Infrared pressure spectra in two- and three-dimensional isotropic incompressible turbulence

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Abstract
We show, using a quasinormal or Eddy Damped Quasinormal Markovianised (EDQNM) approximation to evaluate fourth-order velocity correlations in Fourier space, that the pressure spectrum in three-dimensional isotropic incompressible turbulence is proportional to k2 in the limit k→0. This result is independent of both the infrared kinetic-energy spectrum and Reynolds number. Afterwards, direct numerical simulations and large-eddy simulations (LES) of decaying isotropic turbulence are performed: they agree with this prediction, and show a fast pressure-spectrum decay in this range. LES predict an asymptotic collapse of the infrared pressure spectrum as Epp(k,t)≈0.3∫0kC[E2(q,t)/q2]dqk2, where E(k,t) is the kinetic-energy spectrum. This permits us to predict theoretically that the pressure variance is exactly proportional to the squared kinetic energy, which we check numerically. The same QN/EDQNM analysis carried out in two dimensions predicts pressure spectra slopes of k, k−7/3, and k−5 in the infrared, inverse energy-cascade (in case of forcing), and enstrophy-cascade ranges, respectively.