scholarly journals Direct numerical simulation of free convection over a heated plate

2012 ◽  
Vol 712 ◽  
pp. 418-450 ◽  
Author(s):  
Juan Pedro Mellado
Keyword(s):  
Boundary Layer ◽  
Free Convection ◽  
Boundary Conditions ◽  
Similarity Theory ◽  
Buoyancy Flux ◽  
Heated Plate ◽  
Molecular Diffusivity ◽  
Rayleigh Numbers ◽  
Rayleigh Benard ◽  
Overlap Region

AbstractDirect numerical simulations of free convection over a smooth, heated plate are used to investigate unbounded, unsteady turbulent convection. Four different boundary conditions are considered: free-slip or no-slip walls, and constant buoyancy or constant buoyancy flux. It is first shown that, after the initial transient, the vertical structure agrees with observations in the atmospheric boundary layer and predictions from classical similarity theory. A quasi-steady inner layer and a self-preserving outer layer are clearly distinguished, with an overlap region between them of constant turbulent buoyancy flux. The extension of the overlap region reached in our simulations is more than 100 wall units$ \mathop{ ({\kappa }^{3} / {B}_{s} )}\nolimits ^{1/ 4} $, where${B}_{s} $is the surface buoyancy flux and$\kappa $the corresponding molecular diffusivity (the Prandtl number is one). The buoyancy fluctuation inside the overlap region already exhibits the$\ensuremath{-} 1/ 3$power-law scaling with height for the four types of boundary conditions, as expected in the local, free-convection regime. However, the mean buoyancy gradient and the vertical velocity fluctuation are still evolving toward the corresponding power laws predicted by the similarity theory. The second major result is that the relation between the Nusselt and Rayleigh numbers agrees with that reported in Rayleigh–Bénard convection when the heated plate is interpreted as half a convection cell. The range of Rayleigh numbers covered in the simulations is then$5\ensuremath{\times} 1{0}^{7} \text{{\ndash}} 1{0}^{9} $. Further analogies between the two problems indicate that knowledge can be transferred between steady Rayleigh–Bénard and unsteady convection. Last, we find that the inner scaling based on$\{ {B}_{s} , \hspace{0.167em} \kappa \} $reduces the effect of the boundary conditions to, mainly, the diffusive wall layer, the first 10 wall units. There, near the plate, free-slip conditions allow stronger mixing than no-slip ones, which results in 30 % less buoyancy difference between the surface and the overlap region and 30–40 % thinner diffusive sublayers. However, this local effect also entails one global, substantial effect: with an imposed buoyancy, free-slip systems develop a surface flux 60 % higher than that obtained with no-slip walls, which implies more intense turbulent fluctuations across the whole boundary layer and a faster growth.

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 Parameters for the Collapse of Turbulence in the Stratified Plane Couette Flow

2018 ◽  
Vol 75 (9) ◽  
pp. 3211-3231 ◽  
Author(s):  
Ivo G. S. van Hooijdonk ◽  
Herman J. H. Clercx ◽  
Cedrick Ansorge ◽  
Arnold F. Moene ◽  
Bas J. H. van de Wiel
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Conditions ◽  
Couette Flow ◽  
Neumann Boundary ◽  
Surface Flux ◽  
Shear Capacity ◽  
Buoyancy Flux ◽  
Dirichlet Boundary Conditions ◽  
Obukhov Length

Abstract We perform direct numerical simulation of the Couette flow as a model for the stable boundary layer. The flow evolution is investigated for combinations of the (bulk) Reynolds number and the imposed surface buoyancy flux. First, we establish what the similarities and differences are between applying a fixed buoyancy difference (Dirichlet) and a fixed buoyancy flux (Neumann) as boundary conditions. Moreover, two distinct parameters were recently proposed for the turbulent-to-laminar transition: the Reynolds number based on the Obukhov length and the “shear capacity,” a velocity-scale ratio based on the buoyancy flux maximum. We study how these parameters relate to each other and to the atmospheric boundary layer. The results show that in a weakly stratified equilibrium state, the flow statistics are virtually the same between the different types of boundary conditions. However, at stronger stratification and, more generally, in nonequilibrium conditions, the flow statistics do depend on the type of boundary condition imposed. In the case of Neumann boundary conditions, a clear sensitivity to the initial stratification strength is observed because of the existence of multiple equilibriums, while for Dirichlet boundary conditions, only one statistically steady turbulent equilibrium exists for a particular set of boundary conditions. As in previous studies, we find that when the imposed surface flux is larger than the maximum buoyancy flux, no turbulent steady state occurs. Analytical investigation and simulation data indicate that this maximum buoyancy flux converges for increasing Reynolds numbers, which suggests a possible extrapolation to the atmospheric case.

Instability in Boundary-Layer Free Convection along an inclined Plate

1972 ◽  
Vol 1 (4) ◽  
pp. 197-204 ◽  
Author(s):  
J.B. Lee ◽  
G.S.H. Lock
Keyword(s):  
Boundary Layer ◽  
Free Convection ◽  
Plane Surface ◽  
Convection Flow ◽  
Neutral Stability ◽  
Inclined Plate ◽  
Free Convection Flow ◽  
Roll Vortices ◽  
The Stability ◽  
Rayleigh Numbers

This paper gives theoretical consideration to the problem of the stability of laminar, boundary-layer, free-convection flow of air along a long, inclined plane surface heated isothermally. The analysis considers two forms of small disturbance: a two-dimensional wave disturbance, and a set of longitudinal roll vortices. Development of the appropriate disturbance equations is followed by their numerical solution. The effect of inclination on the neutral stability curves for both disturbance forms is presented graphically along with a comparison of the critical Rayleigh numbers obtained from both disturbance forms.

Similarity models for unsteady free convection flows along a differentially cooled horizontal surface

2013 ◽  
Vol 736 ◽  
pp. 444-463 ◽  
Author(s):  
Alan Shapiro ◽  
Evgeni Fedorovich
Keyword(s):  
Boundary Layer ◽  
Free Convection ◽  
Equations Of Motion ◽  
Single Step ◽  
Step Change ◽  
Horizontal Surface ◽  
Buoyancy Flux ◽  
Return Flow ◽  
Similarity Model ◽  
Surface Buoyancy

AbstractA class of unsteady free convection flows over a differentially cooled horizontal surface is considered. The cooling, specified in terms of an imposed negative buoyancy or buoyancy flux, varies laterally as a step function with a single step change. As thermal boundary layers develop on either side of the step change, an intrinsically unsteady, boundary-layer-like flow arises in the transition zone between them. Self-similarity model solutions of the Boussinesq equations of motion, thermal energy, and mass conservation, within a boundary-layer approximation, are obtained for flows of unstratified fluids driven by a surface buoyancy or buoyancy flux, and flows of stably stratified fluids driven by a surface buoyancy flux. The motion is characterized by a shallow, primarily horizontal flow capped by a weak return flow. Stratification weakens the primary flow and strengthens the return flow. The flows intensify as the step change in surface forcing increases or as the Prandtl number decreases. Simple formulas are obtained for the propagation speeds, trajectories and the evolution of velocity maxima and other local extrema. Similarity-model predictions are verified through numerical simulations in which no boundary-layer approximations are made.

Resolving urban canopy flows: Scales, turbulence and boundary conditions in obstacle-resolving numerical models

2021 ◽  
Author(s):  
Benedikt Seitzer ◽  
Bernd Leitl ◽  
Frank Harms
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Atmospheric Boundary Layer ◽  
Similarity Theory ◽  
Urban Canopy ◽  
Large Eddy Simulations ◽  
Roughness Sublayer ◽  
Boundary Layer Meteorol ◽  
Large Eddy ◽  
Near Wall

<p>Large-eddy simulations are increasingly used for studying the atmospheric boundary layer. With increasing computational resources even obstacle-resolving Large-eddy simulations became possible and will be used in urban climate studies more frequently. In these applications, grid sizes are in the order of a few meters. Whereas major urban structures can be resolved in general, details like aerodynamically rough surface structures can not be resolved explicitly. Based on the original fields of application, boundary conditions in Large-eddy simulations were initially formulated for surfaces of homogeneous roughness and for wall-distances much larger than the roughness sublayer height (Hultmark et al., 2013). The height of the roughness sublayer depends on the size of small-scale obstacles present on the surface exposed to the flow (Raupach et al., 1991). Typically, boundary conditions are evaluated between the surface and the first grid level. Thus, grid resolution in obstacle-resolved Large-Eddy simulations should also be a question of scales and therefore has to be chosen carefully (Basu and Lacser, 2017; Maronga et al., 2020). <br />In several wind tunnel experiments presented here, we measured the near-wall influence of differently scaled and shaped objects on a flow and its turbulence characteristics. Experimental setups were replicated numerically using the PALM model (Maronga et al. 2019). In a first, more generic experiment, the flow over horizontally homogeneous surfaces of different roughness was investigated. In a second experiment, the spatial separation of the turbulence scales was investigated in a more complex flow case. These experiments lead to considerations on model grid sizes in urban type Large-eddy simulations. The limitations of interpreting simulation results within the urban canopy layer are highlighted. There is an urgent need to reconsider how near-wall results of urban large-eddy simulations are generated and interpreted in the context of practical applications like flow and transport modelling in urban canopies. <br /><br /><em><strong>References</strong></em><br /><em>Basu, S. and Lacser, A. (2017). A Cautionary Note on the Use of Monin–Obukhov Similarity Theory in Very High-Resolution Large-Eddy Simulations. Boundary-Layer Meteorol, 163(2):351–355.</em></p> <p><em>Hultmark, M., Calaf, M., and Parlange, M. B. (2013). A new wall shearstress model for atmospheric boundary layer simulations. J Atmos Sci,70(11):3460–3470.</em></p> <p><em>Maronga, B., et al. (2020). Overview of the PALM model system 6.0. Geosci Model Dev Discussions, 06(June):1–63.</em></p> <p><em>Maronga, B., Knigge, C., and Raasch, S. (2020). An Improved Surface Boundary Condition for Large-Eddy Simulations Based on Monin–Obukhov Similarity Theory: Evaluation and Consequences forGrid Convergence in Neutral and Stable Conditions. Boundary-Layer Meteorol, 174(2):297–325.</em></p> <p><em>Raupach, M. R., Antonia, R. A., and Rajagopalan, S. (1991). Rough-wall turbulent boundary layers. Appl Mech Rev, 44(1):1–25</em></p>

Similarity Solution for Unsteady Free Convection From a Vertical Plate at Constant Temperature to Power Law Fluids

2012 ◽  
Vol 134 (10) ◽  
Author(s):  
J. Abolfazli Esfahani ◽  
B. Bagherian
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Free Convection ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Power Law ◽  
Thermal Boundary Layer ◽  
Theoretic Approach ◽  
Local Skin ◽  
Power Law Fluids

The transformation group theoretic approach is applied to perform an analysis of unsteady free convection flow over a vertical flat plate immersed in a power law fluid. The thermal boundary layer induced within a vertical semi-infinite layer of Boussinseq fluid. The system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions via two-parameter group theory. The obtained ordinary differential equations are solved numerically for velocity and temperature using the fourth order Runge-Kutta and shooting method. The effect of Prandtl number and viscosity index (n) on the thermal boundary-layer, velocity boundary-layer, local Nusselt number, and local skin-friction were studied.

Horizontal convection dynamics: insights from transient adjustment

2013 ◽  
Vol 726 ◽  
pp. 559-595 ◽  
Author(s):  
Ross W. Griffiths ◽  
Graham O. Hughes ◽  
Bishakhdatta Gayen
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Time Scales ◽  
Time Scale ◽  
Mechanical Energy ◽  
Buoyancy Flux ◽  
Small Scale ◽  
Vertical Diffusion ◽  
Practical Applications ◽  
Horizontal Convection

AbstractThe dynamics of horizontal convection are revealed by examining transient adjustment toward thermal equilibrium. We restrict attention to high Rayleigh numbers (of $O(1{0}^{12} )$) and a Prandtl number ${\sim }5$ that characterize many practical applications, and consider responses to small changes in the thermal boundary conditions, using laboratory experiments, three-dimensional direct numerical simulations (DNS) and simple theoretical models. The experiments and the mechanical energy budget from the DNS demonstrate that unsteady forcing can produce flow dramatically more active than horizontal convection under steady forcing. The physical mechanisms at work are indicated by the time scales of approach to the new equilibrium, and we show that these can range over two orders of magnitude depending on the imposed change in boundary conditions. Changes that lead to a net destabilizing buoyancy flux give rapid adjustments: for applied heat flux conditions the whole of the circulation is controlled by conduction through the stable portion of the boundary layer, whereas for applied temperature difference the circulation is controlled by small-scale convection within the unstable part of the boundary layer. The experiments, DNS and models are in close agreement and show that the time scale under applied temperatures is as small as 0.01 vertical diffusion time scales, a factor of four smaller than for imposed flux. Both cases give adjustments too rapid for diffusion in the interior to play a significant role, at least through 99 % of the adjustment, and we conclude that diffusion through the full depth is not significant in setting the equilibrium state. Boundary changes leading to a net stabilizing buoyancy flux give a very different response, causing the convection to quickly form a shallow circulation cell, followed eventually by a return to full-depth overturning through a combination of penetrative convection and conduction. The time scale again varies by orders of magnitude, depending on the boundary conditions and the location of the imposed boundary perturbation.

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