Linear Stability Threshold Research Articles

Development of a novel nodalized reduced order model for stability analysis of supercritical fluid in a heated channel

A Novel Nodalized Reduced Order Model (NNROM) is developed in this paper to analyze the linear stability phenomena in a heated channel with supercritical water as a coolant. The existing models are based on finite volume approach, leading to a large number of non-linear time-dependent ODEs, making linear stability analysis (for infinitesimally perturbation) computationally expensive and tedious. Moreover, the non-linear stability analysis considers the effect of small but finite perturbations which becomes even more difficult. It is pointed out that the accuracy of the reduced order model developed here is not compromised, as the comparisons of the model results, with existing studies show good agreement. In ordered to develop the NNROM, the heated channel is divided into N number of nodes. The one-dimensional mass, energy and momentum conservation partial differential equations are converted into the corresponding time-dependent non-linear ordinary differential equations (ODEs) by applying the weighted residual method. The linear stability threshold of the system is determined by analyzing the eigenvalues of the Jacobian matrix at the steady states of the set of ODEs. Moreover, the linear stability boundary (Hopf bifurcation line) is represented in terms of trans-pseudo-critical phase change number(Ntpc), and pseudo-subcooling number(Nspc). A parametric study is done to identify the change in linear stability behavior of the system with the design parameters. Furthermore, non-linear stability analysis is carried out to identify Generalized Hopf (GH) bifurcation points in the Ntpc−Nspc space. The GH points divide the stability boundary into sub-critical Hopf and super-critical Hopf parts, which is further varify by the numerical simulations. The identification of sub-critical region is quite important as it shows unstable limit cycles in the (linearly) stable region.

Suppression of the Kapitza instability in confined falling liquid films

We revisit the linear stability of a falling liquid film flowing through an inclined narrow channel in interaction with a gas phase. We focus on a particular region of parameter space, small inclination and very strong confinement, where we have found the gas to strongly stabilize the film, up to the point of fully suppressing the long-wave interfacial instability attributed to Kapitza (Zh. Eksp. Teor. Fiz., vol. 18 (1), 1948, pp. 3–28). The stabilization occurs both when the gas is merely subject to an aerostatic pressure difference, i.e. when the pressure difference balances the weight of the gas column, and when it flows counter-currently. In the latter case, the degree of stabilization increases with the gas velocity. Our investigation is based on a numerical solution of the Orr–Sommerfeld temporal linear stability problem as well as stability experiments that clearly confirm the observed effect. We quantify the degree of stabilization by comparing the linear stability threshold with its passive-gas limit, and perform a parametric study, varying the relative confinement, the Reynolds number, the inclination angle and the Kapitza number. For example, we find a 30 % reduction of the cutoff wavenumber of the instability for a water film in contact with air, flowing through a channel inclined at$3^{\circ }$and of height 2.8 times the film thickness. We also identify the critical conditions for the full suppression of the instability in terms of the governing parameters. The stabilization is caused by the strong confinement of the gas, which produces perturbations of the adverse interfacial tangential shear stress that are shifted by half a wavelength with respect to the wavy film surface. This tends to reduce flow-rate variations within the film, thus attenuating the inertia-based driving mechanism of the Kapitza instability.

Double-diffusive Soret convection phenomenon in porous media: effect of Vadasz inertia term

The onset of double-diffusive convection in horizontal porous layers for the thermo-diffusive Soret phenomenon in the case of a generalized Darcy model including inertia term is investigated. Via a linearization principle recently introduced in Rionero (Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur 28:21–47, 2017) the coincidence between linear and nonlinear (global) stability thresholds of the thermo-solutal conduction solution, is proved. Necessary and sufficient conditions guaranteeing the onset of steady or oscillatory convection in a closed algebraic form are obtained.

Tridispersive thermal convection

We derive the linear instability and nonlinear stability thresholds for a problem of thermal convection in a tridispersive porous medium with a single temperature. Importantly we demonstrate that the nonlinear stability threshold is the same as the linear instability one. The significance of this is that the linear theory is capturing completely the physics of the onset of thermal convection. This result is different to the general theory of thermal convection in a tridispersive porous material where the temperatures in the macropores, mesopores and micropores are allowed to be different. In that case the coincidence of the nonlinear stability and linear instability boundaries has not been proved.

Quasi-two-dimensional nonlinear evolution of helical magnetorotational instability in a magnetized Taylor–Couette flow

Magnetorotational instability (MRI) is one of the fundamental processes in astrophysics, driving angular momentum transport and mass accretion in a wide variety of cosmic objects. Despite much theoretical/numerical and experimental efforts over the last decades, its saturation mechanism and amplitude, which sets the angular momentum transport rate, remains not well understood, especially in the limit of high resistivity, or small magnetic Prandtl numbers typical to interiors (dead zones) of protoplanetary disks, liquid cores of planets and liquid metals in laboratory. Using direct numerical simulations, in this paper we investigate the nonlinear development and saturation properties of the helical magnetorotational instability (HMRI)—a relative of the standard MRI—in a magnetized Taylor–Couette flow at very low magnetic Prandtl number (correspondingly at low magnetic Reynolds number) relevant to liquid metals. For simplicity, the ratio of azimuthal field to axial field is kept fixed. From the linear theory of HMRI, it is known that the Elsasser number, or interaction parameter determines its growth rate and plays a special role in the dynamics. We show that this parameter is also important in the nonlinear problem. By increasing its value, a sudden transition from weakly nonlinear, where the system is slightly above the linear stability threshold, to strongly nonlinear, or turbulent regime occurs. We calculate the azimuthal and axial energy spectra corresponding to these two regimes and show that they differ qualitatively. Remarkably, the nonlinear state remains in all cases nearly axisymmetric suggesting that this HMRI-driven turbulence is quasi two-dimensional in nature. Although the contribution of non-axisymmetric modes increases moderately with the Elsasser number, their total energy remains much smaller than that of the axisymmetric ones.

Linear and Nonlinear Stability of Double-Diffusive Convection with the Soret Effect

The onset of double-diffusive convection in a horizontal fluid-saturated porous layer is examined by taking the Soret effect into consideration. The linear and nonlinear stability analyses are derived, and the corresponding eigenvalue problems are solved. The nonlinear stability analysis is achieved by using the energy method. In both the cases of linear and nonlinear stability theories, the onset criterion for all possible modes is derived analytically. For numerical computations of the eigenvalue problem, the Chebyshev tau method is employed. It is observed that the effect of stabilization or destabilization caused by the Soret parameter is significant for the Soret parameters which are less than \(Sr = 2\). In the absence of the Soret effect, the linear and nonlinear stability thresholds coincide.

Torque in Taylor–Couette flow of viscoelastic polymer solutions

When a polymer solution is sheared between concentric cylinders, with the inner rotating and the outer fixed, the torque on the inner cylinder is modified compared to a Newtonian fluid of the same viscosity revealing the different flow patterns that emerge above the linear stability threshold for circular Couette flow. Here, mixtures of relatively short and long linear polymers in dilute and semi-dilute concentrations were considered. Their shear viscosity and extensional relaxation time are quantified. The stability of the flow is monitored through torque measurements and flow visualisations for a constant rate of acceleration and deceleration of the inner cylinder in a wide range of polymer concentrations. The torque exhibits an hysteretic behaviour, typical of subcritical transition. For large concentrations, six different numbers of steady solitary pairs of vortices, called diwhirls, were observed depending on the deceleration rate and their torque contribution is reported.

Bidispersive thermal convection

We obtain the linear instability and nonlinear stability thresholds for a problem of thermal convection in a bidispersive porous medium with a single temperature. It is important to note that we show that the linear instability threshold is the same as the nonlinear stability one. This means that the linear theory is capturing completely the physics of the onset of thermal convection. This result contrasts with the general theory of thermal convection in a bidispersive porous material where the temperatures in the macropores and micropores are allowed be different. In that case the coincidence of the stability boundaries has not been proved.

Rayleigh–Bénard stability and the validity of quasi-Boussinesq or quasi-anelastic liquid approximations

The linear stability threshold of the Rayleigh–Bénard configuration is analysed with compressible effects taken into account. It is assumed that the fluid under investigation obeys a Newtonian rheology and Fourier’s law of thermal transport with constant, uniform (dynamic) viscosity and thermal conductivity in a uniform gravity field. Top and bottom boundaries are maintained at different constant temperatures and we consider here mechanical boundary conditions of zero tangential stress and impermeable walls. Under these conditions, and with the Boussinesq approximation, Rayleigh (Phil. Mag., vol. 32 (192), 1916, pp. 529–546) first obtained analytically the critical value $27\unicode[STIX]{x03C0}^{4}/4$ for a dimensionless parameter, now known as the Rayleigh number, at the onset of convection. This paper describes the changes of the critical Rayleigh number due to the compressibility of the fluid, measured by the dimensionless dissipation parameter ${\mathcal{D}}$ and due to a finite temperature difference between the hot and cold boundaries, measured by a dimensionless temperature gradient $a$. Different equations of state are examined: ideal gas equation, Murnaghan’s model (often used to describe the interiors of solid but convective planets) and a generic equation of state with adjustable parameters, which can represent any possible equation of state. In the perspective to assess approximations often made in convective models, we also consider two variations of this stability analysis. In a so-called quasi-Boussinesq model, we consider that density perturbations are solely due to temperature perturbations. In a so-called quasi-anelastic liquid approximation model, we consider that entropy perturbations are solely due to temperature perturbations. In addition to the numerical Chebyshev-based stability analysis, an analytical approximation is obtained when temperature fluctuations are written as a combination of only two modes, one being the original symmetrical (between top and bottom) mode introduced by Rayleigh, the other one being antisymmetrical. The analytical solution allows us to show that the antisymmetrical part of the critical eigenmode increases linearly with the parameters $a$ and ${\mathcal{D}}$, while the superadiabatic critical Rayleigh number departs quadratically in $a$ and ${\mathcal{D}}$ from $27\unicode[STIX]{x03C0}^{4}/4$. For any arbitrary equation of state, the coefficients of the quadratic departure are determined analytically from the coefficients of the expansion of density up to degree three in terms of pressure and temperature.

Fast-ion transport by Alfvén eigenmodes above a critical gradient threshold

Experiments on the DIII-D tokamak have identified how multiple simultaneous Alfvén eigenmodes (AEs) lead to overlapping wave-particle resonances and stochastic fast-ion transport in fusion grade plasmas [C. S. Collins et al., Phys. Rev. Lett. 116, 095001 (2016)]. The behavior results in a sudden increase in fast-ion transport at a threshold that is well above the linear stability threshold for Alfvén instability. A novel beam modulation technique [W. W. Heidbrink et al., Nucl. Fusion 56, 112011 (2016)], in conjunction with an array of fast-ion diagnostics, probes the transport by measuring the fast-ion flux in different phase-space volumes. Well above the threshold, simulations that utilize the measured mode amplitudes and structures predict a hollow fast-ion profile that resembles the profile measured by fast-ion Dα spectroscopy; the modelling also successfully reproduces the temporal response of neutral-particle signals to beam modulation. The use of different modulated sources probes the details of phase-space transport by populating different regions in phase space and by altering the amplitude of the AEs. Both effects modulate the phase-space flows.

Transition to blow-up in a reaction–diffusion model with localized spike solutions

For certain singularly perturbed two-component reaction–diffusion systems, the bifurcation diagram of steady-state spike solutions is characterized by a saddle-node behaviour in terms of some parameter in the system. For some such systems, such as the Gray–Scott model, a spike self-replication behaviour is observed as the parameter varies across the saddle-node point. We demonstrate and analyse a qualitatively new type of transition as a parameter is slowly decreased below the saddle node value, which is characterized by a finite-time blow-up of the spike solution. More specifically, we use a blend of asymptotic analysis, linear stability theory, and full numerical computations to analyse a wide variety of dynamical instabilities, and ultimately finite-time blow-up behaviour, for localized spike solutions that occur as a parameter β is slowly ramped in time below various linear stability and existence thresholds associated with steady-state spike solutions. The transition or route to an ultimate finite-time blow-up can include spike nucleation, spike annihilation, or spike amplitude oscillation, depending on the specific parameter regime. Our detailed analysis of the existence and linear stability of multi-spike patterns, through the analysis of an explicitly solvable non-local eigenvalue problem, provides a theoretical guide for predicting which transition will be realized. Finally, we analyse the blow-up profile for a shadow limit of the reaction–diffusion system. For the resulting non-local scalar parabolic problem, we derive an explicit expression for the blow-up rate near the parameter range where blow-up is predicted. This blow-up rate is confirmed with full numerical simulations of the full PDE. Moreover, we analyse the linear stability of this solution that blows up in finite time.

Erratum to: Three-Dimensional Simulations for Convection in a Porous Medium with Internal Heat Source and Variable Gravity Effects

The problem of convection in a fluid-saturated porous layer which is heated internally and where the gravitational field varies with distance through the layer is studied. The accuracy of both the linear instability and global nonlinear energy stability thresholds is tested using a three-dimensional simulation. Our results support the assertion that the linear theory is very accurate in predicting the onset of convective motion, and thus, regions of stability.

Nonhomogeneous Porosity and Thermal Diffusivity Effects on a Double-Diffusive Convection in Anisotropic Porous Media

Abstract A model for double-diffusive convection in anisotropic and inhomogeneous porous media has been analysed. In particular, the effects of variable permeability, thermal diffusivity and variable gravity with respect to the vertical direction, have been studied. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using three dimensional simulation. Our results show that the linear theory produce a good predicts on the onset of instability in the basic steady state. It is known that as R c ${R_c}$ increases the onset of convection is more likely to be via oscillatory convection as opposed to steady convection, and the three dimensional simulation results show that as R c $Rc$ increases, the actual threshold moving toward the nonlinear stability threshold and the behaviour of the perturbation of the solutions becomes more oscillated.

On the stationary and oscillatory modes of triply resonant penetrative convection

Purpose – The purpose of this paper is to explore a model for thermal convection in a plane layer when the density-temperature relation in the buoyancy term is quadratic. A heat source/sink varying in a linear fashion with a vertical height expressed as z was allowed, functioning as a heat sink in an area of the layer and as a heat source in the remainder. Design/methodology/approach – First, the authors present the governing equations of motion and derive the associated perturbation equations. Second, the authors introduce the linear and nonlinear analysis of the system. Third, the authors transform the system to velocity-vorticity-potential formulation and introduce a numerical study of the problem in three dimensions. Findings – First, the linear instability and nonlinear stability thresholds are derived. Second, the linear instability thresholds accurately predict the onset of instability. Third, the required time to arrive at the steady state increases as Ra tends to RaL . Fourth, the authors find that the convection has three different interesting patterns. Originality/value – With the modernday need for heat transfer or insulation devices in industry, particularly those connected with nanotechnology, the usefulness of a mathematical analysis of such resonance became apparent. Thus, this study is believed to be of value.

Double-diffusive Hadley–Prats flow in a porous medium subject to gravitational variation

A model for double-diffusive porous Hadley–Prats flow under the effect of a variable gravity field is explored. Both linear instability and nonlinear stability analyses are performed to assess the suitability of linear theory to predict the destabilisation of the flow. The resultant eigenvalue problems for both theories are solved using shooting and fourth order Runga–Kutta methods for various flow governing parameters. The results indicate that both the linear and nonlinear results undergo quantitative changes subject to horizontal thermal and solutal gradients along with the presence and absence of gravity and mass flow variations. For certain parameter ranges there are also substantial regions of potential subcritical instability, such that instabilities may arise before the linear stability threshold is reached.

Instabilities of Layers of Deposited Molecules on Chemically Stripe Patterned Substrates: Ridges versus Drops.

A mesoscopic continuum model is employed to analyze the transport mechanisms and structure formation during the redistribution stage of deposition experiments where organic molecules are deposited on a solid substrate with periodic stripe-like wettability patterns. Transversally invariant ridges located on the more wettable stripes are identified as very important transient states and their linear stability is analyzed accompanied by direct numerical simulations of the fully nonlinear evolution equation for two-dimensional substrates. It is found that there exist two different instability modes that lead to different nonlinear evolutions that result (i) at large ridge volume in the formation of bulges that spill from the more wettable stripes onto the less wettable bare substrate and (ii) at small ridge volume in the formation of small droplets located on the more wettable stripes. In addition, the influence of different transport mechanisms during redistribution is investigated focusing on the cases of convective transport with no-slip at the substrate, transport via diffusion in the film bulk and via diffusion at the film surface. In particular, it is shown that the transport process does neither influence the linear stability thresholds nor the sequence of morphologies observed in the time simulation, but only the ratio of the time scales of the different process phases.

Effects of Vertical Throughflow and Variable Gravity on Hadley–Prats Flow in a Porous Medium

The stability of convection in a horizontal porous layer which is saturated with fluid and induced by horizontal temperature gradients subjected to horizontal mass flow is investigated by means of linear and nonlinear stability analysis. The effects of variable gravity field and vertical throughflow are also considered in this analysis. The nonlinear stability analysis part has been developed via energy functional. Shooting and Runge–Kutta methods have been used to solve eigenvalue problem in both cases. Critical vertical thermal Rayleigh numbers for both linear and nonlinear analyses \(R_\mathrm{L}\) and \(R_\mathrm{E}\) are evaluated for different values of horizontal Rayleigh number \(R_x\), horizontal Peclet number Pe, vertical Peclet number \(Q_\mathrm{v}\) and variable gravity parameter \(\eta \). Comparison is made between linear and nonlinear stability results. It has been observed that linear stability results are overpredicting the onset of convection compared with nonlinear theory, and hence subcritical instabilities would arise before one gets the onset of linear stability threshold.

Three dimensional simulations for convection induced by the selective absorption of radiation for the Brinkman model

Convection induced by the selective absorption of radiation in a porous medium is studied analytically and numerically using the Brinkman model. Both linear instability analysis and nonlinear stability analysis are employed. Then, the validity of both the linear instability and global nonlinear energy stability thresholds are tested using three dimensional simulation. Our results show that the linear theory produce a good predicts on the onset of instability in the basic steady state.

Magnetic field and throughflow effects on double-diffusive convection in internally heated anisotropic porous media

A model for double-diffusive convection in an anisotropic porous layer with a constant throughflow is explored, with penetrative convection being simulated via an internal heat source and subjected to a vertical magnetic field and variable gravity effect. The validity of both the linear instability and global nonlinear stability thresholds are tested using three dimensional simulation. Our results show that the linear theory produce a good prediction on the onset of instability in the steady state throughflow. It is known that as Rc increases the onset of convection is more likely to be via oscillatory convection as opposed to steady convection, and the three dimensional simulation results show that as Rc increases, the actual threshold moving toward the nonlinear stability threshold and the behaviour of the perturbation of the solutions becomes more oscillated.

Chemical reaction effect on double diffusive convection in porous media with magnetic and variable gravity effects

We study the problem of double diffusive convective movement of a reacting solute in a viscous incompressible occupying a plane layer in a saturated porous medium and subjected to a vertical magnetic field. The thresholds for linear instability are found and compared to those derived by a global nonlinear energy stability analysis. Then, the accuracy of both the linear instability and global nonlinear energy stability thresholds are tested using a three dimensional simulation. The strong stabilizing effect of gravity field and magnetic field is shown. Moreover, the results support the assertion that the linear theory, in general, is accurate in predicting the onset of convective motion, and thus, regions of stability.