non-Oberbeck Boussinesq Effects Research Articles

Symmetry breaking of rotating convection due to non-Oberbeck–Boussinesq effects

The non-Oberbeck–Boussinesq (NOB) effects arising from variations in thermal expansivity are theoretically and numerically studied in the context of rotating Rayleigh–Bénard convection in forms of two-dimensional rolls. The thermal expansivity increases with pressure (depth), and its variation is measured by a dimensionless factor ε. Utilizing an asymptotic expansion with weak nonlinearity, we derive an amplitude equation, revealing that NOB effects amplify the magnitude of convection. An ε2-order NOB correction leads to a symmetry breaking about the horizontal mid-plane, manifested in the strengthening of convection near the bottom and its weakening near the top, forming bottom-heavy profiles. At ε3-order, the conjunction of NOB effects and nonlinear advection leads to a horizontal symmetry breaking. The values of Taylor number and Prandtl number determine whether upward or downward plumes are stronger. Numerical calculations validate the theoretical analyses in weakly nonlinear regime. This work advances our understanding of hydrothermal plumes in some winter lakes on Earth and in the subglacial oceans on icy moons as well as tracer transport from the seafloor to the ice shell.

Non-Oberbeck–Boussinesq effects on a differentially heated vertical cavity with high-Prandtl number fluid

This study explores the non-Oberbeck–Boussinesq (NOB) effects on hydrodynamics and heat transport in two-dimensional glycerol-filled differentially heated vertical cavity (DHVC). The simulations span Rayleigh numbers (Ra) from 2×103 to 5×109 and temperature difference (Δθ̃) up to 50 K at a Prandtl number (Pr) of 2547. We showed the emergence of stratified flow structures, delineated the NOB effects on temperature distribution symmetry, and analyzed the scaling behaviors of the Nusselt number (Nu), Reynolds number (Re), and thermal boundary layer (BL) thicknesses (λ¯hθ and λ¯cθ) against Ra. For Ra≥3×105, the stratification number (S) shows reduced sensitivity to changes in Ra, stabilizing around 0.5. Additionally, the center temperature (θcen) appears to be unaffected by Ra and increases linearly with Δθ̃ for Ra>106, satisfying θcen≈2.99×10−3K−1Δθ̃. Our results also revealed that Nu∼Raγ Nu and Re∼RaγRe with 0.2649≤γ Nu≤0.2654 and 0.3633≤γ Re≤0.3643, respectively, where γ Nu and γ Re exhibit a monotonic decrease as NOB effects intensify. For all investigated Ra values, NuNOB/NuOB<1 and ReNOB/ReOB>1 hold consistently, with deviations from OB predictions capped at 6.38% and 2.63% for Ra≥108, respectively. The analysis of thermal BL thickness reveals distinct scaling behaviors, characterized by λ¯h,cθ∼Raγ λ¯ h, c, with scaling exponents ranging from −0.2690 to −0.2669 for both OB and NOB scenarios. Notably, it reveals a divergence from water-based DHVC trends, showing linear decreases in the hot wall's scaling exponent and increases for the cold wall.

Non-Oberbeck–Boussinesq effects on the linear stability of a vertical natural convection boundary layer

The non-Oberbeck–Boussinesq effects on the stability of a vertical natural convection boundary layer are investigated using the linearised disturbance equations for air flows up to a temperature difference of $\Delta T=100\,{\rm K}$ . Based on the linear stability results, the neutral curve is shown to be sensitive to the choice of reference temperature. When evaluated using the film temperature $T_f$ , a lower film Grashof number is required to trigger the linear instability for larger $\Delta T$ . The relative contributions of shear and buoyant production to the perturbation kinetic energy budget reveals that the marginally unstable modes are amplified based on different mechanisms: for lower wavenumbers at relatively small Grashof number, the instability is driven by buoyancy; whereas for higher wavenumbers and larger Grashof number, the flow becomes unstable due to a shear instability. The use of reference temperature is found to scale the shear- and buoyant-driven instabilities differently so that no single reference temperature definition would collapse the neutral curves. The linear stability result further demonstrates that at a given Grashof number a higher temperature difference would give a larger amplification rate of the perturbation, which then leads to an earlier onset of the nonlinearities when evaluated at $T_f$ . Finally, by comparing the amplification rates obtained from direct numerical simulation and the linear stability results, the extent of the linear regime is determined for $\Delta T = 100\,{\rm K}$ .

Effects of inclination angle on Rayleigh–Bénard convection under non-Oberbeck–Boussinesq approximation in air

This study employs direct numerical simulations to explore the combined effects of inclination angles (ϕ) and non-Oberbeck–Boussinesq (NOB) conditions on heat transfer and flow patterns in inclined Rayleigh–Bénard convection (RBC). The simulations, conducted in air, classify flow behavior into seven regimes based on roll number and flow state. Heat-transport efficiency is found to be higher in the single-roll state. The study investigates the influence of NOB and inclination on top-bottom symmetry in a fluid-filled cavity, with systematic analysis revealing complex non-monotonic behaviors of the Nusselt number (Nu) and Reynolds number (Re) under different conditions. The effects of ϕ on heat transfer are significant in the modest to moderate range of Rayleigh numbers, becoming less pronounced as these values increase. The study finds that power-law scalings of Nu and Re are sensitive to the inclination angle and robust against NOB effects for ϕ≥40∘. Moreover, in NOB scenarios, the distributions of the local thermal boundary layers (BLs) along the hot and cold walls exhibit a lack of antisymmetry. As a result, there is an obvious deviation between the hot and cold global thermal BLs, which is further influenced by the inclination angle.

Modulation of Rayleigh–Bénard convection with a large temperature difference by inertial nonisothermal particles

This study investigates the modulation by inertial nonisothermal particles in two-dimensional Rayleigh–Bénard (RB) convection with non-Oberbeck–Boussinesq effects due to a large temperature difference. Direct numerical simulations combined with a Lagrangian point-particle method are performed for 1×106≤Ra≤1×108 and 6.1×10−3≤Stf≤1.2, where the Rayleigh number Ra and Stokes number Stf measure the vigor of convection and particle response time, respectively. The typical aspect ratio Γ = 1 is of primary concern. We find that a horizontally arranged double-roll flow pattern prevails at intermediate Stokes numbers with optimal heat transfer efficiency, which has never been reported before. Compared to the single-phase cases, the heat transfer efficiency is enhanced by a factor of two or three. For micro Stokes numbers, unlike cases in the Oberbeck–Boussinesq limit where the addition of particles causes a small amount of flow structure changes, in this study, it is observed that a tiny volume load of particles could actually induce significant flow oscillations or trigger fluid instability for Ra=106; conversely, for medium Rayleigh numbers (Ra=107), it is found that flow reversal is slightly suppressed by small particles. For intermediate Stokes numbers, where particle–fluid couplings are strongest and a wealth of new phenomena emerge, special attention is paid. Considering different aspect ratios, after the addition of particles, it is found that closed RB systems tend to contain an even number of convection rolls rather than odd ones. Quantitatively, heat transfer also improves significantly for various aspect ratios for intermediate Stokes numbers. Subsequent investigations reveal that the narrowing of the horizontal size of convection rolls cannot fully explain the significant enhancement; instead, it should also be attributed to strong couplings between particles and fluid dynamics. Moreover, it is found that both momentum and thermal couplings play crucial roles in enhancing heat transfer efficiency.

Effects of aspect ratio on Rayleigh–Bénard convection under non-Oberbeck–Boussinesq effects in glycerol

Effects of aspect ratio on Rayleigh–Bénard convection under non-Oberbeck–Boussinesq effects in glycerol

Non-Oberbeck–Boussinesq effects on a water-filled differentially heated vertical cavity

In this study, we examined non-Oberbeck–Boussinesq (NOB) effects on a water-filled differentially heated vertical cavity through two-dimensional direct numerical simulations. The simulations encompassed a Rayleigh number (Ra) span of 107–1010, temperature difference (Δθ̃) up to 60 K, and a Prandtl number (Pr) fixed at 4.4. The center temperature (θcen) was found to be independent of Ra and to increase linearly with Δθ̃, as presented by θcen≈1.18×10−3 K−1Δθ̃. The thermal boundary layer (BL) thicknesses near the hot and cold walls (λ¯hθ and λ¯cθ, respectively) are found to scale as λ¯h,cθ∼Raγ λ¯h,c, where the scaling exponent γ λ¯h,c ranges from −0.264 to −0.262. For more detail, the scaling exponent γ λ¯h displays an increasing trend, while γ λ¯c demonstrates a decreasing trend. However, the sum of the hot and cold thermal BL thicknesses was found to be constant at a fixed Ra in the presence of NOB effects. Our detailed investigation of the Nusselt number (Nu) and Reynolds number (Re) revealed that Nu∼Ra0.258 and Re∼Ra0.364, showing insensitivity to NOB effects. These exponents were smaller than those for Rayleigh–Bénard convection. The NOB modifications on Nu and Re were less than 1.2% and 2.5%, respectively, even at Δθ̃=60 K. Our results also revealed that key parameters such as θcen and normalized ratios [(λ¯NOBθ/λ¯OBθ)h,c, NuNOB/NuOB, and ReNOB/ReOB] exhibit universal correlations with Δθ̃. Remarkably, these relationships are consistent across varying Ra values. This observation underscored the influence of NOB effects on these parameters could be confidently forecasted using just the temperature difference (Δθ̃) for Ra∈[107,1010].

Non-Oberbeck–Boussinesq effects in two-dimensional Rayleigh–Bénard convection of different fluids

Non-Oberbeck–Boussinesq (NOB) effects in three representative fluids are quantitatively investigated in two-dimensional Rayleigh–Bénard convection. Numerical simulations are conducted in air, water, and glycerol with Prandtl numbers of Pr=0.71,4.4, and 2547, respectively. We consider Rayleigh number Ra∈[106,109] involving temperature difference (Δθ̃) of up to 60 K. The velocity and temperature profiles are found to be top-bottom antisymmetric under NOB conditions. As Pr increases, the time-averaged temperature of the cavity center ⟨θc⟩t increases under NOB conditions and the value of ⟨θc⟩t is only weakly influenced by Ra for all fluids. For Pr = 4.4 and 2547, with the enhancement of NOB effects, ⟨θc⟩t linearly increases and the maximum θ rms decreases/increases, and its location shifts toward/away from the wall near the bottom/top wall. Dispersed ⟨θc⟩t points and opposite phenomenon are observed in Pr = 0.71. The Nusselt number (Nu) and thermal boundary layer thickness at hot and cold walls (λ¯h,cθ) of the three fluids are comparable, and the Reynolds number (Re) significantly decreases as Pr increases. Under the NOB conditions with Pr = 4.4 and 2547, Nu decreases, Re increases, and λ¯hθ (λ¯cθ) thins (thickens) in an approximately linear fashion. Furthermore, the NOB effects on Nu, Re, and λ¯h,cθ are relatively small for Pr = 0.71 and 4.4, whereas the modifications caused by NOB effects at Pr = 2547 are more significant. The power-law scaling factors of Nu, Re, and λ¯h,cθ are demonstrated to be robust to Pr, as well as NOB effects.

A numerical framework for low-speed flows with large thermal variations

In the numerical simulation of fluid dynamic problems there are situations in which low-speed flows exhibit a non-negligible density variation driven by thermal or compositional distributions. A new method for solving the Low Mach Number equations governing such flows is developed. The basis of the formulation is the semi-implicit scheme build on the Non-oscillatory Forward-in-Time integrator encompassing Multidimensional Positive Definite Advection Transport Algorithm and a robust Krylov solver. The spatial discretisation is constructed using the median dual finite volumes and the method is second-order-accurate in space and time. A collocated arrangement of all flow variables is implemented. The validity of the presented method is evaluated using the Differentially Heated Cavity flows. Three-dimensional simulations demonstrate its efficacy and ability to capture the Non-Oberbeck–Boussinesq effects. Ideally, a numerical method should be able to treat different flow regimes without strong limitations. When implemented, it is shown that in the limit of small density variations the new formulation correctly reproduces simulations of incompressible flows subject to the Oberbeck–Boussinesq approximation. Furthermore, the three-dimensional computations set to the quasi-two-dimensional cases are in excellent agreement with the established two-dimensional Differentially Heated Cavity benchmarks.

Beyond the Oberbeck–Boussinesq and long wavelength approximation

We present the first simulations of a reduced magnetized plasma model that incorporates both arbitrary wavelength polarization and non-Oberbeck–Boussinesq effects. Significant influence of these two effects on the density, electric potential and vorticity and non-linear dynamics of interchange blobs are reported. Arbitrary wavelength polarization implicates so-called gyro-amplification that compared to a long wavelength approximation leads to highly amplified small-scale vorticity fluctuations. These strongly increase the coherence and lifetime of blobs and alter the motion of the blobs through a slower blob-disintegration. Non-Oberbeck–Boussinesq effects incorporate plasma inertia, which substantially decreases the growth rate and linear acceleration of high amplitude blobs, while the maximum blob velocity is not affected. Finally, we generalize and numerically verify unified scaling laws for blob velocity, acceleration and growth rate that include both ion temperature and arbitrary blob amplitude dependence.

Monolithic projection-based method with staggered time discretization for solving non-Oberbeck–Boussinesq natural convection flows

This paper presents an efficient monolithic projection-based method with staggered time discretization (MPM-STD) to examine the non-Oberbeck–Boussinesq (NOB) effects in several natural convection problems involving dramatic temperature-dependent changes in fluid properties. The proposed approach employs the Crank–Nicolson scheme along with staggered time discretization to discretize the momentum and energy equations. The momentum and energy equations are decoupled by evaluating the velocity vector at integral time levels (n+1), and the scalar variables (pressure and temperature) at half-integral time levels (n+12). The observed density variations in all terms result in a variable-coefficient Poisson equation, which is difficult to solve efficiently. The convergence is accelerated via adoption of an appropriate pressure-correction scheme that transforms the aforementioned Poisson equation to a constant-coefficient form. The numerical simulations concerning two-dimensional (2D) periodic NOB Rayleigh–Bénard convection (RBC) in glycerol and 2D differentially heated cavity (DHC) problem in air confirmed the second-order temporal and spatial accuracies of the proposed method. By simulating the 2D DHC problem in air and the RBC problem in liquid (water or glycerol) considering NOB effects, it is concluded that the proposed MPM-STD significantly mitigates the time-step restriction, thereby increasing the computational efficiency, which exceeds that of existing semi-implicit and explicit schemes. Moreover, the potential of the proposed approach with regard to solving challenging three-dimensional (3D) turbulent problems is demonstrated by performing direct simulations of turbulent RBCs under NOB effects involving temperature differences up to 60 K with a corresponding Rayleigh number (Ra=106).

Complex interplay of power-law rheology and non-Oberbeck-Boussinesq effects on natural convection heat transfer in a confined domain

As the first endeavor, the classical differentially heated square cavity problem is revisited to study the non-Oberbeck-Boussinesq (NOB) effects on laminar natural convection heat transfer to non-Newtonian power-law fluids. The augmentation and diminution in both momentum and heat transfer characteristics due to the interplay of power-law rheology and NOB effects are investigated numerically using a finite-volume based open-source solver OpenFOAM®. Moreover, significant non-intuitive changes in thermal and flow fields, due to the temperature-dependent fluid properties, are studied for practical temperature difference ranges. In the present study, both shear-thinning and shear-thickening power-law fluids are taken into account. The NOB results are compared with the corresponding Oberbeck-Boussinesq (OB) results to ascertain qualitative and quantitative inevitable changes in the intricate phenomena. The accelerating NOB effect on heat transfer trend augmented due to shear-thinning fluid behavior, whereas it diminished by the shear-thickening fluids, with reference to a Newtonian case. In particular, for the range of power-law index and temperature difference considered in this study, the average Nusselt number enhances in excess of 40% due to the consequent NOB effects at the extreme case. Results without ascertaining NOB effects would be quite misleading and thus, it should not be neglected in practical scenarios and for their implications. The primary objective of this work is to investigate the natural convection in a square cavity accounting the NOB effects and ascertain the regimes of validity of the Boussinesq approximation. This work finds application in thermal processing of polymeric melts and solutions in several process industries.

Relative importance of temperature-dependent properties in non-Newtonian natural convection around curved surfaces

Natural convection of non-Newtonian power-law fluids around curved surfaces is a ubiquitous phenomenon in industrial applications and research. In this context, the importance of temperature-dependent properties combined with the power-law behavior of the fluid was studied numerically to elucidate the extent and significance of non-Boussinesq effects. An order-of-magnitude-analysis (OMA) indicated that both the thermal expansion coefficient and consistency index had significant temperature dependencies for the class of fluids considered. These dependencies were incorporated in the governing equations applicable over practical temperature ranges. It was found that the effect of curvature on the local shear rate distribution for flow around the object lends additional significance to the non-Oberbeck-Boussinesq (NOB) effects augmented or diminished by power-law behavior. The importance of using actual temperature differences in the analysis because of a complete reversal of trends at very low temperatures was demonstrated. The influence of NOB effects in studies where the fluid and geometrical parameters are optimized was found to be significant. Comprehensive numerical studies to estimate the extent of NOB effects due to the inadequacy of OMA were found necessary. The solutions provided additional insight to the relative importance of thermo-dependent viscosity and thermal expansion coefficient for the class of power-law fluids considered.

Turbulent Rayleigh-Bénard convection under strong non-Oberbeck-Boussinesq conditions

We report on Rayleigh-Benard convection with strongly varying fluid properties experimentally and theoretically. Using pressurized sulfur-hexafluoride (SF6) above its critical point, we are able to make measurements at mean temperatures (Tm) and pressures (Pm) along Prandtl-number isolines in the (T,P) parameter space. This allows us to keep the mean Rayleigh- (Ram) and Prandtl number (Prm) constant while changing the temperature dependences of the fluid properties independently, e.g., probing the liquidlike or gaslike region that are left and right of the supercritical isochore. Hence, non-Oberbeck-Boussinesq (NOB) effects can be measured and analyzed cleanly. We measure the temperature at midheight (Tc) as well as the global vertical heat flux. We observe a significant heat transport enhancement of up to 112% under strong NOB conditions. Furthermore, we develop a theoretical model for the global vertical heat flux based on ideas of Grossmann and Lohse (GL) in OB systems, adjusted for nonconstant fluid properties. In this model, the NOB effects influence the boundary layer and hence Tc, but the change of the heat flux is predominantly due to a change of the fluid properties in the bulk, in particular the heat capacity cp and density ρ. Predictions from our model are consistent with our experimental results as well as with previous measurements carried out in pressurized ethane and cryogenic helium.

On non-Oberbeck–Boussinesq effects in Rayleigh–Bénard convection of air for large temperature differences

We present direct numerical simulations of non-Oberbeck–Boussinesq (NOB) Rayleigh–Benard (RB) convection due to large temperature differences in two-dimensional (2-D) and three-dimensional (3-D) cells. Perfect air is chosen as the operating fluid and the Prandtl number ( ) is fixed to 0.71 for the reference state . In the present system, we consider large temperature differences ranging from 60 K to 240 K, and relatively strong NOB effects are induced at moderate Rayleigh numbers ( ) in the range . The large temperature difference also induces the turbulence system with large density variation. Due to top-down symmetry breaking under NOB conditions, an increase of the centre temperature is found compared to the arithmetic mean temperature of the top and bottom plates, and the shift of is strongly dependent on Rayleigh number and temperature differential . The NOB effects on the Nusselt number ( ) are quite small ( ). The power-law scalings of versus are robust against NOB effects, even for the extremely large temperature difference 240 K, which has never been reached in previous experiments and simulations. The Reynolds numbers , as well as the scalings of versus , are also insensitive to NOB effects. It is noteworthy that the influence of NOB effects on and in 3-D RB flow are weaker than its 2-D counterpart. Furthermore, the extended laminar boundary layer (BL) equations are developed based on the low-Mach-number Navier–Stokes equations, which qualitatively predicts the NOB effects on velocity profiles. Direct numerical simulation results indicate that the top and bottom thermal BLs can compensate each other much better than the velocity BLs under NOB conditions, which contribute to the robustness of .

Augmentation and diminution of non-Boussinesq effects due to non-Newtonian power-law behavior in natural convection

This study investigates the extent and significance of augmentation and diminution of non-Oberbeck–Boussinesq (NOB) effects due to power-law rheology in natural convection. The relative significance of each temperature dependent thermophysical property of a non-Newtonian power-law fluid was first evaluated using an order of magnitude analysis. The limiting criteria were derived for a class of fluids having an Arrhenius-type thermodependent consistency index, and a constant power-law index. Significant fluid property dependencies were identified, incorporated into the conservation equations, and solved numerically. The benchmark problem of natural convection around an isothermal vertical flat plate, immersed in a quiescent power-law fluid, was re-investigated from the viewpoint of NOB effects. It was found that power-law rheology significantly augmented or diminished the acceleration of flow caused by NOB effects. Compared to the Newtonian case, strong shear thinning behavior more than doubled the NOB acceleration of the flow field, and shear thickening inhibited the acceleration by nearly half. Numerical solutions facilitated the visualization of velocity and temperature distributions, and thus provided insights to the underlying physics. It is demonstrated that the cumulative effect of the power-law behavior and temperature dependence of properties is more than a mere superposition of the individual effects. Important contributions from the present work include modifications to the OB approximation’s limiting criteria applicable to power-law fluids, insights into the effect of temperature dependent properties on the flow field and consequent heat transfer, and their correspondence to the results of the order of magnitude analysis. The present investigation of the underlying physics of augmentation and diminution facilitate a better understanding for future studies of NOB effects combined with non-Newtonian behavior.

Direct numerical simulations of Rayleigh–Bénard convection in water with non-Oberbeck–Boussinesq effects

Rayleigh–Bénard convection in water is studied by means of direct numerical simulations, taking into account the variation of properties. The simulations considered a three-dimensional (3-D) cavity with a square cross-section and its two-dimensional (2-D) equivalent, covering a Rayleigh number range of $10^{6}\leqslant Ra\leqslant 10^{9}$ and using temperature differences up to 60 K. The main objectives of this study are (i) to investigate and report differences obtained by 2-D and 3-D simulations and (ii) to provide a first appreciation of the non-Oberbeck–Boussinesq (NOB) effects on the near-wall time-averaged and root-mean-squared (r.m.s.) temperature fields. The Nusselt number and the thermal boundary layer thickness exhibit the most pronounced differences when calculated in two dimensions and three dimensions, even though the $Ra$ scaling exponents are similar. These differences are closely related to the modification of the large-scale circulation pattern and become less pronounced when the NOB values are normalised with the respective Oberbeck–Boussinesq (OB) values. It is also demonstrated that NOB effects modify the near-wall temperature statistics, promoting the breaking of the top–bottom symmetry which characterises the OB approximation. The most prominent NOB effect in the near-wall region is the modification of the maximum r.m.s. values of temperature, which are found to increase at the top and decrease at the bottom of the cavity.

Significance of non-Oberbeck-Boussinesq effects augmented by power-law rheology in natural convection studies around fins

The augmentation and diminution of non-Oberbeck-Boussinesq (NOB) effects due to power-law rheology cause significant changes in the results and associated implications of natural convection studies. This study focuses on the combined effect of spatial arrangement with NOB and power-law effects. Non-intuitive changes in heat transfer trends are caused by the additional effect on the shear rate distribution due to spatial arrangement of objects, represented here by an array of fins. An order of magnitude analysis was used to derive Oberbeck-Boussinesq type equations for a class of power-law fluids with all properties considered as linear functions of temperature and pressure. Significant temperature dependent properties were identified, and an explicit criterion to neglect viscous dissipation for power-law fluids in pure natural convection was derived. The identified temperature dependencies were incorporated into NOB equations and solved numerically to investigate their effect on flow field and heat transfer trends. Shear thinning significantly augmented (more than doubled) the accelerating NOB effect, while shear thickening diminished (nearly halved) it. The tendency of power-law rheology to augment or diminish NOB effects was demonstrated to considerably increase the sensitivity of results to temperature dependent properties, over and above that for the Newtonian case. Investigations to note their practical implications revealed that optimization results without NOB effects could be quite misleading for the fin array problem, due to the differing cumulative extents of augmentation. Additionally, correlation studies may be inaccurate as the nature of trends was changed fundamentally due to NOB augmentation.

Elusive transition to the ultimate regime of turbulent Rayleigh-Bénard convection

By using cryogenic ^{4}He gas as the working fluid in a cylindrical cell 0.3 m in both height and diameter, we study the influence of non-Oberbeck-Boussinesq (NOB) effects on the heat transfer in turbulent Rayleigh-Bénard convection (RBC). We show that the NOB effects increase the heat transfer efficiency when the top plate temperature closely approaches the saturation vapor curve even far away from the critical point. Viewed in this light, our analysis points to the likelihood that the claim of having observed the transition to Kraichnan's ultimate regime, under nominally similar conditions in the experiments with SF_{6} [Phys. Rev. Lett. 108, 024502 (2012)PRLTAO0031-900710.1103/PhysRevLett.108.024502], is probably an NOB effect and the important issue of the transition to the ultimate state of RBC remains open.

Non-Oberbeck-Boussinesq effects due to large temperature differences in a differentially heated square cavity filled with air

We numerically investigate non-Oberbeck-Boussinesq (NOB) effects due to large temperature differences in a two-dimensional differentially heated square cavity using low-Mach-number equations. The working fluid is air and the Prandtl number Pr is 0.71 for the reference state. The considered Rayleigh numbers Ra range from 105 to 109. Various temperature differences ΔT̂ between the hot and cold plates are considered and the maximum value is up to 360K. The critical Rayleigh number for the onset of unsteadiness decrease with increasing temperature differences. The NOB effects on the temperature and velocity fields are investigated. It is found that both the thermal and velocity boundary layers become thicker near the hot plate while they get thinner near the cold plates under NOB conditions. The central temperature is increased compared to Oberbeck-Boussinesq (OB) cases considering NOB effects. The normalized center temperature θc roughly increases linearly with increasing temperature differential ∊. The horizontal velocity near the top plate is normally enhanced by NOB effects while it is normally decreased near the bottom plate under NOB conditions. In spite of these marked qualitative differences in the NOB flow relative to OB flow, the overall integral quantities like the Nusselt number Nu and Reynolds number Re are insensitive to NOB effects and retain their Ra-scaling exponents. The Nu has a minor decrease under NOB conditions. The maximum decrease is only about 3% for temperature difference as large as 360K. Within OB approximation, the Nusselt number Nu is found to scale as ∼Ra0.27, and the inverse thermal boundary layer thickness λθ-1 also scale as ∼Ra0.27. These scaling exponents do not change under NOB conditions. The Reynolds number based on the root mean square (r.m.s) velocity Rerms increases slightly for NOB cases. We also find Rerms∼Ra0.37,Rew∼Ra0.50,Rev∼Ra0.45 for OB cases, where Rew/Rev are Reynolds number based on maximum magnitude of vertical/horizontal velocity over the whole cell. The NOB effects almost have no influence on these scaling exponents.