Shallow Fluid Layer Research Articles

Asymmetric vertical transport in weakly forced shallow flows

In this paper, we report on an investigation of the vertical transport of tracer particles released within a shallow, continuously-forced flow by means of numerical simulations. The investigation is motivated by the shallow flows encountered in many environmental situations and inspired by the laboratory experiments conducted in electromagnetically forced shallow fluid layers. The flow is confined to a thin fluid layer by stress-free top and no-slip bottom walls. The dynamics and the transport properties of the shallow flow are investigated under various flow conditions characterized by a Reynolds number related to the forcing, ReF, and the aspect ratio of vertical and horizontal length scales δ. The forcing generates an array of vortices that becomes unsteady when ReFδ2≳10. These vortices are accompanied by upwellings in their cores which are surrounded by narrower, stronger downwellings. Hence, upwellings occur where the horizontal flow is vorticity-dominated, while downwellings where it is strain-dominated. The magnitude of the asymmetry in strength and size of the vertical flows and their correlation with horizontal structures depends on the flow conditions and significantly influences the vertical spreading of particles within the fluid volume. Under conditions leading to a large asymmetry, particles within updrafts are transported slowly upwards, while particles within downdrafts rapidly move downwards. In addition, particles are trapped for longer within the updrafts than downdrafts because of their correlation with vorticity-dominated regions. However, when the flow becomes fully three-dimensional and highly unsteady for large ReFδ2 values, this transport asymmetry subsides because the updrafts and downdrafts exhibit similar strength and size in such flow conditions. Consequently, similar amounts of particles are transported upwards and downwards at similar rates.

INSTABILITY INVESTIGATION OF THERMO-BIOCONVECTION OF OXYTACTIC MICROORGANISM IN JEFFREY NANOLIQUID WITH EFFECTS OF INTERNAL HEAT SOURCE

The intent of this article is to investigate the influence of internal heat source on the stability of a suspension containing oxytactic microorganisms in a shallow horizontal porous fluid layer saturated by Jeffrey nanoliquid. The Jeffrey-Buongiorno model governs the nanofluid bioconvection flow. Normal mode analysis is utilized, and the principle of exchange of stability is invoked due to the absence of opposing agencies. The stability criteria is defined in terms of critical thermal Rayleigh number as a function of various flow governing parameters by using the weighted residual Galerkin method. It is perceived that the increment of uniform heat supply and the presence of oxytactic microorganism cells as well as the nanoparticles enhances heat transfer and constitutes an unstable system that hastens bioconvection. The nanoparticle Lewis number is found to have a dual impact on the system stability that relies on the nanoparticle Rayleigh number and exhibits destabilizing nature for top-heavy nanoparticle concentration. It is also observed that the Jeffrey parameter produces nonoscillatory instability in the system.

The fluid dynamics of propagating fronts with solutal and thermal coupling

We numerically explore the propagation of reacting fronts in a shallow and horizontal layer of fluid. We focus on fronts that couple with the fluid due to density differences between the products and reactants and also due to heat release from the reaction. We explore fronts where this solutal and thermal coupling is cooperative or antagonistic. We quantify the fluid motion induced by the front and investigate the interactions of the front with the fluid as it propagates through quiescent, cellular and chaotic flow fields. The solutal coupling induces an extended convection roll that travels with the front, the thermal coupling due to heat release from the reaction generates a pair of convection rolls that travels with the front, and when both couplings are present there is a complex signature of these contributions. The details of the front dynamics depend significantly upon the interactions of the front-induced flow field with the fluid ahead of the front.

Dynamo and the Adiabatic Invariant

Abstract The paper considers a dynamo generated by a shallow fluid layer in a celestial body (planet or star). This dynamo is based on the extra invariant for interacting magnetic Rossby waves.

Integrability, bilinearization, solitons and exact three wave solutions for a forced Korteweg-de Vries equation

The Korteweg-de Vries (KdV) equation with a forcing term is considered which demonstrates the mathematical modelling of the physics of a shallow layer of fluid due to external forcing. The characteristic of integrability is investigated via Painlevè analysis and an auto Bäcklund transformation is fabricated via the truncated Painlevè expansion. Using the concept of Bell polynomial approach the bilinear Bäcklund transformation, Lax pair and infinite sequence of conservation laws are constructed. The one-soliton, two-soliton and three-soliton solutions are obtained via simplified Hirota method. We demonstrate the propagation and interaction of soliton solutions graphically. Moreover, a new form of ansatz in the exact three wave method (ETW) along with symbolic computation is applied and some new periodic type of three wave solutions which includes periodic two solitary wave solution, doubly periodic solitary wave solution and breather type of two solitary wave solution are obtained for the forced KdV equation.

An overstability analysis of vertically vibrated suspension of active swimmers subjected to thermal stratification

The present article focuses on the analytical approach to discuss the thermo–vibrational convection in a suspension of the active (gyrotactic) swimmers. The onset of instability criterion is investigated for the stationary and oscillatory modes of convection in a shallow fluid layer with no–slip and rigid–free walls. The eigenvalue problem is tackled by Galerkin scheme to get the desired stability diagram and the correlation between the critical Rayleigh numbers. The overstability in suspension is possible when the unstable density gradient of the gyrotactic particles is opposed by the density variation due to thermo–vibrational influence. The suspension is destabilized due to gyrotactic up–swimming while the increase in Péclet number stabilizes the system. The stabilizing influence of vertical vibration is considerably affected due to thermal gradient which destabilizes the suspension. An interesting result of this study is the influence of thermo–vibrational parameter which is associated with applied thermal and vibrational properties. We reported that the destabilizing nature of thermo–vibrational parameter becomes thermally or vibrationally governed when the suspension is heated or cooled from below. When compared to the rigid–rigid boundaries, the displayed profiles for rigid–free walls yielded less stableness in the suspension.

Slippage effect on the flow stability induced by an inclined temperature gradient

The effect of slippage on the linear hydrodynamic flow stability of a shallow fluid layer induced by an inclined temperature gradient was studied for fluids with Prandtl number of 0.1, 0.7, 10 and 500, related with applications. The basic flow is induced by the horizontal component of the inclined temperature gradient. The slippage in the horizontal walls provide changes in the instability modes giving a completely different map of the stability diagrams. The results suggest that slippage has an destabilizer effect in comparison with the case of no slip boundary condition, which can be seen in the reduction of stable areas for most of the assessed cases. However, for moderate and high Prandtl numbers, the slippage promotes the appearance of the oscillatory transverse mode To, giving as a result, additional codimension-two points.

The validity of two-dimensional models of a rotating shallow fluid layer

Abstract

Propagating fronts in fluids with solutal feedback.

We numerically study the propagation of reacting fronts in a shallow and horizontal layer of fluid with solutal feedback and in the presence of a thermally driven flow field of counterrotating convection rolls. We solve the Boussinesq equations along with a reaction-convection-diffusion equation for the concentration field where the products of the nonlinear autocatalytic reaction are less dense than the reactants. For small values of the solutal Rayleigh number the characteristic fluid velocity scales linearly, and the front velocity and mixing length scale quadratically, with increasing solutal Rayleigh number. For small solutal Rayleigh numbers the front geometry is described by a curve that is nearly antisymmetric about the horizontal midplane. For large values of the solutal Rayleigh number the characteristic fluid velocity, the front velocity, and the mixing length exhibit square-root scaling and the front shape collapses onto an asymmetric self-similar curve. In the presence of counterrotating convection rolls, the mixing length decreases while the front velocity increases. The complexity of the front geometry increases when both the solutal and convective contributions are significant and the dynamics can exhibit chemical oscillations in time for certain parameter values. Last, we discuss the spatiotemporal features of the complex fronts that form over a range of solutal and thermal driving.

On the regularity of the Green–Naghdi equations for a rotating shallow fluid layer

The Green–Naghdi equations are an extension of the shallow-water equations that capture the effects of finite fluid depth at arbitrary order in the characteristic height to width aspect ratio $H/L$. The shallow-water equations capture these effects to first order only, resulting in a relatively simple two-dimensional fluid-dynamical model for the layer horizontal velocity and depth. The Green–Naghdi equations, like the shallow-water equations, are two-dimensional fluid equations expressing momentum and mass conservation. There are different ‘levels’ of the Green–Naghdi equations of rapidly increasing complexity. In the present paper we focus on the behaviour of the lowest-level Green–Naghdi equations for a rotating shallow fluid layer, paying close attention to the flow structure at small spatial scales. We compare directly with the shallow-water equations and study the differences arising in their solutions. By recasting the equations into a form which both explicitly conserves Rossby–Ertel potential vorticity and represents the leading-order departure from geostrophic–hydrostatic balance, we are able to accurately describe both the ‘slow’ predominantly sub-inertial balanced dynamics and the ‘fast’ residual imbalanced dynamics. This decomposition has proved fruitful in studies of shallow-water dynamics but appears not to have been used before in studies of Green–Naghdi dynamics. Importantly, we find that this decomposition exposes a fundamental inconsistency in the Green–Naghdi equations for horizontal scales less than the mean fluid depth, scales for which the Green–Naghdi equations are supposed to more accurately model. Such scales exhibit pronounced activity compared to the shallow-water equations, and in particular spectra for certain fields like the divergence are flat or rising at high wavenumbers. This indicates a lack of convergence at small scales, and is also consistent with the poor convergence of total energy with resolution compared to the shallow-water equations. We suggest a mathematical reformulation of the Green–Naghdi equations which may improve convergence at small scales.

Unstable equilibria and invariant manifolds in quasi-two-dimensional Kolmogorov-like flow.

Recent studies suggest that unstable, nonchaotic solutions of the Navier-Stokes equation may provide deep insights into fluid turbulence. In this article, we present a combined experimental and numerical study exploring the dynamical role of unstable equilibrium solutions and their invariant manifolds in a weakly turbulent, electromagnetically driven, shallow fluid layer. Identifying instants when turbulent evolution slows down, we compute 31 unstable equilibria of a realistic two-dimensional model of the flow. We establish the dynamical relevance of these unstable equilibria by showing that they are closely visited by the turbulent flow. We also establish the dynamical relevance of unstable manifolds by verifying that they are shadowed by turbulent trajectories departing from the neighborhoods of unstable equilibria over large distances in state space.

Top-to-bottom Ekman layer and its implications for shallow rotating flows

The analytical solution is derived for rotational frictional flow in a shallow layer of fluid in which the top and bottom Ekman layers join without leaving a frictionless interior. This vertical structure has significant implications for the horizontal flow. In particular, for a layer of water subjected to both a surface wind stress and bottom friction, the vorticity of the horizontal flow is a function not only of the curl of the wind stress (the classical result for deep water known as Ekman pumping) but also of its divergence. The importance of this divergence term peaks for a water depth around 3 times the Ekman layer thickness. This means that a curl-free but non-uniform wind stress on a shallow sea or lake can, through the dual action of rotation and friction, generate vorticity in the wind-driven currents. We also find that the reduction of three-dimensional dynamics to a two-dimensional model is more subtle than one could have anticipated and needs to be approached with utmost care. Taking the bottom stress as dependent solely on the depth-averaged flow, even with some veering, is not appropriate. The bottom stress ought to include a component proportional to the surface stress, which is negligible for large depths but increases with decreasing water depth.

Orbitally shaken shallow fluid layers. I. Regime classification

Orbital shakers are simple devices that provide mixing, aeration, and shear stress at multiple scales and high throughput. For this reason, they are extensively used in a wide range of applications from protein production to bacterial biofilms and endothelial cell experiments. This study focuses on the behaviour of orbitally shaken shallow fluid layers in cylindrical containers. In order to investigate the behaviour over a wide range of different conditions, a significant number of numerical simulations are carried out under different configuration parameters. We demonstrate that potential theory—despite the relatively low Reynolds number of the system—describes the free-surface amplitude well and the velocity field reasonably well, except when the forcing frequency is close to a natural frequency and resonance occurs. By classifying the simulations into non-breaking, breaking, and breaking with part of the bottom uncovered, it is shown that the onset of wave breaking is well described by Δh/(2R) = 0.7Γ, ...

Orbitally shaken shallow fluid layers. II. An improved wall shear stress model

A new model for the analytical prediction of wall shear stress distributions at the base of orbitally shaken shallow fluid layers is developed. This model is a generalisation of the classical extended Stokes solution and will be referred to as the potential theory-Stokes model. The model is validated using a large set of numerical simulations covering a wide range of flow regimes representative of those used in laboratory experiments. It is demonstrated that the model is in much better agreement with the simulation data than the classical Stokes solution, improving the prediction in 63% of the studied cases. The central assumption of the model—which is to link the wall shear stress with the surface velocity—is shown to hold remarkably well over all regimes covered.

Analytical and Numerical Study of Soret and Dufour Effects on Double Diffusive Convection in a Shallow Horizontal Binary Fluid Layer Submitted to Uniform Fluxes of Heat and Mass

Combined Soret and Dufour effects on thermosolutal convection induced in a horizontal layer filled with a binary fluid and subject to constant heat and mass fluxes are investigated analytically and numerically. The thresholds marking the onset of supercritical and subcritical convection are predicted analytically and explicitly versus the governing parameters. The present investigation shows that different regions exist in the N-Du plane corresponding to different parallel flow regimes. The number, the extent, and the locations of these regions depend on whether SrDu>-(1+Le2)/2Le2=f(Le) or SrDu<-(1+Le2)/2Le2. Conjugate effects of cross-phenomena on thresholds of fluid flow and heat and mass transfer characteristics are illustrated and discussed.

Chaotic thermal convection of couple-stress fluid layer

In this work we study the pattern of bifurcations and intermittent-chaos of non-Newtonian couple-stress shallow fluid layer subject to heating from below. The couple-stress parameter delays onset of convection, synchronizes chaotic behavior, and decreases the heat transfer . Some global aspects of the dynamics such as homoclinic bifurcations and transition to chaos are explored. The effects of particle size on the intermittent-chaos regime at particular normalized Rayleigh number, say $$r=166.1$$ , are investigated. With the increase in couple-stress parameter, the present Lorenz-like system synchronizes to a steady state via a series of periodic solutions interspersed with intervals of chaotic behaviors.

Lissajous trajectories in electromagnetically driven vortices

An experimental and theoretical study of laminar vortical flows driven by oscillating electromagnetic forces that act in orthogonal directions in a shallow electrolytic fluid layer is presented. Forces are generated by the interaction of the field of a dipolar permanent magnet and two imposed alternating electric currents perpendicular to each other with independent frequencies varying in the range of 10–30 mHz. Velocity fields of the time-dependent flow are obtained using particle image velocimetry, while particle tracking allows exploration of the Lagrangian trajectories and time maps. An approximate two-dimensional analytical solution is obtained for the laminar creeping regime so that Lagrangian trajectories are integrated explicitly. These trajectories resemble Lissajous figures with the usual property that, when the ratio of the frequencies of the imposed currents is rational, closed paths are found, while non-closed paths occur when this ratio is irrational. Deviations of this regime that account for slight increase of inertial effects are explored through a quasi-two-dimensional numerical simulation. In this case, non-closed paths are found even for rational frequency ratios. This case was observed in the experiment. Lagrangian trajectories calculated numerically show a qualitative agreement with experimental particle tracking. Furthermore, numerical time maps obtained for increasing inertial effects and rational frequency ratios reveal a chaotic behaviour. Some features of the Lagrangian trajectories are validated experimentally. In particular, topological properties of the calculated and observed time maps are in qualitative agreement. In a characteristic case, a partial time map calculated numerically is compared with the section acquired from the experimental tracking of one particle.

Double-diffusive convection and baroclinic instability in a differentially heated and initially stratified rotating system: the barostrat instability

A water-filled differentially heated rotating annulus with initially prepared stable vertical salinity profiles is studied in the laboratory. Based on two-dimensional horizontal particle image velocimetry data and infrared camera visualizations, we describe the appearance and the characteristics of the baroclinic instability in this original configuration. First, we show that when the salinity profile is linear and confined between two non-stratified layers at top and bottom, only two separate shallow fluid layers can be destabilized. These unstable layers appear nearby the top and the bottom of the tank with a stratified motionless zone between them. This laboratory arrangement is thus particularly interesting to model geophysical or astrophysical situations where stratified regions are often juxtaposed to convective ones. Then, for more general but stable initial density profiles, statistical measures are introduced to quantify the extent of the baroclinic instability at given depths and to analyze the connections between this depth-dependence and the vertical salinity profiles. We find that, although the presence of stable stratification generally hinders full-depth overturning, double-diffusive convection can lead to development of multicellular sideways convection in shallow layers and subsequently to a multilayered baroclinic instability. Therefore we conclude that by decreasing the characteristic vertical scale of the flow, stratification may even enhance the formation of cyclonic and anticyclonic eddies (and thus, mixing) in a local sense.

A kinematic conservation law in free surface flow

The Green–Naghdi system is used to model highly nonlinear weakly dispersive waves propagating at the surface of a shallow layer of a perfect fluid. The system has three associated conservation laws which describe the conservation of mass, momentum, and energy due to the surface wave motion. In addition, the system features a fourth conservation law which is the main focus of this note. It will be shown how this fourth conservation law can be interpreted in terms of a concrete kinematic quantity connected to the evolution of the tangent velocity at the free surface. The equation for the tangent velocity is first derived for the full Euler equations in both two and three dimensional flows, and in both cases, it gives rise to an approximate balance law in the Green–Naghdi theory which turns out to be identical to the fourth conservation law for this system. It is also shown that the conservation equation for the tangent velocity at the free surface appears as an endpoint case of a more general conservation equation for tangent velocities along material surfaces in the body of the fluid.

LARGE-SCALE QUASI-GEOSTROPHIC MAGNETOHYDRODYNAMICS

We consider the ideal magnetohydrodynamics (MHD) of a shallow fluid layer on a rapidly rotating planet or star. The presence of a background toroidal magnetic field is assumed, and the "shallow water" beta-plane approximation is used. We derive a single equation for the slow large length scale dynamics. The range of validity of this equation fits the MHD of the lighter fluid at the top of Earth's outer core. The form of this equation is similar to the quasi-geostrophic (Q-G) equation (for usual ocean or atmosphere), but the parameters are essentially different. Our equation also implies the inverse cascade; but contrary to the usual Q-G situation, the energy cascades to smaller length scales, while the enstrophy cascades to the larger scales. We find the Kolmogorov-type spectrum for the inverse cascade. The spectrum indicates the energy accumulation in larger scales. In addition to the energy and enstrophy, the obtained equation possesses an extra (adiabatic-type) invariant. Its presence implies energy accumulation in the 30° sector around zonal direction. With some special energy input, the extra invariant can lead to the accumulation of energy in zonal magnetic field; this happens if the input of the extra invariant is small, while the energy input is considerable.