The Asymptotic State of Decaying Turbulence

The long-time evolution of decaying homogeneous turbulence is a fundamental problem that awaits a full understanding. We investigate this problem using a comprehensive suite of Direct Numerical Simulations. The simulations cover initial Reynolds numbers from Re_λ = 30 to 145, with multiple independent realizations obtained at each Re_λ to ensure statistical robustness. All flows are initialized with a Birkhoff-Saffman energy spectrum (E(k)∽ k² for small k, where k is the wavenumber) and evolved for an unprecedented duration of over 20,000 initial eddy-turnover times. After initial transients, the turbulent kinetic energy (TKE) exhibits a power-law decay, TKE∽ t^-n, with a nearly constant exponent n. The inertial range becomes progressively narrower, with increasing departures from the initially-prescribed classical k^-5/3 scaling. Our analysis reveals that the decay exponent n is sensitive to the dynamics at the highest wavenumbers, even though these scales contain very little energy. These findings are compared with predictions from Migdal's theory for decaying turbulence (Phys. Fluids 36, 095161, 2024).