Eur. Phys. J. B 35, 133–141 (2003) DOI: 10.1140/epjb/e2003-00264-6 T HE EUROPEAN P HYSICAL JOURNAL B Turbulent thermal convection in a closed domain: viscous boundary layer and mean ﬂow eﬀects R. Verzicco a Politecnico di Bari, DIMeG and CEMeC, Via Re David 200, 70125, Bari, Italia Received 7 March 2003 Published online 22 September 2003 – c  EDP Sciences, Societ` a Italiana di Fisica, Springer-Verlag 2003 Abstract. In this paper the eﬀects of viscous boundary layers and mean ﬂow structures on the heat transfer of a ﬂow in a slender cylindrical cell are analysed using the direct numerical simulation of the Navier– Stokes equations with the Boussinesq approximation. Ideal ﬂows are produced by suppressing the viscous boundary layers and by artiﬁcially enforcing the ﬂow axisymmetry with the aim of checking some proposed explanations for the Nusselt number dependence on the Rayleigh number. The emerging picture suggests that, in this slender geometry,the presence of the viscous boundary layers does not have appreciable impact on the slope of the Nu vs. Ra relation while a transition of the mean ﬂow is most likely the reason for the slope increase observed around Ra =2 × 10 9 . PACS. 47.27.Te Convection turbulent ﬂows – 47.32.-y Fluid ﬂow buoyant – 44.25.+f Heat transfer convective 1 Introduction One of the main advantages of numerical simulation is the possibility to perform ideal experiments with the aim of verifying conjectures or stressing the hypotheses of a the- ory. Indeed, in thermal convection a numerical simulation is always an ideal experiment owing to the precise assign- ment of the temperature boundary conditions, the possi- bility of having plates with inﬁnite heat capacity, the ab- sence of conductive side–wall eﬀects and the unconditional validity of the Boussinesq approximation that in labora- tory experiments might be cause of concern ([1–5]). Nev- ertheless, in this context, by ideal experiments we mean ﬂows that can not be realized, even in an approximate way, by a real experimental apparatus. In particular we will consider the ﬂow developing in a cylindrical cell of aspect ratio (diameter over height) Γ =1/2 heated from below and cooled from above with an adiabatic side wall and with free–slip boundaries, thus preventing the for- mation of viscous boundary layers. The results will be then compared with previous and new simulations with the same temperature boundary conditions but with or- dinary no–slip walls. Another ideal experiment consists of simulations in which the ﬂow is forced to remain axisym- metric; this strongly alters the structure of the mean ﬂow with respect to the full three–dimensional conﬁguration thus allowing the analysis of the eﬀect of coherent large scales on the heat transfer. The motivation for this study comes from several pa- pers in which the roles of mean ﬂow and viscous boundary a e-mail: verzicco@poliba.it layers (and the kinetic energy therein dissipated) are con- sidered for the heat transfer as function of the maximum temperature diﬀerence in the system. These parameters are expressed in non dimensional form by the Nusselt Nu and Rayleigh Ra numbers (see next section for the deﬁni- tions) which can be considered, respectively, as response and input of the system. The seminal paper by Castaing et al. [6] showed the 2/7 power law in the Nu vs. Ra rela- tion for gaseous helium and, even if it is now clear that a simple power law does not ﬁt the whole curve, several the- ories have been proposed for the explanation of the scaling. Shraiman and Siggia [7], for example, were able to derive the correct power law by assuming a thermal boundary layer contained within the viscous one and a linear veloc- ity proﬁle induced by the latter; it can be shown that this is equivalent to assume that the most of the kinetic energy dissipation of the system occurs in the viscous boundary layers. The scenario was complicated by recent experiments in gaseous helium ([3,8–10]) in which the Nu vs. Ra re- lation showed a transition above Ra ≃ 10 11 toward a steeper power law, perhaps indicating the occurrence of the Kraichnan [11] asymptotic regime Nu ∼ Ra 1/2 . Ap- parently, all this complex dynamics is uniﬁed under the theory by Grossmann and Lohse [12,13] which classiﬁes the ﬂows in the Ra–Pr phase space according to the dom- inant contribution (bulk or boundary layer) to the ki- netic energy and temperature variance dissipation rates; the theory is also successful in predicting the Nu vs. Pr dependence and the Reynolds number variation with Ra and Pr (Pr is the Prandtl number deﬁned in the next