaneously integrate many copies of the tangent space equations in order to compute the covariant Lyapunov vectors. We explore chaotic dynamics with a fractal dimension of nearly 50 and we compute over 150 covariant Lyapunov vectors. Using the Lyapunov vectors we quantify the hyperbolicity of the dynamics, the degree of Oseledec splitting, and explore the temporal, spatial, and spectral dynamics of the Lyapunov vectors. Our results indicate that the dynamics undergoes a hyperbolic to non-hyperbolic transition as the Rayleigh number is increased. Our results yield that all of the Lyapunov vectors computed have near tangencies with neighboring vectors. A closer look at the vectors suggests that the dynamics are composed of physical modes that are connected with tangled spurious modes that extend to all of the covariant Lyapunov vectors we have computed. [Preview Abstract] |
Sunday, November 22, 2015 4:07PM - 4:20PM |
D11.00010: Near-optimal source placement in forced convection Piyush Grover, Saleh Nabi We consider the problem of optimal source placement, given the velocity field and sink distribution in forced convection. We use the ratio of scalar concentration variance in presence of advection compared to the purely diffusive case as our cost function. Using a semidefinite relaxation of the original optimization problem, we obtain near-optimal source distributions for several classes of velocity fields in 2D. We compare the numerical results with bounds obtained by analysis of space-time averaged advection-diffusion equation. The dependence of optimal variance reduction on the Lagrangian properties of the velocity field and shape of sink distribution are made explicit in this analysis. This work extends the earlier work in (Thiffeault, J. L., \& Pavliotis, G. A. (2008). Optimizing the source distribution in fluid mixing. Physica D: Nonlinear Phenomena, 237(7), 918-929), and provides a systematic framework that can be extended to more realistic models of forced and mixed convection. [Preview Abstract] |
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