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A proper orthogonal decomposition (POD) analysis and low-dimensional modelling of thermally driven two-dimensional flow of air in a horizontal rotating cylinder, subject to the Boussinesq approximation, is considered. The problem is unsteady due to the harmonic nature of the gravitational buoyancy force with respect to the rotating observer and is characterized by four dimensionless numbers: gravitational Rayleigh number (Rag), the rotational Rayleigh number (RaΩ), the Taylor number (Ta) and Prandtl number (Pr). The data for the POD analysis are obtained by numerical integration of the governing equations of mass, momentum and energy. The POD is applied to the computational data for RaΩ varying in the range 102–106 while Rag and Pr are fixed at 105 and 0.71 respectively. The ratio of Ta to RaΩ is fixed at 100 so that the results apply to physically realistic situations. A new criterion, in the form of appropriately defined error norms, for assessing the truncation error of the POD expansion is proposed. It is shown that these error norms reflect the accuracy of the POD-based reconstructions of a given data ensemble better than the widely employed average energy criterion. The translational symmetry in both space and time of the pair of modes having degenerate (equal) eigenvalues confirms the presence of travelling waves in the flow for several different RaΩ values. The shifts in space and time of the structure of the degenerate modes are utilized to estimate the wave speeds in a given direction. The governing equations for the fluctuations are derived and low-dimensional models are constructed by employing a Galerkin procedure. For each of the five values of RaΩ, the low-dimensional models yield accurate qualitative as well as quantitative behaviour of the system. Sufficient modes are included in the low-dimensional models so that the modelling of the unresolved scales of motion is not needed to stabilize their solution. Not more than 20 modes are required in the low-dimensional models to accurately model the system dynamics. The ability of low-dimensional models to accurately predict the system behaviour for a set of parameters different from those from which they were constructed is also examined.

The proper orthogonal decomposition (POD) has become a very useful tool in the analysis and lowdimensional modelling of flows. It provides an objective way of identifying the 'coherent' structures in a turbulent flow. The application of POD to the case of a thermally driven two-dimensional flow of air in a horizontal rotating cylinder is presented. The data for the POD analysis are obtained by numerical integrations of the governing equations of mass, momentum and energy. The decomposition based on POD modes or eigenfunctions is shown to converge to within 5% deviation of the computational data for a maximum of 15 modes for the different cases. The presence of degenerate eigenvalues is an indicator of travelling waves in the flow, and this is confirmed by symmetry in both space and time for the corresponding eigenfunctions. Wave speeds are also determined for these travelling waves. Furthermore, low-dimensional models are constructed employing a Galerkin procedure. The low-dimensional models yield accurate qualitative as well as quantitative behaviour of the system. Not more than 20 modes are required in the low-dimensional models to accurately model the system dynamics. The ability of low-dimensional models to accurately predict the system behaviour for the set of parameters different from the one they were constructed from is also examined.

In this work a numerical investigation of two-dimensional steady and unsteady natural convection in a circular enclosure whose lower half is nonuniformly heated and upper half is maintained at a constant lower temperature has been carried out. An explicit finite difference method on a nonstaggered rectangular grid is used to solve the momentum and energy equations subject to Boussinesq approximation. The study is carried out for a range of Rayleigh number ͑Ra͒ varying between 10 2 and 10 6 at a fixed Prandtl number ͑Pr͒ taken as 0.71. The numerical experiments reveal that for Raഛ 8500, the flow always attains a steady state. In the steady regime, at very low Rayleigh numbers ͑RaϽ 300͒, it is shown that the velocity field is very weak and the heat transfer is predominantly by conduction. A series solution for the temperature field obtained by neglecting the fluid velocities is shown to agree well with the computed data for RaϽ 300. The convection takes place in the form of two cells with their interface aligned along the vertical diameter. As Ra is increased further, the isotherms distort to form a plume-like structure of hot fluid rising from the hottest point on the lower half of the cylinder wall. Local Nusselt number distribution over the wall shows that only a portion of the nonuniformly heated bottom half of the cylinder wall is responsible for heating the fluid. For Raജ 8900, the numerical simulations show that the steady flow looses its stability and the flow undergoes bifurcations to periodic and quasiperiodic states. On the basis of the data on the amplitude of the periodic flows obtained for a set of Ra slightly greater than 8900, it is shown that the steady flow undergoes a supercritical Hopf bifurcation at RaϷ 8830. An analysis using the proper orthogonal decomposition shows that the instability is in the form of a standing wave. The structure of the unstable mode is examined via empirical eigenfunctions obtained by the method of snapshots. In the unsteady regime ͑RaϾ 8.9ϫ 10 3 ͒, the cells start to swing their interface in an oscillatory manner with time. As Ra is increased further, the character of flow changes from periodic to quasiperiodic.

The present study involves a numerical investigation of buoyancy induced two-dimensional fluid motion in a horizontal, circular, steadily rotating cylinder whose wall is subjected to a periodic distribution of temperature. The axis of rotation is perpendicular to gravity. The governing equations of mass, momentum and energy, for a frame rotating with the enclosure, subject to Boussinesq approximation, have been solved using the Finite Difference Method on a Cartesian colocated grid utilizing a semi-implicit pressure correction approach. The problem is characterized by four dimensionless parameters: (1) Gravitational Rayleigh number Rag; (2) Rotational Rayleigh number RaΩ; (3) Taylor number Ta; and (4) Prandtl number Pr. The investigations have been carried out for a fixed Pr=0.71 and a fixed Rag=105 while RaΩ is varied from 102 to 107. From the practical point of view, RaΩ and Ta are not independent for a given fluid and size of the enclosure. Thus they are varied simultaneously. Further, an observer attached to the rotating cylinder, is stationary while the “g” vector rotates resulting in profound changes in the flow structure and even the flow direction at low enough flow rates RaΩ<105 with phase “ϕg” of the “g” vector. For RaΩ⩾105, the global spatial structure of the flow is characterized by two counter-rotating rolls in the rotating frame while the flow structure does not alter significantly with the phase of the rotating “g” vector. The frequency of oscillation of Nusselt number over the heated portion of the cylinder wall is found to be very close to the rotation frequency of the cylinder for RaΩ⩽105 whereas multiple frequencies are found to exist for RaΩ>105. The time mean Nusselt number for the heated portion of the wall undergoes a nonmonotonic variation with RaΩ, depending upon the relative magnitudes of the different body forces involved.

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