Infinite Fluid Layer Research Articles

Imperfectly conducting eigenflows in a vertical fluid layer

The stability of buoyant flow in an infinite vertical fluid layer bounded by imperfectly conducting rigid walls, called imperfectly conducting eigenflows, is discussed. The third kind boundary conditions describing heat transfer to the external environment are applied to perturbations in temperature. The linear stability analysis is carried out numerically by employing the Chebyshev collocation method. Instability arises when the Grashof number G exceeds its critical value, which depends on the Prandtl number Pr and the Biot number Bi. It is found that the onset of instability changes dramatically depending on the magnitude of Prandtl and Biot numbers particularly when the instability is through the traveling-wave mode. The numerical results show that the Biot number plays a pivotal role in determining the transition Prandtl number PrT at which the instability switches over from one mode to another mode. The novel outcomes suggest the presence of a single PrT for Bi<2.1739 while three distinct values of PrT for Bi≥2.1739. The departure from the conventional single value typically observed at isothermal boundaries signifies the complexity of the instability mechanism.

Multiple wave scattering by submerged obstacles in an infinite channel of finite depth with surface pressure excess

The objective is to study the combined effect of an incident wave, a surface pressure excess and a finite number of submerged obstacles, in the phenomenon of power transfer to an infinite fluid layer of finite depth. The incident wave and the surface pressure excess have the same harmonic time dependence, a fact that allows to eliminate time altogether and consider only steady-state solutions. The surface pressure excess simulates the effect of winds blowing above the water surface in oceans. The technique used in a first part of the paper relying upon the use of finite Fourier transform and separation of variables is extended here to this end. The method allows to separate local perturbations from progressive or standing wave. Our formulae yield the exact solution in closed form in the absence of obstacles, and provide a clearer insight into the flow properties, as compared to previous investigations. Applications are given for discontinuous surface pressure functions. We put in evidence solutions with no outgoing waves, for which the energy transmitted by the surface pressure is exhausted in generating a standing wave, together with local perturbations. Two numerical applications without/with obstacles, for a parabolic surface pressure profile, allow to assess the energy transfer from the pressure-obstacles system to the fluid. The results may be of interest in the field of oscillating water columns and, generally, water power converting technology.

Unidirectional steady-state inhomogeneous Couette flow with a quadratic velocity profile along a horizontal coordinate

The paper presents an exact solution to the boundary value problem describing the steady-state unidirectional flow of a viscous incompressible fluid. The fluid moves in an infinite horizontal strip (infinite fluid layer). The fulfillment of the no-slip condition is postulated at the lower boundary of the viscous fluid layer. At the upper boundary, which is assumed to be rigid, non-uniform velocity distribution is specified. The deformation of the free boundary is neglected due to the use of the rigid-lid boundary condition. The exact solution to the equations of the hydrodynamics of incompressible fluids automatically satisfies the continuity equation (the incompressibility equation). The velocity function is harmonic in this case. The simplest exact solution satisfying the Laplace equation is constructed, which takes into account the features of the velocity field along the transverse (vertical) coordinate and one of the longitudinal (horizontal) coordinates. The paper analyzes the topological properties of the velocity field, the tangential stress field, the vorticity vector, specific kinetic energy, and specific helicity.

Hydrodynamic Forces on a Submerged Circular Cylinder in Two-layer Fluid with an Ice-cover

We consider problems based on linear water wave theory concerning the interaction of wave with horizontal circular cylinder submerged in two-layer ocean consisting of a upper layer of finite depth bounded above by an ice-cover and below by an infinite layer of fluid of greater density, the ice-cover being modelled as an elastic plate of very small thickness. Using the method of multipoles, we formulate the problems of hydrodynamic forces on a submerged cylinder in either the upper or the lower layer. The vertical and horizontal forces on the circular cylinder are obtained and depicted graphically against the wave number for various values of flexural rigidity of ice-cover to show the effect of the presence of ice-cover on these quantities. Also when the flexural rigidity and surface density of the ice-cover are taken to be zero, the ice-cover tends to a free-surface. Then all the forces are the same as in the case of two-layer fluid with free surface.

Added Mass of Fluid and Fundamental Frequencies of a Horizontal Elastic Circular Plate Vibrating in Fluid of Constant Depth

Abstract The paper deals with free vibrations of a horizontal thin elastic circular plate submerged in an infinite layer of fluid of constant depth. The motion of the plate is accompanied by the fluid motion, and thus, the pressure load on this plate results from displacements of the plate in time. The plate and fluid motions depend on boundary conditions, and, in particular, the pressure load depends on the gap between the plate and the fluid bottom. In theoretical description of this phenomenon, we deal with a coupled problem of hydrodynamics in which the plate and fluid motions are coupled through boundary conditions at the plate surfaces. This coupling leads to the so-called co-vibrating (added) mass of fluid, which significantly changes the fundamental frequencies (eigenfrequencies) of the plate. In formulation of the problem, a linear theory of small deflections of the plate is employed. At the same time, one assumes the potential fluid motion with the potential function satisfying Laplace’s equation within the fluid domain and appropriate boundary conditions at fluid boundaries. In order to solve the problem, the infinite fluid domain is divided into sub-domains of simple geometry, and the solution of problem equations is constructed separately for each of these domains. Numerical experiments have been conducted to illustrate the formulation developed in this paper.

Vibrations of a Horizontal Elastic Band Plate Submerged in Fluid of Constant Depth

Abstract The paper deals with free and forced vibrations of a horizontal thin elastic plate submerged in an infinite layer of fluid of constant depth. In free vibrations, the pressure load on the plate results from assumed displacements of the plate. In forced vibrations, the fluid pressure is mainly induced by water waves arriving at the plate. In both cases, we have a coupled problem of hydrodynamics in which the plate and fluid motions are coupled through boundary conditions at the plate surface. At the same time, the pressure load on the plate depends on the gap between the plate and the fluid bottom. The motion of the plate is accompanied by the fluid motion. This leads to the so-called co-vibrating mass of fluid, which strongly changes the eigenfrequencies of the plate. In formulation of this problem, a linear theory of small deflections of the plate is employed. In order to calculate the fluid pressure, a solution of Laplace’s equation is constructed in the doubly connected infinite fluid domain. To this end, this infinite domain is divided into sub-domains of simple geometry, and the solution of the problem equation is constructed separately for each of these domains. Numerical experiments are conducted to illustrate the formulation developed in this paper.

Wave scattering by a horizontal circular cylinder in a three-layer fluid

Using linear water-wave theory, wave scattering by a horizontal circular cylinder submerged in a three-layer ocean consisting of a layer of finite depth bounded above by finite depth water with free surface and below by an infinite layer of fluid of greater density is considered. The cylinder is submerged in either of the three layers. In such a situation time-harmonic waves with given frequency can propagate with three different wave numbers. Employing the method of multipoles the problem is reduced to an infinite system of linear equations which are solved numerically by standard technique after truncation. The transmission and reflection coefficients are obtained and depicted graphically against the wave number for all cases. In a two-layer fluid there are energy identities that exist connecting the transmission and reflection coefficients that arise. These energy identities are systematically extended to the three-fluid cases which are obtained.

Вибрационные механизмы транспорта примеси в конвективных системах

The equations of vibrational convection for non-uniform molecular fluid mixtures have been derived when the diffusion coefficient depends on concentration of components. Equation system is received with the help of the averaging method in the limiting case of high frequencies and small values of amplitude. In a definite sense this one is identical mathematically to well-known equations system of thermal vibrational convection in an approximation of Zen’kovskaya and Simonenko. The problem of concentration field evolution in a plane infinite fluid layer has been solved numerically by the method of finite differences. The modeling situation with homogeneous one-dimensional source distribution of a heavy component on the solid boundary was considered. It was found that the high frequency vibrations can induce mean convective flow in a cavity even in the absence of a temperature inhomogeneity and in the case of weightlessness. At the same time vibrations cause the flow in the form of rolls which occupy the whole volume. The process of admixture ablation into the fluid has been considered for arbitrary oriented vibrations. It has been shown that in dependence on the direction of the vibration axes the process of admixture redistribution passes with various rates. First of all it is determined by the intensity degree of the averaged vibrational flow in a fluid. The modeling calculations were carried out for different values of the non-dimensional parameter which describes the dependence of diffusion coefficient on concentration. Numerical simulation demonstrates that most strong effect of admixture ablation takes place for longitudinal vibrations. On the other hand vibrations which are transversal to layer damps averaged flow. In this case the mass transfer has diffusive character. Also the dependence of mass transfer on diffusion coefficient is most pronounced for tangential vibrations.Received 27.07.2016; accepted 18.08.2016

Wall effects on the stability of convection in an infinite vertical layer

The stability of buoyancy-driven convection in a vertical infinite fluid layer between two rigid walls with different thermal conductivities and thicknesses is presented. Analytical solutions are derived for parallel base flow, for which linear stability analysis predicts the growth of two-dimensional disturbance. The resulting eigenvalue problem was solved using finite-elements method. Neutral stability curves and associated critical Grashof numbers and wavenumbers are supplied for different characteristic parameters of the flow. It is shown that thermal conductivity and thickness of the walls has a weak influence on hydrodynamic modes but a strong influence on thermal modes which become critical for lower values of Prandtl number than in the case of perfect conducting walls.

Bicritical states in temperature-modulated Rayleigh-Bénard convection.

We investigate the Rayleigh-Bénard convection under sinusoidally varying temperatures of the horizontal rigid planes bounding a laterally infinite fluid layer for the bicritical states. The problem is analogous to the well studied Faraday instability and Rayleigh-Bénard convection under gravity modulation. Under modulation, the neutral instability curve is found to alternate between the conventional harmonic and subharmonic tongues in the space of the dimensionless wave number of disturbance and the control parameter. The transition between harmonic and subharmonic critical instability responses is found to occur via a bicritical state, where the two instability responses coexist with different wave numbers. These bicritical states are found to depend upon the modulation parameters and the Prandtl number.

Nonlinear stable steady solutions to the Ostroumov problem

We investigate the linear and weakly nonlinear solutions to a convection problem that was first studied by Ostroumov in 1947. The problem pertains to the stability of the equations governing convective motion in an infinite vertical fluid layer that is heated from below. Ostroumov’s linear stability analysis yields instability threshold conditions that are characterized by zero wavenumber for the Fourier mode in the vertical direction and by eigenfunctions that are independent of the vertical coordinate. Thus, any undertaking at determining the supercritical nonlinear solutions and their stability through a small amplitude expansion fails. This failure is attributed to the fact that the nonlinear interaction of the linear modes vanish identically. In this paper, we put forth exact and stable similarity type solutions to the Ostroumov problem. These solutions are characterized by the same linear threshold conditions as Ostroumov’s solutions. Moreover, we are able to extend the analysis to the supercritical regime through a small amplitude analysis to obtain steady two-dimensional solutions for a small range of Prandtl numbers. These solutions are found to be stable to general two-dimensional, time-dependent disturbances. Furthermore, when the analysis is extended to the case where the fluid layer thickness is also allowed to be infinite, we find that the infinite two-dimensional fluid region becomes linearly unstable through a Batchelor–Nitsche (BN) instability mechanism. Thus, the nonlinear solutions are obtained through a long wavelength expansion, and consequently our analysis provides the nonlinear development of the BN instability for the case of an unstable and uniform density profile. Finally, numerical solutions of the nonlinear problem are also presented which shed light on the flow patterns and temperature distributions for both the vertical channel and the infinitely extended stratified region problems.

Wave scattering by a thin vertical barrier in a two-layer fluid

The present paper is concerned with scattering of surface and interface waves by a vertical plate in a fluid consisting of a layer of finite depth bounded above by a free surface and below by an infinite layer of fluid of density greater than the upper layer. For such a situation time-harmonic waves can propagate with two different wavenumbers K and v(>K) along the free surface and the interface respectively. The problems are formulated in terms of hypersingular integral equations by suitable applications of Green’s integral theorem in terms of difference of potential function across the barrier. These integral equations are solved by a collocation method using a finite series involving Chebyshev polynomials. Reflection and transmission coefficients for incident waves of wavenumbers K and v are computed numerically and depicted graphically in a number of figures for various values of different parameters. The energy identities are used as a partial check on the correctness of the numerical results.

Effect of thermal conductivity and thickness of the walls in the convection of a viscoelastic Maxwell fluid layer

Results for the linear thermoconvective stability of a layer of viscoelastic Maxwell fluid are presented. The stability problem is characterized by taking into account the lower and upper wall thermal conductivities as well as their thicknesses. This allows more realistic theoretical boundary conditions. A system consisting of a horizontal infinite Maxwell fluid layer confined between two parallel walls perpendicular to gravity is considered. The critical Rayleigh number R c , the frequency of oscillation ω c and the wavenumber k c were determined for fixed values of the relaxation time constant F and the Prandtl number Pr. The results are given for a range of wall thermal conductivities and thicknesses. Analytical and numerical solutions were calculated. Some unexpected results were found in comparison to those of the Newtonian fluid where the criticality curves become more unstable when the conductivities of the walls change from very good conductors to very bad conductors.

Stability of impulsively-driven natural convection with unsteady base state: implications of an adiabatic boundary

Stability conditions of a quiescent, horizontally infinite fluid layer with adiabatic bottom subject to sudden cooling from above are studied. Here, at difference from Rayleigh–Bénard convection, the temperature base state is never steady. Instability limits are studied using linear analysis while stability is analyzed using the energy method. Critical stability curves in terms of Rayleigh numbers and convection onset times were obtained for several kinematic boundary conditions. Stability curves resulting from energy and linear approaches exhibit the same temporal growth rate for large values of time, suggesting a bound for the temporal asymptotic behavior of the energy method.

Instability of an oscillatory fluid layer with insoluble surfactants

The linear stability of an infinite fluid layer with a deformable free surface covered by an insoluble surfactant and bounded below by a horizontal rigid plate oscillating in its own plane is studied based on the Floquet theory. The differential system governing the stability problem for perturbations of arbitrary wavenumbers is solved numerically by a Chebyshev collocation method. Stability boundaries are obtained in a wide range of amplitude and frequency of the modulation as well as surfactant elasticity. Results show that the presence of the surfactant may significantly stabilize (destabilize) the flow by raising (lowering) the critical Reynolds number associated with the onset of instability. The effect of the surfactant plays a stabilizing role for small surfactant elasticity and a destabilizing one for relatively large surfactant elasticity. The destabilizing effect of the surfactant on the stability of flows with a zero-shear surface is found for the first time. The disturbance modes in the form of travelling waves may be induced by the surfactant and dominate the instability of the flow.

Slow waves in fractures filled with viscous fluid

Stoneley guided waves in a fluid-filled fracture generally have larger amplitudes than other waves; therefore, their properties need to be incorporated into more realistic models. A fracture is modeled as an infinite layer of viscous fluid bounded by two elastic half-spaces with identical parameters. For small fracture thickness, a simple dispersion equation for wave-propagation velocity is obtained. This velocity is much smaller than the velocity of a fluid wave in a Biot-type solution, in which fracture walls are assumed to be rigid. At seismic prospecting frequencies and realistic fracture thicknesses, the Stoneley guided wave has wavelengths on the order of several meters and a quality factor [Formula: see text] exceeding 10, which indicates the possibility of resonance excitation in fluid-bearing rocks. The velocity and attenuation of Stoneley guided waves are distinctly different at low frequencies for water and for oil. The predominant role of fractures in fluid flow at field scales is supported by permeability data, showing an increase of several orders of magnitude when compared with values obtained at laboratory scales. The data suggest that Stoneley guided waves should be taken into account in theories describing seismic wave propagation in fluid-saturated rocks.

Large Eddy Simulation of turbulent thermal convection using a Mixed Scale Diffusivity Model

The Mixed Scale Diffusivity Model, originally developed in the case of the differentially heated cavity, is applied to compute turbulent Rayleigh-Benard flow in an infinite fluid layer at Pr = 0.71 for a large range of Rayleigh numbers (6.3 × 105 – 2 × 1011). The effect of this SGS modelling, which adjusts locally the SGS diffusivity to the thermal scales of the flow and results in variable PrSGS like the dynamic approach, is emphasised by comparison with LES and TRANS literature data. A single scaling regime is found in a range of Rayleigh numbers 6.3 × 105 – 2 × 1010, whose properties include the Ra0.302 scaling law for the Nusselt number and for the thermal boundary layer thickness, in agreement with the experimental correlation of Niemela et al. (2000). The first indication of a transition towards a new regime appears above Ra = 1011.

The destabilizing effect of boundary slip on Bénard convection

AbstractWe investigate the influence of slip boundary conditions on the onset of Bénard convection in an infinite fluid layer. It is shown that the critical Rayleigh number is a decreasing function of the slip length, and therefore boundary slip is seen to have a destabilizing effect. Chebyshev‐tau and compound matrix formulations for solving the eigenvalue problem are presented. Copyright © 2005 John Wiley & Sons, Ltd.

Non-linear dynamics and pattern formation in a vertical fluid layer heated from the side

We study both experimentally and numerically the convective flow in a tall vertical slot with differently heated walls. The flow is investigated for the fluid with the Prandtl number Pr=26, which is large enough to ensure the traveling waves as primary instability and small enough to prevent boundary layer convection. The flow evolution is determined on the base of the visual observations, power spectra and amplitude analysis. In the numerical simulations of two- and three-dimensional flows, we accept an assumption of an infinite fluid layer. The satisfactory agreement with experiment is observed, and the sequence of convection states is discovered. It starts with a plane-parallel flow as primary solution, which becomes unstable to two counter-propagating waves. It is followed by a tertiary three-dimensional flow in the form of wavy traveling waves. As the Grashof number is increased even further, a chaotically oscillating cellular pattern consisting of the pieces of broken waves arises. The formation of a structure in the form of the vertical rolls chaotically modulated along axes concludes this complicated picture.

Convective heat transport in a rotating fluid layer of infinite Prandtl number: optimum fields and upper bounds on Nusselt number.

By means of the Howard-Busse method of the optimum theory of turbulence we investigate numerically upper bounds on convective heat transport for the case of infinite fluid layer with stress-free vertical boundaries rotating about a vertical axis. We discuss the case of infinite Prandtl number, 1-alpha solution of the obtained variational problem and optimum fields possessing internal, intermediate, and boundary layers. We investigate regions of Rayleigh and Taylor numbers R and Ta, where no analytical bounds can be derived, and compare the analytical and numerical bounds for these regions of R and Ta where such comparison is possible. The increasing rotation has a different influence on the rescaled optimum fields of velocity w(1), temperature theta(1) and the vertical component of the vorticity f(1). The increasing Ta for fixed R leads to vanishing of the boundary layers of w(1) and theta(1). Opposite to this, the increasing Ta leads first to a formation of boundary layers of the field f(1) but further increasing the rotation causes vanishing of these boundary layers. We obtain optimum profiles of the horizontal averaged total temperature field which could be used as hints for construction of the background fields when applying Doering-Constantin method to the problems of rotating convection. The wave number alpha(1) corresponding to the optimum fields follows the asymptotic relationship alpha(1)=(R/5)(1/4) for intermediate Rayleigh numbers. However, when R becomes large with respect to Ta, after a transition region, the power law for alpha(1) becomes close to the power law for the case without rotation. The Nusselt number Nu is close to the nonrotational bound 0.32R(1/3) for the case of large R and small Ta. Nu decreases with increasing Taylor number. Thus, the upper bounds reflect the tendency of inhibiting thermal convection by increasing rotation for a fixed Rayleigh number. For the regions of Rayleigh and Taylor numbers where the numerical and asymptotic bounds on Nu can be compared, the numerical bounds are about 70% lower than the asymptotic bounds.