Turbulent Binary Fluids: A Shell Model Study

We introduce a shell (``GOY'') model for turbulent binary fluids. The variation in the concentration between the two fluids acts as an active scalar leading to a redefined conservation law for the energy, which is incorporated into the model together with a conservation law for the scalar. The model is studied numerically at very high values of the Prandtl and Reynolds numbers and we investigate the properties close to the critical point of the miscibility gap where the diffusivity vanishes. A peak develops in the spectrum of the scalar, showing that a strongly turbulent flow leads to an increase in the mixing time. The peak is, however, not very pronounced. The mixing time diverges with the Prandtl number as a power law with an exponent of approximately 0.9. The continuum limit of the shell equations leads to a set of equations which can be solved by a scaling ansatz, consistent with an exact scaling of the Navier-Stokes equations in the inertial range. In this case a weak peak also persists for a certain time in the spectrum of the scalar. Exact analytic solutions of the continuous shell equations are derived in the inertial range. Starting with fluids at rest, from an initial variation of the concentration difference, one can provoke a ``spontaneous'' generation of a velocity field, analogous to MHD in the early universe.