Asymptotic theory of wall-attached convection in a horizontal fluid layer with a vertical magnetic field

A horizontal fluid layer heated from below in the presence of a vertical magnetic field is considered. A simple asymptotic analysis is presented which demonstrates that a convection mode attached to the side walls of the layer sets in at Rayleigh numbers much below those required for the onset of convection in the bulk of the layer. The analysis complements an earlier analysis by Houchens [J. Fluid Mech. 469, 189 (2002)] which derived expressions for the critical Rayleigh number for the onset of convection in a vertical cylinder with an axial magnetic field in the cases of two aspect ratios. © 2008 American Institute of Physics.

Interfacial instabilities of a fluid annulus in a rotating Hele-Shaw cell

We have studied the interfacial instabilities experienced by a liquid annulus as it moves radially in a circular Hele-Shaw cell rotating with angular velocity Omega. The instability of the leading interface (oil displacing air) is driven by the density difference in the presence of centrifugal forcing, while the instability of the trailing interface (air displacing oil) is driven by the large viscosity contrast. A linear stability analysis shows that the stability of the two interfaces is coupled through the pressure field already at a linear level. We have performed experiments in a dry cell and in a cell coated with a thin fluid layer on each plate, and found that the stability depends substantially on the wetting conditions at the leading interface. Our experimental results of the number of fingers resulting from the instability compare well with the predictions obtained through a numerical integration of the coupled equations derived from a linear stability analysis. Deep in the nonlinear regime we observe the emission of liquid droplets through the formation of thin filaments at the tip of outgrowing fingers.

Acoustic scattering by an inhomogeneous layer on a rigid plate

The problem of scattering of time-harmonic acoustic waves by an inhomogeneous fluid layer on a rigid plate in R2 is considered. The density is assumed to be unity in the media: within the layer the sound speed is assumed to be an arbitrary bounded measurable function. The problem is modelled by the reduced wave equation with variable wavenumber in the layer and a Neumann condition on the plate. To formulate the problem and prove uniqueness of solution a radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as a system of two second kind integral equations over the layer and the plate. Under additional assumptions on the wavenumber in the layer, uniqueness of solution is proved and the nonexistence of guided wave solutions of the homogeneous problem established. General results on the solvability of systems of integral equations on unbounded domains are used to establish existence and continuous dependence in a weighted norm of the solution on the given data.

Transition in inclined internally heated fluid layers

Non-linear solutions and studies of their stability are presented for flows in a homogeneously heated fluid layer under the influence of a constant pressure gradient or when the mass flux across any lateral cross-section of the channel is required to vanish. The critical Grashof number is determined by a linear stability analysis of the basic state which depends only on the z-coordinate perpendicular to the boundary. Bifurcating longitudinal rolls as well as secondary solutions depending on the streamwise x-coordinate are investigated and their amplitudes are determined as functions of the supercritical Grashof number for various Prandtl numbers and angles of inclination of the layer. Solutions that emerge from a Hopf bifurcation assume the form of propagating waves and can thus be considered as steady flows relative to an appropriately moving frame of reference. The stability of these solutions with respect to three-dimensional disturbances is also analyzed in order to identify possible bifurcation points for evolving tertiary flows.

Jupiter's and Saturn's convectively driven banded jets in the laboratory

The banded patterns of cloud and wind are among the most striking features of the atmospheres of Jupiter and Saturn, but their dynamical origin remains poorly understood. Most approaches towards understanding zonation so far (also in the terrestrial oceans) have used highly idealized models to show that it might originate from dynamical anisotropy in a shallow turbulent fluid layer due to the planetary β-effect. Here we report the results of laboratory experiments, conducted on a 14-m diameter turntable, which quantitatively confirm that multiple zonal jets may indeed be generated and maintained by this mechanism in the presence of deep convection and a topographic β-effect. At the very small values of Ekman number (≤2 × 10−5) and large local Reynolds numbers (≥2000, based on jet scales) achieved, the kinetic energy spectra suggest the presence of both energy-cascading and enstrophy-cascading inertial ranges in addition to the zonation near twice the Rhines wave number.

Surface agents' influence on the flexural strength of bilaminated ceramics

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Patch and whole of surface mass-transfer rate measurements on a stepped rotating cylinder electrode

A recently developed whole of surface electroplating technique was used to obtain mass-transfer rates in the separated flow region of a stepped rotating cylinder electrode. These data are compared with previously reported mass-transfer rates obtained with a patch electrode. It was found that the two methods yield different results, where at lower Reynolds numbers, the mass-transfer rate enhancement was noticeably higher for the whole of the surface electrode than for the patch electrode. The location of the peak mass transfer behind the step, as measured with a patch electrode, was reported to be independent of the Reynolds number in previous studies, whereas the whole of the surface electrode shows a definite Reynolds number dependence. Large eddy simulation results for the recirculating region behind a step are used in this work to show that this difference in behavior is related to the existence of a much thinner fluid layer at the wall for which the velocity is a linear junction of distance from the wall. Consequently, the diffusion layer no longer lies well within a laminar sublayer. It is concluded that the patch electrode responds to the wall shear stress for smooth wall flow as well as for the disturbed flow region behind the step. When the whole of the surface is electro-active, the response is to mass transfer even when this is not a sole function of wall shear stress. The results demonstrate that the choice of the mass-transfer measurement technique in corrosion studies can have a significant effect on the results obtained from empirical data.

Transition in homogeneously heated inclined plane parallel shear flows

The transition of internally heated inclined plane parallel shear flows is examined numerically for the case of finite values of the Prandtl number Pr. We show that as the strength of the homogeneously distributed heat source is increased the basic flow loses stability to two-dimensional perturbations of the transverse roll type in a Hopf bifurcation for the vertical orientation of the fluid layer, whereas perturbations of the longitudinal roll type are most dangerous for a wide range of the value of the angle of inclination. In the case of the horizontal inclination transverse roll and longitudinal roll perturbations share the responsibility for the prime instability. Following the linear stability analysis for the general inclination of the fluid layer our attention is focused on a numerical study of the finite amplitude secondary travelling-wave solutions (TW) that develop from the perturbations of the transverse roll type for the vertical inclination of the fluid layer. The stability of the secondary TW against three-dimensional perturbations is also examined and our study shows that for Pr=0.71 the secondary instability sets in as a quasi-periodic mode, while for Pr=7 it is phase-locked to the secondary TW. The present study complements and extends the recent study by Nagata and Generalis (2002) in the case of vertical inclination for Pr=0.

Range of validity of weakly non-linear theory in Rayleigh-Bénard problem

In this paper we examine the equilibrium states of finite amplitude flow in a horizontal fluid layer with differential heating between the two rigid boundaries. The solutions to the Navier-Stokes equations are obtained by means of a perturbation method for evaluating the Landau constants and through a Newton-Raphson iterative method that results from the Fourier expansion of the solutions that bifurcate above the linear stability threshold of infinitesimal disturbances. The results obtained from these two different methods of evaluating the convective flow are compared in the neighborhood of the critical Rayleigh number. We find that for small Prandtl numbers the discrepancy of the two methods is noticeable. © 2009 The Physical Society of Japan.

Transition in plane parallel shear flows heated internally

The stability of internally heated inclined plane parallel shear flows is examined numerically for the case of finite value of the Prandtl number, Pr. The transition in a vertical channel has already been studied for 0≤Pr≤100 with or without the application of an external pressure gradient, where the secondary flow takes the form of travelling waves (TWs) that are spanwise-independent (see works of Nagata and Generalis). In this work, in contrast to work already reported (J. Heat Trans. T. ASME 124 (2002) 635-642), we examine transition where the secondary flow takes the form of longitudinal rolls (LRs), which are independent of the steamwise direction, for Pr=7 and for a specific value of the angle of inclination of the fluid layer without the application of an external pressure gradient. We find possible bifurcation points of the secondary flow by performing a linear stability analysis that determines the neutral curve, where the basic flow, which can have two inflection points, loses stability. The linear stability of the secondary flow against three-dimensional perturbations is also examined numerically for the same value of the angle of inclination by employing Floquet theory. We identify possible bifurcation points for the tertiary flow and show that the bifurcation can be either monotone or oscillatory. © 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Bifurcation contrasts between plane Poiseuille flow and plane Magnetohydrodynamic flow

The stability characteristics of an incompressible viscous pressure-driven flow of an electrically conducting fluid between two parallel boundaries in the presence of a transverse magnetic field are compared and contrasted with those of Plane Poiseuille flow (PPF). Assuming that the outer regions adjacent to the fluid layer are perfectly electrically insulating, the appropriate boundary conditions are applied. The eigenvalue problems are then solved numerically to obtain the critical Reynolds number Rec and the critical wave number ac in the limit of small Hartmann number (M) range to produce the curves of marginal stability. The non-linear two-dimensional travelling waves that bifurcate by way of a Hopf bifurcation from the neutral curves are approximated by a truncated Fourier series in the streamwise direction. Two and three dimensional secondary disturbances are applied to both the constant pressure and constant flux equilibrium solutions using Floquet theory as this is believed to be the generic mechanism of instability in shear flows. The change in shape of the undisturbed velocity profile caused by the magnetic field is found to be the dominant factor. Consequently the critical Reynolds number is found to increase rapidly with increasing M so the transverse magnetic field has a powerful stabilising effect on this type of flow.

Effect of aspect ratio on thermal convection in open and closed systems

Internally heated fluids are found across the nuclear fuel cycle. In certain situations the motion of the fluid is driven by the decay heat (i.e. corium melt pools in severe accidents, the shutdown of liquid metal reactors, molten salt and the passive control of light water reactors) as well as normal operation (i.e. intermediate waste storage and generation IV reactor designs). This can in the long-term affect reactor vessel integrity or lead to localized hot spots and accumulation of solid wastes that may prompt local increases in activity. Two approaches to the modeling of internally heated convection are presented here. These are based on numerical analysis using codes developed in-house and simulations using widely available computational fluid dynamics solvers. Open and closed fluid layers at around the transition between conduction and convection of various aspect ratios are considered. We determine optimum domain aspect ratio (1:7:7 up to 1:24:24 for open systems and 5:5:1, 1:10:10 and 1:20:20 for closed systems), mesh resolutions and turbulence models required to accurately and efficiently capture the convection structures that evolve when perturbing the conductive state of the fluid layer. Note that the open and closed fluid layers we study here are bounded by a conducting surface over an insulating surface. Conclusions will be drawn on the influence of the periodic boundary conditions on the flow patterns observed. We have also examined the stability of the nonlinear solutions that we found with the aim of identifying the bifurcation sequence of these solutions en route to turbulence.

Slot Film Cooling: A Comprehensive Experimental Characterization

When components of a propulsion system are exposed to elevated flow temperatures there is a risk for catastrophic failure if the components are not properly protected from the thermal loads. Among several strategies, slot film cooling is one of the most commonly used, yet poorly understood active cooling techniques. Tangential injection of a relatively cool fluid layer protects the surface(s) in question, but the turbulent mixing between the hot mainstream and cooler film along with the presence of the wall presents an inherently complex problem where kinematics, thermal transport and multimodal heat transfer are coupled. Furthermore, new propulsion designs rely heavily on CFD analysis to verify their viability. These CFD models require validation of their results, and the current literature does not provide a comprehensive data set for film cooling that meets all the demands for proper validation, namely a comprehensive (kinematic, thermal and boundary condition data) data set obtained over a wide range of conditions. This body of work aims at solving the fundamental issue of validation by providing high quality comprehensive film cooling data (kinematics, thermal mixing, heat transfer). 3 distinct velocity ratios (VR=uc/u∞) are examined corresponding to wall-wake (VR~0.5), min-shear (VR ~ 1.0), and wall-jet (VR~2.0) type flows at injection, while the temperature ratio TR= T∞/Tc is approximately 1.5 for all cases. Turbulence intensities at injection are 2-4% for the mainstream (urms/u∞, vrms/u∞,), and on the order of 8-10% for the coolant (urms/uc, vrms/uc,). A special emphasis is placed on inlet characterization, since inlet data in the literature is often incomplete or is of relatively low quality for CFD development. The data reveals that min-shear injection provides the best performance, followed by the wall-jet. The wall-wake case is comparably poor in performance. The comprehensive data suggests that this relative performance is due to the mixing strength of each case, as well as the location of regions of strong mixing with respect to the wall. Kinematic and thermal data show that strong mixing occurs in the wall-jet away from the wall (y/s>1), while strong mixing in the wall-wake occurs much closer to the wall (y/s<1). Min-shear cases exhibit noticeably weaker mixing confined to about y/s=1. Additionally to these general observations, the experimental data obtained in this work is analyzed to reveal scaling laws for the inlets, near-wall scaling, detecting and characterizing coherent structures in the flow as well as to provide data reduction strategies for comparison to CFD models (RANS and LES).

An approximate solution method for boundary layer flow of a power law fluid over a flat plate

The work in this paper deals with the development of momentum and thermal boundary layers when a power law fluid flows over a flat plate. At the plate we impose either constant temperature, constant flux or a Newton cooling condition. The problem is analysed using similarity solutions, integral momentum and energy equations and an approximation technique which is a form of the Heat Balance Integral Method. The fluid properties are assumed to be independent of temperature, hence the momentum equation uncouples from the thermal problem. We first derive the similarity equations for the velocity and present exact solutions for the case where the power law index n = 2. The similarity solutions are used to validate the new approximation method. This new technique is then applied to the thermal boundary layer, where a similarity solution can only be obtained for the case n = 1.