Finite Volume vs.vs. Streaming-based Lattice Boltzmann algorithm for fluid-dynamics simulations: a one-to-one accuracy and performance study

A new finite volume (FV) discretisation method for the Lattice Boltzmann (LB) equation which combines high accuracy with limited computational cost is presented. In order to assess the performance of the FV method we carry out a systematic comparison, focused on accuracy and computational performances, with the standard $streaming$ (ST) Lattice Boltzmann equation algorithm. To our knowledge such a systematic comparison has never been previously reported. In particular we aim at clarifying whether and in which conditions the proposed algorithm, and more generally any FV algorithm, can be taken as the method of choice in fluid-dynamics LB simulations. For this reason the comparative analysis is further extended to the case of realistic flows, in particular thermally driven flows in turbulent conditions. We report the first successful simulation of high-Rayleigh number convective flow performed by a Lattice Boltzmann FV based algorithm with wall grid refinement.Comment: 15 pages, 14 figures (discussion changes, improved figure readability

Theories of convection and the spectrum of turbulence in the solar photosphere

Classical theories of turbulence do not describe accurately inertial range scaling laws in turbulent convection and notably fail to model the shape of the turbulent spectrum of solar photospheric convection. To understand these discrepancies, a detailed study of scale-by-scale budgets in turbulent Rayleigh-B\'enard convection is presented, with particular emphasis placed on anisotropy and inhomogeneity. A generalized Kolmogorov equation applying to convection is derived and its various terms are computed using numerical simulations of turbulent Boussinesq convection. The analysis of the isotropic part of the equation shows that the third-order velocity structure function is significantly affected by buoyancy forcing and large-scale inhomogeneities. Anisotropic contributions to this equation are also shown to be comparable to their isotropic counterpart at moderate to large scales. Implications of these results for convection in the solar photosphere, mesogranulation and supergranulation are discussed.Comment: 6 pages, 3 figures -- To appear in the Proceedings of Symposium no. 239 "Convection in Astrophysics", International Astronomical Union., held 21-25 August, 2006 in Prague, Czech Republi

Universality of anisotropic fluctuations from numerical simulations of turbulent flows

We present new results from a direct numerical simulation of a three dimensional homogeneous Rayleigh-Benard system (HRB), i.e. a convective cell with an imposed linear mean temperature profile along the vertical direction. We measure the SO(3)-decomposition of both velocity structure functions and buoyancy terms. We give a dimensional prediction for the values of the anisotropic scaling exponents in this Rayleigh-Benard systems. Measured scaling does not follow dimensional estimate, while a better agreement can be found with the anisotropic scaling of a different system, the random-Kolmogorov-flow (RKF). Our findings support the conclusion that scaling properties of anisotropic fluctuations are universal, i.e. independent of the forcing mechanism sustaining the turbulent flow.Comment: 4 pages, 3 figure

The computational modeling of inferential and referential competence

Abstract

Rayleigh and Prandtl number scaling in the bulk of Rayleigh-Benard turbulence

The Rayleigh (Ra) and Prandtl (Pr) number scaling of the Nusselt number Nu, the Reynolds number Re, the temperature fluctuations, and the kinetic and thermal dissipation rates is studied for (numerical) homogeneous Rayleigh-Benard turbulence, i.e., Rayleigh-Benard turbulence with periodic boundary conditions in all directions and a volume forcing of the temperature field by a mean gradient. This system serves as model system for the bulk of Rayleigh-Benard flow and therefore as model for the so called ``ultimate regime of thermal convection''. With respect to the Ra dependence of Nu and Re we confirm our earlier results \cite{loh03} which are consistent with the Kraichnan theory \cite{kra62} and the Grossmann-Lohse (GL) theory \cite{gro00,gro01,gro02,gro04}, which both predict $Nu \sim Ra^{1/2}$ and $Re \sim Ra^{1/2}$. However the Pr dependence within these two theories is different. Here we show that the numerical data are consistent with the GL theory $Nu \sim Pr^{1/2}$, $Re \sim Pr^{-1/2}$. For the thermal and kinetic dissipation rates we find \eps_\theta/(\kappa \Delta^{2}L^{-2}) \sim (Re Pr)^{0.87} and \eps_u/(\nu^3 L^{-4}) \sim Re^{2.77}, also both consistent with the GL theory, whereas the temperature fluctuations do not depend on Ra and Pr. Finally, the dynamics of the heat transport is studied and put into the context of a recent theoretical finding by Doering et al. \cite{doe05}.Comment: 8 pages, 9 figure

Non-Oberbeck-Boussinesq effects in two-dimensional Rayleigh-Benard convection in glycerol

We numerically analyze Non-Oberbeck-Boussinesq (NOB) effects in two-dimensional Rayleigh-Benard flow in glycerol, which shows a dramatic change in the viscosity with temperature. The results are presented both as functions of the Rayleigh number (Ra) up to $10^8$ (for fixed temperature difference between the top and bottom plates) and as functions of "non-Oberbeck-Boussinesqness'' or "NOBness'' ($\Delta$) up to 50 K (for fixed Ra). For this large NOBness the center temperature $T_c$ is more than 5 K larger than the arithmetic mean temperature $T_m$ between top and bottom plate and only weakly depends on Ra. To physically account for the NOB deviations of the Nusselt numbers from its Oberbeck-Boussinesq values, we apply the decomposition of $Nu_{NOB}/Nu_{OB}$ into the product of two effects, namely first the change in the sum of the top and bottom thermal BL thicknesses, and second the shift of the center temperature $T_c$ as compared to $T_m$. While for water the origin of the $Nu$ deviation is totally dominated by the second effect (cf. Ahlers et al., J. Fluid Mech. 569, pp. 409 (2006)) for glycerol the first effect is dominating, in spite of the large increase of $T_c$ as compared to $T_m$.Comment: 6 pages, 7 figure

Velocity gradients statistics along particle trajectories in turbulent flows: the refined similarity hypothesis in the Lagrangian frame

We present an investigation of the statistics of velocity gradient related quantities, in particluar energy dissipation rate and enstrophy, along the trajectories of fluid tracers and of heavy/light particles advected by a homogeneous and isotropic turbulent flow. The Refined Similarity Hypothesis (RSH) proposed by Kolmogorov and Oboukhov in 1962 is rephrased in the Lagrangian context and then tested along the particle trajectories. The study is performed on state-of-the-art numerical data resulting from numerical simulations up to Re~400 with 2048^3 collocation points. When particles have small inertia, we show that the Lagrangian formulation of the RSH is well verified for time lags larger than the typical response time of the particle. In contrast, in the large inertia limit when the particle response time approaches the integral-time-scale of the flow, particles behave nearly ballistic, and the Eulerian formulation of RSH holds in the inertial-range.Comment: 7 pages, 7 figures; Physical Review E (accepted Dec 7, 2009

Lagrangian single particle turbulent statistics through the Hilbert-Huang Transform

The Hilbert-Huang transform is applied to analyze single particle Lagrangian velocity data from numerical simulations of hydrodynamic turbulence. The velocity trajectory is described in terms of a set of intrinsic mode functions, C_{i}(t), and of their instantaneous frequency, \omega_{i}(t). On the basis of this decomposition we define the \omega-conditioned statistical moments of the C_{i} modes, named q-order Hilbert Spectra (HS). We show that such new quantities have enhanced scaling properties as compared to traditional Fourier transform- or correlation-based (Structure Functions) statistical indicators, thus providing better insights into the turbulent energy transfer process. We present a clear empirical evidence that the energy-like quantity, i.e. the second-order HS, displays a linear scaling in time in the inertial range, as expected from dimensional analysis and never observed before. We also measure high order moment scaling exponents in a direct way, without resorting the Extended Self Similarity (ESS) procedure. This leads to a new estimate of the Lagrangian structure functions exponents which are consistent with the multifractal prediction in the Lagrangian frame as proposed in [Biferale et al., Phys. Rev. Lett. vol. 93, 064502 (2004)].Comment: 5 pages, 5 figure

Impact of trailing wake drag on the statistical properties and dynamics of finite-sized particle in turbulence

We study by means of an Eulerian-Lagrangian model the statistical properties of velocity and acceleration of a neutrally-buoyant finite-sized particle in a turbulent flow statistically homogeneous and isotropic. The particle equation of motion, beside added mass and steady Stokes drag, keeps into account the unsteady Stokes drag force - known as Basset-Boussinesq history force - and the non-Stokesian drag based on Schiller-Naumann parametrization, together with the finite-size Faxen corrections. We focus on the case of flow at low Taylor-Reynolds number, Re_lambda ~ 31, for which fully resolved numerical data which can be taken as a reference are available (Homann & Bec 651 81-91 J. Fluid Mech. (2010)). Remarkably, we show that while drag forces have always minor effects on the acceleration statistics, their role is important on the velocity behavior. We propose also that the scaling relations for the particle velocity variance as a function of its size, which have been first detected in fully resolved simulations, does not originate from inertial-scale properties of the background turbulent flow but it is likely to arise from the non-Stokesian component of the drag produced by the wake behind the particle. Furthermore, by means of comparison with fully resolved simulations, we show that the Faxen correction to the added mass has a dominant role in the particle acceleration statistics even for particle with size in the inertial range.Comment: 9 pages, 9 figure