A comparison of routes to a strange attractor in one-dimensional local models of turbulent free and forced convection

An unfiltered additive turbulent decomposition is presented and applied to two model one-dimensional problems representing, respectively, free and forced convection analogous to the Burgers' equation problem. Only the small-scale part of the decomposition is considered in the present study in order to elucidate, and compare, the steps leading to the chaotic solutions associated with a strange attractor (and, presumably, turbulence) in the two convection models. This requires prescribing additional parameter values for local gradients of pressure, velocity, and temperature, which would be directly calculated in the large-scale part of a complete solution. The parameters are shown to influence the precise nature of the solutions for any given values of the main bifurcation parameters, the Rayleigh, Reynolds, and Prandtl numbers. In particular, their variation for fixed values of the main bifurcation parameters can lead to sequences of fiburcations embedded within the main bifurcation sequences. A local Galerkin procedure is utilized to solve the small-scale equations; results for the various flow regimes (e.g., periodic, chaotic) are displayed in terms of power spectra, velocity and temperature versus time, and Poincare maps. Significant qualitative differences between the routes to a strange attractor are observed in the free and forced convection cases. Moreover, the bifurcation sequence exhibited by the free convection model is remarkably similar to that observed in laboratory studies of Rayleigh-Benard convection.