Compressible Convection Research Articles

Upward Overshooting in Turbulent Compressible Convection. I. Effects of the Relative Stability Parameter, the Prandtl Number, and the Péclet Number

Abstract In this paper, we investigate the upward overshooting by three-dimensional numerical simulations. We find that the above convectively stable zone can be partitioned into three layers: the thermal adjustment layer (mixing both entropy and material), the turbulent dissipation layer (mixing material but not entropy), and the thermal dissipation layer (mixing neither entropy nor material). The turbulent dissipation layer is separated from the thermal adjustment layer and the thermal dissipation layer by the first and second zero-points of the vertical velocity correlation. The simulation results are in good agreement with the prediction of the one-dimensional turbulent Reynolds stress model. First, the layer structure is similar. Second, the upper boundary of the thermal adjustment layer is close to the peak of the magnitude of the temperature perturbation. Third, the Péclet number at the upper boundary of the turbulent dissipation layer is close to 1. In addition, we have studied the scalings of the overshooting distance on the relative stability parameter S, the Prandtl number Pr, and the Péclet number Pe. The scaling on S is not unique. The trend is that the overshooting distance decreases with S. Fitting on Pr shows that the overshooting distance increases with Pr. Fitting on Pe shows that the overshooting distance decreases with Pe. Finally, we calculate the ratio of the thickness of the turbulent dissipation layer to that of the thermal adjustment layer. The ratio remains almost constant, with an approximate value of 2.4.

Overshooting in simulations of compressible convection

Context.Convective motions that overshoot into regions that are formally convectively stable cause extended mixing.Aims.We aim to determine the scaling of the overshooting depth (dos) at the base of the convection zone as a function of imposed energy flux (ℱn) and to estimate the extent of overshooting at the base of the solar convection zone.Methods.Three-dimensional Cartesian simulations of hydrodynamic compressible non-rotating convection with unstable and stable layers were used. The simulations used either a fixed heat conduction profile or a temperature- and density-dependent formulation based on Kramers opacity law. The simulations covered a range of almost four orders of magnitude in the imposed flux, and the sub-grid scale diffusivities were varied so as to maintain approximately constant supercriticality at each flux.Results.A smooth heat conduction profile (either fixed or through Kramers opacity law) leads to a relatively shallow power law withdos∝ ℱn0.08for low ℱn. A fixed step-profile of the heat conductivity at the bottom of the convection zone leads to a somewhat steeper dependency ondos∝ ℱn0.12in the same regime. Experiments with and without subgrid-scale entropy diffusion revealed a strong dependence on the effective Prandtl number, which is likely to explain the steep power laws as a function of ℱnreported in the literature. Furthermore, changing the heat conductivity artificially in the radiative and overshoot layers to speed up thermal saturation is shown to lead to a substantial underestimation of the overshooting depth.Conclusions.Extrapolating from the results obtained with smooth heat conductivity profiles, which are the most realistic set-up we considered, suggest that the overshooting depth for the solar energy flux is about 20% of the pressure scale height at the base of the convection zone. This is two to four times higher than the estimates from helioseismology. However, the current simulations do not include rotation or magnetic fields, which are known to reduce convective overshooting.

A well-balanced scheme for the simulation tool-kit A-MaZe: implementation, tests, and first applications to stellar structure

Characterizing stellar convection in multiple dimensions is a topic at the forefront of stellar astrophysics. Numerical simulations are an essential tool for this task. We present an extension of the existing numerical tool-kit A-MaZe that enables such simulations of stratified flows in a gravitational field. The finite-volume based, cell-centered, and time-explicit hydrodynamics solver of A-MaZe was extended such that the scheme is now well-balanced in both momentum and energy. The algorithm maintains an initially static balance between gravity and pressure to machine precision. Quasi-stationary convection in slab-geometry preserves gas energy (internal plus kinetic) on average, despite strong local up- and down-drafts. By contrast, a more standard numerical scheme is demonstrated to result in substantial gains of energy within a short time on purely numerical grounds. The test is further used to point out the role of dimensionality, viscosity, and Rayleigh number for compressible convection. Applications to a young sun in 2D and 3D, covering a part of the inner radiative zone, as well as the outer convective zone, demonstrate that the scheme meets its initial design goal. Comparison with results obtained for a physically identical setup with a time-implicit code show qualitative agreement.

CFD simulation for designing a chlorine gaseous sanitizer treatment system

A 3D numerical model using COMSOL software was developed to simulate the profile of ClO2 gas concentration and bacterial inactivation in treatment chambers with different gas inlets and outlets. The multiphysics simulation based on a finite element method (FEM) included compressible flow, convection and diffusion, and especially, pathogen inactivation codes using Java language were embedded into the program. Finally, each component of the model was coupled and simulated in a transient state. The main fluid stream reached a steady state in a short time because the input speed of ClO2 gas was fast compared to the size of the chamber. Though the chamber was a closed system, compressible air was not observed inside the chamber. Various scenarios with different gas inlet and outlet locations can influence the uniformity profile of increasing ClO2 gas concentration and bacterial inactivation. The optimal configuration was finally found by means of the simulation, and there was a good correspondence in pathogen inactivation profiles between CFD simulation and experiment. This study revealed the computational fluid dynamics (CFD) codes which can help the food processing industry in constructing a scaled-up system and establishing optimum sanitizing treatment conditions.

Onset of fully compressible convection in a rapidly rotating spherical shell

The onset of thermal convection in a rapidly rotating spherical shell is studied by linear stability analysis based on the fully compressible Navier–Stokes equations. Compressibility is quantified by the number of density scale heights $N_{\unicode[STIX]{x1D70C}}$, which measures the intensity of density stratification of the motionless, polytropic base state. The nearly adiabatic flow with polytropic index $n=1.499<n_{a}=1.5$ is considered, where $n_{a}$ is the adiabatic polytropic index. By investigating the stability of the base state with respect to the disturbance of specified wavenumber, the instability process is found to be sensitive to the Prandtl number $Pr$ and to $N_{\unicode[STIX]{x1D70C}}$. For large $Pr$ and small $N_{\unicode[STIX]{x1D70C}}$, the quasi-geostrophic columnar mode loses stability first; while for relatively small $Pr$ a new quasi-geostrophic compressible mode is identified, which becomes unstable first under strong density stratification. The inertial mode can also occur first for relatively small $Pr$ and a certain intensity of density stratification in the parameter range considered. Although the Rayleigh numbers $Ra$ for the onsets of the quasi-geostrophic compressible mode and columnar mode are different by several orders of magnitude, we find that they follow very similar scaling laws with the Taylor number. The critical $Ra$ for convection onset is found to be always positive, in contrast with previous results based on the widely used anelastic model that convection can occur at negative $Ra$. By evaluating the relative magnitude of the time derivative of density perturbation in the continuity equation, we show that the anelastic approximation in the present system cannot be applied in the small-$Ra$ and large-$N_{\unicode[STIX]{x1D70C}}$ regime.

Numerical solutions of compressible convection with an infinite Prandtl number: comparison of the anelastic and anelastic liquid models with the exact equations

We developed a numerical method for the set of equations governing fully compressible convection in the limit of infinite Prandtl numbers. Reduced models have also been analysed, such as the anelastic approximation and the anelastic liquid approximation. The tests of our numerical schemes against self-consistent criteria have shown that our numerical simulations are consistent from the point of view of energy dissipation, heat transfer and entropy budget. The equation of state of an ideal gas has been considered in this work. Specific effects arising because of the compressibility of the fluid are studied, like the scaling of viscous dissipation and the scaling of the heat flux contribution due to the mechanical power exerted by viscous forces. We analysed the solutions obtained with each model (fully compressible model, anelastic and anelastic liquid approximations) in a wide range of dimensionless parameters and determined the errors induced by each approximation with respect to the fully compressible solutions. Based on a rationale on the development of the thermal boundary layers, we can explain reasonably well the differences between the fully compressible and anelastic models, in terms of both the heat transfer and viscous dissipation dependence on compressibility. This could be mostly an effect of density variations on thermal diffusivity. Based on the different forms of entropy balance between exact and anelastic models, we find that a necessary condition for convergence of the anelastic results to the exact solutions is that the product $\unicode[STIX]{x1D716}q$ must be small compared to unity, where $\unicode[STIX]{x1D716}$ is the ratio of the superadiabatic temperature difference to the adiabatic difference, and $q$ is the ratio of the superadiabatic heat flux to the heat flux conducted along the adiabat. The same condition seems also to be associated with a convergence of the computed heat fluxes. Concerning the anelastic liquid approximation, we confirm previous estimates by Anufriev et al. (Phys. Earth Planet. Inter., vol. 152, 2005, pp. 163–190) and find that its results become generally close to those of the fully compressible model when $\unicode[STIX]{x1D6FC}T{\mathcal{D}}$ is small compared to unity, where $\unicode[STIX]{x1D6FC}$ is the isobaric thermal expansion coefficient, $T$ is the temperature (here $\unicode[STIX]{x1D6FC}T=1$ for an ideal gas) and ${\mathcal{D}}$ is the dissipation number.

A High-performance and Portable All-Mach Regime Flow Solver Code with Well-balanced Gravity. Application to Compressible Convection

Abstract Convection is an important physical process in astrophysics well-studied using numerical simulations under the Boussinesq and/or anelastic approximations. However, these approaches reach their limits when compressible effects are important in the high-Mach flow regime, e.g., in stellar atmospheres or in the presence of accretion shocks. In order to tackle these issues, we propose a new high-performance and portable code called “ARK” with a numerical solver well suited for the stratified compressible Navier–Stokes equations. We take a finite-volume approach with machine precision conservation of mass, transverse momentum, and total energy. Based on previous works in applied mathematics, we propose the use of a low-Mach correction to achieve a good precision in both low and high-Mach regimes. The gravity source term is discretized using a well-balanced scheme in order to reach machine precision hydrostatic balance. This new solver is implemented using the Kokkos library in order to achieve high-performance computing and portability across different architectures (e.g., multi-core, many-core, and GP-GPU). We show that the low-Mach correction allows to reach the low-Mach regime with a much better accuracy than a standard Godunov-type approach. The combined well-balanced property and the low-Mach correction allowed us to trigger Rayleigh–Bénard convective modes close to the critical Rayleigh number. Furthermore, we present 3D turbulent Rayleigh–Bénard convection with low diffusion using the low-Mach correction leading to a higher kinetic energy power spectrum. These results are very promising for future studies of high Mach and highly stratified convective problems in astrophysics.

Quantifying the Correlation Between Mobile Continents and Elevated Temperatures in the Subcontinental Mantle

AbstractContinents influence the mantle's convective wavelength and the heat flow escaping from the planet's surface. Over the last few decades, many numerical and analytical studies have contributed to the debate about whether the continents can warm up the subcontinental mantle or not and if they do, then to what extent? However, a consensus regarding the exact nature and magnitude of this correlation between continents and elevated temperatures in the subcontinental mantle remains to be achieved. By conducting a systematic parameter study using 2‐D global mantle convection simulations with mobile continents, we provide qualitative and quantitative observations on the nature of this correlation. In our incompressible and compressible convection models, we observe the general processes of downwellings bringing cold material into the mantle along continental margins and a subsequent buildup of warm thermal anomalies underneath the continents. We compute the amplitude and degree of this correlation using spectral decomposition of the temperature and composition fields. The dominant degree of correlation evolves with time and changes with continental configuration. Using simple empirical fits, we observe that this correlation decreases with increasing core temperature, number of continents, internal heating, or decreasing reference viscosity. We also report simple regressions of the time dependence of this correlation. Additionally, we show that decompression melting as a result of a mantle upwelling or small‐scale sublithospheric convection leads to voluminous volcanism. The emplacement of this dense basalt‐eclogite material breaks the continents apart and destroys the correlation.

Numerical solutions of 2-D steady compressible natural convection using high-order flux reconstruction

Low Mach number flows are common and typical in industrial applications. When simulating these flows, performance of traditional compressible flow solvers can deteriorate in terms of both efficiency and accuracy. In this paper, a new high-order numerical method for two-dimensional (2-D) state low Mach number flows is proposed by combining flux reconstruction (FR) and preconditioning. Firstly, a Couette flow problem is used to assess the efficiency and accuracy of preconditioned FR. It is found that the FR scheme with preconditioning is much more efficient than the original FR scheme. Meanwhile, this improvement still preserves the numerical accuracy. Using this new method and without the Boussinesq assumption, classic natural convection is directly simulated for cases of small and large temperature differences. For the small temperature difference, a p and h refinement study is conducted to verify the grid convergence and accuracy. Then, the influence of the Rayleigh number (Ra) is analyzed. By comparing with the reference results, the numerical results of preconditioned FR is very close to that calculated by incompressible solvers. Furthermore, a large temperature difference test case is calculated and analyzed, indicating this method is not limited by the Boussinesq assumption and is also applicable to heat convection with large temperature differences.

The adjoint equations for thermochemical compressible mantle convection: derivation and verification by twin experiments.

The adjoint method is an efficient way to obtain gradient information in a mantle convection model relative to past flow structure, allowing one to retrodict mantle flow from observations of the present-day mantle state. While adjoint equations for isochemical mantle flow have been derived for both incompressible and compressible flows, here we extend the method to thermochemical mantle flow models, and present thermochemical adjoint equations in the elastic-liquid approximation. We verify the method with twin experiments, and retrodict the flow history of a thermochemical reference model (reference twin) assuming for the final state, either a consistent thermochemical interpretation, using the thermochemical adjoint equations, or an inconsistent purely thermal interpretation, using the isochemical adjoint equations. The consistent simulation correctly retrodicts the flow evolution of the reference twin. The inconsistent case, instead, restores a false flow history whereby internal buoyancy forces and convectively maintained topography are overestimated. Because the cost function is reduced in either case, our results suggest that the adjoint method can be used to link assumptions on the role of chemical mantle heterogeneity to geologic inferences of dynamic topography, thus providing additional means to test hypotheses on mantle composition and dynamics.

Validation of a two-dimensional gas-kinetic scheme for compressible natural convection on structured and unstructured meshes

Gas-kinetic schemes (GKS) have been developed as a kinetic Finite-Volume approach to computational fluid dynamics. The GKS a priori allows to obtain approximate solutions of the fully compressible Navier-Stokes equations. In our contribution we show simulation results of compressible natural convection at large temperature differences and low Mach numbers beyond the applicable range of the Boussinesq approximation. The simulations were performed on non-uniform quadrilateral and unstructured triangular grids. The dependence of the critical Rayleigh number on the temperature difference for compressible Rayleigh-Bénard convection, predicted by theory, is accurately reproduced. Moreover, heat transfer in a buoyancy driven square cavity with differentially heated sides at large Rayleigh numbers and large temperature differences is investigated. Temperature and velocity profiles as well as Nusselt numbers show good agreement with benchmark results in literature. After validating the scheme for thermal compressible convection, we investigate unsteady natural convection at a Rayleigh number of Ra=5⋅109 and at a large temperature difference of Th/Tc=4. We find that compressibility has a leading influence on the stability of the boundary layers, such that the flow at the heated wall becomes unstable, whereas the flow at the cooled wall remains stable. This phenomenon has not yet received much attention.

Numerical investigation of laminar compressible natural convection flow in asymmetrically and isothermally heated open-ended inclined channel

This study presents a numerical investigation of laminar compressible natural convection flow induced under high-temperature difference in an open-ended inclined parallel-walled channel heated isothermally and asymmetrically at the top wall (hot plate facing downwards) while the lower plate is considered adiabatic. The investigation is carried out with air (Pr=0.72) as the working fluid for a temperature difference of 110 °C from the ambient, over of range of modified Rayleigh number 6.2×101 to 1.6×104 and inclination of channel varying between 10 and 90° with respect to horizontal position. For the simulation, compressible governing equations without Boussinesq approximation is solved by employing new modified all-speed Roe scheme with matching preconditioning method, dual time-stepping technique and utilizing the modified Local One-dimensional Inviscid (LODI) relations suitable for compressible natural convection flow as non-reflecting boundary conditions at the inlet and outlet of the channel. Average Nusselt number based on inter-plate spacing increases with the increase of modified Rayleigh number and also with the increase of inclination from the horizontal position. This increase in average Nusselt number can be described by the variation of the level of thermal saturation inside the channel. Significant reduction in average Nusselt number is observed in the combined zone of lower angle of inclination and lower modified Rayleigh number. Visualization of fluid flow (temperature contour and velocity magnitude contour) indicates the existence of both fully developed flow and developing flow regime within the span of investigated modified Rayleigh number. Based on the investigation, a composite correlation between average Nusselt number and product of modified Rayleigh number and the sine of the angle of inclination considering as a single parameter is presented which can be conveniently used for designing of engineering application dealing with natural convection heat transfer under high-temperature difference.

Investigation on optimization of the thermal performance for compressible laminar natural convection flow in open-ended vertical channel

The present paper focuses on the optimization of the thermal performance for compressible laminar natural convection flow induced under high-temperature difference in an open-ended vertical channel by optimizing the channel inter-plate spacing using numerical simulation. The present investigation is conducted for a wide range of Rayleigh number (Ra) 104 to 107 in channel heated asymmetrically by uniform surface temperature with air (Pr0.72) as working fluid. Several values of channel gap between plates, new modified preconditioned all-speed Roe scheme along with dual time stepping technique and modified Local One-dimensional Inviscid (LODI) relations as channel inlet and outlet boundary conditions suitable for compressible laminar natural convection is employed for the current simulation. Heat transfer rate in terms of average Nusselt number is obtained for all Rayleigh number and channel aspect ratio is obtained. Variation of thermal and velocity profiles, the variation of average Nusselt number and mass flow rate into the channel for the combination of each Rayleigh number and channel aspect ratio is reported. From the results obtained, a correlation for optimum aspect ratio with Rayleigh number which maximizes the heat transfer within the channel is presented.

Numerical investigation of buoyancy-driven compressible laminar flow using new method preconditioned all-speed roe scheme

In this study, numerical simulation is performed with a focus on the application of new modified preconditioned all-speed Roe scheme to simulate natural convection flows. The new modified preconditioned all-speed Roe Scheme primarily utilizes the local flow parameters for calculation of the coefficients of numerator in dissipation term of the scheme instead of using the global cut-off Mach number strategy (which is advantageous for natural convection flows) and has been shown to have better accuracy than preconditioned Roe Scheme for Low Mach number flows. For the present simulation, the compressible governing equation in conservation form, new modified preconditioned all-speed Roe scheme and dual time stepping are employed. The validation of numerical algorithm is divided into two investigations a) natural convection flow within differentially heated enclosed square cavity and b) open-ended vertical channel asymmetrically heated for a wide range of Rayleigh number with air (Pr 0.72) as working fluid. Visualization of fluid flow dynamics conducted for both classes of geometries for all range of Rayleigh number show similar phenomena in accordance with previous literatures and compared data also show very good agreement with previous literatures. All results indicate that the new modified preconditioned all-speed Roe scheme is very much competent and accurate for simulation of buoyancy-induced compressible convection flows without relying on the correct prediction of global cut-off Mach number.

Nonlinear convective instability in the compressible magnetic convection problem without heat conductivity

We investigate the convective instability problem for a full compressible viscous magnetohydrodynamic (MHD) fluid with zero resistivity and zero heat conductivity in the presence of a uniform gravitational force in a bounded domain Ω∈R3. First, we establish an instability criterion for the magnetic convection problem, and construct unstable solutions of linearized magnetic convection problem that grow in time in the Sobolev space H3. Then, based on the linearly unstable solutions, we further modify the initial data of the linearly unstable solutions to be ones of the original magnetic convection problem. Finally, using the local well-posedness of classical solutions to the original magnetic convection problem, and a modified bootstrap instability method, we can construct unstable solutions for the original magnetic convection problem in the sense of Hadamard under the instability criterion.

A generalized Characteristic-Based Split projection method for Navier-Stokes with real fluids

A new computational approach for the general numerical simulation of fluid flows is presented and demonstrated for compressible natural convection flows with real fluid properties. The Characteristic-Based Split finite element method is generalized, and abbreviated as GCBS, in the present work to introduce both pressure-expansion effects in addition to thermal-expansion effects on density within the projection step, in contrast to recent works that preserve only one of those two expansion effects. Moreover, we keep the fluid in thermodynamic equilibrium to support a real equation of state and fluid model. The method retains the full Navier-Stokes equations, transitioning smoothly between compressible and incompressible flows, without special artificial-compressibility methods, special low-Mach-number methods, or other modeling treatments. Neither isothermal or isobaric assumptions are made. The non-linear solution obtained is validated by the linearized solution during the early time period of the simulation. Natural convection in a cavity is explored for its convenience in testing across known benchmark flow regimes, from classical Boussinesq flow to the full Navier-Stokes equations with real fluid models. Results for natural convection in a cavity across Rayleigh numbers, non-dimensional adiabatic temperature gradients, and temperature differences, are obtained across flow regimes with the GCBS method. Classical Boussinesq, Thermodynamic Boussinesq, low-Mach number, and real-gas flows are presented, demonstrating the generalization of the method across flow regimes. Thus, our contribution is to generalize the Characteristic-Based Split method to capture the full Navier-Stokes equations while demonstrating the capability with compressible natural convection.

Large-scale dynamos in rapidly rotating plane layer convection

Context.Convectively driven flows play a crucial role in the dynamo processes that are responsible for producing magnetic activity in stars and planets. It is still not fully understood why many astrophysical magnetic fields have a significant large-scale component.Aims.Our aim is to investigate the dynamo properties of compressible convection in a rapidly rotating Cartesian domain, focusing upon a parameter regime in which the underlying hydrodynamic flow is known to be unstable to a large-scale vortex instability.Methods.The governing equations of three-dimensional non-linear magnetohydrodynamics (MHD) are solved numerically. Different numerical schemes are compared and we propose a possible benchmark case for other similar codes.Results.In keeping with previous related studies, we find that convection in this parameter regime can drive a large-scale dynamo. The components of the mean horizontal magnetic field oscillate, leading to a continuous overall rotation of the mean field. Whilst the large-scale vortex instability dominates the early evolution of the system, the large-scale vortex is suppressed by the magnetic field and makes a negligible contribution to the mean electromotive force that is responsible for driving the large-scale dynamo. The cycle period of the dynamo is comparable to the ohmic decay time, with longer cycles for dynamos in convective systems that are closer to onset. In these particular simulations, large-scale dynamo action is found only when vertical magnetic field boundary conditions are adopted at the upper and lower boundaries. Strongly modulated large-scale dynamos are found at higher Rayleigh numbers, with periods of reduced activity (grand minima-like events) occurring during transient phases in which the large-scale vortex temporarily re-establishes itself, before being suppressed again by the magnetic field.

Using the CIFIST grid of CO5BOLD 3D model atmospheres to study the effects of stellar granulation on photometric colours

Context. The atmospheres of cool stars are temporally and spatially inhomogeneous due to the effects of convection. The influence of this inhomogeneity, referred to as granulation, on colours has never been investigated over a large range of effective temperatures and gravities.Aim. We aim to study, in a quantitative way, the impact of granulation on colours.Methods. We use the CIFIST (Cosmological Impact of the FIrst Stars) grid of CO5BOLD (COnservative COde for the COmputation of COmpressible COnvection in a BOx of L Dimensions, L = 2, 3) hydrodynamical models to compute emerging fluxes. These in turn are used to compute theoretical colours in the UBV RI, 2MASS, HIPPARCOS, Gaia and SDSS systems. Every CO5BOLD model has a corresponding one dimensional (1D) plane-parallel LHD (Lagrangian HydroDynamics) model computed for the same atmospheric parameters, which we used to define a “3D correction” that can be applied to colours computed from fluxes computed from any 1D model atmosphere code. As an example, we illustrate these corrections applied to colours computed from ATLAS models.Results. The 3D corrections on colours are generally small, of the order of a few hundredths of a magnitude, yet they are far from negligible. We find that ignoring granulation effects can lead to underestimation of Teff by up to 200 K and overestimation of gravity by up to 0.5 dex, when using colours as diagnostics. We have identified a major shortcoming in how scattering is treated in the current version of the CIFIST grid, which could lead to offsets of the order 0.01 mag, especially for colours involving blue and UV bands. We have investigated the Gaia and HIPPARCOS photometric systems and found that the (G − Hp), (BP − RP) diagram is immune to the effects of granulation. In addition, we point to the potential of the RVS photometry as a metallicity diagnostic.Conclusions. Our investigation shows that the effects of granulation should not be neglected if one wants to use colours as diagnostics of the stellar parameters of F, G, K stars. A limitation is that scattering is treated as true absorption in our current computations, thus our 3D corrections are likely an upper limit to the true effect. We are already computing the next generation of the CIFIST grid, using an approximate treatment of scattering.

On the oscillatory modes of compressible thermal convection in inclined differentially heated cavities

We consider the case of a tall differentially heated cavity (containing air) for which the buoyancy flow is in the so-called boundary-layer regime (displaying thin regions featured by strong temperature gradients adjacent to the vertical walls and thermally stratified fluid in the centre). By using a numerical method able to account for compressibility and variable viscosity effects, we investigate the response of such a system to the application of temperature differences in a range of characteristic numbers for which the resulting flow is expected to be time-periodic or highly time-dependent. More precisely, the problem is examined by allowing both the Rayleigh number and the inclination (θ) of this configuration with respect to the vertical direction to span relatively wide intervals (1.8 × 105 ≤ Ra ≤ 3 × 106, −90° ≤ θ ≤ 90°). Insights into the influence of variable thermodynamic properties are obtained by comparing such numerical results with equivalent simulations based on the classical Boussinesq approximation. The evolution of the considered system towards a fully turbulent state is also discussed to a certain extent.

Compressible Coupled Natural Convection and Volumetric Radiation in a Square Cavity

Coupled thermal convection with radiation in the cavity containing a participating medium has been reexamined numerically in order to determine the conditions under which the Boussinesq approximation is applicable. It is part of this approximation that density variations appear solely in the buoyancy term of the momentum equation and are neglected in all other terms. The coupled momentum, energy, and radiative transfer equations are solved by an iterative procedure based on coupled discrete ordinates and finite volume methods in the context of low-Mach-number approximation of the Navier–Stokes equation. Based on the results of this study, it is found that, in the presence of the participating medium, the Boussinesq approximation is true for temperature differences approximately less than 120 K.