Onset of convective stability in a fluid-saturated porous layer subjected to time-dependent heating

A theoretical analysis of thermal instability driven by buoyancy forces in transient temperature fields is conducted in an initially quiescent, fluid-saturated, horizontal porous layer. Darcy's law is used to explain characteristics of fluid motion and linear stability theory is employed to predict the onset of buoyancy-driven motion. Under the principle of exchange of stabilities, the stability analysis is performed on the basis of the linear amplification theory. The result predicts the critical condition of onset of buoyancy-driven convection, which is governed by the Darcy-Rayleigh number. The present stability criteria predict the experimental data quite well.     

Nonlinear dynamic behavior of non-convective zone in salt gradient solar pond

Non-Convective Zone (NCZ) of salt gradient solar pond is a typical double diffusive system of salinity and temperature, and it is subjected to instable effects of adverse temperature gradient. The onset of instability may occur as an oscillatory motion because of the stabilizing effect of the salinity. In this paper, the marginal state between the steady state and the convection of the NCZ is studied. The stability of the Boussinesq approximation of the Navier-Stokes equations is analyzed by a perturbation approach. The marginal states for the onset of convection are obtained by analytical method, which is based on the linearization of the ordinary differential equations, and then numerical method is used to solve the nonlinear ordinary differential equations. Numerical results provide the trajectories of the temperature and velocity coefficients in the three-dimensional phase space, as well as the two-dimensional temperature, salinity and velocity fields in NCZ. The results demonstrate that the numerical study is in agreement with the marginal stability and the critical Rayleigh number derived from linear stability analysis. Both the linear and nonlinear studies indicate that oscillation is a narrow region above the stable region; however, the nonlinear numerical results indicate that the linear stability analysis leans to a larger upper boundary in the oscillatory regions.     

A low‐order model for analysing effects of blade fatigue load control

A new low‐order mathematical model is introduced to analyse blade dynamics and blade load‐reducing control strategies for wind turbines. The model consists of a typical wing section model combined with a rotor speed model, leading to four structural degrees of freedom (flapwise, edgewise and torsional blade oscillations and rotor speed). The aerodynamics is described by an unsteady aerodynamic model. The equations of motion are derived in non‐linear and linear form. The linear equations of motion are used for stability analysis and control design. The non‐linear equations of motion are used for time simulations to evaluate control performance. The stability analysis shows that the model is capable of predicting classical flutter and stall‐induced vibrations. The results from the stability analysis are compared with known results, showing good agreement. The model is used to compare the performance of one proportional–integral–derivative controller and two full‐state feedback controllers. Copyright © 2006 John Wiley & Sons, Ltd.     

Linear stability analysis of a fluid-saturated porous layer subjected to time-dependent heating

A theoretical analysis of thermal instability driven by buoyancy forces under the transient temperature fields is conducted in an initially quiescent, fluid-saturated and horizontal porous layer. Darcy’s law is used to explain the characteristics of fluid motion and under the principle of exchange of stabilities, the linear stability theory is employed to derive stability equations. The stability equations are analyzed by the initial value approach with the proper initial conditions. Two stability limits, τ_s and τ_r are obtained under the strong and the relative stability concepts. The critical condition of onset of buoyancy-driven convection is governed by the Darcy–Rayleigh number, as expected. The growth period for disturbances to grow is seemed to be required until the instabilities are detected experimentally. The convective motion can be detected experimentally from a certain time τ₀  4τ_r.     

Comparative study of sinusoidal and non-sinusoidal two-frequency internal heat modulation in a Rayleigh-Bénard system

A linear stability analysis is assented to investigate the effect of two-frequency internal heat modulation at the onset of convection in a Newtonian liquid. The correction Rayleigh number and wave number for small amplitudes is calculated using the Venezian approach. Under two-frequency internal heat modulation, the motion is found to be subcritical. To quantify heat transfer in the system, the three-mode Lorenz model is solved numerically. Various combinations of sinusoidal and non-sinusoidal waveforms influence the onset of convection and heat transfer in the system due to two-frequency internal heat modulation. The parameters' influence on heat transfer is seen to be dependent on the presence of a heat source or sink.     

Onset of Rayleigh–Bénard MHD convection in a micropolar fluid

The present work is concerned with the effect of a uniform magnetic field on the onset of convection in an electrically conducting micropolar fluid. A flat fluid layer bounded by horizontal rigid boundaries, subjected to thermal boundary conditions of the Neumann type, is considered. The parallel flow approximation is used to predict analytically the critical Rayleigh number for the onset of convection. The onset of motion is found to depend on the Hartmann number Ha, materials parameters K, B, $\lambda$, and the micro-rotation boundary condition n. A linear stability analysis is carried out to study numerically the onset of convection. The predictions of the analytical model are found to be in good agreement with the numerical solution. The above results are also compared with those obtained numerically for the case of a system subject to Dirichlet thermal boundary conditions.     

On the onset of natural convection in differentially heated shallow fluid layers with internal heat generation

This paper reports the results of an analytical and numerical investigation to determine the effect of internal heat generation on the onset of convection, in a differentially heated shallow fluid layer. The case with the bottom plate at a temperature higher than the top plate mimics the classical Rayleigh Benard convection. However, internal heat generation adds a new dimension to the problem. Linear stability analysis is first carried out for the case of an infinitely wide cavity. The effect of aspect ratio on the onset of convection is studied by solving the full Navier–Stokes equations and the equation of energy and observing the temperature contours. A bisection algorithm is used for an accurate prediction of the onset. The numerical results are used to plot the stability curves for eight different aspect ratios. A general correlation is developed to determine the onset of convection in a differentially heated cavity for various aspect ratios. For an aspect ratio of 10, it is seen that the cavity approaches the limit of an infinite cavity. Analytical results obtained by using linear stability analysis agree very well with the “full” CFD simulations, for the above aspect ratio.     

Onset of Marangoni instability of a two-component evaporating droplet

The temperature and solute concentration reductions across a thin boundary layer near the free surface of an evaporating droplet may induce cellular flow motion in the droplet because of Marangoni instability. The present study is aimed at investigating theoretically the onset of Marangoni instability due to the evaporation of a two-component evaporating droplet.

With the quasi-steady approximation which means that the surrounding gas motion is asymptotically steady, the size change of the droplet is negligible, and the temperature and concentration distributions of the droplet are temporarily frozen at each specified instant of interest, the onset condition for Marangoni instability is obtained through the linear stability analysis.

By assuming the surface tension is a monotonically decreasing function of both temperature and concentration of the higher-volatility substance, the thermocapillary and diffuso-capillary effects augment each other. Therefore, the theoretical analysis predicts a linear relation, with a negative slope, between the onset thermal Marangoni number, Ma_T, and the onset solute Marangoni number, Ma_S. Moreover, when liquid Lewis number Le_l>1, the critical wave number, l_c, may possess different values depending on the variation of the thermocapillary effect and diffuso-capillary effect. In addition, Le_l has a stronger effect on the critical solute Marangoni number Ma_S,C, than on the critical thermal Marangoni number Ma_T,C. That is, as Le_l decreases, Ma_T,C decreases mildly while Ma_S,C increases drastically.     

Thermoconvective instability in a horizontal porous cavity saturated with cold water

The onset of convection in a horizontal porous cavity with regard to the density maximum of water at 3.98°C is studied using a linear stability analysis. In the formulation of the problem use is made of the Brinkman-extended Darcy model which is relevant to sparsely packed porous media. A parabolic density-temperature relationship is used to model the effect of density inversion. The perturbation equations are solved with the aid of the Galerkin and finite element methods. The onset of motion is found to be dependent of the aspect ratio A of the cavity, the Darcy number Da, the inversion parameter γ and the hydrodynamic boundary conditions applied on the horizontal walls of the porous layer. The results for a viscous fluid (Da→∞) and the Darcy porous medium (Da→0) emerge from the present analysis as limiting cases. Numerical results for finite amplitude convection, obtained by solving the full governing equations, indicate that subcritical convection is possible when the upper stable layer extends over more than the half depth. Also, the existence of multiple solutions for a given range of governing parameters is demonstrated.     

The effect of stable thermal stratification on the onset of double-diffusive convection in electrochemical systems

The onset time of double-diffusive convection in time-dependent, nonlinear concentration fields was investigated theoretically and experimentally. The stability analysis was conducted on the basis of the propagation theory. Under the linear stability theory, the linearized perturbation equations were transformed similarly by using the concentration penetration depth as a length-scaling factor. The newly derived stability equations were solved numerically. Also the onset time was determined experimentally by employing an electrochemical technique where 0.03–0.3 M CuSO₄ + 1.5 M H₂SO₄ aqueous solutions were adopted as electrolyte. The onset time of double-diffusive convection was delayed or shortened depending on the degree of stable stratification.     

Stability analysis of double-diffusive convection in superposed fluid and porous layers using a one-equation model

Linear stability analysis has been carried out to predict the onset of double-diffusive convection in superposed fluid and porous layers using a one-equation model. The eigenvalue problem is solved numerically by a finite difference scheme. Results have been obtained for the thermal convection and salt-finger cases. Comparing with the results obtained for the same problems by Chen and Chen [F. Chen, C.F. Chen, J. Heat Transfer 110 (1988) 403–409] using a two-equation model, we find that these two methods give the same general characteristics of the marginal stability curves, however, there are differences in the critical conditions and the flow streamlines at onset. Carefully conducted experiments are needed to determine which model gives the more realistic results.     

Thermocapillary instability in a horizontal liquid layer subjected to a transverse magnetic field and irradiation

With the presence of a transverse magnetic field and external incident radiation, thermocapillary instability in a horizontal liquid layer is investigated using linear stability theory. Assuming that the neutral state is stationary, the critical conditions leading to the onset of convective motion are determined numerically. This article examines the effects of the magnetic field, non-uniform volumetric energy source and radiative surface properties on convective instability. The dependence of the stability characteristics on the optical thickness, surface reflectivity, the magnetic field intensity, and the external radiative sources is analyzed in detail.     

Thermal stability of a nonuniformly heat generating annular fluid layer

The thermal stability of two dimensional parallel convective motion in a fluid confined between two vertical cylinders is studied. The convective motion is induced by nonuniformly distributed heat sources in the fluid layer. A spectral collocation method is employed to solve the axisymmetric perturbed equations arising in the linear stability analysis. The stability boundary depends on the source distribution parameter, source strength parameter, Prandtl number and radius ratio. In general the flow remains least stable when the source distribution parameter is maximum in the middle of the two bounding cylinders. Thermal buoyancy mode of instability is introduced by the source distribution parameter for lower values of the radius ratio.     

The onset of natural convection in a horizontal nanofluid layer heated from below

A linear stability analysis is performed for the onset of natural convection in a horizontal nanofluid layer heated from below. The motion of nanoparticles is characterized by both the thermophoresis and Brownian diffusion effects. Different from previous studies in the literature, both the dependences of thermophoresis on nanoparticle volume fraction and Brownian motion on temperature are taken into consideration in the theoretical model. The result reveals that the base flow is mainly dominated by the effect of thermophoresis and the Brownian diffusion coefficient can be treated as a constant reasonably when a finite temperature difference is imposed across the nanofluid layer. Accordingly, a novel base solution of nanoparticle volume fraction is derived. It is found that the profile of nanoparticle concentration depends heavily on the magnitude of thermophoretic diffusion, which may exhibit a nonlinear distribution across the nanofluid layer once the effect of thermophoresis is significant. The suspended nanoparticles produce a strong destabilizing effect and a tiny volume fraction of nanoparticles is sufficient to trigger the onset of convection and make the nanofluid layer become unconditionally unstable. The dispersion spectra of unstable modes are demonstrated and the most unstable mode with the maximum growth rate is explored. The growth rate of the most unstable mode is found to increase significantly with increasing nanoparticle concentration, while the influence of heat capacity ratio of nanoparticle to base fluid on the behavior of thermal convection is negligible.     

On the onset of thermal convection in rotating nanofluid layer saturating a Darcy–Brinkman porous medium

In this paper, the effect of rotation on the onset of thermal convection in a horizontal layer of nanofluid saturated by a Darcy–Brinkman porous medium is considered. A linear stability analysis based upon normal mode is used to find solution of the fluid layer confined between two free boundaries. The onset criterion for stationary and oscillatory convection is derived analytically and graphically. The effects of the concentration Rayleigh number, Taylor number, Lewis number, Darcy number and modified diffusivity ratio on the stability of the system are investigated. The sufficient conditions for the non-existence of overstability are also derived.     

Flap/lead–lag aeroelastic stability of wind turbine blades

A numerical tool for investigating the aeroelastic stability of a single wind turbine blade subjected to combined flap/lead–lag motion is presented. Its development is motivated by recent concern about destructive edgewise vibrations of modern stall‐controlled blades. The stability tool employs a finite element formulation to discretize in space the structural and aerodynamic governing equations. Unsteady aerodynamics is considered by means of the extended ONERA lift and drag models. The mathematical form of these models allows for a combined treatment of dynamics and aerodynamics through the introduction of a so‐called ‘aeroelastic beam element’. This is an extended two‐node beam element having both deformation and aerodynamic degrees of freedom. Several linear and non‐linear versions of the stability tool are available, differing in the way that instantaneous lift and/or drag is treated. In the linear case, stability is investigated through eigenvalue analysis. Time domain integration is employed for non‐linear stability analysis. Results are presented and discussed for a 17 m stall‐controlled blade. Copyright © 2001 John Wiley & Sons, Ltd.     

Stability Investigation on the Thin Films in Capillary

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Effect of anisotropic permeability on double‐diffusive bidisperse porous medium

The model of thermosolutal convection in a fluid‐saturated bidisperse porous medium of Darcy type is studied in this paper. The permeability is allowed to be horizontally isotropic for both the macro‐ and microphases. The linear instability and nonlinear stability are analyzed by taking the Soret effect into account. Furthermore, the effect of anisotropy parameter, Soret coefficient, and other physical parameters on the stability of the system are investigated. It is shown that the linear instability boundaries and the energy stability boundaries do not coincide when the layer is heated and salted from below, where a region of potential subcritical instability occurs. The results reveal that the horizontal to vertical permeability ratio plays a crucial role in the stability of the system. It is also observed that for large values of the salt Rayleigh number, the onset of thermal convection is more likely to be via oscillatory convection rather than stationary convection. Furthermore, the onset of stationary convection is significantly influenced by the presence of the Soret coefficient.     

Effects of heat source distribution on natural convection induced by internal heating

The effects of a heat source distribution on natural convection induced by internal heating are studied by using simplified models of the distribution. A linear stability analysis is made to study the effects on critical Rayleigh number and critical wavenumber. The total amount of heat generation to set convection and the asymmetry in the convective motion are discussed for two extreme cases of heat source distribution. Effect of additional bottom wall heating is also investigated on the critical condition and the asymmetry of the convective motion.     

A linear and non-linear analysis on interfacial instability of gas-liquid two-phase flow through a circular pipe

A linear instability analysis was conducted firstly on the interface of a stratified gas-liquid two-phase flow in a circular piper employing a two-fluid model. The constitutive equations simulation technique was discussed, and the dispersive equation of interfacial waves was derived. The effects of flow rates of gas and liquid, liquid viscosity, surface tension and tube inclination on the stability of interface were investigated. A set of non-linear hyperbolic governing equations was deduced from the complete two-fluid model equation by omitting the effect of the surface tension and assuming a quasi-steady-state for the gas phase. Using characteristic line and finite difference, the propagation and growth of the interfacial disturbances were investigated in terms of gas and liquid superficial velocities. Then the results of the non-linear stability analysis were compared with those obtained by the linear stability analysis and experimental data. The non-linear stability analysis not only confirms the conclusions reached by the linear instability analysis, but also gives an insight into the growth and propagation of the interfacial disturbances on the interface of a gas-liquid two-phase flow.