Parallel Rolls Research Articles

Influence of Rotation and Viscosity on Parallel Rolls of Electrically Conducting Fluid

Rayleigh–Bénard convection is a fundamental fluid dynamics phenomenon that significantly influences heat transfer in various natural and industrial processes, such as geophysical dynamics in the Earth’s liquid core and the performance of heat exchangers. Understanding the behavior of conductive fluids under the influence of heating, rotation, and magnetic fields is critical for improving thermal management systems. Utilizing the Boussinesq approximation, this study theoretically examines the nonlinear convection of a planar layer of conductive liquid that is heated from below and subjected to rotation about a vertical axis in the presence of a magnetic field. We focus on the onset of stationary convection as the temperature difference applied across the planar layer increases. Our theoretical approach investigates the formation of parallel rolls aligned with the magnetic field under free–free boundary conditions. To analyze the system of nonlinear equations, we expand the dependent variables in a series of orthogonal functions and express the coefficients of these functions as power series in a parameter ϵ. A solution for this nonlinear problem is derived through Fourier analysis of perturbations, extending to O(ϵ8), which allows for a detailed visualization of the parallel rolls. Graphical results are presented to explore the dependence of the Nusselt number on the Rayleigh number (R) and Ekman number (E). We observe that both the local Nusselt number and average Nusselt number increase as the Ekman number decreases. Furthermore, the flow appears to become more deformed as E decreases, suggesting an increased influence of external factors such as rotation. This deformation may enhance mixing within the fluid, thereby improving heat transfer between different regions.

Analytical solution to calendering in eccentric cylindrical coordinates

Calendering is the process in which molten material is dragged through the nip region to produce a film or sheet. By nip region, we mean the area between two corotating rolls. Here, we analyze the calendering problem in eccentric cylindrical coordinates with the simplest fluid, Newtonian. We first assume the velocity profile as vθ(ξ,θ). We arrive at the analytical solution for the velocity profile and pressure distribution when the fluid passes between parallel rolls. We then get the flow rate (and, thus, the sheet thickness) by integrating the velocity profile between the parallel rolls. We include a worked example to teach how to use our main result.

Vibroconvective Patterns in a Layer under Translational Vibrations of Circular Polarization

This article experimentally investigates thermal vibrational convection in horizontal layers, subject to circular translational oscillations in the horizontal plane. The definite direction of translational vibrations lacks investigation, and the case of a layer heated from above is considered. At large negative values of the gravitational Rayleigh number, the thermovibrational convection appears in a threshold manner with an increase in the vibration intensity. Our results show that in the case of strong gravitational stabilization, thermovibrational convection develops in the form of patterns with strong anisotropy of spatial periods in orthogonal directions. The vibroconvective patterns have the form of parallel rolls divided along their length into relatively short segments. The layer thickness determines the distance between the rolls, and the longitudinal wavelength, depends on the Rayleigh number. Convective cells are studied using the noninvasive thermohromic methodic. It is found that when using the tracers for flow visualization, the concentration and type of the visualizer particles have a serious impact on the shape of the observed vibroconvective structures. In particular, the presence of even a small number of tracers (used in the study of velocity fields by the PIV method) generates flows and intensifies the heat transfer below the threshold of thermovibrational convection excitation.

Small atlas of residual stresses trajectories during surfacing or how to make steel ‘transparent’

ABSTRACT The article shows the unique capabilities of the physical non-destructive magnetoelastic method by the example of creating a small atlas of the principal stresses trajectories fields (isostat) during electric arc deposition. The object of the research is two plates from steel St3 with dimensions of 500x400x8 and 400x200x8 mm, attached along the edges. Six deposition variants are carried out on their surface using a semiautomatic device in a carbon dioxide environment: ‘Parallel rolls’, ‘Circles’, ‘Fork’, ‘Arcs’, ‘Cross’, ‘Angle’, imitating common weld seams. The work presents the sequence of the isostat detection process: drawing a coordinate net on a sample, determining the inclination angles values of tangents related to the trajectories of the greatest principal stresses in the net knots, graphic constructions. The magnetoelastic method is implemented through the use mechanical stress metre – MSM-4 M with a sensor base of 5 mm, an operating frequency of 1000 Hz, an inclinometer error of ±2 degrees. An atlas of typical isostat fields is created on the basis of control at 1900 net knots on the surface of a compact steel sample. Visualization of power flows is provided, which makes the steel appear to be ‘transparent’ and opens up additional opportunities for researchers and production engineers to optimize deposition and welding technologies, and to evaluate the effectiveness of reducing residual stresses in various ways.

Rayleigh–Bénard convection in viscoelastic liquid bridges

Abstract

Onset of Rayleigh-Bénard convection for intermediate aspect ratio cylindrical containers

The convection patterns that occur at and slightly above the onset of convection in cylindrical containers were determined as a function of aspect ratio, using simulations of Rayleigh-Bénard convection and linear stability analysis. The study focused primarily on aspect ratios 6≤Γ≤20, where Γ = diameter/depth, with conducting or insulating, and no-slip boundary conditions and Prandtl numbers Pr = 0.7 and 28.9. Simulations demonstrate azimuthally pure Fourier mode patterns at onset consistent with what is expected from bifurcation theory, with an m = 1 mode, for even values of Γ, and a concentric roll pattern, or m = 0 mode, for odd values of Γ. For Rayleigh numbers slightly higher than onset other pure or mixed mode patterns were found and then for even higher Rayleigh numbers, straight parallel rolls were found. A linear stability analysis was used to determine the critical Rayleigh number, Rac, and flow pattern for a large range of aspect ratios and was found to agree with the simulation results.

Numerical simulations of a shock-filament interaction

We present 3D hydrodynamic adiabatic simulations of a shock interacting with a dense, elongated cloud. We compare how the nature of the interaction changes with the filament's length and its orientation to the shock, and with the shock Mach number and the density contrast of the filament. We then examine the differences with respect to 3D spherical-cloud calculations. We find significant differences in the morphology of the interaction when M = 10 and χ = 102: in many cases, three parallel rolls are formed, and spread furthermore apart with time, and periodic vortex shedding can occur off the ends of oblique filaments. Sideways-on filaments are accelerated more quickly, and initially lose mass more quickly than spherical clouds due to their greater surface area to volume ratio. However, at late stages, they lose mass more slowly, due to the reduced relative speed between the filament and the post-shock flow. The acceleration and mixing time-scales can vary by a factor of 2 as the filament orientation changes. Oblique filaments can achieve transverse velocities up to 10 per cent of the shock speed. Some aspects of our simulations are compared against experimental and numerical work on rigid cylinders.

Resonance patterns in spatially forced Rayleigh–Bénard convection

AbstractWe report on the influence of a quasi-one-dimensional periodic forcing on the pattern selection process in Rayleigh–Bénard convection (RBC). The forcing was introduced by a lithographically fabricated periodic texture on the bottom plate. We study the convection patterns as a function of the Rayleigh number ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Ra}$) and the dimensionless forcing wavenumber ($q_f$). For small $\mathit{Ra}$, convection takes the form of straight parallel rolls that are locked to the underlying forcing pattern. With increasing $\mathit{Ra}$, these rolls give way to more complex patterns, due to a secondary instability. The forcing wavenumber $q_f$ was varied in the experiment over the range of $0.6q_c<q_f<1.4q_c$, with $q_c$ being the critical wavenumber of the unforced system. We investigate the stability of straight rolls as a function of $q_f$ and report patterns that arise due to a secondary instability.

Oscillatory instability and fluid patterns in low-Prandtl-number Rayleigh-Bénard convection with uniform rotation

We present the results of direct numerical simulations of flow patterns in a low-Prandtl-number (Pr = 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 (Lx × Ly × 1) and a wide range of Taylor numbers (750 ⩽ Ta ⩽ 5000) for the purpose. The horizontal aspect ratio η = Ly/Lx 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 Ta < 2700, but observed a set of parallel rolls in the form of standing waves for Ta ⩾ 2700. The three-dimensional convection was temporally quasiperiodic for 750 ⩽ Ta ⩽ 1114, chaotic for 1115 ⩽ Ta < 1125, and quasiperiodic once again for 1125 ⩽ Ta < 2700. The three-dimensional quasiperiodic flow bifurcated into a two-dimensional periodic flow for Ta ⩾ 2700. The convection at the primary instability were three-dimensional and quasiperiodic in time for all values of Ta (⩽3000) in a box with η = 2. Küppers-Lortz type of convection was observed at the primary instability in longer boxes (4 ⩽ η ⩽ 10). The convective structures consisted of the appearance and disappearance of straight rolls, rhombic patterns, and wavy rolls inclined at an angle \documentclass[12pt]{minimal}\begin{document}$\phi = \frac{\pi }{2} - \arctan {(\eta ^{-1})}$\end{document}ϕ=π2−arctan(η−1) with the straight rolls. We also present a low-dimensional model constructed for a square box (η = 1).

Onset of Rayleigh-Bénard convection in cylindrical containers

We determined the critical Rayleigh numbers Ra{c} for the onset of convection in cylindrical containers with aspect ratios 1 approximately <Gamma [ triple bond] D/L approximately <9 ( D is the diameter and L the height) and the patterns that form just above Ra{c}, both from experiment and by direct numerical simulation (DNS). Results for Ra{c} agree well with the linear stability analysis by Buell and Catton for containers with finite sidewall conductivity. For Gamma<or=1.58+/-0.10 , we found that the patterns correspond to an azimuthal Fourier mode with mode number m=1 , corresponding to a single convection roll. For 1.58 approximately < Gamma approximately <3.26+/-0.02 , the pattern was a concentric roll, corresponding to m=0 . For 3.26<or=Gamma approximately >4, an m=1 mode was found again, but near Gamma=4 either m=1 or m=2 was observed in different runs. These results are consistent with the marginal stability curves calculated by Buell and Catton in the sense that the mode that is the first as a function of Ra to acquire a positive growth rate is the one that is observed. For Gamma approximately >4, the theoretical marginal curves for the four lowest modes lie very close together. There we found patterns near onset that corresponded to various modes, including m=2 and 4. At relatively large Gamma approximately > 6, we observed parallel straight rolls quite close to onset. Our patterns agree with several DNS investigations by others, but at some Gamma values differ from those observed experimentally by Stork and Müller. Some results for the pattern evolution with increasing Ra are reported as well.

Extreme multiplicity in cylindrical Rayleigh-Bénard convection. II. Bifurcation diagram and symmetry classification

A large number of flows with distinctive patterns have been observed in experiments and simulations of Rayleigh-Bénard convection in a water-filled cylinder whose radius is twice the height. We have adapted a time-dependent pseudospectral code, first, to carry out Newton's method and branch continuation and, second, to carry out the exponential power method and Arnoldi iteration to calculate leading eigenpairs and determine the stability of the steady states. The resulting bifurcation diagram represents a compromise between the tendency in the bulk toward parallel rolls and the requirement imposed by the boundary conditions that primary bifurcations be toward states whose azimuthal dependence is trigonometric. The diagram contains 17 branches of stable and unstable steady states. These can be classified geometrically as roll states containing two, three, and four rolls; axisymmetric patterns with one or two tori; threefold-symmetric patterns called Mercedes, Mitsubishi, marigold, and cloverleaf; trigonometric patterns called dipole and pizza; and less symmetric patterns called CO and asymmetric three rolls. The convective branches are connected to the conductive state and to each other by 16 primary and secondary pitchfork bifurcations and turning points. In order to better understand this complicated bifurcation diagram, we have partitioned it according to azimuthal symmetry. We have been able to determine the bifurcation-theoretic origin from the conductive state of all the branches observed at high Rayleigh number.

Rossby waves and α-effect

Rossby waves drifting in the azimuthal direction are a common feature at the onset of thermal convective instability in a rapidly rotating spherical shell. They can also result from the destabilization of a Stewartson shear layer produced by differential rotation as expected in the liquid sodium experiment (Derviche Tourner Sodium) working in Grenoble, France. A usual way to explain why Rossby waves can contribute to the dynamo process goes back to Busse [A model of the Geodynamo. Geophys. J. R. Astron. Soc. 1975, 42, 437–459]. In his picture, the flow geometry is a cylindrical array of parallel rolls aligned with the rotation axis. The axial flow component (parallel to the rotation axis) is maximum in the middle of each roll and changes its sign from one roll to the next (case (i)). It is produced by the Ekman pumping at the fluid containing shell boundary. The corresponding dynamo mechanism can be explained in terms of an α-tensor with non-zero diagonal coefficients. It corresponds to the heuristic picture given by Busse (1975). In rapidly rotating objects like the Earth's core (or in a fast rotating experiment), Rossby waves occur in the limit of small Ekman number (). In that case, the main source of the axial flow component is not the Ekman pumping but rather the “geometrical slope effect” due to the spherical shape of the fluid containing shell. This implies that the axial flow component is maximum at the borders of the rolls and not at the centres (case (ii)). If assumed to be stationary, such rolls would lead to zero coefficients on the diagonal of the α-tensor, making the dynamo probably less efficient if possible at all. Actually, the rolls are drifting as a wave, and we show that this drift implies non-zero coefficients on the diagonal of the α-tensor. These new coefficients are, in essence, very different from the ones obtained in case (i) and cannot be interpreted in terms of the heuristic picture of Busse (1975). They were interpreted as higher-order effects in Busse (1975). In addition, we consider rolls not only drifting but also having an arbitrary radial phase shift as expected in real objects.

Numerical simulation of thermoconvective flows and more uniform depositions in a cold wall rectangular APCVD reactor

At atmospheric pressure, the usual flow conditions in the cold wall horizontal rectangular thermal CVD reactors correspond to steady longitudinal thermoconvective rolls that make non-uniform vapour depositions, in shape of longitudinal parallel ridges. In order to get more uniform depositions, the pressure is generally lowered under the atmospheric pressure to promote forced convection flows, instead of mixed convection ones. In the present paper, using three-dimensional direct numerical simulations, we propose and analyse a method to get uniform deposition without lowering the pressure in the reactor. It consists in adequately exciting the parallel thermoconvective rolls at channel inlet to make them unsteady, periodic and sinuous in order to get a uniform time average of the deposition. This method is shown to be adapted for the horizontal and rectangular APCVD reactors with large longitudinal and transversal aspect ratios, when the Reynolds number of the gas flow is O(100), whatever the value of the surface Damköhler number. This situation is encountered in the online or scrolling APCVD reactors used to deposit coatings on float glass in the flat glass industry for instance. The simulations are based on simplified models for the transport equations (Boussinesq model) and the kinetics of the heterogeneous reactions (deposition model of silicon from hydrogen and silane: SiH 4→Si+2H 2).

Convective Roll Dynamics in LiquidHe4near the Onset of Convection

We present results of experiments on Rayleigh-Bénard convection in liquid 4He at several temperatures. We show visually that with carefully defined boundary conditions the basic convection state consists of parallel rolls which are aligned in one of two directions, the angle thus defined as being temperature dependent and we attempt to explain this behavior. We also show directly the skew-varicose instability acting on the basic state and correlate it with fluctuations in the temperature difference across the fluid layer.

Electroconvection in nematic liquid crystals with positive dielectric and negative conductivity anisotropy.

Electroconvection in an unusual nematic compound with strongly positive dielectric anisotropy and negative anisotropy of the conductivity is investigated. For homeotropic alignment, where one has a direct transition to rolls or squares depending on the frequency of the applied voltage, we present a quantitative theory. From the comparison we infer values for some viscosities, which are rather unusual, but not unreasonable in view of the vicinity of the nematic-smectic transition. For planar alignment, electroconvection sets in above a splay Freedericksz transition with "parallel rolls," which is also captured by the theory.

Example of a chaotic crystal: the labyrinth.

Labyrinthine structures often appear as the final steady state of pattern forming systems. Being disordered, they exhibit the same kind of short range positional order as the Newell-Pomeau turbulent crystal. Labyrinths can be seen as a limit case of the texture of disordered rolls with a coherence length of the same order as the wavelength. In the various two-dimensional model equations we looked at, labyrinths and parallel rolls are steady states for the same parameters, their occurrence depending on the initial conditions. Comparing the stability of these two structures, we find that in variational models their energy is very close, rolls always being more stable than labyrinths. For the nonvariational model we propose a numerical experiment which displays a well defined bifurcation from parallel rolls to labyrinths as the more stable state.

Dynamic Parallel Roll Alignment

A new method of checking the alignment of parallel rolls, while they are rotating, is described in this paper. The method utilizes the autoreflection technique, along with a strobe light and photoelectric trigger, to determine the orientation of the normal to a mirror fixed to the end of a roll. Three or more observations of the normal at different rotational positions of the roll allow for the computation of the roll’s rotation-axis orientation. The method was applied to a small machine in both the dynamic (operating) and static (nonoperating) modes. From this application, it was determined that standard deviations of orientations can be expected to be in the order of 4 arcs for the dynamic mode and 2 arcs for the static mode.

Eulerian and Lagrangian Langmuir circulation patterns

An idealized Langmuir circulation pattern in the oceanic surface layer consists of parallel rolls with a fixed spatial periodicity. Observed surface manifestations of the circulation patterns, formed as collections of floating material and known as windrows, are roughly parallel to the surface wind direction, but form a network of intersecting lines. Field observations of the windrow intersections, referred to as “Y-junctions,” indicate a preferred orientation in which the two forks of the “Y” coalesce when viewed from upwind (the Y-junctions are said to point downwind). Weakly nonlinear theory yields patterns in the instantaneous Eulerian fields, for example contours of surface convergence, that show joinings and bifurcations at locations that may be termed pattern defects. These take the form of downwind-pointing Y-junctions occurring at negative defects and their obverse at positive defects with no preference for either type. We show that the preferential orientation of the Y-junctions can be explained as a Lagrangian effect. A simplified model for studying Y-junction formation was developed and shows that downwind-pointing Y-junctions are created by negative defects when the basic state surface velocity component transverse to the rolls is less than or equal to the perturbation velocity. Positive defects do not create any windrow intersections. Surface tracer simulations using the weakly nonlinear theory confirm predictions of the simplified model.

NUMERICAL STUDY OF RAYLEIGH-BE´NARD CONVECTION IN A CYLINDER

Numerical solutions of the three-dimensional equations for Rayleigh-Bénard convection in a vertical cylinder are presented. The energy and vorticity transport equations are solved using the Samarskii-Andreyev alternating direction implicit (ADI) scheme. A fast Fourier transform algorithm is used to solve for the vector potential. Solutions are presented for aspect ratios of 2 and 4, a Prandtl number Pr = 7, and Rayleigh numbers 12 h Ra h 37500. A conductive (no motion) state exists when Ra h Ra c = 1860. For Ra > Ra c , there are four main types of flow structures. These are concentric, radial, parallel, and cross rolls. The Nusselt number depends on the type of flow structure.

Visualization of roll patterns in Rayleigh–Bénard convection of air in a rectangular shallow cavity

Flow visualization is conducted here to study the vortex flow patterns associated with the Rayleigh–Bénard convection in a horizontal shallow cavity of air. The cavity is a rectangular enclosure characterized by the aspect ratios A x =16 and A z =20. Attention is focused on the convection rolls driven at slightly supercritical and subcritical buoyancies. The results show that at slightly subcritical Ra the induced vortex flow is in the form of rectangular rolls along the cavity sides and short straight parallel rolls in the cavity core. At slightly higher Ra near Ra c ∞ (=1708) more rectangular rolls appear and the short straight rolls in the cavity core merge together to form a serpentine roll. At slightly supercritical buoyancy with 2000⩽ Ra⩽3000 the entire cavity is filled with the straight rolls all parallel to the short sides of the cavity. At an even higher Ra of 4000 the vortex rolls become irregular and time dependent. Moreover, the processes through which various vortex flow structures evolve during the transient stage are shown to be rather complicate and the vortex flow patterns during the flow formation are significantly affected by the heating rate in raising the buoyancy force. Furthermore, the wavenumber reduction at higher Ra for the parallel vortex roll pattern was noted to mainly result from the splitting of some rectangular rolls into cells and the subsequent merging of the cells into bigger rolls at the intermediate stage of the flow formation.