Renormalized eddy viscosity and Kolmogorov’s constant in forced Navier-Stokes turbulence

The free-decay renormalization-group theory for Navier-Stokes turbulence [Zhou, Vahala, and Hossain, Phys. Rev. A 37, 2590 (1988)] is extended to the case of forced turbulence. An eddy-damping function is obtained, which is nonlocal in time and space. Using a multitime scale perturbation analysis, a time-local renormalized eddy viscosity is determined as a fixed point of an integro-difference recursion relation. It exhibits a mild cusp behavior for the particular forcing exponent that gives the Kolmogorov energy spectrum, similar to that for free-decaying turbulence. As in the free-decay theory, the triple nonlinearity in the renormalized Navier-Stokes equation is essential for the cusp to occur near the boundary between the unresolvable and resolvable scales. Unlike the ε-expansion theory, however, the renormalized eddy viscosity exhibits a wave-number dependence in the supergrid scales. The standard inertial wave-number scaling of the eddy viscosity is recovered for the Kolmogorov energy spectrum. A numerical value for the Kolmogorov constant is obtained using the Yakhot-Orszag equivalence assumption $C_K$=1.44. .AE