Critical Grashof Number Research Articles

Soret-driven convection of Maxwell-Cattaneo fluids in a vertical channel

The Soret effect, also known as thermal diffusion, plays a crucial role in the phenomenon of double diffusion convection in liquids. This study investigates Soret-driven convection within a vertical double-diffusive layer of Maxwell-Cattaneo (M-C) fluids, where the boundaries maintain constant temperatures and solute concentrations that are distinct from each other. The heat transfer equation for Maxwell-Cattaneo fluids is governed by a hyperbolic rule of heat conduction, rather than the typical Fourier parabolic one. The Chebyshev collocation method is employed to solve the corresponding stability eigenvalue problem. The neutral stability curve shows significant fluctuation responses due to the M-C effect. When the Cattaneo number (C) reaches 0.02, multiple local minima appear in the critical Grashof number (Gr). The instability the thermal convection is found to be amplified by the combined effects of Maxwell-Cattaneo and Soret, along with the Grashof number, while the double diffusion effect appears to suppress the instability of convective system. The influence of Soret effect on convective instability will diminish dramatically as the Gr number rises above 8200.

Study on the steady-oscillatory transition of three-dimensional (3D) natural convection via Hopf bifurcation

The transition from steady-state flow to periodic oscillatory flow for the natural convection by Hopf bifurcation is investigated in a three-dimensional (3D) cavity. The spectral collocation method (SCM) in combination with the artificial compressibility method (ACM), which is developed by ourselves as a numerical method SCM-ACM with high accuracy, is employed to solve the governing equations directly instead of linear stability analysis method that is commonly used for the research on flow instability. The results show that the amplitude decays exponentially with time and the decay rate is linear with the Grashof number (Gr). The critical Grashof number for steady-oscillatory transition is obtained as Grcr = 3.423 × 106. The dimensionless angular frequency ωcr = 0.24 is also determined by Fourier analysis. In this work, we also examine the heat-momentum interactions within the boundary layers, visualize the periodic oscillations of temperature and velocity amplitudes, and analyze the origin of instability from multiple angles. The results show that large oscillations of velocity and temperature are observed near the isothermal walls. The oscillation is enhanced by the increase of thermal boundary layer thickness and flow velocity at both ends of isothermal walls. The maximum velocity and temperature amplitudes appear at the lower left and upper right corners of the mid-plane (Z = 0.5), where are the origin of instability, and the spanwise walls are almost independent of oscillations. The oscillatory flow of natural convection in three-dimensional cavity originates from the continuously increasing buoyancy force, and its transition occurs by Hopf bifurcation. Moreover, the temperature amplitude exhibits a wavy distribution on the mid-plane (X = 0.5) and strongly depends on the depth Z. These results provide benchmark data for future numerical studies and engineering application.

Linear instability and nonlinear flow states in a horizontal pipe flow under bottom heating and transverse magnetic field

Linear stabilities of the liquid metal mixed convection in a horizontal pipe under bottom heating and transverse magnetic field are studied through linear global stability analyses. Three branches of the linear stability boundary curves are determined by the eigenvalue computation of the most unstable modes. One branch is located in the region of large Hartmann number and determined by the linear unstable mode which was first revealed by the numerical simulations of Zikanov, Listratov & Sviridov (J. Fluid Mech., vol. 720, 2013, pp. 486–516). This branch curve shows that the global unstable mode exists above a threshold of Hartmann number, which agrees with the experiment of Genin et al. (Temperature fluctuations in a heated horizontal tube affected by transverse magnetic field. In Proc. 8th PAMIR Conf. Fund. Appl. MHD, Borgo, Corsica, France, pp. 37–41, 2011). The other two branch curves determined by two different long-wave unstable modes intersect with each other in the region of small Hartmann number. The critical Grashof number on these two curves increases exponentially with the increase of the Hartmann number. Through energy budget analyses at the critical thresholds of these unstable modes, it is found that, for the unstable mode at large Hartmann numbers, buoyancy is the dominant destabilizing term which demonstrates the hypothetical explanation of Zikanov et al. (2013) who regard natural convection as a destabilization mechanism. It is further revealed that, with respect to the unstable modes on the critical stability curves of small Hartmann numbers, the dominant destabilization comes from the streamwise shear of the basic flow. Finally, within the linear unstable region, fully developed nonlinear flow states of the mixed convection are investigated by direct numerical simulations (DNS) with several sets of selected dimensionless parameters. The spatio-temporal structures of these nonlinear flow states are discussed in detail with comparison with the linear unstable global modes.

Radiative Heat Transfer on Optically Thick Fluid Past an Oscillating Vertical Plate with Variable Temperature

A theoretical study of radiation heat transfer with reference to an optically thick fluid past an oscillating vertical flat plate with variable temperature in the presence of convection and radiation has been presented. The fluid is considered to be a gray, absorbing-emitting radiation but non- scattering medium. The Rosseland flux approximation plays an important role in determining the effect of radiation heat transfer contribution. This problem is an improvement of Stoke’s first and second problem to justify the physical signifance on this problem. This problem is solved by employing Laplace transfrom method. Numerical results of velocity and temperature distributions are depicted graphically. Also, numerical results of frictional shearing stress and critical Grashof number are presented in tables.

The bimodal instability of thermal convection in a tall vertical annulus

We pursue a three-dimensional linear stability analysis to investigate the convective stability in a cylindrical annulus imposed with a horizontal temperature gradient. The investigation focuses on the stability of both axisymmetric and asymmetric modes in a wide range of Prandtl number Pr and the radii ratio η between the inner to outer cylinders. Results show that, for the axisymmetric mode, the stability strongly depends on the Prandtl number. For 0≤Pr≤1.4, the instability sets in as the shear mode that the instability enhances as η increases while it is virtually independent of Pr. The critical Grashof number can be approximated by Grc(η)≈103×4η−0.68+3. For 1.5≤Pr≤12.5, the buoyant mode appears and competes with the shear mode to predominate the stability under various Pr and η, namely, the bimodal instability occurs. The transition between the two modes occurs at the specific radii ratio η1=0.21Pr0.727−0.31. For Pr≥12.6, the shear mode vanishes and the buoyant mode prevails. The asymmetric modes are less prevailing than the axisymmetric mode for small Prandtl numbers, except that the mode of an azimuthal wavenumber equal to 1 becomes dominant in two small regions of η. When the Prandtl number is large, the axisymmetric mode predominates over the asymmetric mode except when η is very small. Otherwise, the asymmetric mode of a larger azimuthal wavenumber is less prevailing.

Modelling Settling-Driven Gravitational Instabilities at the Base of Volcanic Clouds Using the Lattice Boltzmann Method

Field observations and laboratory experiments have shown that ash sedimentation can be significantly affected by collective settling mechanisms that promote premature ash deposition, with important implications for dispersal and associated impacts. Among these mechanisms, settling-driven gravitational instabilities result from the formation of a gravitationally-unstable particle boundary layer (PBL) that grows between volcanic ash clouds and the underlying atmosphere. The PBL destabilises once it reaches a critical thickness characterised by a dimensionless Grashof number, triggering the formation of rapid, downward-moving ash fingers that remain poorly characterised. We simulate this process by coupling a Lattice Boltzmann model, which solves the Navier-Stokes equations for the fluid phase, with a Weighted Essentially Non Oscillatory (WENO) finite difference scheme which solves the advection-diffusion-settling equation describing particle transport. Since the physical problem is advection dominated, the use of the WENO scheme reduces numerical diffusivity and ensures accurate tracking of the temporal evolution of the interface between the layers. We have validated the new model by showing that the simulated early-time growth rate of the instability is in very good agreement with that predicted by linear stability analysis, whilst the modelled late-stage behaviour also successfully reproduces quantitative results from published laboratory experiments. The results show that the model is capable of reproducing both the growth of the unstable PBL and the non-linear dependence of the fingers’ vertical velocity on both the initial particle concentration and the particle diameter. Our validated model is used to expand the parameter space explored experimentally and provides key insights into field studies. Our simulations reveal that the critical Grashof number for the instability is about ten times larger than expected by analogy with thermal convection. Moreover, as in the experiments, we found that instabilities do not develop above a given particle threshold. Finally, we quantify the evolution of the mass of particles deposited at the base of the numerical domain and demonstrate that the accumulation rate increases with time, while it is expected to be constant if particles settle individually. This suggests that real-time measurements of sedimentation rate from volcanic clouds may be able to distinguish finger sedimentation from individual particle settling.

Free convection development caused by bed shrinkage in a vacuum-induced smoldering reactor

In smoldering reactors, a simple change in the way air is supplied to the system directly impacts the entire combustion process. Buoyancy forces may compete with the airflow entering the reactor, resulting in recirculation zones and stagnating the smoldering front. This paper assesses the competition between free convection and vacuum-induced airflow in a shrinking smoldering bed through dimensionless analysis guided by reliable combustion experiments and numerical heat transfer simulations for 22 scenarios. The dimensional analysis reduces the problem to the knowledge of Grashof (106−108), Darcy (10−10−10−9), Bejan (109−1011), and Prandtl (2.5−5.0) numbers. Parallel streamlines indicate negligible effects of free convection when the shrinkage is meager. From the moment the thermal plume starts to grow, the boundary-layer is squeezed to the reactor wall, gradually quenching the combustion front. The dominance of free convection is established for a critical Grashof number around 108. The intrinsic transient aspect of the problem limits the ratio of free convection velocity relative to the imposed flow velocity. Still, if used, it must be compared with the average velocity taken inside the boundary-layer. A developed correlation for the Nusselt number expresses the energy available in the fluid–porous interface.

Linear Stability of a Steady Convective Flow between Permeable Cylinders

Linear stability analysis of a steady convective flow in a tall vertical annulus caused by nonlinear heat sources is conducted in the paper. Heat sources are generated as a result of a chemical reaction. The effect of radial cross-flow through permeable porous walls of the annulus is analyzed. The problem is relevant to biomass thermal conversion. The base flow solution is obtained by solving nonlinear boundary value problem. Linear stability analysis is performed, using collocation method. The calculations show that radial inward or outward flow has a stabilizing effect on the flow, while the increase in the Frank–Kamenetskii parameter (proportional to the intensity of the chemical reaction) destabilizes the flow. The increase in the Reynolds number based on the radial velocity leads to the appearance of the second minimum on the marginal stability curves. The rate of increase in the critical Grashof number with respect to the Reynolds number is different for inward and outward radial flows.

Inertial migration of a non-neutrally buoyant particle in a linear shear flow with thermal convection

Inertial migration of a nonneutrally buoyant disk particle in a linear shear flow under thermal convection (TC) is studied numerically. Particle dynamics and equilibrium position depend on competition between the inertial lift force, the gravitational force (buoyancy), and the TC lift force. When the particle is heavier than the fluid, inertial lift-off is observed and a modified phase diagram is established in terms of the channel Reynolds number Re and density ratio \ensuremath{\sigma}. When the particle is lighter than the fluid, TC induced re-suspension and sedimentation are observed, where the normalized critical Grashof number, Gr${}_{\mathrm{crit}}$/\ensuremath{\sigma}, depends linearly on Re for all particle densities studied.

Inertial migration of a neutrally buoyant circular particle in a planar Poiseuille flow with thermal fluids

This work presents an extensive numerical study on the inertial migration of a neutrally buoyant circular particle in a planar Poiseuille flow with thermal convection using the lattice Boltzmann method. The numerical analysis confirms that, once the channel Grashof number Gr reaches a critical value Grc, the particle lateral equilibrium position transits from above the centerline to below. Moreover, the critical Grashof number Grc depends on the channel Reynolds number and the particle-to-channel size ratio. A force scaling analysis is performed, from which a unified Grashof number Gr* is introduced to consider the synergetic effects of thermal convection and particle size. In addition, it is also found that the occurrence of equilibrium position transition is not sensitive to the initial conditions, but the exact transition point depends on the initial conditions, such as the initial position of the particle and the fluid flow state.

Linear global stability of liquid metal mixed convection in a horizontal bottom-heating duct under strong transverse magnetic field

Two-dimensional steady-state solutions of liquid metal mixed convection in a horizontal bottom-heating duct under a strong magnetic field are first computed numerically by the Newton iteration method along with the spatial discretization of the Taylor–Hood finite element. Two branches of steady solutions with symmetrical rolls and a pair of asymmetrical solutions with a single roll are identified and can be regarded as the base flow for linear global stability analysis. The symmetrical steady solution for the first branch has a nearly uniform distribution for the temperature field in the transverse direction, while the second branch occurs at much larger Grashof numbers and the temperature field becomes nonuniform transversely. The linear stability analysis is performed for a fixed Reynolds number and Prandtl number with Re = 5000 and Pr = 0.0321. For the symmetrical rolls of the first branch, with an increase in the Grashof number, two-dimensional stationary instabilities first occur at small Hartmann numbers, while three-dimensional oscillatory instabilities first appear at moderate or large Hartmann numbers. From the further study of the two-dimensional instabilities, it is revealed that the asymmetrical solution is actually bifurcated supercritically from the symmetrical solution at a two-dimensional critical Grashof number. In addition, the critical curve of the Grashof number with respect to the Hartmann number for the three-dimensional oscillatory mode shows that there exists a minimum critical Grashof number, which occurs at a moderate Hartmann number. The critical curves of the one-roll asymmetrical solution are also exhibited and determined by two three-dimensional oscillatory unstable modes. It is revealed that there exists a minimum Hartmann number below which the asymmetrical steady-state can always remain stable for all Grashof numbers (5 × 105–107). The energy analyses at the oscillatory critical thresholds with different Hartmann numbers are performed to exhibit that buoyancy is the dominant destabilizing term, and the magnetic force is always the main stabilization term for both symmetrical and asymmetrical solutions. In addition, both streamwise and cross-sectional shears of the basic flow are important for the determination of the linear stability boundary of the asymmetrical solution.

Linear stability analysis of the convective flow in a spherical gap with rotating inner surface

We consider the flow in the spherical gap with the radii ratio η=R1/R2=0.5, which is caused by the axial buoyancy force and rotation of the inner surface. The inner surface is warmer than the outer surface (T1 > T2). The basic flow is axisymmetric but has a completely different structure in the northern and southern hemispheres. While buoyancy and Ekman pumping forces compete with each other in the upper hemisphere, they have the same direction in the lower hemisphere. For the fluid, a linear stability analysis with the Prandtl number Pr=0.015 was performed. We calculate the critical Grashof number as a function of the Reynolds number Grc(Re). We have also found that the basic flow becomes unstable with respect to the non-axisymmetric perturbations. The bifurcation sets in as oscillating. We also present some examples of the fully developed three-dimensional flow.

Fully-developed natural convection in a periodic array of pin-fins

The pin-fin heat exchangers/sinks show better performance over other conventional heat exchangers/sinks due to its high surface area to volume ratio and finds application in the field of thermal management of modern electronic components. The present study deals with the two-dimensional numerical investigation of natural convection in an array of periodically in-line mounted square pin-fins. The effect of variation in geometric parameters on the fluid flow and heat transfer has been investigated for different Grashof number. Both the Navier-Stokes and energy equations have been solved using second order spatial and temporal discretizations in the framework of finite difference method. Periodic boundary conditions have been employed for velocity, temperature and pressure. Numerical simulation has been carried out using MAC algorithm for a range of geometrical pitch ranging between 1.5 and 3.0. A Prandtl number of 0.7 (corresponding to air) has been used. The results show that the flow becomes unsteady and periodic oscillatory above a critical Grashof number. This critical value of Grashof number is found to decrease with increasing fin spacing. The Nusselt number is also found to increase with increase in Grashof number for a particular pitch whereas a decrease in Nusselt number is observed with increase in pitch at constant Grashof number.

Thermal Radiation on Transient Laminar Gray Gas Flow Past an Oscillating Vertical Plate with Variable Temperature

Authors have presented an analytical approach on transient laminar gray gas flow past an oscillating vertical flat plate in the presence of natural convection and radiation. The fluid is considered to be a gray, absorbing-emitting radiation but non- scattering medium. The Rosseland flux approximation is employed to simulate radiation heat transfer contribution. We have also studied Stoke’s first and second problem to justify the physical significance on this problem. This problem is solved by employing Laplace Transform method. Numerical results of velocity and temperature distributions are depicted graphically. Also, numerical results of frictional shearing stress and critical Grashof number are presented in tables.

A Numerical Study of Fluid Flow and Heat Transfer in Carbon Dioxide Enclosures on Mars

In order to support the future thermal control and energy conservation design for the Mars rover, numerical studies on natural convection in CO2 enclosures on Mars’ surface were conducted for both horizontal and vertical enclosures. The parameters are as follows: the atmospheric pressure was 1000 Pa, the gravitational acceleration was 3.62 m/s2, and the Prandtl number was 0.77. The heat flux, temperature, and velocity fields of the CO2 enclosures were obtained with the aspect ratio ranging from 5.56 to 200 and the Grashof number ranging from 430 to 2.6 × 104. It was found that natural convection formed more easily in the horizontal enclosures than that in the vertical enclosures when the enclosures had same thickness. With the increasing thickness of the enclosures, Rayleigh–Bénard convections formed in the horizontal enclosures, while only single-cell convections formed in the vertical enclosures. The heat flux through the horizontal enclosures was greater than that through the vertical enclosures with the same thickness when natural convection formed. The maximum difference between them reached 35.26%, which was illustrated by the field synergy principle. A hysteresis phenomenon of the natural convection dominating the heat transfer was found in the vertical enclosure on Mars’ surface. New values for the critical Grashof number and correlations for the average Nusselt number for both the horizontal and vertical CO2 enclosures on Mars’ surface were also developed.

Stability of natural convection in a vertical non-Newtonian fluid layer with an imposed magnetic field

A numerical study has been conducted to analyze the influence of a uniform horizontal magnetic field on the stability of buoyancy driven parallel shear flow in a differentially heated vertical layer of an electrically conducting couple stress fluid; a type of non-Newtonian fluid. Within the framework of linear stability theory, the resulting complex generalized eigenvalue problem is solved numerically using the Chebyshev collocation method with QZ algorithm. The critical Grashof number $$G_{c}$$ and the corresponding wave number $$\alpha_{c}$$ and wave speed $$c_{c}$$ are computed for a wide range of couple stress parameter $$\varLambda_{c}$$ , Prandtl number $$Pr$$ and Hartmann number $$M$$ . It is found that the value of $$Pr$$ at which the instability switches over from stationary to travelling-wave mode increases with increasing $$M$$ and decreasing $$\varLambda_{c}$$ . The effect of magnetic field is to delay the onset of instability while an opposite kind of behavior is observed with increasing $$\varLambda_{c}$$ . The streamlines presented herein demonstrate the development of complex dynamics at the transition mode.

Hopf/steady-state mode interaction in a vertical slot—Effect of 1:4 resonance

Hopf/steady-state mode interaction in a vertical slot with side-wall heating is revisited. Linear stability analysis reveals that a pair of Hopf modes and a steady mode share the critical Grashof number at a crossover Prandtl number P*=12.454 256 44. About a quarter century ago, the non-resonant mode interaction between the critical modes was investigated for a Prandtl number close to P*. In the present paper, we claim that the critical Hopf modes and the steady mode have wavenumbers approximately in a ratio of 1:4. We then examine the effect of 1:4 resonance on the Hopf/steady mode interaction although the 1:4 resonance arises at the quartic-order approximation. This implies that the resonance is so weak that its effect of phase coupling is expected to be negligibly small compared with the effect of coupling acting through the moduli of amplitudes arising at the cubic order. We find, however, that the resonance is indispensable for the bifurcation of symmetric mixed modes with phase differences of 0 and π, from which an asymmetric mixed mode bifurcates.

Analysis of mixed convection in the Czochralski model in a wide range of Prandtl numbers

The results of the calculations and analysis of the effect of separate and joint rotation of the crystal and the crucible on the flow stability are presented for a wide range of Prandtl numbers (from 0.01 to 10). The regimes with a high stability threshold are determined for different combinations of the rotation velocities. It is shown that for high Prandtl numbers, simultaneous rotation of the crystal and the crucible makes it possible to increase the critical Grashof number in 9–12 times. A resultant diagram (map) of the limiting regimes of natural and mixed convection is constructed. Themethodology of control and analysis of 2D and 3D instability modes is discussed.

Non-Boussinesq stability analysis of natural-convection gaseous flow on inclined hot plates

The buoyancy-driven boundary-layer flow that develops over a semi-infinite inclined hot plate is known to become unstable at a finite distance from the leading edge, characterized by a critical value of the Grashof number Gr based on the local boundary-layer thickness. The nature of the resulting instability depends on the inclination angle ϕ, measured from the vertical direction. For values of ϕ below a critical value ϕc the instability is characterized by the appearance of spanwise traveling waves, whereas for ϕ>ϕc the bifurcated flow displays Görtler-like streamwise vortices. The Boussinesq approximation, employed in previous linear stability analyses, ceases to be valid for gaseous flow when the wall-to-ambient temperature ratio Θw is not close to unity. The corresponding non-Boussinesq analysis is presented here, accounting also for the variation with temperature of the different transport properties. A temporal stability analysis including nonparallel effects of the base flow is used to determine curves of neutral stability, which are then employed to delineate the dependences of the critical Grashof number and of its associated wave length on the inclination angle ϕ and on the temperature ratio Θw for the two instability modes, giving quantitative information of interest for configurations with Θw-1∼1. The analysis provides in particular the predicted dependence of the crossover inclination angle ϕc on Θw, indicating that for gaseous flow with Θw-1∼1 spanwise traveling waves are predominant over a range of inclination angles 0⩽ϕ⩽ϕc that is significantly wider than that predicted in the Boussinesq approximation.

A numerical study on the migration of a neutrally buoyant particle in a Poiseuille flow with thermal convection

The migration of a neutrally buoyant particle in a Poiseuille flow with thermal convection is investigated with the double-population lattice Boltzmann method. Compared with the isothermal case, the migration of the particle is notably affected by thermal convection. Five competing mechanisms are responsible for the lateral migration of the particle, i.e., wall repulsion due to lubrication, inertial lift related to shear slip, a lift due to particle rotation, a lift due to the curvature of the undisturbed velocity profile and a lift induced by thermal convection. Within the parameter range examined (Gr<2000,Rec<200), a critical Grashof number Grc is identified, beyond which the equilibrium position is insensitive to the initial position. Furthermore, a power law dependence of the critical Grashof number Grc on the channel Reynolds number Rec is discovered. In addition, with the channel Reynolds number Rec increasing, the forced flow relative to the particle is becoming stronger, leading to a declining effect of thermal convection.