Transient natural convection in a cavity with time-varying thermal forcing on a sidewall

Abstract

Motivated by the environmental application of the diurnal heating and cooling effect of a water wall on a building, the present thesis considers natural convection flows in rectangular enclosures subjected to a temporal thermal forcing. The rectangular enclosure has all walls non-slip, and the upper and lower boundaries and one wall are adiabatic. The other wall is at a spatially uniform temperature or heat flux that varies with time as a sine wave. Direct numerical simulations have been conducted, and the fully developed stage of the flow, when the flow is periodic, is of interest.

To obtain an accurate solution, a new fourth-order compact finite difference spatial discretisation is proposed. Unlike the standard high order approach for local methods, such as finite-difference or finite-volume, which produces large stencils and thus introduces complexity in the boundary treatment and parallelisation, the proposed fourth-order compact scheme iteratively applies a low order spatial discretisation method to achieve higher accuracy. The scheme allows for a simple application of boundary conditions, can be applied on a non-uniform grid and allows a standard parallelisation approach to be used. The scheme is implemented and tested in an unsteady finite-difference heat equation solver and benchmarked against the analytical solution to validate the order of accuracy. It has been included in a full fractional-step Navier-Stokes solver and validated for the lid-driven cavity and natural convection problems.

Using water and air as working fluids, the time-varying temperature forcing on the wall produces an alternating direction vertical natural convection boundary layer that rises in the heating phase and falls in the cooling phase, entraining fluid from the cavity interior and discharging it alternatively at the top and bottom of the cavity, while the boundary layer is bi-directional during the transition from heating to cooling with a falling flow in the upper near wall region and a rising flow in the lower near wall region, and vice-versa for the cooling to heating transition. A stable stratification is maintained in the cavity core. The flow behaviour is dependent on the forcing frequency with low and high frequency regimes and an intermediate region, based on three characteristic time scales, the forcing period, the development time for the boundary layer and the filling time of the cavity. In the low frequency regime the filling time is less than the forcing period and the time average stratification S ̅ is well approximated by S ̅~f^(4/5). In the high frequency regime the forcing period is smaller than the boundary layer development time and S ̅~f^(-2). The maximum S ̅ occurs in the intermediate region, between the low and high frequency regimes. The maximum value of S ̅ increases with increasing Rayleigh number, approaching S ̅≅1.0 for the highest values of Rayleigh number considered. Scaling analysis has been conducted for Pr>1 fluid in a square enclosure. Scaling relations are obtained for the development of the natural convection boundary layer, the passage of the outflow intrusion, the filling of the cavity with the boundary layer outflow and resulting thermal stratification based on the Rayleigh number and forcing frequency in the low and high frequency regimes. Numerical simulations are carried out with the Rayleigh number and non-dimensional forcing frequency in the ranges 1×10^4≪Ra≪1×10^8 and 0.0001≪f≪10, and the results show that the proposed scaling relations give good predictions of the flow behaviours in the two regimes. The flow is also dependent on the aspect ratio of the cavity, with the transition frequency from the low frequency regime to the intermediate region increasing with increasing aspect ratio, while the boundary between the intermediate region and high frequency regime is not dependent on the aspect ratio.

Natural convection flow in an air-filled cavity subjected to a periodic temperature boundary condition exhibits an increased non-linearity and instability, especially at a high Rayleigh number Ra=1×10^8. The stratification parameter responds at a dominant frequency of two times the forcing frequency, together with its super-harmonics at four times, six times, eight times the forcing frequency. At f≅0.125 a resonant effect is observed with an amplification of the stratification variation at the response frequency of f_strat=0.25, and the appearance of super- and sub-harmonic modes. This effect is believed to be a result of the interaction of the forcing mode and the mode one internal wave. The resonance can also be observed in shallow cavities, featuring seiching motions of isotherms, supporting the hypothesis that the internal wave motions are excited by the forcing mode.

In addition to the temperature boundary condition, a heat flux boundary condition has also been applied. The flow structure is similar to the temperature boundary condition case with an alternating direction boundary layer and a stratified core. Instead of a spatially uniform temperature on the wall for the temperature boundary condition, the temperature on the wall increases with increasing y location for the heat flux boundary condition. The boundary layer is always uni-directional. Scaling analysis has been conducted for the low and high frequency regimes for Pr>1, with the characteristic time scales being the forcing period and the boundary layer development time. The scaling relations are then verified using the simulations, with the results showing overall good agreement with the derived scaling relations.