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In scalar turbulence it is sometimes the case that the scalar diffusivity issmaller than the viscous diffusivity. The thermally-driven turbulent convectionin water is a typical example. In such a case the smallest scale in the problemis the Batchelor scale $l_b$, rather than the Kolmogorov scale $l_k$, as $l_b =l_k/Sc^{1/2}$, where Sc is the Schmidt number (or Prandtl number in the case oftemperature). In the numerical studies of such scalar turbulence, theconventional approach is to use a single grid for both the velocity and scalarfields. Such single-resolution scheme often over-resolves the velocity fieldbecause the resolution requirement for scalar is higher than that for thevelocity field, since $l_b<l_k$ for $Sc>1$. In this paper we put forward analgorithm that implements the so-called multiple-resolution method with afinite-volume code. In this scheme, the velocity and pressure fields are solvedin a coarse grid, while the scalar field is solved in a dense grid. The centralidea is to implement the interpolation scheme on the framework of finite-volumeto reconstruct the divergence-free velocity from the coarse to the dense grid.We demonstrate our method using a canonical model system of fluid turbulence,the Rayleigh-B\'enard convection. We show that, with the tailored mesh design,considerable speed-up for simulating scalar turbulence can be achieved,especially for large Schmidt (Prandtl) numbers. In the same time, sufficientaccuracy of the scalar and velocity fields can be achived by thismultiple-resolution scheme. Although our algorithm is demonstrated with a caseof an active scalar, it can be readily applied to passive scalar turbulentflows.

Multiple-resolution scheme in finite-volume code for active or passive scalar turbulence