Mohammad S. Emran

Aspect ratio dependence of heat transfer and large-scale flow in turbulent convection

The heat transport and corresponding changes in the large-scale circulation (LSC) in turbulent Rayleigh–Benard convection are studied by means of three-dimensional direct numerical simulations as a function of the aspect ratio Γ of a closed cylindrical cell and the Rayleigh number Ra . The Prandtl number is Pr = 0.7 throughout the study. The aspect ratio Γ is varied between 0.5 and 12 for a Rayleigh number range between 10 7 and 10 9 . The Nusselt number Nu is the dimensionless measure of the global turbulent heat transfer. For small and moderate aspect ratios, the global heat transfer law Nu = A × Ra β shows a power law dependence of both fit coefficients A and β on the aspect ratio. A minimum of Nu (Γ) is found at Γ ≈ 2.5 and Γ ≈ 2.25 for Ra = 10 7 and Ra = 10 8 , respectively. This is the point where the LSC undergoes a transition from a single-roll to a double-roll pattern. With increasing aspect ratio, we detect complex multi-roll LSC configurations in the convection cell. For larger aspect ratios Γ ≳ 8, our data indicate that the heat transfer becomes independent of the aspect ratio of the cylindrical cell. The aspect ratio dependence of the turbulent heat transfer for small and moderate Γ is in line with a varying amount of energy contained in the LSC, as quantified by the Karhunen–Loeve or proper orthogonal decomposition (POD) analysis of the turbulent convection field. The POD analysis is conducted here by the snapshot method for at least 100 independent realizations of the turbulent fields. The primary POD mode, which replicates the time-averaged LSC patterns, transports about 50% of the global heat for Γ ≥ 1. The snapshot analysis enables a systematic disentanglement of the contributions of POD modes to the global turbulent heat transfer. Although the smallest scale – the Kolmogorov scale η K – and the largest scale – the cell height H – are widely separated in a turbulent flow field, the LSC patterns in fully turbulent fields exhibit strikingly similar texture to those in the weakly nonlinear regime right above the onset of convection. Pentagonal or hexagonal circulation cells are observed preferentially if the aspect ratio is sufficiently large (Γ ≳ 8).

Fine-scale statistics of temperature and its derivatives in convective turbulence

We study the fine-scale statistics of temperature and its derivatives in turbulent Rayleigh-Benard convection. Direct numerical simulations are carried out in a cylindrical cell with unit aspect ratio filled with a fluid with Prandtl number equal to 0.7 for Rayleigh numbers between 10 7 and 10 9 . The probability density function of the temperature or its fluctuations is found to be always non-Gaussian. The asymmetry and strength of deviations from the Gaussian distribution are quantified as a function of the cell height. The deviations of the temperature fluctuations from the local isotropy, as measured by the skewness of the vertical derivative of the temperature fluctuations, decrease in the bulk, but increase in the thermal boundary layer for growing Rayleigh number, respectively. Similarly to the passive scalar mixing, the probability density function of the thermal dissipation rate deviates significantly from a log-normal distribution. The distribution is fitted well by a stretched exponential form. The tails become more extended with increasing Rayleigh number which displays an increasing degree of small-scale intermittency of the thermal dissipation field for both the bulk and the thermal boundary layer. We find that the thermal dissipation rate due to the temperature fluctuations is not only dominant in the bulk of the convection cell, but also yields a significant contribution to the total thermal dissipation in the thermal boundary layer. This is in contrast to the ansatz used in scaling theories and can explain the differences in the scaling of the total thermal dissipation rate with respect to the Rayleigh number.

Boundary layer structure in turbulent Rayleigh–Bénard convection

The structure of the boundary layers in turbulent Rayleigh‐B´ enard convection is studied by means of three-dimensional direct numerical simulations. We consider convection in a cylindrical cell at aspect ratio one for Rayleigh numbers of RaD 3 10 9 and 3 10 10 at fixed Prandtl number PrD 0:7. Similar to the experimental results in the same setup and for the same Prandtl number, the structure of the laminar boundary layers of the velocity and temperature fields is found to deviate from the prediction of Prandtl‐Blasius‐Pohlhausen theory. Deviations decrease when a dynamical rescaling of the data with an instantaneously defined boundary layer thickness is performed and the analysis plane is aligned with the instantaneous direction of the large-scale circulation in the closed cell. Our numerical results demonstrate that important assumptions of existing classical laminar boundary layer theories for forced and natural convection are violated, such as the strict twodimensionality of the dynamics or the steadiness of the fluid motion. The boundary layer dynamics consists of two essential local dynamical building blocks, a plume detachment and a post-plume phase. The former is associated with larger variations of the instantaneous thickness of velocity and temperature boundary layer and a fully three-dimensional local flow. The post-plume dynamics is connected with the large-scale circulation in the cell that penetrates the boundary region from above. The mean turbulence profiles taken in localized sections of the boundary layer for each dynamical phase are also compared with solutions of perturbation expansions of the boundary layer equations of forced or natural convection towards mixed convection. Our analysis of both boundary layers shows that the near-wall dynamics combines elements of forced Blasius-type and natural convection.

Resolving the fine-scale structure in turbulent Rayleigh-Benard convection

We present high-resolution direct numerical simulation studies of turbulent Rayleigh–Benard convection in a closed cylindrical cell with an aspect ratio of one. The focus of our analysis is on the finest scales of convective turbulence, in particular the statistics of the kinetic energy and thermal dissipation rates in the bulk and the whole cell. The fluctuations of the energy dissipation field can directly be translated into a fluctuating local dissipation scale which is found to develop ever finer fluctuations with increasing Rayleigh number. The range of these scales as well as the probability of high-amplitude dissipation events decreases with increasing Prandtl number. In addition, we examine the joint statistics of the two dissipation fields and the consequences of high-amplitude events. We have also investigated the convergence properties of our spectral element method and have found that both dissipation fields are very sensitive to insufficient resolution. We demonstrate that global transport properties, such as the Nusselt number, and the energy balances are partly insensitive to insufficient resolution and yield correct results even when the dissipation fields are under-resolved. Our present numerical framework is also compared with high-resolution simulations which use a finite difference method. For most of the compared quantities the agreement is found to be satisfactory.

Conditional statistics of thermal dissipation rate in turbulent Rayleigh-Bénard convection

AbstractThe statistical properties of the thermal dissipation rate in turbulent Rayleigh-Bénard convection in a cylindrical cell are studied by means of three-dimensional direct numerical simulations for a fixed Prandtl number Pr = 0.7 and aspect ratio Γ = 1. The Rayleigh numbers Ra are between 107 and 3 × 1010. We apply a criterion that decomposes the cell volume into two disjoint subsets: the plume-dominated part and the turbulent background part. The plume-dominated set extends over the whole cell volume and is not confined to the boundary layers. It forms a complex spatial skeleton on which the heat is transported in the convection cell and its volume fraction decreases with increasing Rayleigh number. The latter finding holds also when the threshold, which separates both subvolumes, is varied. The Rayleigh number dependence of the mean moments and probability density functions of the thermal dissipation are analyzed on the subvolumes and related to other possible divisions of the convection volume, such as into boundary layer and bulk. The largest thermal dissipation events are always found in the plume-dominated subset.

Large-scale mean patterns in turbulent convection

Large-scale patterns, which are well-known from the spiral defect chaos regime of thermal convection at Rayleigh numbers

Lagrangian tracer dynamics in a closed cylindrical turbulent convection cell.

Ra 10^5

Lagrangian analysis of turbulent convection

. They are uncovered 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

Turbulent heat transfer and large-scale flow in convection cells with aspect ratio Γ > 1

Ra_{\ast}

Temperature boundary layer profiles in turbulent Rayleigh-Benard convection

, that describes the convection of the mean patterns, is indeed in the spiral defect chaos range. The turbulent Prandtl numbers are smaller than one with