Natural convection in an infinite horizontal layer subject to periodic heating along the lower wall has been investigated using a combination of numerical and asymptotic techniques. The heating maintains the same mean temperatures at both walls while producing sinusoidal temperature variations along one horizontal direction, with its spatial distribution characterized by the wavenumber α and the amplitude expressed in terms of a Rayleigh number Ra p . The primary response of the system takes the form of stationary convection consisting of rolls with the axis orthogonal to the heating wave vector and structure determined by the particular values of Ra p and α. It is shown that for sufficiently large α convection is limited to a thin layer adjacent to the lower wall with a uniform conduction zone emerging above it; the temperature in this zone becomes independent of the heating pattern and varies in the vertical direction only. Linear stability of the above system has been considered and conditions leading to the emergence of secondary convection have been identified. Secondary convection gives rise to either longitudinal rolls, transverse rolls or oblique rolls at the onset, depending on α. The longitudinal rolls are parallel to the primary rolls and the transverse rolls are orthogonal to the primary rolls, and both result in striped patterns. The oblique rolls lead to the formation of convection cells with aspect ratio dictated by their inclination angle and formation of rhombic patterns. Two mechanisms of instability have been identified. In the case of α = O(1), parametric resonance dominates and leads to a pattern of instability that is locked in with the pattern of heating according to the relation δ cr = α/2, where δ cr denotes the component of the critical disturbance wave vector parallel to the heating wave vector. The second mechanism, the Rayleigh-Bénard (RB) mechanism, dominates for large α, where the instability is driven by the uniform mean vertical temperature gradient created by the primary convection, with the critical disturbance wave vector δ cr → 1.56 for α → ∞ and the fluid response becoming similar to that found in the case of a uniformly heated wall. Competition between these mechanisms gives rise to non-commensurable states in the case of longitudinal rolls and the appearance of soliton lattices, to the formation of distorted transverse rolls, and to the appearance of the wave vector component in the direction perpendicular to the forcing direction. A rapid stabilization is observed when the heating wavenumber is reduced below α ≈ 2.2 and no instability is found when α < 1.6 in the range of Ra p considered. It is shown that α plays the † Email address for correspondence: mfloryan@eng.uwo.ca ‡ Current address: role of an effective pattern control parameter and its judicious selection provides a means for the creation of a wide range of flow responses.
It is demonstrated that a significant drag reduction for pressure-driven flows can be realized by applying spatially distributed heating. The heating creates separation bubbles that separate the stream from the bounding walls and, at the same time, alter the distribution of the Reynolds stress, thereby providing a propulsive force. The strength of this effect is of practical interest for heating with wavenumbers $\ensuremath{\alpha} = O(1)$ and for flows with small Reynolds numbers and, thus, it is of potential interest for applications in micro-channels. Explicit results given for a very simple sinusoidal heating demonstrate that the drag-reducing effect increases proportionally to the second power of the heating intensity. This increase saturates if the heating becomes too intense. Drag reduction decreases as ${\ensuremath{\alpha} }^{4} $ when the heating wavenumber becomes too small, and as ${\ensuremath{\alpha} }^{\ensuremath{-} 7} $ when the heating wavenumber becomes too large; this decrease is due to the reduction in the magnitude of the Reynolds stress. The drag reduction can reach up to 87 % for the heating intensities of interest and heating patterns corresponding to the most effective heating wavenumber.
Pressure-gradient-driven flows in grooved horizontal channels were investigated. The results show that a significant reduction in pressure losses can be achieved by exposing such channels to spatially distributed heating. The system response strongly depends on the characterization of both patterns and on their relative position, leading to a pattern interaction problem. Mismatch and misplacement of both patterns may result in a significant increase in pressure losses or may have no effect on such losses. The reduction in pressure loss is associated with the formation of convection rolls on the bounding surfaces due to spatially distributed buoyancy along the streamwise direction. The pressure-gradient-reducing effect is active only in small Reynolds number flows. Explicit results are given for fluids with the Prandtl number Pr = 0.71, representing air.
Mixed convection in a channel with flow driven by a pressure gradient and subject to spatially periodic heating along one of the walls has been studied. The pattern of the heating is characterized by the wavenumber ${\it\alpha}$ and its intensity is expressed in terms of the Rayleigh number $\mathit{Ra}_{p}$. The primary convection has the form of counter-rotating rolls with the wavevector parallel to the wavevector of the heating. The resulting net heat flow between the walls increases proportionally to $\mathit{Ra}_{p}$ but the growth saturates when $\mathit{Ra}_{p}=O(10^{3})$. The most effective heating pattern corresponds to ${\it\alpha}\approx 1$, as this leads to the most intense transverse motion. The primary convection is subject to transition to secondary states with the onset conditions depending on ${\it\alpha}$. The conditions leading to transition between different forms of secondary motion have been determined using the linear stability theory. Three patterns of secondary motion may occur at small Reynolds numbers $\mathit{Re}$, i.e. longitudinal rolls, transverse rolls and oblique rolls, with the critical conditions varying significantly as a function of ${\it\alpha}$. An increase of ${\it\alpha}$ leads to the elimination of the longitudinal rolls and, eventually, to the elimination of the oblique rolls, with the transverse rolls assuming the dominant role. For large ${\it\alpha}$, the transition is driven by the Rayleigh–Bénard mechanism; while for ${\it\alpha}=O(1)$, the spatial parametric resonance dominates. The global flow characteristics are identical regardless of whether the heating is applied at the lower or the upper wall.
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