Dynamic model with scale-dependent coefficients in the viscous range

The standard dynamic procedure is based on the scale-invariance assumption that the model coefficient C is the same at the grid and test-filter levels. In many applications this condition is not met, e.g. when the filter-length, delta, approaches the Kolmogorov scale, and C(delta approaches eta) approaches O. Using a priori tests, we show that the standard dynamic model yields the coefficient corresponding to the test-filter scale (alpha delta) instead of the grid-scale (delta). Several approaches to account for scale dependence are examined and/or tested in large eddy simulation of isotropic turbulence: (a) take the limit alpha approaches 1; (b) solve for two unknown coefficients C(Delta) and C(alpha delta) in the least-square-error formulation; (c) the 'bi-dynamic model', in which two test-filters (e.g. at scales 2(delta) and 4(delta) are employed to gain additional information on possible scale-dependence of the coefficient, and an improved estimate for the grid-level coefficient is obtained by extrapolation, (d) use theoretical predictions for the ratio C(alpha delta)/C(delta) and dynamically solve for C(delta). None of these options is found to be entirely satisfactory, although the last approach appears applicable to the viscous range.