Low-dimensional model of turbulent mixed convection in a complex domain

We construct a low-dimensional model (LDM) of turbulent mixed convection in a Cartesian cell with in- and outlets and local sources of heat which is narrow in one of the two horizontal space directions. The basis is a high-resolution three-dimensional direct numerical simulation (DNS) record. The model is derived with basis functions, which have been obtained by a proper orthogonal decomposition (POD) using the snapshot method. The POD analysis is applied for a sequence of three-dimensional snapshots as well as for data which are bulk-averaged in the direction of narrow extension. This step is taken since the flow is found to have no significant dependence along this direction in the cell. We compare the three-dimensional and two-dimensional POD modes. This simplification reduces the complexity of the problem significantly and allows us to construct and run a two-dimensional LDM with a small number of degrees of freedom. We study the long-time dynamical behavior of this system using a closure of the LDM based on a mode-dependent viscosity and diffusivity. The LDM has been optimized in terms of the standard deviation of the energy spectrum and the transient energy for different numbers of degrees of freedom by comparison with the original DNS data. We find that the evolution of the coherent structures of flow and temperature agrees well with the two-dimensional original data and determine their contribution to the global transfer of heat. Root-mean-square profiles of the fluctuations of the turbulent fields agree qualitatively well with the original simulation data, but deviate slightly in amplitude. We conclude that the reduction in the dimensionality and the number of degrees of freedom can reproduce the gross features of the mixed convection flow in this particular setup well.