Rigid Horizontal Boundaries Research Articles

Onset of Oscillatory Convection of a Chemically Reacting Fluid with Rigid Horizontal Boundaries

Onset of Oscillatory Convection of a Chemically Reacting Fluid with Rigid Horizontal Boundaries

Laminar-turbulent transitions during thermal gravitational-capillary convection in horizontal liquid layers heated from below

The evolution of the spatial form of the flow versus the Rayleigh number in layers with two rigid horizontal boundaries and layers with a free surface is experimentally investigated depending on the Rayleigh and Marangoni numbers. The experiments were carried out with layers of ethyl alcohol and water. A thermal imager was used to measure temperature fields on free surfaces of liquids.

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Abstract

Spontaneous inertia–gravity wave emission in the differentially heated rotating annulus experiment

The source mechanism of inertia–gravity waves (IGWs) observed in numerical simulations of the differentially heated rotating annulus experiment is investigated. The focus is on the wave generation from the balanced part of the flow, a process presumably contributing significantly to the atmospheric IGW field. Direct numerical simulations are performed for an atmosphere-like configuration of the annulus and possible regions of IGW activity are characterised by a Hilbert-transform algorithm. In addition, the flow is separated into a balanced and unbalanced part, assuming the limit of a small Rossby number, and the forcing of IGWs by the balanced part of the flow is derived rigorously. Tangent-linear simulations are then used to identify the part of the IGW signal that is rather due to radiation by the internal balanced flow than to boundary-layer instabilities at the side walls. An idealised fluid set-up without rigid horizontal boundaries is considered as well, to investigate the effect of the identified balanced forcing unmasked by boundary-layer effects. The direct simulations of the realistic and idealised fluid set-ups show a clear baroclinic-wave structure exhibiting a jet–front system similar to its atmospheric counterparts, superimposed by four distinct IGW packets. The subsequent tangent-linear analysis indicates that three wave packets are radiated from the internal flow and a fourth one is probably caused by boundary-layer instabilities. The forcing by the balanced part of the flow is found to play a significant role in the generation of IGWs, so it supplements boundary-layer instabilities as a key factor in the IGW emission in the differentially heated rotating annulus.

Onset of oscillatory Rayleigh-Bénard magnetoconvection with rigid horizontal boundaries

We present the results of linear stability analysis of oscillatory Rayleigh-Bénard magnetoconvection with rigid and thermally conducting boundaries. We have investigated two types of horizontal surfaces: (i) electrically conducting and (ii) boundaries which do not allow any outward current normal to the surface (magnetic vacuum conditions). For the case of electrically conducting boundaries, the critical Rayleigh number Rao(Q,Pr,Pm), the critical wave number ko(Q,Pr,Pm), and the frequency at the instability onset ω(Q,Pr,Pm) increase as the Chandrasekhar number Q is raised for fixed non-zero values of thermal Prandtl Pr and magnetic Prandtl number Pm. For small values of Pr, the frequency of oscillation ω at the primary instability shows a rapid increase with Pm for very small values of Pm followed by a decrease at relatively larger values of Pm. In the limit of Pr→0, Rao and ko are found to be independent of Q. However, the frequency ω increases with Q, but decreases with Pm in this limit. The oscillatory instability is possible at the onset of magnetoconvection if and only if Chandrasekhar's criterion is valid (i.e., Pm > Pr) and Q is raised above a critical value Qc(Pr,Pm) such that the product Pm*Qc≈91 for large Pm. For the stellar interior of an astrophysical body (Pm≈10−4 and Pr≈10−8), the value of this product Pm*Qc≈230. The boundary conditions for magnetic vacuum change the critical values of Rayleigh number, wave number, and frequency of oscillation at the onset. The oscillatory magnetoconvection occurs in this case, if Q>Qc, where Pm*Qc≈42 for large Pm. For steller interior, this value is approximately 64. A low-dimensional model is also constructed to study various patterns near the onset of oscillatory convection for rigid, thermally and electrically conducting boundaries. The model shows standing and drifting fluid patterns in addition to flow reversal close to the onset of magnetoconvection.

The relaxation time for viscous and porous gravity currents following a change in flux

The equilibration time $\unicode[STIX]{x1D70F}$ in response to a change in flux from $Q$ to $\unicode[STIX]{x1D6EC}Q$ after an injection period $T$ applied to either a low-Reynolds-number gravity current or one propagating through a porous medium, in both axisymmetric and one-dimensional geometries, is shown to be of the form $\unicode[STIX]{x1D70F}=Tf(\unicode[STIX]{x1D6EC})$, independent of all the remaining physical parameters. Numerical solutions are used to investigate $f(\unicode[STIX]{x1D6EC})$ for each of these situations and compare very well with experimental results in the case of an axisymmetric current propagating over a rigid horizontal boundary. Analysis of the relaxation towards self-similarity provides an illuminating connection between the excess (deficit) volume from early times and an asymptotically equivalent shift in time origin, and hence a good quantitative estimate of $\unicode[STIX]{x1D70F}$. The case $\unicode[STIX]{x1D6EC}=0$ of equilibration after ceasing injection at time $T$ is a singular limit. Extensions to high-Reynolds-number currents and to the case of a constant-volume release followed by constant-flux injection are discussed briefly.

A model for Rayleigh-Bénard magnetoconvection

A model for three-dimensional Rayleigh-B\'{e}nard convection in low-Prandtl-number fluids near onset with rigid horizontal boundaries in the presence of a uniform vertical magnetic field is constructed and analyzed in detail. The kinetic energy $K$, the convective entropy $\Phi$ and the convective heat flux ($Nu-1$) show scaling behaviour with $\epsilon = r-1$ near onset of convection, where $r$ is the reduced Rayleigh number. The model is also used to investigate various magneto-convective structures close to the onset. Straight rolls, which appear at the primary instability, become unstable with increase in $r$ and bifurcate to three-dimensional structures. The straight rolls become periodically varying wavy rolls or quasiperiodically varying structures in time with increase in $r$ depending on the values of Prandtl number $Pr$. They become irregular in time, with increase in $r$. These standing wave solutions bifurcate first to periodic and then quasiperiodic traveling wave solutions, as $r$ is raised further. The variations of the critical Rayleigh number $Ra_{os}$ and the frequency $\omega_{os}$ at the onset of the secondary instability with $Pr$ are also studied for different values of Chandrasekhar's number $Q$.

Numerical stability of symmetric solitary-wave-like waves of a two-layer fluid—Forced modified KdV equation

Forced internal waves at the interface of a two-layer incompressible fluid in a two-dimensional domain with rigid horizontal boundaries are studied. The lower boundary is assumed to have a small obstruction. We derive a time-dependent forced modified KdV equation when the KdV theory fails and study the stabilities of four types of symmetric time-independent solitary-wave-like solutions numerically.

Convective instability of an Ohmic liquid layer in an unsteady thermal field

The linear stability problem for an incompressible viscous poorly conducting Ohmic liquid between two rigid horizontal boundaries with time-periodic temperature distribution and under steady transverse electric field is considered. The free charge in the liquid is due only to the nonuniform electroconduction. The electrohydrodynamic (EHD) approximation is used. Floquet theory is applied for finding various instability thresholds in linear approximation. The influence of time-dependent temperature field modulation on the liquid layer behavior is studied with and without an additional steady component. The boundaries of instability and the characteristics of critical disturbances are determined. Depending on the frequency and amplitude of modulation, the temperature gradient can destabilize the equilibrium of the liquid or stabilize otherwise unstable base state. In addition to synchronous and subharmonic responses to external modulation, the instability can be associated with quasiperiodic disturbances. The critical electric Rayleigh number is given as a function of frequency and heating level. The limit of low frequency modulation is studied by an asymptotic method.

Nonlinear analysis and solitary waves for two superposed streaming electrified fluids of uniform depths with rigid boundaries

The nonlinear modulation of the interfacial waves of two superposed dielectric fluids with uniform depths and rigid horizontal boundaries, under the influence of constant normal electric fields and uniform horizontal velocities, is investigated using the multiple-time scales method. It is found that the behavior of small perturbations superimposed on traveling wave trains can be described by a nonlinear Schrodinger equation in a frame of reference moving with the group velocity. Wave-like solutions to this equation are examined, and different types of localized excitations (envelope solitary waves) are shown to exist. It is shown that when these perturbations are neutrally stable and sufficiently long, solutions to the nonlinear Schrodinger equation may be approximated by the well-known Korteweg-de Vries equation. The speed of the solitary on the interface is seen to be reduced by the electric field. It is found that there are two critical values of the applied voltage that lead to (i) breaking up of the solitary waves, and (ii) bifurcation of solutions of the governing equations. On the other hand, the complex amplitude of standing wave trains near the marginal state is governed by a similar type of nonlinear Schrodinger equation in which the roles of time and space are interchanged. This equation, under a suitable transformation, is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solutions with variable speed. Using the tangent hyperbolic method, it is observed that the wave speed increases as well as decreases, with the increase of electric field values, according to the chosen wavenumbers range. Finally, the nonlinear stability analysis is discussed in view of the coefficients of nonlinear Schrodinger equation to show the effects of various physical parameters, and also to recover the some limiting cases studied earlier in the literature.

Transition to turbulent convection in a fluid layer heated from below at moderate aspect ratio

Numerical simulations of Rayleigh–Bénard convection in a fluid layer heated from below between two rigid horizontal boundaries have been performed for Rayleigh numbers Ra up to 107, Prandtl numbers in the range between 0.7 and 60 and for aspect ratios up to 20. Periodic boundary conditions in the horizontal plane have been used. To a considerable extent, the evolution towards turbulent convection at high values of Ra is governed by processes exhibited by instabilities of steady or time periodic forms of convection at lower Rayleigh numbers. With increasing Ra, the properties of convection are increasingly determined by the thermal boundary layers. The role of mean flows which may be symmetric or antisymmetric with respect to the mid-plane of the layer is emphasized.

The effects of hydraulic resistance on dam-break and other shallow inertial flows

The effects of resistive forces on unsteady shallow flows over rigid horizontal boundaries are investigated theoretically. The dynamics of this type of motion are driven by the streamwise gradient of the hydrostatic pressure, which balances the inertia of the fluid and the basal resistance. Drag forces are often negligible provided the fluid is sufficiently deep. However, close to the front of some flows where the depth of the moving layer becomes small, it is possible for drag to substantially influence the motion. Here we consider three aspects of unsteady shallow flows. First we consider a regime in which the drag, inertia and buoyancy (pressure gradient) are formally of the same magnitude throughout the entire current and we construct a new class of similarity solutions for the motion. This reveals the range of solution types possible, which includes those with continuous profiles, those with discontinuous profiles and weak shocks and those which are continuous but have critical points of transition at which the gradients may be discontinuous. Next we analyse one-dimensional dam-break flow and calculate how drag slows the motion. There is always a region close to the front in which drag forces are not negligible. We employ matched asymptotic expansions to combine the flow at the front with the flow in the bulk of the domain and derive theoretical predictions that are compared to laboratory measurements of dam-break flows. Finally we investigate a modified form of dam-break flow in which the vertical profile of the horizontal velocity field is no longer assumed to be uniform. It is found that in the absence of drag it is no longer possible to find a kinematically consistent front of the fluid motion. However the inclusion of drag forces within the region close to the front resolves this difficulty. We calculate velocity and depth profiles within the drag-affected region, and obtain the leading-order expression for the rate at which the fluid propagates when the magnitude of the drag force is modelled using Chae#169;zy, Newtonian and power-law fluid closures; this compares well with experimental data and provide new insights into dam-break flows.

Synoptic Development in a Two-and-One-Half-Layer Contour Dynamics System

Synoptic development is studied using a two-and-one-half-layer contour dynamics model in full spherical geometry. The model has three isentropic layers: two lower layers that are dynamically active and one upper layer that is kept motionless. The isentropic layers cover the whole sphere and are confined in the vertical direction by two rigid horizontal boundaries. The model is assumed to be in hydrostatic and geostrophic equilibrium; forcing/friction and diabatic heating/cooling are neglected. In both active layers, the horizontal structure is represented by a piecewise-uniform distribution of potential vorticity with a single front in each layer. The potential vorticity front in the upper active layer can be associated with the tropopause or, to be more specific, with the sudden change in height of the tropopause at the jet stream. The potential vorticity front in the lower active layer enhances the baroclinicity of the system and may be associated with the polar front. Because of the assumption of hydrostatic and geostrophic equilibrium, the model atmosphere is completely defined by the instantaneous positions of the contours and can be integrated in time using the technique of contour dynamics. It is shown that realistic zonal flows can be obtained by a suitable choice of parameters. A linear stability analysis reveals that small-amplitude perturbations of given planetary wavenumber may grow for only specific latitudinal positions of the potential vorticity fronts. The maximum growth rates and highest planetary wavenumbers are found for potential vorticity fronts that are located at approximately the same latitude. Because of the conceptual simplicity of the model, in which the potential vorticity structure is represented by only two contours, the instability mechanism expresses itself in a clear way. The contour dynamics model also captures the nonlinear stages of cyclogenesis remarkably well, as is evident from the time evolution of the fronts and the time evolution of a passive tracer in a numerical simulation of a cutoff cyclone.

3D seismic response of a limited valley via BEM using 2.5D analytical Green's functions for an infinite free-rigid layer

This paper presents analytical solutions for computing the 3D displacements in a flat solid elastic stratum bounded by a rigid base, when it is subjected to spatially sinusoidal harmonic line loads. These functions are also used as Greens functions in a boundary element method code that simulates the seismic wave propagation in a confined or semi-confined 2D valley, avoiding the discretization of the free and rigid horizontal boundaries. The models developed are then used to simulate wave propagation within a rigid stratum and valleys with different dimensions and geometries, when struck by a spatially sinusoidal harmonic vertical line load. Simulations are performed in the frequency domain, for varying spatial wave numbers in the axial direction of the valley. Time results are obtained by means of inverse Fourier transforms, to help understand how the geometry of the valley may affect the variation of the displacement field.

Normal modes for a stratified viscous fluid layer

We consider internal gravity waves in a stratified fluid layer with rigid horizontal boundaries and periodic boundary conditions on the sides at constant temperature with a small constant viscosity, modelled using the incompressible Navier-Stokes equations. Using operator-theoretic methods to study the damping rates of internal waves we prove there are non-oscillatory wave modes with arbitrarily small damping rates. We provide an asymptotic approximation for these non-oscillatory modes. Additionally, we find that the eigenvalues for damped oscillations are in an explicitly describable half-ring.

Nonlinear EHD stability of the travelling and standing waves of two superposed dielectric bounded fluids in relative motion

The nonlinear modulation of the interfacial waves of two superposed dielectric fluids with uniform depths and rigid horizontal boundaries, under the influence of constant normal electric fields and uniform horizontal velocities, is investigated using the multiple-time scales method. It is found that the quasi-monochromatic travelling waves can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. The stability of uniform and periodic solution of the nonlinear Schrödinger equation is tested by means of three-mode model, involving the interaction of a finite number of Fourier components. A set of coupled differential equations is obtained, describing the effect of sideband modulations. The Benjamin–Feir instability appears clearly related to the linearized technique, and is the first stage of the nonlinear recurrence phenomenon. On the other hand, the complex amplitude of quasi-monochromatic standing waves near the cut-off wavenumber is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation makes it possible to estimate the nonlinear effect on the linear cut-off wavenumber.

Decay rates of internal waves in a fluid near the liquid-vapor critical point

We study the damping of internal waves in a viscous fluid near the liquid-vapor critical point. Such a fluid becomes strongly stratified by gravity due to its large compressibility. Using the variable-density incompressible Navier-Stokes equations, we model an infinite fluid layer with rigid horizontal boundaries and periodic side boundary conditions. We present operator-theoretic results that predict the existence of internal-wave modes with arbitrarily small damping rates. We also solve the eigenvalue problem numerically using a compound matrix shooting method and a second method based on a matched-asymptotic perturbation expansion for small viscosity. At temperatures far above the critical point, the damping of the internal waves is substantially influenced by both boundary layer and volumetric effects. The boundary layer effect is caused by horizontal shearing layers near the two fixed horizontal boundaries. As the temperature approaches the critical temperature, an additional internal shearing layer develops as the density stratification curve steepens on approach to the two-phase regime. Numerical calculations show that for some of the internal-wave modes this causes a dramatic increase in the damping rate that dominates the boundary layer effects.

Double-diffusive parallel flow induced in a horizontal Brinkman porous layer subjected to constant heat and mass fluxes: analytical and numerical studies

Fluid flow and heat and mass transfer induced by double-diffusive natural convection in a horizontal porous layer subjected to vertical gradients of temperature and concentration are studied analytically and numerically using the Brinkman-extended Darcy model. Both cases of rigid and free horizontal boundaries are examined in the present study. The parameters governing the problem are the Rayleigh number RT, the Lewis number Le, the buoyancy ratio N, the Darcy number Da and the aspect ratio Ar. The analytical solution is based on the parallel flow approximation. The critical Rayleigh number corresponding to the onset of the parallel flow in this system is determined analytically as a function of Le, N and Da. For sufficiently small Da, both free and rigid boundaries yield results which are identical to those predicted by the Darcy model. The present investigation shows that there exists a region in the plane (N, Le) where the convective flow is not possible in the layer regardless of the Rayleigh and Darcy numbers considered.

Numerical study of convection in the horizontal Bridgman configuration under the action of a constant magnetic field. Part 2. Three-dimensional flow

The effects of a constant magnetic field on electrically conducting liquid-metal flows in a parallelepiped cavity are investigated using a spectral numerical method involving direct numerical solution of the Navier–Stokes and Ohm equations for three-dimensional flows. Three horizontal Bridgman configurations are studied: buoyancy-driven convection in a confined cavity and in a cavity where the top boundary is a stress-free surface and thirdly, thermocapillary-driven flow in a cavity where the upper boundary is subjected to effects of surface tension. The results of varying the Hartmann number (Ha) are described for a cavity with Ax = L/H = 4 and Ay = W/H = 1, where L is the length, W is the width and H is the height of the cavity. In general, an increase in the strength of the applied magnetic field leads to several fundamental changes in the properties of thermal convection. The convective circulation progressively loses its intensity and when Ha reaches a certain critical value, which is found to depend on the direction (longitudinal or vertical) of the applied magnetic field, decrease of the flow intensity takes on an asymptotic form with important changes in the structure of the flow circulation. The flow structure may be separated into three regions: the core flow, Hartmann layers which develop in the immediate vicinity of the rigid horizontal boundaries or at the endwalls, and parallel layers appearing in the vicinity of the sidewalls. The behaviour of the maxima of velocity and of the overall flow circulation is found to depend on both the boundary conditions used and the direction of the applied magnetic field. Furthermore, the interaction of the electric current density with the applied magnetic field which leads to the structural reorganization described above can also create more subtle flow modifications, such as flow inversions which are observed mainly in the central region of the cavity.

Effects of additive noise at the onset of Rayleigh-Bénard convection.

The effects of additive Gaussian white noise on the onset of Rayleigh-B\'enard convection are studied by means of a phenomenological model, the stochastic Swift-Hohenberg equation. The strength of the noise term arising from thermal fluctuations is given for both free-slip and rigid horizontal boundaries. As was already pointed out by previous authors this term contains the small parameter ${\mathit{k}}_{\mathit{B}}$T/\ensuremath{\rho}d${\ensuremath{\nu}}^{2}$, where \ensuremath{\rho} is the mass density, d the plate separation, and \ensuremath{\nu} the kinematic viscosity. For typical liquids this parameter is of order ${10}^{\mathrm{\ensuremath{-}}9}$. Experiments involving fluctuation effects may be interpreted in terms of this model if the noise strength is treated as an adjustable parameter, which turns out to be larger than the typical thermal value by four orders of magnitude. The effects of fluctuations on the bifurcation of an infinite system are studied, and the earlier arguments of the present authors leading to a first-order transition are reviewed [Swift and Hohenberg, Phys. Rev. A 15, 319 (1977)]. The conditions under which the multimode model can be approximated by a single-mode stochastic amplitude equation are investigated, and an earlier analytic approximation scheme for calculating the response to a time-dependent Rayleigh number is applied to the multimode model. A comparison with available experimental and numerical simulation data is presented.