Erwin P. van der Poel

The unifying theory of scaling in thermal convection: the updated prefactors

The unifying theory of scaling in thermal convection (Grossmann & Lohse, J. Fluid. Mech., vol. 407, 2000, pp. 27–56; henceforth the GL theory) suggests that there are no pure power laws for the Nusselt and Reynolds numbers as function of the Rayleigh and Prandtl numbers in the experimentally accessible parameter regime. In Grossmann & Lohse (Phys. Rev. Lett., vol. 86, 2001, pp. 3316–3319) the dimensionless parameters of the theory were fitted to 155 experimental data points by Ahlers & Xu (Phys. Rev. Lett., vol. 86, 2001, pp. 3320–3323) in the regime 3×107≤Ra≤3×109 and 4≤Pr≤34 and Grossmann & Lohse (Phys. Rev. E, vol. 66, 2002, p. 016305) used the experimental data point from Qiu & Tong (Phys. Rev. E, vol. 64, 2001, p. 036304) and the fact that Nu(Ra,Pr) is independent of the parameter a, which relates the dimensionless kinetic boundary thickness with the square root of the wind Reynolds number, to fix the Reynolds number dependence. Meanwhile the theory is, on the one hand, well-confirmed through various new experiments and numerical simulations; on the other hand, these new data points provide the basis for an updated fit in a much larger parameter space. Here we pick four well-established (and sufficiently distant) Nu(Ra,Pr) data points and show that the resulting Nu(Ra,Pr) function is in agreement with almost all established experimental and numerical data up to the ultimate regime of thermal convection, whose onset also follows from the theory. One extra Re(Ra,Pr) data point is used to fix Re(Ra,Pr). As Re can depend on the definition and the aspect ratio, the transformation properties of the GL equations are discussed in order to show how the GL coefficients can easily be adapted to new Reynolds number data while keeping Nu(Ra,Pr) unchanged

Exploring the phase diagram of fully turbulent Taylor–Couette flow

Direct numerical simulations of Taylor–Couette flow, i.e. the flow between two coaxial and independently rotating cylinders, were performed. Shear Reynolds numbers of up to 3×10 5 , corresponding to Taylor numbers of Ta=4.6×10 10 , were reached. Effective scaling laws for the torque are presented. The transition to the ultimate regime, in which asymptotic scaling laws (with logarithmic corrections) for the torque are expected to hold up to arbitrarily high driving, is analysed for different radius ratios, different aspect ratios and different rotation ratios. It is shown that the transition is approximately independent of the aspect and rotation ratios, but depends significantly on the radius ratio. We furthermore calculate the local angular velocity profiles and visualize different flow regimes that depend both on the shearing of the flow, and the Coriolis force originating from the outer cylinder rotation. Two main regimes are distinguished, based on the magnitude of the Coriolis force, namely the co-rotating and weakly counter-rotating regime dominated by Rayleigh-unstable regions, and the strongly counter-rotating regime where a mixture of Rayleigh-stable and Rayleigh-unstable regions exist. Furthermore, an analogy between radius ratio and outer-cylinder rotation is revealed, namely that smaller gaps behave like a wider gap with co-rotating cylinders, and that wider gaps behave like smaller gaps with weakly counter-rotating cylinders. Finally, the effect of the aspect ratio on the effective torque versus Taylor number scaling is analysed and it is shown that different branches of the torque-versus-Taylor relationships associated to different aspect ratios are found to cross within 15 % of the Reynolds number associated to the transition to the ultimate regime. The paper culminates in phase diagram in the inner versus outer Reynolds number parameter space and in the Taylor versus inverse Rossby number parameter space, which can be seen as the extension of the Andereck et al. (J. Fluid Mech., vol. 164, 1986, pp. 155–183) phase diagram towards the ultimate regime.

A pencil distributed finite difference code for strongly turbulent wall-bounded flows

We present a numerical scheme geared for high performance computation of wall-bounded turbulent flows. The number of all-to-all communications is decreased to only six instances by using a two-dimensional (pencil) domain decomposition and utilizing the favourable scaling of the CFL time-step constraint as compared to the diffusive time-step constraint. As the CFL condition is more restrictive at high driving, implicit time integration of the viscous terms in the wall-parallel directions is no longer required. This avoids the communication of non-local information to a process for the computation of implicit derivatives in these directions. We explain in detail the numerical scheme used for the integration of the equations, and the underlying parallelization. The code is shown to have very good strong and weak scaling to at least 64K cores.

Boundary layer dynamics at the transition between the classical and the ultimate regime of Taylor-Couette flow

Direct numerical simulations of turbulent Taylor-Couette flow are performed up to inner cylinder Reynolds numbers of Re i = 105 for a radius ratio of η = r i /r o = 0.714 between the inner and outer cylinders. With increasing Re i , the flow undergoes transitions between three different regimes: (i) a flow dominated by large coherent structures, (ii) an intermediate transitional regime, and (iii) a flow with developed turbulence. In the first regime the large-scale rolls completely drive the meridional flow, while in the second one the coherent structures recover only on average. The presence of a mean flow allows for the coexistence of laminar and turbulent boundary layer dynamics. In the third regime, the mean flow effects fade away and the flow becomes dominated by plumes. The effect of the local driving on the azimuthal and angular velocity profiles is quantified, in particular, we show when and where those profiles develop.

Comparison between two- and three-dimensional Rayleigh- Bénard convection

Two-dimensional and three-dimensional Rayleigh–Benard convection is compared using results from direct numerical simulations and previous experiments. The phase diagrams for both cases are reviewed. The differences and similarities between two- and three-dimensional convection are studied using Nu(Ra) for Pr=4.38 and Pr=0.7 and Nu(Pr) for Ra up to 108. In the Nu(Ra) scaling at higher Pr, two- and three-dimensional convection is very similar, differing only by a constant factor up to Ra=1010. In contrast, the difference is large at lower Pr, due to the strong roll state dependence of Nu in two dimensions. The behaviour of Nu(Pr) is similar in two and three dimensions at large Pr. However, it differs significantly around Pr=1. The Reynolds number values are consistently higher in two dimensions and additionally converge at large Pr. Finally, the thermal boundary layer profiles are compared in two and three dimensions

Connecting flow structures and heat flux in turbulent Rayleigh-Bénard convection.

The aspect ratio (Γ) dependence of the heat transfer (Nusselt number Nu in dimensionless form) in turbulent (two-dimensional) Rayleigh-Bénard convection is numerically studied in the regime 0.4≤Γ≤1.25 for Rayleigh numbers 10(7)≤Ra≤Ra(9) and Prandtl numbers Pr=0.7 (gas) and 4.3 (water). Nu(Γ) shows a very rich structure with sudden jumps and sharp transitions. We connect these structures to the way the flow organizes itself in the sample and explain why the aspect ratio dependence of Nu is more pronounced for small Pr. Even for fixed Γ different turbulent states (with different resulting Nu) can exist, between which the flow can or cannot switch. In the latter case the heat transfer thus depends on the initial conditions.

Flow states in two-dimensional Rayleigh-Bénard convection as a function of aspect-ratio and Rayleigh number

In this numerical study on two-dimensional Rayleigh-Benard convection we consider 107 ⩽ Ra ⩽ 1012 in aspect-ratio 0.23 ⩽ Γ ⩽ 13 samples. We focus on several cases. First, we consider small aspect-ratio cells, where at high Ra number we find a sharp transition from a low Ra number branch towards a high Ra number branch, due to changes in the flow structure. Subsequently, we show that the influence of the aspect-ratio on the heat transport decreases with increasing aspect-ratio, although even at very large aspect-ratio of Γ ≈ 10 variations up to 2.5% in the heat transport as a function of Γ are observed. Finally, we observe long-lived transients up to at least Ra = 109, as in certain aspect-ratio cells we observe different flow states that are stable for thousands of turnover times.

Transition to geostrophic convection : the role of the boundary conditions

We conduct computations of rotating Rayleigh–Benard convection in the so-called geostrophic regime, characterized by strong thermal forcing (high Rayleigh numbers) and strong rotation (small Ekman numbers). We employ the full Navier–Stokes equations in our computations and compare no-slip and stress-free boundaries for the plates. The Ekman boundary layers, that exist in the no-slip case but not for stress-free, enhance convective heat transfer and prevent the formation of large-scale flow structures.

Translational and rotational dynamics of a large buoyant sphere in turbulence

We report experimental measurements of the translational and rotational dynamics of a large buoyant sphere in isotropic turbulence. We introduce an efficient method to simultaneously determine the position and (absolute) orientation of a spherical body from visual observation. The method employs a minimization algorithm to obtain the orientation from the 2D projection of a specific pattern drawn onto the surface of the sphere. This has the advantages that it does not require a database of reference images, is easily scalable using parallel processing, and enables accurate absolute orientation reference. Analysis of the sphere’s translational dynamics reveals clear differences between the streamwise and transverse directions. The translational autocorrelations and PDFs provide evidence for periodicity in the particle’s dynamics even under turbulent conditions. The angular autocorrelations show weak periodicity. The angular accelerations exhibit wide tails, however without a directional dependence.

Comparison of computational codes for direct numerical simulations of turbulent Rayleigh–Bénard convection

Abstract Computational codes for direct numerical simulations of Rayleigh–Benard (RB) convection are compared in terms of computational cost and quality of the solution. As a benchmark case, RB convection at Ra = 10 8 and Pr = 1 in a periodic domain, in cubic and cylindrical containers is considered. A dedicated second-order finite-difference code ( AFID / RBflow ) and a specialized fourth-order finite-volume code ( Goldfish ) are compared with a general purpose finite-volume approach ( OpenFOAM ) and a general purpose spectral-element code ( Nek5000 ). Reassuringly, all codes provide predictions of the average heat transfer that converge to the same values. The computational costs, however, are found to differ considerably. The specialized codes AFID / RBflow and Goldfish are found to excel in efficiency, outperforming the general purpose flow solvers Nek5000 and OpenFOAM by an order of magnitude with an error on the Nusselt number Nu below 5%. However, we find that Nu alone is not sufficient to assess the quality of the numerical results: in fact, instantaneous snapshots of the temperature field from a near wall region obtained for deliberately under-resolved simulations using Nek5000 clearly indicate inadequate flow resolution even when Nu is converged. Overall, dedicated special purpose codes for RB convection are found to be more efficient than general purpose codes.