Freely decaying turbulence in a finite domain at finite Reynolds number

Abstract

We perform direct numerical simulations to study the effects of the finite Reynolds number and domain size on the decay law of Saffman turbulence. We observe that the invariant for Saffman turbulence, u2ℓ3, and non-dimensional dissipation coefficient, Cϵ = ϵ/(u3/ℓ), are sensitive to finite domain size; here, u is the rms velocity, ℓ is the integral length scale, and ϵ is the energy dissipation rate. Consequently, the exponent n in the decay law u2 ∼ t−n for Saffman turbulence deviates from 6/5. Due to the finite Reynolds number and the domain size, Saffman turbulence decays at a faster rate (i.e., n > 6/5). However, the exponent n = 6/5 is more sensitive to the domain size than to the Reynolds number. From the simulations, we find that n remains close to 6/5 as long as Rλ ≳ 10 and ℓ ≲ 0.3Lbox; here, Rλ is the Reynolds number based on the Taylor microscale and Lbox is the domain size. We also notice that n becomes slightly lower than 6/5 for a part of the decay period. Interestingly, this trend n < 6/5 is also observed earlier in freely decaying grid-generated turbulence.