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The K-type and H-type transitions of a natural convection boundary layer of a fluid of Prandtl number 7 adjacent to an isothermally heated vertical surface are investigated by means of three-dimensional direct numerical simulation (DNS). These two types of transitions refer to different flow features at the transitional stage from laminar to turbulence caused by two different types of perturbations. To excite the K-type transition, superimposed Tollmien–Schlichting (TS) and oblique waves of the same frequency are introduced into the boundary layer. It is found that a three-layer longitudinal vortex structure is present in the boundary layer undergoing the K-type transition. The typical aligned $\wedge$-shaped vortices characterizing the K-type transition are observed for the first time in pure natural convection boundary layers. For exciting the H-type transition, superimposed TS and oblique waves of different frequencies, with the frequency of the oblique waves being half of the frequency of the TS waves, are introduced into the boundary layer. Unlike the three-layer longitudinal vortex structure observed in the K-type transition, a double-layer longitudinal vortex structure is observed in the boundary layer undergoing the H-type transition. The successively staggered $\wedge$-shaped vortices characterizing the H-type transition are also observed in the downstream boundary layer. The staggered pattern of $\wedge$-shaped vortices is considered to be caused by temporal modulation of the TS and oblique waves. Interestingly the flow structures of both the K-type and H-type transitions observed in the natural convection boundary layer are qualitatively similar to those observed in Blasius boundary layers. However, an analysis of turbulence energy production suggests that the turbulence energy production by buoyancy rather than Reynolds stresses dominates the K-type and H-type transitions. In contrast, the turbulence energy production by Reynolds stresses is the only factor contributing to the transition in Blasius boundary layers.
This study presents a detailed scaling analysis quantifying the transient behaviour of natural convection in a reservoir model induced by iso-flux surface heating. It is found that horizontal conduction, which has often been neglected in previous analyses, plays an important role in the development of the flow. Depending on the Rayleigh number, three possible pathways through which the flow develops towards the final steady state are identified. A thermal boundary layer initially grows downwards from the surface. When the thermal boundary layer reaches the sloping bottom and becomes indistinct, a horizontal temperature gradient establishes due to the increasing water depth in the offshore direction. A flow is then driven towards the offshore direction by a buoyancy-induced horizontal pressure gradient, which convects away the heat input from the water surface. On the other hand, the horizontal temperature gradient also conducts heat away. The flow behaviour is determined by the interaction between the horizontal conduction and convection. An interesting flow feature revealed by the present scaling analysis is that the region across which the thermal boundary layer encompasses the full water depth shrinks over time at a certain stage of the flow development. The shrinking process eventually stops when this region coincides with a conduction-dominated subregion. The present scaling results are verified by corresponding numerical simulations.
This study considers the natural convection flow in a water body subjected to heating by solar radiation. The investigation into this type of natural convection flow has been motivated by the fact that it is known to play a crucial role in the daytime heat and mass transfer in shallow regions of natural water reservoirs and lakes, with a resultant impact on biological activity. An analytical solution for temperature in such an internally heated system shows that the temperature stratification consists of an upper stable stratification and a lower unstable stratification. One of the important consequences of such a nonlinear temperature stratification is the limitation of the mixing driven by rising thermal plumes with the penetration length scale of the plumes determining the lower mixed layer thickness. A theoretical analysis conducted in the present study suggests that in relatively deep waters, the lower mixed layer thickness is equal to the attenuation length of the radiation, which has important implications for water quality, including the transport of pollutants and nutrients in the water body. Scalings are also obtained for the quasi-steady boundary layer. The theoretical analysis is validated against numerical simulations.
The instability characteristics and resonance of a natural convection boundary layer adjacent to an isothermally heated vertical surface are investigated using direct stability analyses. The detailed streamwise evolution of the boundary-layer frequencies is visualized via the power spectra of the temperature time series in the thermal boundary layer. It is found that the entire thermal boundary layer may be divided into three distinct regions according to the frequency profile, which include an upstream low-frequency region, a transitional region (with both low- and high-frequency bands) and a downstream high-frequency region. The high-frequency band in the downstream region determines the resonance characteristics of the thermal boundary layer, which can be triggered by a single-mode perturbation at frequencies within the high-frequency band. The single-mode perturbation experiments further reveal that the maximum resonance of the thermal boundary layer is triggered by a perturbation at the characteristic frequency of the boundary layer. For the boundary-layer flow at $\mathit{Ra}= 3. 6\times 1{0}^{10} $ and $\mathit{Pr}= 7$, a net heat transfer enhancement of up to 44 % is achieved by triggering resonance of the boundary layer. This significant enhancement of heat transfer is due to the resonance-induced advancement of the laminar–turbulent transition, which is found to be dependent on the perturbation frequency and amplitude. Evidence from different perspectives revealing the same position of the transition are provided and discussed. The outcomes of this investigation demonstrate the prospect of a resonance-based approach for enhancing heat transfer.
The present investigation is concerned with natural convection in a wedge-shaped domain induced by constant isothermal heating at the water surface. Complementary to the study of daytime heating by solar radiation relevant to nearshore regions of lakes and reservoirs previously reported by the same authors, this study focuses on sensible heating imposed by the atmosphere when it is warmer than the water body. A semi-analytical approach coupled with scaling analysis and numerical simulation is adopted to resolve the problem. Two flow regimes are identified depending on the comparison between the Rayleigh number and the inverse of the square of the bottom slope. For the lower Rayleigh number regime, the entire flow domain eventually becomes isothermal and stationary. For the higher Rayleigh number regime, the flow domain is composed of two distinct subregions, a conductive subregion near the shore and a convective subregion offshore. Within the conductive subregion, the maximum local flow velocity occurs when the thermal boundary layer reaches the local bottom, and the subregion eventually becomes isothermal and stationary. In the offshore convective subregion, a steady state is reached with a distinct thermal boundary layer below the surface and a steady flow velocity. The dividing position between the two subregions and the major time and velocity scales governing the flow development in both subregions are proposed by the scaling analysis and validated by corresponding numerical simulation.
Natural convection in calm near-shore waters induced by daytime heating or nighttime cooling plays a significant role in cross-shore exchanges with significant biological and environmental implications. Having previously reported an improved scaling analysis on the daytime radiation-induced natural convection, the authors present in this paper a detailed scaling analysis quantifying the flow properties at varying offshore distances induced by nighttime surface cooling. Two critical functions of offshore distance have been derived to identify the distinctness and the stability of the thermal boundary layer. Two flow scenarios are possible depending on the bottom slope. For the relatively large slope scenario, three flow regimes are possible, which are discussed in detail. For each flow regime, all the possible distinctive subregions are identified. Two different sets of scaling incorporating the offshore-distance dependency have been derived for the conduction-dominated region and stable-convection-dominated region respectively. It is found that the scaling for flow in the stable-convection-dominated region also applies to the time-averaged mean flow in the unstable region. The present scaling results are verified by numerical simulations.
Transient natural convection flows around a thin fin on the sidewall of a differentially heated cavity, which includes a lower intrusion under the fin, a starting plume bypassing the fin and a thermal flow entrained into the vertical thermal boundary layer downstream of the fin in a typical case, are investigated using a scaling analysis and direct numerical simulations. The obtained scaling relations show that the thickness and velocity of the transient natural convection flows around the fin are determined by different dynamic and energy balances, which can be either a buoyancy-viscous balance or a buoyancy-inertial balance, depending on the Rayleigh number, the Prandtl number and the fin length. A time scale of the transition from a buoyancy-viscous flow regime to a buoyancy-inertial flow regime is obtained. The major scaling relations quantifying the transient natural convection flows are also validated by direct numerical simulations. In general, there is a good agreement between the scaling predictions and the corresponding numerical results.
The present study is concerned with radiation-induced natural convection in a water-filled triangular enclosure with a sloping bottom, which is directly relevant to buoyancy-driven flows in littoral regions. An improved scaling analysis is carried out to reveal more detailed features of the flow than a previously reported analysis. Two critical functions of the Rayleigh number with respect to the horizontal position are derived from the scaling for identifying the distinctness and stability of the thermal boundary layer. Four flow scenarios are possible, depending on the bottom slope and the maximum water depth. For each flow scenario, the flow domain may be composed of multiple subregions with distinct thermal and flow features, depending on the Rayleigh number. The dividing points between neighbouring subregions are determined by comparisons of the critical functions of the Rayleigh number with the global Rayleigh number. Position-dependent scales have been established to quantify the flow properties in different subregions. The different flow regimes for the case with relatively large bottom slopes and shallow waters are examined in detail. The present scaling results are verified by numerical simulations.
This study considers the convective instability of a water-filled shallow wedge with an absorptive bottom subject to solar radiation. Previous studies have revealed that a thermal boundary layer develops along the sloping bottom due to the absorption of penetrative radiation there, and this boundary layer is potentially unstable to the Rayleigh–Bénard instability. The stability properties of the thermal boundary layer are determined in the present study by perturbing the three-dimensional numerical solution. The result of the direct stability analysis has confirmed previous scaling with respect to the convective instability. Additional features of the thermal layer instability have also been revealed from the direct stability analysis.
The authors have previously reported a model experiment on the unsteady natural
convection in a triangular domain induced by the absorption of solar radiation. This
issue is reconsidered here both analytically and numerically. The present study consists
of two parts: a scaling analysis and a numerical simulation. The scaling analysis for
small bottom slopes reveals that a number of flow regimes are possible depending on
the Rayleigh number and the relative value of certain non-dimensional parameters
describing the flow. In a typical situation, the flow can be classified broadly into a
conductive, a transitional or a convective regime determined merely by the Rayleigh
number. Proper scales have been established to quantify the flow properties in each
of these flow regimes. The numerical simulation has verified the scaling results.
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