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Journal ArticleDOI

Correlation of internal flow structure with heat transfer efficiency in turbulent Rayleigh-B\'enard convection

TL;DR: In this paper, the authors investigated the correlation between different flow modes and the instantaneous Nusselt number in a two-dimensional square Rayleigh-Benard convection cell and showed that the volume-averaged and kinetic energy dissipation based flow modes can better reproduce the correlation of internal flow structures with heat transfer efficiency.
Abstract: To understand how internal flow structures manifest themselves in the global heat transfer, we study the correlation between different flow modes and the instantaneous Nusselt number ($Nu$) in a two-dimensional square Rayleigh-Benard convection cell. High-resolution and long-time direct numerical simulations are carried out for Rayleigh numbers between $10^{7}$ and $10^{9}$ and a Prandtl number of 5.3. The investigated Nusselt numbers include the volume-averaged $Nu_{\text{vol}}$, the wall-averaged $Nu_{\text{wall}}$, the kinetic energy dissipation based $Nu_{\text{kinetic}}$, and the thermal energy dissipation based $Nu_{\text{thermal}}$. The Fourier mode decomposition and proper orthogonal decomposition are adopted to extract the coherent flow structure. Our results show that the single-roll mode, the horizontally stacked double-roll mode, and the quadrupolar flow mode are more efficient for heat transfer on average. In contrast, the vertically stacked double-roll mode is inefficient for heat transfer on average. The volume-averaged $Nu_{\text{vol}}$ and the kinetic energy dissipation based $Nu_{\text{kinetic}}$ can better reproduce the correlation of internal flow structures with heat transfer efficiency than that of the wall-averaged $Nu_{\text{wall}}$ and the thermal energy dissipation based $Nu_{\text{thermal}}$, even though these four Nusselt numbers give consistent time-averaged mean values. The ensemble-averaged time trace of $Nu$ during flow reversal shows that only the volume-averaged $Nu_{\text{vol}}$ can reproduce the overshoot phenomena that is observed in the previous experimental study. Our results reveal that the proper choice of $Nu$ is critical to obtain a meaningful interpretation.
Citations
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Journal ArticleDOI
TL;DR: In this paper, a spatially developing transitional flow in a vertical channel with one side uniformly heated and subjected to random velocity fluctuations at the inlet has been performed, and two characteristic frequency bands are observed in the flow, near the heated wall.
Abstract: Numerical simulations of a spatially developing transitional flow in a vertical channel with one side uniformly heated and subjected to random velocity fluctuations at the inlet have been performed. Two characteristic frequency bands are observed in the flow, near the heated wall. The ability of the Proper Orthogonal Decomposition and the time-domain Spectral Proper Orthogonal Decomposition (SPOD) to decompose the flow is assessed, and SPOD is shown to be a powerful tool, as it is capable of separating the most energetic modes into two great families whose frequency content matches the frequency bands previously identified. The spatial structure of the modes is described, and their contribution to the turbulent heat transfer and velocity-temperature correlation is evaluated. Finally, the modes are linked to coherent structures that are observed in instantaneous visualizations of the flow, and a scenario of the development of the coherent structures in the laminar-turbulent transitional process is proposed.

9 citations

Journal ArticleDOI
TL;DR: In this paper, a cubic cavity under homogeneous and inhomogeneous heating for Rayleigh number Ra = 10 7 and Prandtl number Pr = 6.46 was numerically investigated.

7 citations

Journal ArticleDOI
TL;DR: In this paper, power spectra and structure functions (SFs) of the temperature and radial velocity fields, calculated in the radial and azimuthal directions, in annular centrifugal Rayleigh-B\'{e}nard convection (ACRBC) for Rayleigh number Ra $ \in [{10^8,{10^{11}}]], Prandtl number Pr = 10.7, and inverse Rossby number $\rm{Ro}^{-1} = 16$ using the spatial data obtained by quasi-two-dimensional direct numerical simulation.
Abstract: We analyse the power spectra and structure functions (SFs) of the temperature and radial velocity fields, calculated in the radial and azimuthal directions, in annular centrifugal Rayleigh-B\'{e}nard convection (ACRBC) for Rayleigh number Ra $ \in [{10^8},{10^{11}}]$, Prandtl number Pr = 10.7, and inverse Rossby number $\rm{Ro}^{-1} = 16$ using the spatial data obtained by quasi-two-dimensional direct numerical simulation. Bolgiano and Obukhov-like (BO59-like) scalings for the energy spectrum in both the azimuthal and radial directions and thermal spectrum in the azimuthal direction are observed. The range of BO59-like scaling becomes wider as Ra increases. At $\rm{Ra} = 10^{11}$, it is found that BO59-like scaling ${E_u}({k_r}) \sim {k_r}^{ - 11/5}$ spans nearly two decades for the energy spectrum calculated in the radial direction. Power-law fittings in the range larger than the Bolgiano scales, the scaling exponents of transverse and longitudinal velocity SFs versus the order coincide with the theoretical prediction of BO59 scaling $\zeta _p^u = 3p/5$ basically. The second-order temperature SFs exhibit a gradual transition from the Obukhov-Corrsin behaviour at scales smaller than the Bolgiano scales to the BO59 behaviour at scales larger than the Bolgiano scales. The slopes from the 3rd to 6th-order temperature SFs are similar, which is similar to classical Rayleigh-B\'{e}nard convection and Rayleigh-Taylor turbulence. The probability density functions (p.d.f.) of temperature fluctuations $\delta T/{\sigma _T}$ reveal the cold plumes are strong and the p.d.f. in different regions at high Ra are similar. The stronger turbulent-mixing and larger centrifugal buoyancy in ACRBC may result in the BO59-like scaling.

6 citations

Journal ArticleDOI
TL;DR: In this article, the authors investigate the statistics of dissipation rates in turbulent Rayleigh-Benard convection inside a cubic cell for air (P r = 0.7 ) in the Rayleigh number range 2 × 10 6 ≤ R a ≤ 10 9 using direct numerical simulations.

4 citations

References
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Journal ArticleDOI
TL;DR: An overview of the lattice Boltzmann method, a parallel and efficient algorithm for simulating single-phase and multiphase fluid flows and for incorporating additional physical complexities, is presented.
Abstract: We present an overview of the lattice Boltzmann method (LBM), a parallel and efficient algorithm for simulating single-phase and multiphase fluid flows and for incorporating additional physical complexities. The LBM is especially useful for modeling complicated boundary conditions and multiphase interfaces. Recent extensions of this method are described, including simulations of fluid turbulence, suspension flows, and reaction diffusion systems.

6,565 citations

Journal ArticleDOI
TL;DR: The Navier-Stokes equations are well-known to be a good model for turbulence as discussed by the authors, and the results of well over a century of increasingly sophisticated experiments are available at our disposal.
Abstract: It has often been remarked that turbulence is a subject of great scientific and technological importance, and yet one of the least understood (e.g. McComb 1990). To an outsider this may seem strange, since the basic physical laws of fluid mechanics are well established, an excellent mathematical model is available in the Navier-Stokes equations, and the results of well over a century of increasingly sophisticated experiments are at our disposal. One major difficulty, of course, is that the governing equations are nonlinear and little is known about their solutions at high Reynolds number, even in simple geometries. Even mathematical questions as basic as existence and uniqueness are unsettled in three spatial dimensions (cf Temam 1988). A second problem, more important from the physical viewpoint, is that experiments and the available mathematical evidence all indicate that turbulence involves the interaction of many degrees of freedom over broad ranges of spatial and temporal scales. One of the problems of turbulence is to derive this complex picture from the simple laws of mass and momentum balance enshrined in the NavierStokes equations. It was to this that Ruelle & Takens (1971) contributed with their suggestion that turbulence might be a manifestation in physical

3,721 citations

Journal ArticleDOI
TL;DR: In this article, a theoretical investigation of the spectrum of a turbulent fluid at large wave-numbers is presented, taking into account the two effects of convection with the fluid and molecular diffusion with diffusivity k. Hypotheses of the kind made by Kolmogoroff for the small-scale variations of velocity in a turbulent motion at high Reynolds number are assumed to apply also to small-size variations of θ.
Abstract: When some external agency imposes on a fluid large-scale variations of some dynamically passive, conserved, scalar quantity θ like temperature or concentration of solute, turbulent motion of the fluid generates small-scale variations of θ. This paper describes a theoretical investigation of the form of the spectrum of θ at large wave-numbers, taking into account the two effects of convection with the fluid and molecular diffusion with diffusivity k. Hypotheses of the kind made by Kolmogoroff for the small-scale variations of velocity in a turbulent motion at high Reynolds number are assumed to apply also to small-scale variations of θ.Previous contributions to the problem are reviewed. These have established that the spectrum of θ varies as , the result being given by (4.8). The same result is obtained, using essentially the same approximation about the velocity field, from a different kind of analysis in terms of velocity and θ correlations. Finally, the relation between this work and Townsend's model of the small-scale variations of vorticity in a turbulent fluid is discussed.

1,665 citations

Journal ArticleDOI
TL;DR: This work reviews many significant developments over the past decade of the lattice-Boltzmann method and discusses higherorder boundary conditions and the simulation of microchannel flow with finite Knudsen number.
Abstract: With its roots in kinetic theory and the cellular automaton concept, the lattice-Boltzmann (LB) equation can be used to obtain continuum flow quantities from simple and local update rules based on particle interactions. The simplicity of formulation and its versatility explain the rapid expansion of the LB method to applications in complex and multiscale flows. We review many significant developments over the past decade with specific examples. Some of the most active developments include the entropic LB method and the application of the LB method to turbulent flow, multiphase flow, and deformable particle and fiber suspensions. Hybrid methods based on the combination of the Eulerian lattice with a Lagrangian grid system for the simulation of moving deformable boundaries show promise for more efficient applications to a broader class of problems. We also discuss higherorder boundary conditions and the simulation of microchannel flow with finite Knudsen number. Additionally, the remarkable scalability of the LB method for parallel processing is shown with examples. Teraflop simulations with the LB method are routine, and there is no doubt that this method will be one of the first candidates for petaflop computational fluid dynamics in the near future.

1,585 citations

Book ChapterDOI
01 Jan 1991
TL;DR: In this article, the velocity components at each point P = (xi, x2, x3, t) of the region G under consideration belonging to the four-dimensional space were regarded as random variables in the sense of probability theory.
Abstract: §1. We denote by ua(P) = ua(xl, x2, x3, t), a= 1,2,3, the velocity components at time t at a point with rectangular Cartesian coordi­nates xi, x2, x3. When studying turbulence it is natural to regard the velocity components ua(P) at each point P = (xi, x2, x3, t) of the region G under consideration belonging to the four-dimensional space (xi, x2i X3, t) as random variables in the sense of probability theory (for this approach, see the paper by Millionshchikov [1]).

1,538 citations