Nusselt Number Correlations for Turbulent Natural Convection Flows Using Asymptotic Analysis of the Near-Wall Region

2006 ◽  
Vol 129 (8) ◽  
pp. 1100-1105 ◽  
Author(s):  
C. Balaji ◽  
M. Hölling ◽  
H. Herwig
Keyword(s):  
Nusselt Number ◽  
Plate Boundary ◽  
Core Layer ◽  
Wall Layer ◽  
Wall Region ◽  
Complex Flows ◽  
Logarithmic Term ◽  
Turbulent Natural Convection ◽  
C Complex ◽  
Rayleigh Benard

In this paper, we propose a general methodology by which a universal temperature profile, derived by matching temperature gradients in the overlap region of the wall layer and the core layer, that is valid for high Rayleigh number flows, can be recast into a correlation for the Nusselt number. We then evaluate its adequacy for three types of flows: (a) fully developed flows (e.g., the infinite channel), (b) developing flows (e.g., the vertical flat plate boundary layer), and (c) complex flows (e.g., Rayleigh-Bénard convection and the differentially heated square cavity). The correlation for the Nusselt number, in general, contains a logarithmic term, usually missing in earlier studies, with which we are able to match existing direct numerical simulations or experimental data very well for both fully developed and complex flows.

scholarly journals Transition to ultimate Rayleigh–Bénard turbulence revealed through extended self-similarity scaling analysis of the temperature structure functions

2018 ◽  
Vol 851 ◽  
Author(s):  
Dominik Krug ◽  
Xiaojue Zhu ◽  
Daniel Chung ◽  
Ivan Marusic ◽  
Roberto Verzicco ◽  
...  
Keyword(s):  
Nusselt Number ◽  
Structure Functions ◽  
Wall Turbulence ◽  
Scaling Analysis ◽  
Wall Region ◽  
Temperature Structure ◽  
Self Similarity ◽  
Extended Self ◽  
Scalar Structure ◽  
Rayleigh Benard

In turbulent Rayleigh–Bénard (RB) convection, a transition to the so-called ultimate regime, in which the boundary layers (BL) are of turbulent type, has been postulated. Indeed, at very large Rayleigh number $Ra\approx 10^{13}{-}10^{14}$ a transition in the scaling of the global Nusselt number $Nu$ (the dimensionless heat transfer) and the Reynolds number with $Ra$ has been observed in experiments and very recently in direct numerical simulations (DNS) of two-dimensional (2D) RB convection. In this paper, we analyse the local scaling properties of the lateral temperature structure functions in the BLs of this simulation of 2D RB convection, employing extended self-similarity (ESS) (i.e., plotting the structure functions against each other, rather than only against the scale) in the spirit of the attached-eddy hypothesis, as we have recently introduced for velocity structure functions in wall turbulence (Krug et al., J. Fluid Mech., vol. 830, 2017, pp. 797–819). We find no ESS scaling at $Ra$ below the transition and in the near-wall region. However, beyond the transition and for large enough wall distance $z^{+}>100$, we find clear ESS behaviour, as expected for a scalar in a turbulent boundary layer. In striking correspondence to the $Nu$ scaling, the ESS scaling region is negligible at $Ra=10^{11}$ and well developed at $Ra=10^{14}$, thus providing strong evidence that the observed transition in the global Nusselt number at $Ra\approx 10^{13}$ indeed is the transition from a laminar type BL to a turbulent type BL. Our results further show that the relative slopes for scalar structure functions in the ESS scaling regime are the same as for their velocity counterparts, extending their previously established universality. The findings are confirmed by comparing to scalar structure functions in three-dimensional turbulent channel flow.

Direct numerical simulations of Rayleigh–Bénard convection in water with non-Oberbeck–Boussinesq effects

2019 ◽  
Vol 881 ◽  
pp. 1073-1096 ◽  
Author(s):  
Andreas D. Demou ◽  
Dimokratis G. E. Grigoriadis
Keyword(s):  
Numerical Simulations ◽  
Two Dimensions ◽  
Direct Numerical Simulations ◽  
Wall Region ◽  
Number Range ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Near Wall

Rayleigh–Bénard convection in water is studied by means of direct numerical simulations, taking into account the variation of properties. The simulations considered a three-dimensional (3-D) cavity with a square cross-section and its two-dimensional (2-D) equivalent, covering a Rayleigh number range of $10^{6}\leqslant Ra\leqslant 10^{9}$ and using temperature differences up to 60 K. The main objectives of this study are (i) to investigate and report differences obtained by 2-D and 3-D simulations and (ii) to provide a first appreciation of the non-Oberbeck–Boussinesq (NOB) effects on the near-wall time-averaged and root-mean-squared (r.m.s.) temperature fields. The Nusselt number and the thermal boundary layer thickness exhibit the most pronounced differences when calculated in two dimensions and three dimensions, even though the $Ra$ scaling exponents are similar. These differences are closely related to the modification of the large-scale circulation pattern and become less pronounced when the NOB values are normalised with the respective Oberbeck–Boussinesq (OB) values. It is also demonstrated that NOB effects modify the near-wall temperature statistics, promoting the breaking of the top–bottom symmetry which characterises the OB approximation. The most prominent NOB effect in the near-wall region is the modification of the maximum r.m.s. values of temperature, which are found to increase at the top and decrease at the bottom of the cavity.

Numerical simulations of Rayleigh–Bénard convection for Prandtl numbers between 10−1 and 104 and Rayleigh numbers between 105 and 109

2010 ◽  
Vol 662 ◽  
pp. 409-446 ◽  
Author(s):  
G. SILANO ◽  
K. R. SREENIVASAN ◽  
R. VERZICCO
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Nusselt Number ◽  
Numerical Simulations ◽  
Multiple Solutions ◽  
Large Scale ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

We summarize the results of an extensive campaign of direct numerical simulations of Rayleigh–Bénard convection at moderate and high Prandtl numbers (10−1 ≤ Pr ≤ 104) and moderate Rayleigh numbers (105 ≤ Ra ≤ 109). The computational domain is a cylindrical cell of aspect ratio Γ = 1/2, with the no-slip condition imposed on all boundaries. By scaling the numerical results, we find that the free-fall velocity should be multiplied by $1/\sqrt{{\it Pr}}$ in order to obtain a more appropriate representation of the large-scale velocity at high Pr. We investigate the Nusselt and the Reynolds number dependences on Ra and Pr, comparing the outcome with previous numerical and experimental results. Depending on Pr, we obtain different power laws of the Nusselt number with respect to Ra, ranging from Ra2/7 for Pr = 1 up to Ra0.31 for Pr = 103. The Nusselt number is independent of Pr. The Reynolds number scales as ${\it Re}\,{\sim}\,\sqrt{{\it Ra}}/{\it Pr}$, neglecting logarithmic corrections. We analyse the global and local features of viscous and thermal boundary layers and their scaling behaviours with respect to Ra and Pr, and with respect to the Reynolds and Péclet numbers. We find that the flow approaches a saturation state when Reynolds number decreases below the critical value, Res ≃ 40. The thermal-boundary-layer thickness increases slightly (instead of decreasing) when the Péclet number increases, because of the moderating influence of the viscous boundary layer. The simulated ranges of Ra and Pr contain steady, periodic and turbulent solutions. A rough estimate of the transition from the steady to the unsteady state is obtained by monitoring the time evolution of the system until it reaches stationary solutions. We find multiple solutions as long-term phenomena at Ra = 108 and Pr = 103, which, however, do not result in significantly different Nusselt numbers. One of these multiple solutions, even if stable over a long time interval, shows a break in the mid-plane symmetry of the temperature profile. We analyse the flow structures through the transitional phases by direct visualizations of the temperature and velocity fields. A wide variety of large-scale circulation and plume structures has been found. The single-roll circulation is characteristic only of the steady and periodic solutions. For other regimes at lower Pr, the mean flow generally consists of two opposite toroidal structures; at higher Pr, the flow is organized in the form of multi-jet structures, extending mostly in the vertical direction. At high Pr, plumes mainly detach from sheet-like structures. The signatures of different large-scale structures are generally well reflected in the data trends with respect to Ra, less in those with respect to Pr.

scholarly journals Numerical study of the turbulent natural convection of nanofluids in a partially heated cubic cavity

2021 ◽  
pp. 57-57
Author(s):  
Zakaria Lafdaili ◽  
Sakina El-Hamdani ◽  
Abdelaziz Bendou ◽  
Karim Limam ◽  
Bara El-Hafad
Keyword(s):  
Nusselt Number ◽  
Natural Convection ◽  
Heat Exchange ◽  
Metallic Nanoparticles ◽  
Volume Fraction ◽  
Numerical Study ◽  
Three Dimensional ◽  
Stability Characteristic ◽  
Horizontal Strip ◽  
Turbulent Natural Convection

In this work we study numerically the three-dimensional turbulent natural convection in a partially heated cubic cavity filled with water containing metallic nanoparticles, metallic oxides and others based on carbon.The objective is to study and compare the effect of the addition of nanoparticles studied in water and also the effect of the position of the heated partition on the heat exchange by turbulent natural convection in this type of geometry, which can significantly improve the design of heat exchange systems for better space optimization. For this we have treated numerically for different volume fractions the turbulent natural convection in the two cases where the cavity is heated respectively by a vertical and horizontal strip in the middle of one of the vertical walls. To take into account the effects of turbulence, we used the standard turbulence model ? - ?. The governing equations are discretized by the finite volume method using the power law scheme which offers a good stability characteristic in this type of flow. The results are presented in the form of isothermal lines and current lines. The variation of the mean Nusselt number is calculated for the two positions of the heated partition as a function of the volume fraction of the nanoparticles studied in water for different Rayleigh numbers.The results show that carbon-based nanoparticles intensify heat exchange by convection better and that the position of the heated partition significantly influences heat exchange by natural convection. In fact, an improvement in the average Nusselt number of more than 20% is observed for the case where the heated partition is horizontal.

scholarly journals Bulk scaling in wall-bounded and homogeneous vertical natural convection

2018 ◽  
Vol 841 ◽  
pp. 825-850 ◽  
Author(s):  
Chong Shen Ng ◽  
Andrew Ooi ◽  
Detlef Lohse ◽  
Daniel Chung
Keyword(s):  
Nusselt Number ◽  
Natural Convection ◽  
Temperature Gradient ◽  
Power Law ◽  
Exact Relation ◽  
Bénard Convection ◽  
Benard Convection ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Previous numerical studies on homogeneous Rayleigh–Bénard convection, which is Rayleigh–Bénard convection (RBC) without walls, and therefore without boundary layers, have revealed a scaling regime that is consistent with theoretical predictions of bulk-dominated thermal convection. In this so-called asymptotic regime, previous studies have predicted that the Nusselt number ($\mathit{Nu}$) and the Reynolds number ($\mathit{Re}$) vary with the Rayleigh number ($\mathit{Ra}$) according to $\mathit{Nu}\sim \mathit{Ra}^{1/2}$ and $\mathit{Re}\sim \mathit{Ra}^{1/2}$ at small Prandtl numbers ($\mathit{Pr}$). In this study, we consider a flow that is similar to RBC but with the direction of temperature gradient perpendicular to gravity instead of parallel to it; we refer to this configuration as vertical natural convection (VC). Since the direction of the temperature gradient is different in VC, there is no exact relation for the average kinetic dissipation rate, which makes it necessary to explore alternative definitions for $\mathit{Nu}$, $\mathit{Re}$ and $\mathit{Ra}$ and to find physical arguments for closure, rather than making use of the exact relation between $\mathit{Nu}$ and the dissipation rates as in RBC. Once we remove the walls from VC to obtain the homogeneous set-up, we find that the aforementioned $1/2$-power-law scaling is present, similar to the case of homogeneous RBC. When focusing on the bulk, we find that the Nusselt and Reynolds numbers in the bulk of VC too exhibit the $1/2$-power-law scaling. These results suggest that the $1/2$-power-law scaling may even be found at lower Rayleigh numbers if the appropriate quantities in the turbulent bulk flow are employed for the definitions of $\mathit{Ra}$, $\mathit{Re}$ and $\mathit{Nu}$. From a stability perspective, at low- to moderate-$\mathit{Ra}$, we find that the time evolution of the Nusselt number for homogenous vertical natural convection is unsteady, which is consistent with the nature of the elevator modes reported in previous studies on homogeneous RBC.

Effect of Sidewall Conductance on Nusselt Number for Rayleigh-Bénard Convection: A Semi-Analytical and Experimental Correction

2019 ◽  
Vol 141 (12) ◽  
Author(s):  
U. Madanan ◽  
R. J. Goldstein
Keyword(s):  
Thermal Conductivity ◽  
Nusselt Number ◽  
Analytical Model ◽  
Thermal Conductance ◽  
Bénard Convection ◽  
Benard Convection ◽  
Extended Surfaces ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Good Agreement

Abstract The effect of sidewall conductance on Nusselt number for the Rayleigh-Bénard convection is examined by performing nearly identical sets of experiments with sidewalls made of three different materials. These experimental results are utilized to extrapolate and estimate the Nusselt number for an ideal zero-thermal-conductivity sidewall case, which is the case when the sidewalls are perfectly insulating. A semi-analytical model is proposed, based on the concept of extended surfaces, to compute the discrepancy in Nusselt number caused by the presence of finite thermal conductance of the sidewalls. The predictions obtained using this model are found to be in good agreement with the corresponding experimentally determined values.

Molecular Gas Radiation in a Laminar or Turbulent Pipe Flow

1976 ◽  
Vol 98 (1) ◽  
pp. 101-107 ◽  
Author(s):  
A. T. Wassel ◽  
D. K. Edwards
Keyword(s):  
Nusselt Number ◽  
Internal Heat ◽  
Heat Fluxes ◽  
Turbulent Pipe Flow ◽  
Band Model ◽  
Wall Region ◽  
Major Band ◽  
Turbulent Reynolds Number ◽  
Source Case ◽  
Gas Layer

The temperature profile and the radiative and convective wall-heat fluxes for a hydro-dynamically established flow of a nongray gas in a cylinder are calculated numerically for laminar or turbulent flow. Turbulent eddy viscosity is represented by a two-region model having a near-wall-region and a far-from-wall region. Gas properties are represented by the exponential-winged band model. The controlling parameters are Nrm, the ratio of radiation conductance to molecular conductance, τR,i, the maximum spectral optical depth of the radius for the ith gas band, Wi, a measure of the importance of the ith band, and Rt, the turbulent Reynolds number. Qualitatively the results for a gas with a single major band agree with a previous solution for a cylinder with internal heat generation. Radiative Nusselt number NuR increases nearly linearly with Nrm. Radiative flux increases with τR rapidly at first and then only slowly. Increasing Rt increases markedly the convective Nusselt number NuC and appreciably NuR. The gas layer effective transmissivity was found to increase with increasing Rt and decrease with increasing τR. Quantitative comparisons with the constant-volume-heat-source case show discrepancies of up to 10 percent in NuR and up to 25 percent in NuC.

Numerical simulation of the turbulent Rayleigh–Bénard problem using subgrid modelling

1985 ◽  
Vol 158 ◽  
pp. 245-268 ◽  
Author(s):  
Thomas M. Eidson
Keyword(s):  
Numerical Simulation ◽  
Small Scale ◽  
Direct Simulation ◽  
Turbulent Natural Convection ◽  
Eddy Simulation ◽  
Subgrid Model ◽  
Short Period ◽  
Bénard Problem ◽  
Benard Problem ◽  
Rayleigh Benard

A numerical simulation of turbulent natural convection (the Rayleigh–Bénard problem) has been conducted using large-eddy-simulation (LES) methods and the results compared with several experiments. The development of the LES equation is outlined and discussed. The modelling of the small-scale turbulent motion (called subgrid modelling) is also discussed. The resulting LES equations are solved and data collected over a short period of time in a similar manner to the direct simulation of the governing conservation equations. An explicit, second-order accurate, finite-difference scheme is used to solve the equations. Various average properties of the resulting flow field are calculated from the data and compared with experimental data in the literature. The use of a subgrid model allows a higher value of Ra to be simulated than was previously possible with a direct simulation. The highest Ra successfully simulated was 2.5 × 106. The problems at higher values of Ra are discussed and suggestions for improvements made.

The minimal flow unit in near-wall turbulence

1991 ◽  
Vol 225 ◽  
pp. 213-240 ◽  
Author(s):  
Javier Jiménez ◽  
Parviz Moin
Keyword(s):  
Wall Stress ◽  
Wall Layer ◽  
Flow Unit ◽  
Wall Turbulence ◽  
Longitudinal Vortex ◽  
Spanwise Direction ◽  
Wall Region ◽  
Low Velocity ◽  
Good Agreement ◽  
Near Wall

Direct numerical simulations of unsteady channel flow were performed at low to moderate Reynolds numbers on computational boxes chosen small enough so that the flow consists of a doubly periodic (in x and z) array of identical structures. The goal is to isolate the basic flow unit, to study its morphology and dynamics, and to evaluate its contribution to turbulence in fully developed channels. For boxes wider than approximately 100 wall units in the spanwise direction, the flow is turbulent and the low-order turbulence statistics are in good agreement with experiments in the near-wall region. For a narrow range of widths below that threshold, the flow near only one wall remains turbulent, but its statistics are still in fairly good agreement with experimental data when scaled with the local wall stress. For narrower boxes only laminar solutions are found. In all cases, the elementary box contains a single low-velocity streak, consisting of a longitudinal strip on which a thin layer of spanwise vorticity is lifted away from the wall. A fundamental period of intermittency for the regeneration of turbulence is identified, and that process is observed to consist of the wrapping of the wall-layer vorticity around a single inclined longitudinal vortex.

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