Numerical simulations of Rayleigh–Bénard convection for Prandtl numbers between 10−1 and 104 and Rayleigh numbers between 105 and 109

2010 ◽  
Vol 662 ◽  
pp. 409-446 ◽  
Author(s):  
G. SILANO ◽  
K. R. SREENIVASAN ◽  
R. VERZICCO
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Nusselt Number ◽  
Numerical Simulations ◽  
Multiple Solutions ◽  
Large Scale ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

We summarize the results of an extensive campaign of direct numerical simulations of Rayleigh–Bénard convection at moderate and high Prandtl numbers (10−1 ≤ Pr ≤ 104) and moderate Rayleigh numbers (105 ≤ Ra ≤ 109). The computational domain is a cylindrical cell of aspect ratio Γ = 1/2, with the no-slip condition imposed on all boundaries. By scaling the numerical results, we find that the free-fall velocity should be multiplied by $1/\sqrt{{\it Pr}}$ in order to obtain a more appropriate representation of the large-scale velocity at high Pr. We investigate the Nusselt and the Reynolds number dependences on Ra and Pr, comparing the outcome with previous numerical and experimental results. Depending on Pr, we obtain different power laws of the Nusselt number with respect to Ra, ranging from Ra2/7 for Pr = 1 up to Ra0.31 for Pr = 103. The Nusselt number is independent of Pr. The Reynolds number scales as ${\it Re}\,{\sim}\,\sqrt{{\it Ra}}/{\it Pr}$, neglecting logarithmic corrections. We analyse the global and local features of viscous and thermal boundary layers and their scaling behaviours with respect to Ra and Pr, and with respect to the Reynolds and Péclet numbers. We find that the flow approaches a saturation state when Reynolds number decreases below the critical value, Res ≃ 40. The thermal-boundary-layer thickness increases slightly (instead of decreasing) when the Péclet number increases, because of the moderating influence of the viscous boundary layer. The simulated ranges of Ra and Pr contain steady, periodic and turbulent solutions. A rough estimate of the transition from the steady to the unsteady state is obtained by monitoring the time evolution of the system until it reaches stationary solutions. We find multiple solutions as long-term phenomena at Ra = 108 and Pr = 103, which, however, do not result in significantly different Nusselt numbers. One of these multiple solutions, even if stable over a long time interval, shows a break in the mid-plane symmetry of the temperature profile. We analyse the flow structures through the transitional phases by direct visualizations of the temperature and velocity fields. A wide variety of large-scale circulation and plume structures has been found. The single-roll circulation is characteristic only of the steady and periodic solutions. For other regimes at lower Pr, the mean flow generally consists of two opposite toroidal structures; at higher Pr, the flow is organized in the form of multi-jet structures, extending mostly in the vertical direction. At high Pr, plumes mainly detach from sheet-like structures. The signatures of different large-scale structures are generally well reflected in the data trends with respect to Ra, less in those with respect to Pr.

Thermal convection in inclined cylindrical containers

2016 ◽  
Vol 790 ◽  
Author(s):  
Olga Shishkina ◽  
Susanne Horn
Keyword(s):  
Reynolds Number ◽  
Nusselt Number ◽  
Large Scale ◽  
Direct Numerical Simulations ◽  
Large Scale Circulation ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard ◽  
Small Inclination

By means of direct numerical simulations (DNS) we investigate the effect of a tilt angle ${\it\beta}$, $0\leqslant {\it\beta}\leqslant {\rm\pi}/2$, of a Rayleigh–Bénard convection (RBC) cell of aspect ratio 1, on the Nusselt number $\mathit{Nu}$ and Reynolds number $\mathit{Re}$. The considered Rayleigh numbers $\mathit{Ra}$ range from $10^{6}$ to $10^{8}$, the Prandtl numbers range from 0.1 to 100 and the total number of the studied cases is 108. We show that the $\mathit{Nu}\,({\it\beta})/\mathit{Nu}(0)$ dependence is not universal and is strongly influenced by a combination of $\mathit{Ra}$ and $\mathit{Pr}$. Thus, with a small inclination ${\it\beta}$ of the RBC cell, the Nusselt number can decrease or increase, compared to that in the RBC case, for large and small $\mathit{Pr}$, respectively. A slight cell tilt may not only stabilize the plane of the large-scale circulation (LSC) but can also enforce an LSC for cases when the preferred state in the perfect RBC case is not an LSC but a more complicated multiple-roll state. Close to ${\it\beta}={\rm\pi}/2$, $\mathit{Nu}$ and $\mathit{Re}$ decrease with increasing ${\it\beta}$ in all considered cases. Generally, the $\mathit{Nu}({\it\beta})/\mathit{Nu}(0)$ dependence is a complicated, non-monotonic function of ${\it\beta}$.

scholarly journals Large-scale mean patterns in turbulent convection

2015 ◽  
Vol 776 ◽  
pp. 96-108 ◽  
Author(s):  
Mohammad S. Emran ◽  
Jörg Schumacher
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Mean Flow ◽  
Turbulent Fluctuations ◽  
The Mean ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

Large-scale patterns, which are well-known from the spiral defect chaos (SDC) regime of thermal convection at Rayleigh numbers $\mathit{Ra}<10^{4}$, continue to exist in three-dimensional numerical simulations of turbulent Rayleigh–Bénard convection in extended cylindrical cells with an aspect ratio ${\it\Gamma}=50$ and $\mathit{Ra}>10^{5}$. They are revealed when the turbulent fields are averaged in time and turbulent fluctuations are thus removed. We apply the Boussinesq closure to estimate turbulent viscosities and diffusivities, respectively. The resulting turbulent Rayleigh number $\mathit{Ra}_{\ast }$, that describes the convection of the mean patterns, is indeed in the SDC range. The turbulent Prandtl numbers are smaller than one, with $0.2\leqslant \mathit{Pr}_{\ast }\leqslant 0.4$ for Prandtl numbers $0.7\leqslant \mathit{Pr}\leqslant 10$. Finally, we demonstrate that these mean flow patterns are robust to an additional finite-amplitude sidewall forcing when the level of turbulent fluctuations in the flow is sufficiently high.

scholarly journals Bulk scaling in wall-bounded and homogeneous vertical natural convection

2018 ◽  
Vol 841 ◽  
pp. 825-850 ◽  
Author(s):  
Chong Shen Ng ◽  
Andrew Ooi ◽  
Detlef Lohse ◽  
Daniel Chung
Keyword(s):  
Nusselt Number ◽  
Natural Convection ◽  
Temperature Gradient ◽  
Power Law ◽  
Exact Relation ◽  
Bénard Convection ◽  
Benard Convection ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Previous numerical studies on homogeneous Rayleigh–Bénard convection, which is Rayleigh–Bénard convection (RBC) without walls, and therefore without boundary layers, have revealed a scaling regime that is consistent with theoretical predictions of bulk-dominated thermal convection. In this so-called asymptotic regime, previous studies have predicted that the Nusselt number ($\mathit{Nu}$) and the Reynolds number ($\mathit{Re}$) vary with the Rayleigh number ($\mathit{Ra}$) according to $\mathit{Nu}\sim \mathit{Ra}^{1/2}$ and $\mathit{Re}\sim \mathit{Ra}^{1/2}$ at small Prandtl numbers ($\mathit{Pr}$). In this study, we consider a flow that is similar to RBC but with the direction of temperature gradient perpendicular to gravity instead of parallel to it; we refer to this configuration as vertical natural convection (VC). Since the direction of the temperature gradient is different in VC, there is no exact relation for the average kinetic dissipation rate, which makes it necessary to explore alternative definitions for $\mathit{Nu}$, $\mathit{Re}$ and $\mathit{Ra}$ and to find physical arguments for closure, rather than making use of the exact relation between $\mathit{Nu}$ and the dissipation rates as in RBC. Once we remove the walls from VC to obtain the homogeneous set-up, we find that the aforementioned $1/2$-power-law scaling is present, similar to the case of homogeneous RBC. When focusing on the bulk, we find that the Nusselt and Reynolds numbers in the bulk of VC too exhibit the $1/2$-power-law scaling. These results suggest that the $1/2$-power-law scaling may even be found at lower Rayleigh numbers if the appropriate quantities in the turbulent bulk flow are employed for the definitions of $\mathit{Ra}$, $\mathit{Re}$ and $\mathit{Nu}$. From a stability perspective, at low- to moderate-$\mathit{Ra}$, we find that the time evolution of the Nusselt number for homogenous vertical natural convection is unsteady, which is consistent with the nature of the elevator modes reported in previous studies on homogeneous RBC.

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

2016 ◽  
Vol 791 ◽  
Author(s):  
Xiaozhou He ◽  
Eberhard Bodenschatz ◽  
Guenter Ahlers
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Cylindrical Sample ◽  
Ultimate State ◽  
Large Scale Circulation ◽  
Classical State ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

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’.

scholarly journals Energy and Variance Budgets of a Diffusive Staircase with Implications for Heat Flux Scaling

2016 ◽  
Vol 46 (8) ◽  
pp. 2553-2569 ◽  
Author(s):  
Magnus Hieronymus ◽  
Jeffrey R. Carpenter
Keyword(s):  
Heat Flux ◽  
Numerical Simulations ◽  
Scaling Laws ◽  
State Energy ◽  
Diffusive Regime ◽  
Diffusive Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard ◽  
Double Diffusive

AbstractThe steady-state energy and thermal variance budgets form the basis for most current methods for evaluating turbulent fluxes of buoyancy, heat, and salinity. This study derives these budgets for a double-diffusive staircase and quantifies them using direct numerical simulations; 10 runs with different Rayleigh numbers are considered. The energy budget is found to be well approximated by a simple three-term balance, while the thermal variance budget consists of only two terms. The two budgets are also combined to give an expression for the ratio of the heat and salt fluxes. The heat flux scaling is also studied and found to agree well with earlier estimates based on laboratory experiments and numerical simulations at high Rayleigh numbers. At low Rayleigh numbers, however, the authors find large deviations from earlier scaling laws. Last, the scaling theory of Grossman and Lohse, which was developed for Rayleigh–Bénard convection and is based on the partitioning of the kinetic energy and tracer variance dissipation, is adapted to the diffusive regime of double-diffusive convection. The predicted heat flux scalings are compared to the results from the numerical simulations and earlier estimates.

Has the ultimate state of turbulent thermal convection been observed?

2015 ◽  
Vol 785 ◽  
pp. 270-282 ◽  
Author(s):  
L. Skrbek ◽  
P. Urban
Keyword(s):  
Nusselt Number ◽  
Rayleigh Number ◽  
Ultimate State ◽  
Logarithmic Factor ◽  
Aspect Ratios ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

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.

scholarly journals Vertical natural convection: application of the unifying theory of thermal convection

2015 ◽  
Vol 764 ◽  
pp. 349-361 ◽  
Author(s):  
Chong Shen Ng ◽  
Andrew Ooi ◽  
Detlef Lohse ◽  
Daniel Chung
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Temperature Fields ◽  
Mean Flow ◽  
Bénard Convection ◽  
Benard Convection ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Gl Theory

AbstractResults from direct numerical simulations of vertical natural convection at Rayleigh numbers $1.0\times 10^{5}$–$1.0\times 10^{9}$ and Prandtl number $0.709$ support a generalised applicability of the Grossmann–Lohse (GL) theory, which was originally developed for horizontal natural (Rayleigh–Bénard) convection. In accordance with the GL theory, it is shown that the boundary-layer thicknesses of the velocity and temperature fields in vertical natural convection obey laminar-like Prandtl–Blasius–Pohlhausen scaling. Specifically, the normalised mean boundary-layer thicknesses scale with the $-1/2$-power of a wind-based Reynolds number, where the ‘wind’ of the GL theory is interpreted as the maximum mean velocity. Away from the walls, the dissipation of the turbulent fluctuations, which can be interpreted as the ‘bulk’ or ‘background’ dissipation of the GL theory, is found to obey the Kolmogorov–Obukhov–Corrsin scaling for fully developed turbulence. In contrast to Rayleigh–Bénard convection, the direction of gravity in vertical natural convection is parallel to the mean flow. The orientation of this flow presents an added challenge because there no longer exists an exact relation that links the normalised global dissipations to the Nusselt, Rayleigh and Prandtl numbers. Nevertheless, we show that the unclosed term, namely the global-averaged buoyancy flux that produces the kinetic energy, also exhibits both laminar and turbulent scaling behaviours, consistent with the GL theory. The present results suggest that, similar to Rayleigh–Bénard convection, a pure power-law relationship between the Nusselt, Rayleigh and Prandtl numbers is not the best description for vertical natural convection and existing empirical relationships should be recalibrated to better reflect the underlying physics.

scholarly journals Two-scalar turbulent Rayleigh–Bénard convection: numerical simulations and unifying theory

2018 ◽  
Vol 848 ◽  
pp. 648-659 ◽  
Author(s):  
Yantao Yang ◽  
Roberto Verzicco ◽  
Detlef Lohse
Keyword(s):  
Reynolds Number ◽  
Numerical Simulations ◽  
Parameter Space ◽  
Good Accuracy ◽  
Direct Numerical Simulations ◽  
Turbulent Convection ◽  
Free Parameters ◽  
Buoyancy Driven Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

We conduct direct numerical simulations for turbulent Rayleigh–Bénard (RB) convection, driven simultaneously by two scalar components (say, temperature and concentration) with different molecular diffusivities, and measure the respective fluxes and the Reynolds number. To account for the results, we generalize the Grossmann–Lohse theory for traditional RB convection (Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56; Phys. Rev. Lett., vol. 86 (15), 2001, pp. 3316–3319; Stevens et al., J. Fluid Mech., vol. 730, 2013, pp. 295–308) to this two-scalar turbulent convection. Our numerical results suggest that the generalized theory can successfully capture the overall trends for the fluxes of two scalars and the Reynolds number without introducing any new free parameters. In fact, for most of the parameter space explored here, the theory can even predict the absolute values of the fluxes and the Reynolds number with good accuracy. The current study extends the generality of the Grossmann–Lohse theory in the area of buoyancy-driven convection flows.

scholarly journals How heat transfer efficiencies in turbulent thermal convection depend on internal flow modes

2011 ◽  
Vol 676 ◽  
pp. 1-4 ◽  
Author(s):  
KE-QING XIA
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Thermal Convection ◽  
Large Scale ◽  
Internal Flow ◽  
Global Transport ◽  
Large Scale Flow ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Flow States

How internal flow states can influence the global transport properties in a turbulent system has always been an intriguing question. Weiss & Ahlers (J. Fluid Mech., this issue, vol. 676, 2011, pp. 5–40) have provided an example by measuring the instantaneous Nusselt number in turbulent Rayleigh-Bénard convection and correlating it to the different modes of large-scale flow.

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