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.

scholarly journals Scaling Laws for the Heterogeneously Heated Free Convective Boundary Layer

2014 ◽  
Vol 71 (11) ◽  
pp. 3975-4000 ◽  
Author(s):  
Chiel C. van Heerwaarden ◽  
Juan Pedro Mellado ◽  
Alberto De Lozar
Keyword(s):  
Boundary Layer ◽  
Kinetic Energy ◽  
Convective Boundary Layer ◽  
Scaling Laws ◽  
Fixed Ratio ◽  
Buoyancy Flux ◽  
Convective Boundary ◽  
Optimal State ◽  
Surface Buoyancy ◽  
Surface Buoyancy Flux

Abstract The heterogeneously heated free convective boundary layer (CBL) is investigated by means of dimensional analysis and results from large-eddy simulations (LES) and direct numerical simulations (DNS). The investigated physical model is a CBL that forms in a linearly stratified atmosphere heated from the surface by square patches with a high surface buoyancy flux. Each simulation has been run long enough to show the formation of a peak in kinetic energy, corresponding to the “optimal” heterogeneity size with strong secondary circulations, and the subsequent transition into a horizontally homogeneous CBL. Scaling laws for the time of the optimal state and transition and for the vertically integrated kinetic energy (KE) have been developed. The laws show that the optimal state and transition do not occur at a fixed ratio of the heterogeneity size to the CBL height. Instead, these occur at a higher ratio for simulations with increasing heterogeneity sizes because of the development of structures in the downward-moving air that grow faster than the CBL thickness. The moment of occurrence of the optimal state and transition are strongly related to the heterogeneity amplitude: stronger amplitudes result in an earlier optimal state and a later transition. Furthermore, a decrease in patch size combined with a compensating increase in patch surface buoyancy flux to maintain the energy input results in decreasing KE and a later transition. The simulations suggest that a CBL with a heterogeneity size smaller than the initial CBL height has less entrainment than a horizontally homogeneous CBL, whereas one with a larger heterogeneity size has more.

Buoyancy Effects on a Boundary Layer Along an Infinite Cylinder With a Step Change of Surface Temperature

1983 ◽  
Vol 105 (1) ◽  
pp. 96-101 ◽  
Author(s):  
L. S. Yao
Keyword(s):  
Boundary Layer ◽  
Free Convection ◽  
Surface Temperature ◽  
Forced Convection ◽  
Leading Edge ◽  
Step Change ◽  
Convection Boundary Layer ◽  
Stagnation Line ◽  
Convection Boundary ◽  
Free Convection Boundary Layer

The laminar boundary layer induced by a horizontal forced flow along an infinite vertical cylinder with a step change of surface temperature is studied by a finite-difference method. Close to the thermal leading edge, the buoyancy force induces a strong free-convection boundary layer. Slightly above the thermal leading edge, the boundary layer starts to separate at the rear stagnation line (φ = 180 deg). The region of separated flow grows toward the forward stagnation line and becomes stationary at φ = 104 deg as one moves upward. In other words, free convection dominates the heat transfer along the thermal leading edge. The importance of forced convection increases as one moves vertically from the thermal leading edge and eventually becomes the dominant mode. The numerical results show that the free-convection boundary layer is suppressed at the forward stagnation line and is carried toward the rear stagnation line by the forced convection. The phenomenon shares many similarities with a thermal plume affected by forced convection.

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.

scholarly journals Buoyancy Effects in a Stratified Ekman Layer

2009 ◽  
Vol 39 (10) ◽  
pp. 2581-2599 ◽  
Author(s):  
James C. McWilliams ◽  
Edward Huckle ◽  
Alexander F. Shchepetkin
Keyword(s):  
Boundary Layer ◽  
Ekman Layer ◽  
Buoyancy Flux ◽  
Buoyancy Frequency ◽  
Surface Cooling ◽  
Mean Velocity ◽  
Vertical Scale ◽  
Viscosity Profile ◽  
The Mean ◽  
Surface Buoyancy

Abstract The K-profile parameterization scheme is used to investigate the stratified Ekman layer in a “fair weather” regime of weak mean surface heating, persistently stable density stratification, diurnal solar cycle, and broadband fluctuations in the surface stress and buoyancy flux. In the case of steady forcing, the boundary layer depth typically scales as h ∼ u*/Nf, where u* is the friction velocity, f is the Coriolis frequency, and N is the interior buoyancy frequency that confirms empirical fits. The diurnal cycle of solar forcing acts to deepen the boundary layer because of net interior absorption and compensating surface cooling. Parameterized mesoscale and submesoscale eddy-induced restratification flux compresses the boundary layer. With transient forcing, the mean boundary layer profiles are altered; that is, rectification occurs with a variety of causes and manifestations, including changes in h and in the Ekman profile u(z). Overall, stress fluctuations tend to deepen the mean boundary layer, especially near the inertial frequency. Low- and high-frequency surface buoyancy-flux fluctuations have net shallowing and deepening effects, respectively. Eddy-induced interior profile fluctuations are relatively ineffective as a source of boundary layer rectification. Rectification effects in their various combinations lead to a range of mean velocity and buoyancy profiles. In particular, they lead to a “rotated” effective eddy-viscosity profile with misalignment between the mean turbulent stress and mean shear and to a “flattening” of the velocity profile with a larger vertical scale for the current veering than the speed decay; both of these effects from rectification are consistent with previous measurements.

scholarly journals g-Jitter induced free convection boundary layer on heat transfer from a sphere with constant heat flux

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Sharidan Shafie ◽  
Norsarahaida Amin
Keyword(s):  
Boundary Layer ◽  
Heat Flux ◽  
Free Convection ◽  
Equations Of Motion ◽  
Constant Heat Flux ◽  
Convection Boundary Layer ◽  
Physical Conditions ◽  
Implicit Finite Difference ◽  
Limiting Case ◽  
Free Convection Boundary Layer

The free convection from a sphere, which is subjected to a constant surface heat flux in the presence of g-jitter is theoretically investigated in this paper. The governing equations of motion are first non-dimensionalized and the resulting equations obtained after the introduction of vorticity are solved numerically using an implicit finite difference method for a limiting case Re >> 1 or the boundary layer approximations. Table and graphical results for the skin friction and wall temperature distributions as well as for the velocity and temperature profiles are presented and discussed for various parametric physical conditions Prandtl number, Pr=0.72, 1 and 7. Results indicate that g-jitter induced convective flows is stronger when Pr is small.

On the Growth of Layers of Nonprecipitating Cumulus Convection

2007 ◽  
Vol 64 (8) ◽  
pp. 2916-2931 ◽  
Author(s):  
Bjorn Stevens
Keyword(s):  
Boundary Layer ◽  
Inversion Layer ◽  
Bowen Ratio ◽  
Cloud Layer ◽  
Buoyancy Flux ◽  
Cloud Base ◽  
Moist Convection ◽  
Wide Range ◽  
Surface Buoyancy ◽  
Convective Layer

A prototype problem of a nonprecipitating convective layer growing into a layer of uniform stratification and exponentially decreasing humidity is introduced to study the mechanism by which the cumulus-topped boundary layer grows. The problem naturally admits the surface buoyancy flux, outer layer stratification, and moisture scale as governing parameters. Large-eddy simulations show that many of the well-known properties of the cumulus-topped boundary layer (including a well-mixed subcloud layer, a cloud-base transition layer, a conditionally unstable cloud layer, and an inversion layer) emerge naturally in the simulations. The simulations also quantify the differences between nonprecipitating moist convection and its dry counterpart. Whereas dry penetrative convective layers grow proportionally to the square root of time (diffusively) the cumulus layers grow proportionally to time (ballistically). The associated downward transport of warm, dry air results in a significant decrease in the surface Bowen ratio. The linear-in-time growth of the cloud layer is shown to result from the transport and subsequent evaporation of liquid water into the inversion layer. This process acts as a sink of buoyancy, which acts to imbue the free troposphere with the properties of the cloud layer. A simple model, based on this mechanism, and formulated in terms of an effective dry buoyancy flux (which is constrained by the subcloud layer’s similarity to a dry convective layer), is shown to provide good predictions of the growth of the layer across a wide range of governing parameters.

scholarly journals Growth and Decay of a Convective Boundary Layer over a Surface with a Constant Temperature

2016 ◽  
Vol 73 (5) ◽  
pp. 2165-2177 ◽  
Author(s):  
Chiel C. van Heerwaarden ◽  
Juan Pedro Mellado
Keyword(s):  
Boundary Layer ◽  
Kinetic Energy ◽  
Power Law ◽  
Convective Boundary Layer ◽  
Reynolds Numbers ◽  
Buoyancy Flux ◽  
Surface Model ◽  
Convective Boundary ◽  
Surface Buoyancy ◽  
Surface Buoyancy Flux

Abstract The growth and decay of a convective boundary layer (CBL) over a surface with a constant surface temperature that develops into a linear stratification is studied, and a mathematical model for this system is derived. The study is based on direct numerical simulations with four different Reynolds numbers; the two simulations with the largest Reynolds numbers display Reynolds number similarity, suggesting that the results can be extrapolated to the atmosphere. Because of the interplay of the growing CBL and the gradually decreasing surface buoyancy flux, the system has a complex time evolution in which integrated kinetic energy, buoyancy flux, and dissipation peak and subsequently decay. The derived model provides characteristic scales for bulk properties of the CBL. Even though the system is unsteady, self-similar vertical profiles of buoyancy, buoyancy flux, and velocity variances are recovered. There are two important implications for atmospheric modeling. First, the magnitude of the surface buoyancy flux sets the time scale of the system; thus, over a rough surface the roughness length is a key variable. Therefore, the performance of the surface model is crucial in large-eddy simulations of convection over water surfaces. Second, during the phase in which kinetic energy decays, the integrated kinetic energy never follows a power law, because the buoyancy flux and dissipation balance until the kinetic energy has almost vanished. Therefore, the applicability of power-law decay models to the afternoon transition in the atmospheric boundary layer is questionable; the presented model provides a physically sound alternative.

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