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 $\Delta T = 100\,{\rm K}$ .

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.

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 .

Taguchi Method and Numerical Simulation for Variable Viscosity and Non-Linear Boussinesq Effects on Natural Convection over a Vertical Truncated Cone in Porous Media

This study uses an optimization approach representation and numerical solution for the variable viscosity and non-linear Boussinesq effects on the free convection over a vertical truncated cone in porous media. The surface of the vertical truncated cone is maintained at uniform wall temperature and uniform wall concentration (UWT/UWC). The viscosity of the fluid varies inversely to a linear function of the temperature. The partial differential equation is transformed into a non-similar equation and solved by Keller box method (KBM). Compared with previously published articles, the results are considered to be very consistent. Numerical results for the local Nusselt number and local Sherwood number with the six parameters (1) dimensionless streamwise coordinate ξ, (2) buoyancy ratio N, (3) Lewis number Le, (4) viscosity-variation parameter θ r , (5) non-linear temperature parameter δ 1 , and (6) non-linear concentration parameter δ 2 are expressed in figures and tables. The Taguchi method was used to predict the best point of the maxima of the local Nusselt (Sherwood) number of 3.8636 (5.1156), resulting in ξ (4), N (10), Le (0.5), θ r (−2), δ 1 (2), δ 2 (2) and ξ (4), N (10), Le (2), θ r (−2), δ 1 (2), δ 2 (2), respectively.

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.

Linear and weakly nonlinear analysis of Rayleigh–Bénard convection of perfect gas with non-Oberbeck–Boussinesq effects

The influences of non-Oberbeck–Boussinesq (NOB) effects on flow instabilities and bifurcation characteristics of Rayleigh–Bénard convection are examined. The working fluid is air with reference Prandtl number $Pr=0.71$ and contained in two-dimensional rigid cavities of finite aspect ratios. The fluid flow is governed by the low-Mach-number equations, accounting for the NOB effects due to large temperature difference involving flow compressibility and variations of fluid viscosity and thermal conductivity with temperature. The intensity of NOB effects is measured by the dimensionless temperature differential $\unicode[STIX]{x1D716}$. Linear stability analysis of the thermal conduction state is performed. An $\unicode[STIX]{x1D716}^{2}$ scaling of the leading-order corrections of critical Rayleigh number $Ra_{cr}$ and disturbance growth rate $\unicode[STIX]{x1D70E}$ due to NOB effects is identified, which is a consequence of an intrinsic symmetry of the system. The influences of weak NOB effects on flow instabilities are further studied by perturbation expansion of linear stability equations with regard to $\unicode[STIX]{x1D716}$, and then the influence of aspect ratio $A$ is investigated in detail. NOB effects are found to enhance (weaken) flow stability in large (narrow) cavities. Detailed contributions of compressibility, viscosity and buoyancy actions on disturbance kinetic energy growth are identified quantitatively by energy analysis. Besides, a weakly nonlinear theory is developed based on centre-manifold reduction to investigate the NOB influences on bifurcation characteristics near convection onset, and amplitude equations are constructed for both codimension-one and -two cases. Rich bifurcation regimes are observed based on amplitude equations and also confirmed by direct numerical simulation. Weakly nonlinear analysis is useful for organizing and understanding these simulation results.

Flow reversals in Rayleigh–Bénard convection with non-Oberbeck–Boussinesq effects

Flow reversals in two-dimensional Rayleigh–Bénard convection led by non-Oberbeck–Boussinesq (NOB) effects due to large temperature differences are studied by direct numerical simulation. Perfect gas is chosen as the working fluid and the Prandtl number is 0.71 for the reference state. If NOB effects are included, the flow pattern $P_{11}$ with only one dominant roll often becomes unstable by the growth of the cold corner roll, which sometimes results in cession-led flow reversals. By exploiting the vorticity transport equation, it is found that the asymmetries of buoyancy and viscous forces are responsible for the growth of the cold corner roll because both such asymmetries cause an imbalance between the corner rolls and the large-scale circulation (LSC). The buoyancy force near the cold wall increases and decreases near the hot wall originating from the temperature-dependent isobaric thermal expansion coefficient ${\it\alpha}=1/T$ if NOB effects are included. Moreover, the decreased dissipation due to lower viscosity is favourable for the growth of the cold corner roll, while the increased viscosity further suppresses the growth of the hot corner roll. Finally, it is found that the boundary layer near the cold wall plays an important role in the mass transport from LSC to corner rolls subject to mass conservation.

On the mechanism of self gravitating Rossby interfacial waves in proto-stellar accretion discs

The dynamical response of edge waves under the influence of self-gravity is examined in an idealised two-dimensional model of a proto-stellar disc, characterised in steady state as a rotating vertically infinite cylinder of fluid with constant density except for a single density interface at some radius . The fluid in basic state is prescribed to rotate with a Keplerian profile modified by some additional azimuthal sheared flow. A linear analysis shows that there are two azimuthally propagating edge waves, kin to the familiar Rossby waves and surface gravity waves in terrestrial studies, which move opposite to one another with respect to the local basic state rotation rate at the interface. Instability only occurs if the radial pressure gradient is opposite to that of the density jump (unstably stratified) where self-gravity acts as a wave stabiliser irrespective of the stratification of the system. The propagation properties of the waves are discussed in detail in the language of vorticity edge waves. The roles of both Boussinesq and non-Boussinesq effects upon the stability and propagation of these waves with and without the inclusion of self-gravity are then quantified. The dynamics involved with self-gravity non-Boussinesq effect is shown to be a source of vorticity production where there is a jump in the basic state density In addition, self-gravity also alters the dynamics via the radial main pressure gradient, which is a Boussinesq effect. Further applications of these mechanical insights are presented in the conclusion including the ways in which multiple density jumps or gaps may or may not be stable.

Azimuthal diffusion of the large-scale-circulation plane, and absence of significant non-Boussinesq effects, in turbulent convection near the ultimate-state transition

We present measurements of the orientation ${\it\theta}_{0}$ and temperature amplitude ${\it\delta}$ of the large-scale circulation in a cylindrical sample of turbulent Rayleigh–Bénard convection (RBC) with aspect ratio ${\it\Gamma}\equiv D/L=1.00$ ($D$ and $L$ are the diameter and height respectively) and for the Prandtl number $Pr\simeq 0.8$. The results for ${\it\theta}_{0}$ revealed a preferred orientation with up-flow in the west, consistent with a broken azimuthal invariance due to the Earth’s Coriolis force (see Brown & Ahlers (Phys. Fluids, vol. 18, 2006, 125108)). They yielded the azimuthal diffusivity $D_{{\it\theta}}$ and a corresponding Reynolds number $Re_{{\it\theta}}$ for Rayleigh numbers over the range $2\times 10^{12}\lesssim Ra\lesssim 1.5\times 10^{14}$. In the classical state ($Ra\lesssim 2\times 10^{13}$) the results were consistent with the measurements by Brown & Ahlers (J. Fluid Mech., vol. 568, 2006, pp. 351–386) for $Ra\lesssim 10^{11}$ and $Pr=4.38$, which gave $Re_{{\it\theta}}\propto Ra^{0.28}$, and with the Prandtl-number dependence $Re_{{\it\theta}}\propto Pr^{-1.2}$ as found previously also for the velocity-fluctuation Reynolds number $Re_{V}$ (He et al., New J. Phys., vol. 17, 2015, 063028). At larger $Ra$ the data for $Re_{{\it\theta}}(Ra)$ revealed a transition to a new state, known as the ‘ultimate’ state, which was first seen in the Nusselt number $Nu(Ra)$ and in $Re_{V}(Ra)$ at $Ra_{1}^{\ast }\simeq 2\times 10^{13}$ and $Ra_{2}^{\ast }\simeq 8\times 10^{13}$. In the ultimate state we found $Re_{{\it\theta}}\propto Ra^{0.40\pm 0.03}$. Recently, Skrbek & Urban (J. Fluid Mech., vol. 785, 2015, pp. 270–282) claimed that non-Oberbeck–Boussinesq effects on the Nusselt and Reynolds numbers of turbulent RBC may have been interpreted erroneously as a transition to a new state. We demonstrate that their reasoning is incorrect and that the transition observed in the Göttingen experiments and discussed in the present paper is indeed to a new state of RBC referred to as ‘ultimate’.

Natural convection in shear-thinning fluids: Experimental investigations by MRI

An experimental investigation of the Rayleigh–Bénard convection in shear-thinning fluids using MRI technics is presented. The experimental setup consists on a cylindrical cavity defined by a finite aspect ratio A=D/d=6. Qualitative and quantitative results are provided. Flow structure is determined from velocity mapping for a Newtonian fluid, Glycerol and for shear-thinning fluids, Xanthan gum aqueous solutions with weight concentrations ranging from 0.1% to 0.2%. In the case of the Glycerol and the Xanthan solution at 0.1%, one recovers similar results in terms of criticality with Rac≈1800 and patterns since the convection is characterized by rolls. When the Xanthan concentration is increased, the critical Rayleigh number is not modified, however the onset occurs with hexagonal pattern. Because the critical temperature differences increase with the concentrations due to an increase in viscosity, one can think that hexagonal patterns are due to variations of physical properties with temperature (non Oberbeck–Boussinesq effects). Similarities with some results obtained in the Newtonian case are highlighted. We have observed a transition from hexagonal patterns to rolls by increasing the Rayleigh number. This pattern transition is characterized by a discrepancy in the maximal velocity values. By using shear-thinning fluids, results show an increase in the intensity of convection compared with the Newtonian case.

Has the ultimate state of turbulent thermal convection been observed?

An important question in turbulent Rayleigh–Bénard convection is the scaling of the Nusselt number with the Rayleigh number in the so-called ultimate state, corresponding to asymptotically high Rayleigh numbers. A related but separate question is whether the measurements support the so-called Kraichnan law, according to which the Nusselt number varies as the square root of the Rayleigh number (modulo a logarithmic factor). Although there have been claims that the Kraichnan regime has been observed in laboratory experiments with low aspect ratios, the totality of existing experimental results presents a conflicting picture in the high-Rayleigh-number regime. We analyse the experimental data to show that the claims on the ultimate state leave open an important consideration relating to non-Oberbeck–Boussinesq effects. Thus, the nature of scaling in the ultimate state of Rayleigh–Bénard convection remains open.

On non-Oberbeck–Boussinesq effects in three-dimensional Rayleigh–Bénard convection in glycerol

AbstractRayleigh–Bénard convection in glycerol (Prandtl number $\mathit{Pr}= 2547. 9$) in a cylindrical cell with an aspect ratio of $\Gamma = 1$ was studied by means of three-dimensional direct numerical simulations (DNS). For that purpose, we implemented temperature-dependent material properties into our DNS code, by prescribing polynomial functions up to seventh order for the viscosity, the heat conductivity and the density. We performed simulations with the common Oberbeck–Boussinesq (OB) approximation and with non-Oberbeck–Boussinesq (NOB) effects within a range of Rayleigh numbers of $1{0}^{5} \leq \mathit{Ra}\leq 1{0}^{9} $. For the highest temperature differences, $\Delta = 80~\mathrm{K} $, the viscosity at the top is ${\sim }360\hspace{0.167em} \% $ times higher than at the bottom, while the differences of the other material properties are less than $15\hspace{0.167em} \% $. We analysed the temperature and velocity profiles and the thermal and viscous boundary-layer thicknesses. NOB effects generally lead to a breakdown of the top–bottom symmetry, typical for OB Rayleigh–Bénard convection. Under NOB conditions, the temperature in the centre of the cell ${T}_{c} $ increases with increasing $\Delta $ and can be up to $15~\mathrm{K} $ higher than under OB conditions. The comparison of our findings with several theoretical and empirical models showed that two-dimensional boundary-layer models overestimate the actual ${T}_{c} $, while models based on the temperature or velocity scales predict ${T}_{c} $ very well with a standard deviation of $0. 4~\mathrm{K} $. Furthermore, the obtained temperature profiles bend closer towards the cold top plate and further away from the hot bottom plate. The situation for the velocity profiles is reversed: they bend farther away from the top plate and closer towards to the bottom plate. The top boundary layers are always thicker than the bottom ones. Their ratio is up to 2.5 for the thermal and up to 4.5 for the viscous boundary layers. In addition, the Reynolds number $\mathit{Re}$ and the Nusselt number $\mathit{Nu}$ were investigated: $\mathit{Re}$ is higher and $\mathit{Nu}$ is lower under NOB conditions. The Nusselt number $\mathit{Nu}$ is influenced in a nonlinear way by NOB effects, stronger than was suggested by the two-dimensional simulations. The actual scaling of $\mathit{Nu}$ with $\mathit{Ra}$ in the NOB case is $\mathit{Nu}\propto {\mathit{Ra}}^{0. 298} $ and is in excellent agreement with the experimental data.

Convection in Multiphase Fluid Flows Using Lattice Boltzmann Methods

We present high-resolution numerical simulations of convection in multiphase flows (boiling) using a novel algorithm based on a lattice Boltzmann method. We first study the thermodynamical and kinematic properties of the algorithm. Then, we perform a series of 3D numerical simulations changing the mean properties in the phase diagram and compare convection with and without phase coexistence at Rayleigh number Ra∼10(7). We show that in the presence of nucleating bubbles non-Oberbeck-Boussinesq effects develop, the mean temperature profile becomes asymmetric, and heat-transfer and heat-transfer fluctuations are enhanced, at all Ra studied. We also show that small-scale properties of velocity and temperature fields are strongly affected by the presence of the buoyant bubble leading to high non-gaussian profiles in the bulk.