Linear Instability Thresholds Research Articles

On viscous stratified Darcy–Forchheimer flow in a horizontal porous layer with thermal anisotropy and variable permeability

This study analyzes the effect of anisotropy and the internal heat source in a Darcy–Forchheimer porous layer. It is well known that the variations in viscosity can be attributed to the temperature. Therefore, in the present problem, we consider a linear variation in viscosity with temperature for simplicity. We first derived the linear instability theory and then established global stability using the energy functional approach. In the global stability analysis, we show that working with the L2 norm fails to give a sufficient condition for global stability by exhibiting that the associated maximization problem is unbounded in the underlying stability measure space. Then, we show that a conditional stability bound can be achieved by restricting the internal heat source parameter Q with higher-order norms. The eigenvalue problems obtained in linear and nonlinear theories were integrated numerically. The linear and nonlinear instability thresholds are then compared to identify the potential regions of sub-critical instabilities. It is observed that the system is stabilized when the horizontal component of thermal diffusivity dominates and is unstable when the vertical component of thermal diffusivity dominates. We also found that increasing the variable permeability parameter λ destabilized the system. It is observed that increasing viscosity stabilizes the system, and decreasing viscosity encourages the start of convection. It is also interesting that, in the presence of an internal heat source, the region of subcritical instability increases with increasing viscosity effect but reduces with increasing vertical permeability λ.

Double diffusive convection in bidispersive porous media with couple stresses effect and relatively large macropores

In this study, we have investigated the problem of double-diffusive convection in a porous medium that exhibits bidispersive characteristics and is influenced by couple stresses. The model employed considers the Darcy theory in the micropores, while utilizing the Brinkman theory in the macropores. The system encompasses two competing effects: a temperature gradient that induces instability and a salt gradient that enhances system stability. The study involved conducting analyses of linear instability and nonlinear stability. The eigenvalue systems obtained from both the linear and nonlinear theories were solved using the Chebyshev collocation method. To ensure the accuracy of this numerical method, the results were cross-validated by comparing them with the analytical solution of the eigenvalue system derived from the linear instability theory. Notably, the Chebyshev collocation method exhibited high accuracy even in the presence of high-order derivatives arising from the couple stresses term. Finally, numerical computations were performed to determine and extensively discuss the linear instability thresholds and global nonlinear thresholds. The findings contribute valuable insights into the behavior of the system under investigation. The results show that the stability thresholds of bidisperse flow in porous media are lower than those of single-phase flow. Moreover, the results confirm the stabilizing effect of the Brinkman parameter, couple stresses parameter, and permeability ratio parameter. It has also noted the stabilizing effect of the concentration Rayleigh number if the layer is salted below and the destabilizing effect if the layer is salted above.

Sharp Instability Estimates for Bidisperse Convection with Local Thermal Non-equilibrium

We analyse a theory for thermal convection in a Darcy porous material where the skeletal structure is one with macropores, but also cracks or fissures, giving rise to a series of micropores. This is thus thermal convection in a bidisperse, or double porosity, porous body. The theory allows for non-equilibrium thermal conditions in that the temperature of the solid skeleton is allowed to be different from that of the fluid in the macro- or micropores. The model does, however, allow for independent velocities and pressures of the fluid in the macro- and micropores. The threshold for linear instability is shown to be the same as that for global nonlinear stability. This is a key result because it shows that one may employ linearized theory to ensure that the key physics of the thermal convection problem has been captured. It is important to realize that this has not been shown for other theories of bidisperse media where the temperatures in the macro- and micropores may be different. An analytical expression is obtained for the critical Rayleigh number and numerical results are presented employing realistic parameters for the physical values which arise.Article HighlightsA two-temperature regime for a bidisperse Darcy porous medium is proposed to study the thermal convection problem.The optimal result of coincidence between the linear instability and nonlinear stability critical thresholds is proven.Numerical analysis enhances that the scaled heat transfer coefficient between the fluid and solid and the porosity-weighted conductivity ratio stabilize the problem significantly.

Natural convection in a fluid saturating an anisotropic porous medium in LTNE: effect of depth-dependent viscosity

Thermal convection in a fluid saturating an anisotropic porous medium in local thermal nonequilibrium (LTNE) is investigated, with specific attention to the effect of variable viscosity on the onset of convection. Many fluids show a remarkable dependence of viscosity on temperature that cannot be neglected. For this reason, we take into account a fluid whose viscosity decreases exponentially with depth, according to Straughan (Acta Mech. 61:59–72, 1986), Torrance and Turcotte (J. Fluid Mech. 47(1):113–125, 1971). The novelty of this paper is to highlight how variable viscosity coupled with the LTNE assumption affects the onset of convection. A numerical procedure shows the destabilising effect of depth-dependent viscosity. Moreover, it comes out that the LTNE hypothesis makes the influence of viscosity more intense. Linear instability analysis of the conduction solution is carried out by means of the Chebyshev-tau method coupled to the QZ algorithm, which provides the critical Rayleigh number for the onset of convection in a straightforward way. The energy method is employed in order to study the nonlinear stability. The optimal result of coincidence between the linear instability threshold and the global nonlinear stability threshold is obtained. The influence of anisotropic permeability and conductivity, weighted conductivity ratio, and interaction coefficient on the onset of convection is highlighted.

Falling film on an anisotropic porous medium

The stability and dynamics of a falling liquid film over an anisotropic porous medium are studied using a one-domain approach. Our stability analysis shows a significant departure from the effective no-slip boundary condition in the isotropic case. Anisotropy does not affect the threshold of linear instability. However, a non-trivial dual effect of anisotropy on the film stability is observed depending on the permeability of the porous medium. This dual effect results from the net balance of the enhancement of viscous diffusion at the top Brinkman sublayer and the mitigation of viscous damping in the core Darcy sublayer. Three-equation models have been derived from the lubrication theory approximation in terms of the exact mass balance and averaged momentum balances in the porous and liquid layers. In the nonlinear regime, anisotropy has a dual effect by damping capillary waves at large permeabilities and enhancing them at low permeabilities. Anisotropy also affects wave speeds and shapes, modifies travelling-wave branches of solutions, affects the development of a time-periodic wavetrain by inlet forcing and alters the noise-driven dynamics of the flow. These effects result from the mitigation of mass exchange at the liquid–porous interface and the contribution of the cross-stream permeability in the Brinkman top sublayer to the viscous diffusion.

Predicting the Z-pinch Dimits shift through gyrokinetic tertiary instability analysis of the entropy mode

The Dimits shift, an upshift in the onset of turbulence from the linear instability threshold, caused by self-generated zonal flows, can greatly enhance the performance of magnetic confinement plasma devices. Except in simple cases, using fluid approximations and model magnetic geometries, this phenomenon has proved difficult to understand and quantitatively predict. To bridge the large gap in complexity between simple models and realistic treatment in toroidal magnetic geometries (e.g. tokamaks or stellarators), the present work uses fully gyrokinetic simulations in a Z-pinch geometry to investigate the Dimits shift through the lens of tertiary instability analysis, which describes the emergence of drift waves from a zonally dominated state. Several features of the tertiary instability, previously observed in fluid models, are confirmed to remain. Most significantly, an efficient reduced-mode tertiary model, which previously proved successful in predicting the Dimits shift in a gyrofluid limit (Hallenbert & Plunk, J. Plasma Phys., vol. 87, issue 05, 2021, 905870508), is found to be accurate here, with only slight modifications to account for kinetic effects.

Effects of magnetic island on profile formation in flux-driven ITG turbulence

Full-f gyrokinetic simulations of ion temperature gradient (ITG) turbulence in the presence of a magnetic island are performed. A newly developed method for evaluating the flux-surface average is implemented to treat adiabatic electrons inside the magnetic island precisely. A neoclassical simulation below the threshold for linear ITG instability shows that the density profile does not relax at the O-point, although the ion temperature profile is flattened there. This results from the force balance in the direction of the magnetic field between the pressure gradient related to ion parallel motion and the mean radial electric field. A flux-driven ITG turbulence simulation shows a quasi-periodic transport reduction due to interaction between the background temperature profile and the vortex mode, which is a nonlinearly generated mesoscale structure with the same mode numbers as the magnetic island. These results indicate that not only the parallel streaming but also the equilibrium electric field and turbulence contribute significantly to profile formation around a magnetic island.

Linear and Non-Linear Stability Analysis for Thermal Convection in A Bidispersive Porous Medium with Thermal Non-Equilibrium Effects

The linear instability and nonlinear stability analyses are performed for the model of bidispersive local thermal non-equilibrium flow. The effect of local thermal non-equilibrium on the onset of convection in a bidispersive porous medium of Darcy type is investigated. The temperatures in the macropores and micropores are allowed to be different. The effects of various interaction parameters on the stability of the system are discussed. In particular, the effects of the porosity modified conductivity ratio parameters, and , with the inter-phase momentum transfer parameters and, on the onset of thermal Convection are also considered. Furthermore, the nonlinear stability boundary is found to be below the linear instability threshold. The numerical results are presented for free-free boundary conditions.

Thermal convection with generalized friction

A model for thermal convection with generalized friction is investigated. It is shown that the linear instability threshold is the same as the global stability one. In addition, decay of the energy in the L^2 norm is shown for the perturbation velocity and temperature fields. However, due to the presence of the generalized friction we establish exponential decay in the L^{beta +1} norm for the perturbation temperature, where beta >1.

Saturation physics of threshold heat-flux reduction

The saturation physics of ion-temperature-gradient-driven turbulence is examined in relation to the temperature-gradient variation of the heat flux, which can exhibit an upshift of the critical gradient for significant flux relative to the linear instability threshold. Gyrokinetic measurements of saturation properties and spectral energy transfer, which will be defined in Sec. II, are presented, indicating that the physics of saturation is fundamentally unchanged on either side of the upshifted gradient. To analyze heat transport below and above the upshifted critical gradient, a fluid model for toroidal ion-temperature-gradient turbulence is modified to include the kinetic instability threshold. The model and the heat flux are rendered in the eigenmode decomposition to track the dominant mode-coupling channel of zonal-flow-catalyzed transfer to a conjugate stable mode. Given linear and nonlinear symmetries, the stable mode level and the cross-correlation of the unstable and stable mode amplitudes are related to the unstable mode level via linear physics. The heat flux can then be written in terms of the unstable-mode level, which through a nonlinear balance depends on the eigenmode-dependent coupling coefficients and the triplet correlation time of the dominant coupled modes. Resonance in these quantities leads to suppressed heat flux above the linear threshold, with a nonlinear upshift of the critical gradient set by the resonance broadening of a finite perpendicular wavenumber and collisionality.

Instability of Vertical Throughflows in Bidisperse Porous Media

In this paper, the instability of a vertical fluid motion, or throughflow, is investigated in a horizontal bidisperse porous layer that is uniformly heated from below. By means of the order-1 Galerkin approximation method, the critical Darcy–Rayleigh number for the onset of steady instability is determined in closed form. The coincidence between the linear instability threshold and the global nonlinear stability threshold, in the energy norm, is shown.

Stability of double-diffusive natural convection in a vertical fluid layer

The stability of basic buoyant flow in a vertical fluid layer induced by temperature and solute concentration differences between the vertical boundaries is investigated. The linear dynamics of the perturbed flow is formulated as an eigenvalue problem and solved numerically by employing the Chebyshev collocation method. The validity of Squire's theorem is proved, and therefore, two-dimensional motions are considered. The neutral stability curves defining the threshold of linear instability and the critical values of the thermal Grashof number and wave number at the onset of instability are determined for various values of the Prandtl number Pr, the solute Grashof number GS, and the Lewis number Le. The magnitude of the Prandtl number at which the transition from stationary to travelling-wave mode occurs can be either increased or decreased by tuning the values of GS and Le. For certain combinations of the parameters, there exist one or two closed disconnected travelling-wave neutral curves emphasizing the necessity of multiple thermal Grashof numbers to embark upon the stability of fluid flow, a result of contrast to that of the single-diffusive fluid layer. The mechanism of modal instability is deciphered by using the method of energy budget and four different modes of instability are identified, one of which is new and due entirely to the presence of solutal buoyancy.

The onset of thermal convection in anisotropic and rotating bidisperse porous media

The onset of thermal convection in anisotropic rotating bidisperse porous media is investigated. The optimal result concerning the coincidence between linear instability and nonlinear stability thresholds with respect the L^2-norm is obtained.

Effect of anisotropy on the onset of convection in rotating bi-disperse Brinkman porous media

Thermal convection in a horizontally isotropic bi-disperse porous medium (BDPM) uniformly heated from below is analysed. The combined effects of uniform vertical rotation and Brinkman law on the stability of the steady state of the momentum equations in a BDPM are investigated. Linear and nonlinear stability analysis of the conduction solution is performed, and the coincidence between linear instability and nonlinear stability thresholds in the L^2-norm is obtained.

On the Stability of Convection in a Non-Newtonian Vertical Fluid Layer in the Presence of Gold Nanoparticles: Drug Agent for Thermotherapy

We consider the effect of gold nanoparticles on the stability properties of convection in a vertical fluid layer saturated by a Jeffreys fluid. The vertical boundaries are rigid and hold at uniform but different temperatures. Brownian diffusion and thermophoresis effects are considered. Due to numerous applications in the biomedical industry, such a study is essential. The linear stability is investigated through the normal mode disturbances. The resulting stability problem is an eighth-order ordinary differential complex eigenvalue problem that is solved numerically using the Chebyshev collection method. Its solution provides the neutral stability curves, defining the threshold of linear instability, and the critical parameters at the onset of instability are determined for various values of control parameters. The results for Newtonian fluid and second-grade fluid are delineated as particular cases from the present study. It is shown that the Newtonian fluid has a more stabilizing effect than the second-grade and the Jeffreys fluids in the presence of gold nanoparticles and, Jeffreys fluid is the least stable.

Bidispersive double diffusive convection with relatively large macropores and generalized boundary conditions

This paper is concerned with the question of the beginning of convective motion in a fluid saturated porous layer, containing a salt in solution, heated below and salted above and below. This model has a single temperature and employs the Darcy theory in the micropores, the Brinkman theory, however, being utilized in the macropores. The effect of slip boundary conditions on the stability of the model is also studied. General boundary conditions regarding temperature and salt are also taken into account. It will be shown that the linear instability threshold is the same as that of nonlinear stability if the layer is salted from above, indicating that the linear theory entirely captures the physics of the onset of thermal convection. In the case of salting from below, the behavior of the transition from stationary to oscillatory convection is investigated in detail, as the boundary conditions change from prescribed temperature and salt concentration toward those of prescribed heat flux and salt flux. The nonlinear stability threshold does not coincide with that of linear instability; thus, regions of possible subcritical instability are still present. We believe that the problem presented in this paper has not been addressed before and that its study will have great scientific value and impact.

Bidispersive thermal convection with relatively large macropores and generalized velocity and temperature boundary conditions

In a fluid-saturated bidisperse porous medium at a single temperature, the issue of thermal convection when the Darcy theory is used in the micropores, particularly the effects of slip boundary conditions on the model stability, was examined (whereas the Brinkman theory is used in the macropores). In addition, the effect of general temperature boundary conditions was also addressed. The governing equations of motion are provided, followed by the way in which the related equations of perturbation were derived. In addition, the linear instability and nonlinear stability analyses of the system were introduced, with the numerical approach used to approximate the eigenvalue system resulting from our analysis. The threshold for linear instability was proved to be the same as the one for nonlinear stability, showing that the linear theory accurately captures the mechanism of the onset of thermal convection. The numerical results for stability/instability thresholds were also introduced. The researchers assumed that this was the first time a mixed Darcy–Brinkman model had been used in bidisperse flow with slip boundary conditions.

Thermal convection in rotating anisotropic bidispersive porous layers

Linear and nonlinear stability analyses for thermal convection in a rotating bidispersive porous medium, horizontally isotropic, is performed. Since the coincidence between linear instability and non-linear stability thresholds has been proved, the linear theory captures completely the physics of the problem at stake.

Intermittent fluctuations due to Lorentzian pulses in turbulent thermal convection

Turbulent motions due to flux-driven thermal convection are investigated by numerical simulations and stochastic modeling. Tilting of convection cells leads to the formation of sheared flows and quasi-periodic relaxation oscillations for the energy integrals far from the threshold for linear instability. The probability density function for the temperature and radial velocity fluctuations in the fluid layer changes from a normal distribution at the onset of turbulence to a distribution with an exponential tail for large fluctuation amplitudes for strongly driven systems. The frequency power spectral density has an exponential shape, which is a signature of deterministic chaos. By use of a novel deconvolution method, this is shown to result from the presence of Lorentzian pulses in the underlying time series, demonstrating that exponential frequency spectra can also persist in turbulent flow regimes.

Flow regimes of Rayleigh–Bénard convection in a vertical magnetic field

The effects of a vertical static magnetic field on the flow structure and global transport properties of momentum and heat in liquid metal Rayleigh–Benard convection are investigated. Experiments are conducted in a cylindrical convection cell of unity aspect ratio, filled with the alloy GaInSn at a low Prandtl number of . Changes of the large-scale velocity structure with increasing magnetic field strength are probed systematically using multiple ultrasound Doppler velocimetry sensors and thermocouples for a parameter range that is spanned by Rayleigh numbers of and Hartmann numbers of . Our simultaneous multi-probe temperature and velocity measurements demonstrate how the large-scale circulation is affected by an increasing magnetic field strength (or Hartmann number). Lorentz forces induced in the liquid metal first suppress the oscillations of the large-scale circulation at low , then transform the one-roll structure into a cellular large-scale pattern consisting of multiple up- and downwellings for intermediate , before finally expelling any fluid motion out of the bulk at the highest accessible leaving only a near-wall convective flow that persists even below Chandrasekhar’s linear instability threshold. Our study thus proves experimentally the existence of wall modes in confined magnetoconvection. The magnitude of the transferred heat remains nearly unaffected by the steady decrease of the fluid momentum over a large range of Hartmann numbers. We extend the experimental global transport analysis to momentum transfer and include the dependence of the Reynolds number on the Hartmann number.