Stability Of Convection Research Articles

A new stabilized time-spectral finite element solver for fast simulation of blood flow

The increasing application of cardiorespiratory simulations for diagnosis and surgical planning necessitates the development of computational methods significantly faster than the current technology. To achieve this objective, we leverage the time-periodic nature of these flows by discretizing equations in the frequency domain instead of the time domain. This approach markedly reduces the size of the discrete problem and, consequently, the simulation cost. With this motivation, we introduce a finite element method for simulating time-periodic flows that are physically stable. The proposed time-spectral method is formulated by augmenting the baseline Galerkin’s method with a least-squares penalty term that is weighed by a positive-definite stabilization matrix. An error estimate is established for the convective–diffusive system, showing that the proposed method emulates the behavior of existing standard time methods including optimal convergence rate in diffusive regimes and stability in strong convection. This method is tested on a patient-specific Fontan model at nominal Reynolds and Womersley numbers of 500 and 10, respectively, demonstrating its ability to replicate conventional time simulation results using as few as 7 modes at 11% of the computational cost. Owing to its higher local-to-processor computation density, the proposed method also exhibits improved parallel scalability for fast simulation of time-critical applications.

Influence of Two-Frequency Rotational Modulation on the Dynamics of the Rayleigh–Bénard Convection in Water-Based Nanoliquids with Either AA7072 or AA7075 Nanoparticles

The effect of time-periodic two-frequency rotation modulation on Rayleigh–Bénard convection in water with either AA7072 or AA7075 nanoparticles is investigated. The single-phase description of the Khanafer–Vafai–Lightstone model is used for modeling the nanoliquids. An asymptotic expansion procedure is adopted in the case of the linear stability to obtain the correction (due to modulation) to the Rayleigh number at marginal stability of unmodulated convection. A nonlinear regime of convection is considered with a nonautonomous generalized Lorenz model as the governing equation. The method of multiscales is then employed to obtain the coupled nonautonomous Ginzburg–Landau equations with cubic nonlinearity from the Lorenz model. These equations are presented in the phase-amplitude form and the amplitude is used to quantify the heat transport. The modulation amplitude is considered to be small (of order less than unity) and moderate frequencies of modulation are considered. We found that there is a threshold frequency beyond which the system behavior reverses. At frequencies below the threshold, the mean Nusselt number increases with an increase in the amplitude of modulation while an opposite influence is seen for values above the threshold. Such a behavior is a consequence of what is analogously seen in the case of the critical Rayleigh number. The influence of two-frequency modulation is more pronounced on the results of the linear and nonlinear regimes compared to that of the single-frequency one. The heat transport is enhanced due to the presence of dilute concentration of suspended nanoparticles (either AA7072 or AA7075 nanoalloys) in water. The influence of nanoparticles is to modify the threshold values generating chaos but it does not qualitatively alter the dynamical behavior of the system. The plots of Lyapunov exponents reveal that there is no possibility of hyper-chaos in the generalized Lorenz model when there is a rotational modulation.

Impact of Local Thermal Non-Equilibrium and Gravity Fluctuations on the Onset of a Darcy-Brinkman Porous Convection

The numerical investigation of the Darcy-Brinkman convective problem with gravity fluctuations in a Local Thermal Nonequilibrium (LTNE) model is conducted. Utilizing linear stability analysis, the convective problem is explored, and the numerical values of the Rayleigh and wave numbers for onset convection are computed through the one-term Galerkin approach. Three distinct types of gravity fluctuations (linear, parabolic, and exponential) are considered. The results show the Darcy number and gravity parameter delay the onset of convection. The porosity-scald conductivity ratio and interface heat transfer coefficient have a significant effect on the stability of the configuration. Graphical representations depict the effects of various parameters, highlighting the significant impacts of incorporating gravity fluctuations and non-equilibrium conditions in determining convection stability thresholds.

Stability and instability of thermosolutal convection in a Brinkman–Darcy–Kelvin–Voigt fluid with couple stress effect

In this article, the phenomenon of thermosolutal convection within a fluid characterized by the Brinkman–Darcy–Kelvin–Voigt (BDKV) model is delved into, while the impact of couple stresses on this process is considered. Both linear instability and nonlinear stability analyses are encompassed in our investigation. Several noteworthy observations have been made. When the fluid layer is heated from below and salt is introduced from above, it is found that the points at which stability and instability thresholds are reached coincide. This alignment is supported by the validity of the linear theory in predicting the initiation of convection under these conditions. However, the scenario changes when the layer is salted from the bottom while being heated. In this case, the stability thresholds remain constant, regardless of variations in the salt Rayleigh number. This discrepancy between the thresholds of linear instability and nonlinear stability is deemed significant. To gain a deeper understanding, numerical computations were conducted to identify and thoroughly discuss the thresholds of linear instability. These findings offer valuable insights into the behavior of the system under study. It is indicated by our results that parameters such as Brinkman, couple stresses, and Kelvin–Voigt contribute to stabilizing the system. Additionally, it was noted that the salt Rayleigh number has a stabilizing effect when the layer is salted from below, whereas it has a destabilizing effect when salt is introduced from above.

Heating from above in non-scattering suspension: Phototactic bioconvection under collimated irradiation

Examining phototactic bioconvection in non-scattering suspensions with upper heating and collimated irradiation, this study delves into the intricate dynamics influenced by light and microorganisms. The study focuses on the linear stability of the basic state, examining neutral curves. The numerical analysis involves solving a system of equations using the MATLAB bvp4c solver. The investigation considers the impact of parameters, such as the thermal Rayleigh number, critical total intensity, and Lewis number, on the critical bioconvection Rayleigh number. As the critical total intensity varies, a transition from a stationary to an oscillatory solution (and vice versa) is observed. Phototactic microorganisms are incorporated into the model, and the results show how varying parameters affect convection patterns and stability. The findings reveal interesting phenomena, including Hopf bifurcations and limit cycles.

Numerical Model of Simultaneous Multi-Regime Boiling Quenching of Metals

This work presents a heat transfer and boiling model that computes the evolution of the temperature field in a representative steel workpiece quenched from 850 or 930 °C by immersion in water flowing at average velocities of 0.2 or 0.6 m/s, respectively. Under these conditions, all three boiling regimes were present during cooling: stable vapor film, nucleate boiling, and single-phase convection. The model was based on the numerical solution of the heat conduction equation coupled to the solution of the energy and momentum equations for water. The mixture phase approach was adopted using the Lee model to compute the rates of water evaporation–condensation. Heat flux at the wall was calculated for all regimes using a single semi-mechanistic model. Therefore, the evolution of boiling regimes at every position on the wall surface was automatically determined. Predictions were validated using laboratory results, namely: (a) videorecording the upward motion of the wetting front along the workpiece wall surface; and (b) cooling curves obtained with embedded thermocouples in the steel probe. Wall heat flux calculations were used to determine the importance of the simultaneous presence of all three boiling regimes on the heat flux distribution. It was found that this simultaneous presence leads to high heat flux variations that should be avoided in production lines. In addition, it was determined that the corresponding inverse heat conduction problem to estimate the active heat transfer boundary condition must be set-up for 2D heat flow.

Stability of buoyancy–thermocapillary convection in molten silicon liquid bridge between two disks with different radii under gravity

By employing a linear stability analysis based on the spectral element method, we investigated the impact of radius ratio (Γr) on the stability of buoyancy–thermocapillary convection in a molten silicon liquid bridge (Pr = 0.011). This liquid bridge was located between two coaxial disks with different radii under the influence of gravity. The aspect ratio of the liquid bridge was maintained at Γ = 1, with a volume ratio Γv = 1 and a fixed height. To explore the physical mechanisms behind convection instability, a perturbation energy analysis was adopted. The free surface shape was determined using the Young–Laplace equation, and two distinct heating strategies were employed. In a top-heated liquid bridge, the convection stability under gravity is always stronger than under zero-gravity. However, in a bottom-heated liquid bridge, the convection stability under gravity is not consistently stronger than under zero-gravity; specifically, when 0.522 < Γr < 0.673, the convection stability under gravity is weaker than under zero-gravity. Despite the small height of the liquid bridge (approximately 2 mm), the maximum relative difference of the critical Marangoni number (Mac) between gravity and zero-gravity conditions reaches as high as 227.8%. In a bottom-heated liquid bridge, oscillatory instability occurs at larger radius ratios (Γr = 0.8) compared to the zero-gravity condition. Furthermore, all instabilities for various radius ratios and heating strategies were found to be of hydrodynamic in nature.

Global nonlinear stability of bidispersive porous convection with throughflow and depth-dependent viscosity

The linear instability and the nonlinear stability analyses have been performed to examine the combined impact of a uniform vertical throughflow and a depth-dependent viscosity on bidispersive porous convection using the Darcy theory with a single temperature field. The validity of the principle of exchange of stability is proved. The eigenvalue problems resulting from both linear instability and nonlinear stability analyses with variable coefficients are numerically solved using the Chebyshev pseudo-spectral method. The equivalence of linear instability and nonlinear stability boundaries is established in the absence of throughflow, while in its presence, the subcritical instability is shown to be evident. The stability of the system is independent of the direction of throughflow in the case of constant viscosity, whereas upflow is found to be more stabilizing than downflow when the viscosity is varying with depth. While the viscosity parameter offers a destabilizing influence on the onset of convection in the absence of throughflow, it imparts both stabilizing and destabilizing effects on the same in its presence. The influence of the ratio of permeabilities and the interphase momentum transfer parameter is to make the system more stable. The findings of a mono-disperse porous medium are presented as a specific case within the broader context of this investigation.

The stability of bio-thermal Marangoni convection in a fluid layer containing oxytactic microorganisms

The stability of bio-thermal Marangoni convection in a horizontal layer of fluid containing motile oxytactic microorganisms is investigated. The analysis of the instability caused by the upswimming of oxytactic microorganisms and by the temperature variation in the fluid layer is discussed using a linear stability analysis. The resulting equations of the problem are solved numerically using the Chebyshev spectral method. Results obtained showed that mode of instability can be either stationary or overstability depending on the range of the parameters’ values. Moreover, results showed that there is a correlation between the bioconvection Marangoni number and the thermal Marangoni number.

THE EFFECT OF COUPLE STRESSES ON STABILITY ANALYSIS OF MAGNETIZED FERROFLUID SATURATING A POROUS MEDIUM HEATED FROM BELOW

This work aims to inspect the impact of couple stress forces on the convective stability of magnetized ferrofluid saturating a porous medium for different combinations of bounding surfaces. Both linear and nonlinear analyses are conducted to obtain eigenvalue problems. Normal mode analysis is used for linear analysis, while the energy method is used for nonlinear analysis, and a generalized energy functional is introduced. For solving eigenvalue problems, the Galerkin method is employed. It is found that the Rayleigh numbers for the two analyses did not match, suggesting the existence of a subcritical region. Furthermore, it is observed that the subcritical region decreased as the magnetic parameter increased, whereas an increase in the couple stress parameter increased the subcritical region. The effect of medium permeability has been found to be destabilizing. For this analysis, three combinations of bounding surfaces are considered. It is also observed that fluid confined in the rigid-rigid bounding surface is more thermally stable, which is suitable for convection in ferrofluid.

Effect of Oldroyd-B fluid on thermal convection stability in a bidispersive porous media with relatively large macropores

This study investigates the effect of Oldroyd-B fluids on the thermal convection instability in a bi-disperse porous medium with relatively large macropores. The modified momentum equation of Oldroyd-B fluid in the macro- and micropores is first derived, assuming local thermal equilibrium. The Orr–Sommerfeld (O–S) equation is derived using the new modified incompressible Navier–Stokes equations of Oldroyd-B fluid. The Chebyshev collocation method is employed to solve the generalized eigenvalue problem, and the neutral stability curves and temporal growth rates are obtained. The results demonstrate that the relaxation time enhances the convective instability, while the retardation time suppresses it. Furthermore, the effect of relaxation time is generally small for smaller wave number, but the effect of relaxation time is significant for large wave number, and this effect becomes accentuated as the wave number increases. The retardation time does not have any influence when the wavenumber is small, and the influence of retardation time becomes small with the large relaxation time. Moreover, the convection instability is enhanced with the higher micropermeability and a higher ratio of momentum transfer coefficient between the macro- and microphases will suppress the instability.

CONVECTIVE STABILITY OF HORIZONTAL FILTRATION FLOW THROUGH A CLOSED DOMAIN OF POROUS MEDIA WITH CLOGGING

The present paper is devoted to the study of horizontal filtration flow through a closed porous domain, with the extraction of some impurities from the mixture by immobilizing them. Usually, the filter is damaged after some time of use because of clogging. Here, we generalize the mathematical model for immobilization and clogging. The investigation of the transition of instability modes from monotonic to oscillatory and the influence of clogging on these phenomena are presented. It is shown that the oscillatory mode is observed in long domains or at moderate intensity of the external horizontal flow. At low flow intensities, the convective cells are stationary, and there is no reason for oscillations. At high intensities, the external flow suppresses the convective oscillations. It is found that the interval of flow intensity values, in which oscillations are observed, grows with increasing domain length; and for thin domains, large intensities are needed to excite the oscillatory mode. Clogging leads to the stabilization of horizontal flow with respect to convective perturbations and sometimes to the dumping of the oscillations. The critical curves and instability maps in a wide range of the problem parameters are obtained and analyzed. For the limiting cases, a comparison with the results of the well-known Horton-Rogers-Lapwood problem (HRL) has been made.

Стационарная вращательно-симметричная конвекция при изотахофорезе в круговой цилиндрической области с недеформируемой свободной непроницаемой боковой поверхностью

The results of a study of the isotachophoresis process convective stability are presented. Isotachophoresis is the method of separation of a multicomponent mixture to individual components by an external electric field. It is assumed that the process of isotachophoresis occur in an infinite vertical circular cylinder with free side boundaries. The problem of the stationary (monotonic, neutral) concentration gravitational convection occurrence is transform to the study of the solution stability for the problem linearized on the solution of the original problem corresponding to the mechanical equilibrium of the final stage of the isotachophoresis process, in which the zones of the individual components of the mixture move with the identical constant velocity in a stationary flow. It is indicated that the main contribution to the stability/instability of the equilibrium is made by concentration gradients in the vicinity of the boundaries between the zones of individual components. Due to the complexity of constructing an exact solution corresponding to mechanical equilibrium, the exact solution replaced by an asymptotic solution (piecewise constant concentration distribution), which, in turn, is an exact solution of the asymptotic version of the transport under action of an electric field component equations (diffusion-free model). The choice of piecewise constant mechanical equilibrium leads to concentration gradients in the form of delta functions and a linear boundary value problem with delta-like coefficients to determine the critical parameters of instability. An exact solution of the problem (a system of ordinary differential equations with boundary conditions) and, using the method of one-dimensional perturbations, an exact dispersion relation for determining critical parameters (in the case of vanishingly small diffusion) is constructed.

Instability onset of Rayleigh-Bénard convection for diatomic rarefied gases: Effects of molecular interaction models and flow parameters

Studies on the stability of Rayleigh-Bénard convection in rarefied gases mainly focus on monatomic gases, whereas diatomic gases are the prevailing medium in actual flows. This paper conducts a study on the Rayleigh-Bénard convection stability of diatomic rarefied gases based on the Navier-Stokes (NS) equations modified with slip boundary conditions. We aim to elaborate the effects of different molecular interaction models and flow parameters (Knudsen number Kn, Froude number Fr, Prandtl number Pr, etc.). And several interesting phenomena have been discovered: (1) Compared with monatomic gases, the instability region of diatomic gases expands towards higher Knudsen numbers and smaller Froude numbers; (2) It is generally believed that increasing rarefaction leads to flow stabilization. However, we have found that in the case of small wave numbers within the scope of our research, the rarer the gas, the higher the growth rate, which contradicts traditional cognition; (3) Competition mechanism between buoyancy and gravity in the stability of Rayleigh-Bénard convection exists; (4) When considering the variations of Prandtl number and specific heat ratio with temperature, the relative deviation of the growth rate of the most unstable mode is significant.

Generalized mean state of the isothermal Darcy–Benard problem and its instability onset

Thermally-driven natural convection is theoretically explored in the present work within a porous layer with respect to the Darcy’s law. The usual non circulating basic cellular flow between two infinitely long horizontal Darcy porous plates leads to generalized basic state considering the isothermal boundary conditions leading to the Hadley flow configuration. Such a motion is governed by a circulatory flow together with a temperature both varying along the horizontal axis. Hence, besides the buoyancy Rayleigh parameter, yet another parameter appears dominating the evolution of basic cellular circulation and thermal progress. Instability onset of this basic state is targeted based on the linear stability analysis of small normal modes. The resulting 3D instability system of equations is treated numerically and the stationary modes leading to convective neutral stability are determined. The found eigenvalues are also justified to be those deviating from the well-known no-flow Darcy porous solution through an asymptotic perturbation method in the special circumstances. The bounds of neutral stability curves in regard to the longitudinal, oblique and transverse modes are thoroughly investigated and it is shown that longitudinal modes are always the most unstable ones, while the transverse and oblique modes having finite encapsulated zone of instability can undergo a complete stability after a threshold value of horizontal temperature gradient parameter.

A nonlinear stability analysis for magnetized ferrofluid heated from below in the presence of couple stresses for combination of different bounding surfaces

This work aims to inspect the impact of couple stress forces on the convective stability of magnetized ferrofluid for different combinations of bounding surfaces. Both linear and nonlinear analyses are conducted to obtain eigenvalue problems. Normal mode analysis is used for linear analysis, while the energy method is used for nonlinear analysis, and a generalized energy functional is introduced. For solving eigenvalue problems, the Galerkin method is employed. It is found that the Rayleigh numbers for the two analyses did not match, suggesting the existence of a subcritical instability region. Furthermore, it is observed that the subcritical instability region decreased as the magnetic parameter increased, whereas an increase in the couple stress parameter increased the subcritical instability region. For this analysis, three different combinations of bounding surfaces are considered. It is also observed that fluid confined in the rigid–rigid bounding surface is more thermally stable, which is suitable for convection in ferrofluid.

Horizontal length of finite-amplitude thermal convection cells with temperature-dependent viscosity

Temperature-dependent viscosity convection is investigated for various horizontal wavelengths of the convective cells. Finite-amplitude steady solutions are obtained by the Newton method in a two-dimensional layer for various values of the Rayleigh number and strength of temperature-dependence of viscosity, and their stability is examined through numerical time integrations. The viscosity η of the model varies with temperature T as η∝exp−γT, where the parameter γ denotes the strength of the temperature-dependency of η. Although approximately square convection cells are stable when γ is small, the stable convective structure elongates horizontally as γ increases in the middle range of γ less than about 10. When γ exceeds that range, the stable convection approaches a square cell.Scaling relations for the Nusselt number that include the effect of the horizontal wavelength are developed. The results obtained by the numerical steady solutions are well explained by the proposed novel scaling relations. When the solutions with the maximum Nusselt number are traced using the scaling relations for various γ, we find that the convective cells elongate gradually as γ increases until γ<8.6, and then the convection becomes narrower. The most elongated convection is expected to appear at the threshold with a horizontal length λ of 6.6, which may not depend on the Rayleigh number. Our results suggest that rocky exoplanets (such as super-Earths), which will be studied in detail in the future, may have surface plates with various horizontal scales.

The Onset of Darcy–Brinkman Convection in a Porous Layer with Mutual Impact of Thermal Non-Equilibrium and Non-Uniform Temperature Gradients

The two-field thermal conditions of local thermal nonequilibrium (LTNE) were used to investigate linear stability of thermal convection in a liquid-saturated, porous layer via the extended Brinkman–Darcy model for different non-uniform basic thermal gradients. The critical values were numerically computed by the Galerkin method for rigid isothermal boundaries. The impact of LTNE and different forms of non-uniform basic temperature gradients on the onset of porous convection was examined. The porosity modified conductivity ratio has no influence on system stability at a small inter-phase heat transport coefficient limit. However, for higher values of the inter-phase heat transport coefficient, an increase in the porosity modified conductivity ratio hastens the onset of convection. An increase in the Darcy number delays the convective motions. The results for different basic temperature profiles are symmetric qualitatively. In addition, the possibility of control of convection by a basic temperature profile was studied in detail.

Learning an optimised stable Taylor-Galerkin convection scheme based on a local spectral model for the numerical error dynamics

We present a new spectral framework to design and optimise numerical methods for convection problems termed Local Transfer function Analysis (LTA), which improves the state-of-art model for error dynamics due to numerical discretisation. LTA converts the numerical discretisation into a network of transfer function blocks in the spectral space, where optimisation of the solution error becomes an impedance matching problem. In addition, a machine learning model based on the graph neural network is employed to learn the optimal coefficients of the transfer function leading to the matched impedance by examining both the local physical field and the local mesh topology. We demonstrate applications of this technique to design Taylor-Galerkin finite element numerical schemes for solving linear convection on irregular meshes in 1D. Using the LTA, we identify and constrain the neural network predictions to the numerically stable design space, thus improving the stability and generalisation of the optimal parameters that lead to a matched impedance.

Dynamic mode decomposition of thermocapillary convection in GaAs melt liquid bridge between unequal ends

The effects of the radius ratio on the stability of thermocapillary convection in the GaAs melt (Pr=0.068) liquid bridge, situated between unequal ends, are studied by three-dimensional direct numerical simulation. The simulated results reveal that, for the first time, the GaAs melt flow transforms from a stable axisymmetric flow into a three-dimensional steady flow through the first bifurcation at 0.5≤Γr≤0.9. The bifurcation phenomenon was previously common in the range of small Prandtl numbers(0.001≤Pr≤0.057). Additionally, we observe that the stability of the thermocapillary liquid bridge is initially weakened and then slowly improved as the radius ratio Γr decreases. The second bifurcation occurs with the further increase of Ma, and the flow becomes oscillatory. Remarkably, two different oscillation forms are identified, depending on the difference in the degree of symmetry of the flow field structure during its oscillation process. Once the flow reaches a stable periodic oscillation, the Dynamic Mode Decomposition (DMD) technique is used to study the inherent spatio-temporal evolution of the temperature field. By combining the spectrum obtained from the Fast Fourier Transform (FFT), we provide evidence for the existence of mixed forms with different azimuthal wavenumbers from a dynamic perspective, intuitively displaying the different components of the oscillation and examining the contribution of different modes to the oscillation. The dynamic model analysis demonstrates that oscillation form I is a mixed form with m = 1 and m = 2, while oscillation form II belongs to the form where m = 2 is absolutely dominant.