On the optimum fields and bounds for heat and mass transport in two turbulent flows

The optimum theory of turbulence is one of the few tools for obtaining analytical results for transport of heat, mass or momentum by turbulent flows. This is achieved by asymptotic theory which is valid for large values of the characteristic numbers of the investigated fluid system. For small and intermediate values of the Reynolds, Rayleigh or Taylor numbers we have to solve numerically the Euler-Lagrange equations of the corresponding variational problems. Below we discuss numerical results from the application of the Howard-Busse method of the optimum theory of turbulence to two problems: convective heat transport in non-rotating and rotating fluid layer and mass transport in pipe flow. We obtain profiles of the optimum fields and discuss the evolution of the thickness of the boundary layers as well as present our first results about the lower bound on the mass transport in a pipe flow.