Breaking of scale invariance in the time dependence of correlation functions in isotropic and homogeneous turbulence

Abstract

In this paper, we present theoretical results on the statistical properties of stationary, homogeneous and isotropic turbulence in incompressible flows in three dimensions. Within the framework of the Non-Perturbative Renormalization Group, we derive a closed renormalization flow equation for a generic n-point correlation (and response) function for large wave-numbers with respect to the inverse integral scale. The closure is obtained from a controlled expansion and relies on extended symmetries of the Navier-Stokes field theory. It yields the exact leading behavior of the flow equation at large wave-numbers | pi|, and for arbitrary time differences ti in the stationary state. Furthermore, we obtain the form of the general solution of the corresponding fixed point equation, which yields the analytical form of the leading wave-number and time dependence of n-point correlation functions, for large wave-numbers and both for small ti and in the limit ti → ∞. At small ti, the leading contribution at large wave-number is logarithmically equivalent to −α(εL) 2/3 | ti pi| 2 , where α is a non universal constant, L the integral scale and ε the mean energy injection rate. For the 2-point function, the (tp) 2 dependence is known to originate from the sweeping effect. The derived formula embodies the generalization of the effect of sweeping to n−point correlation functions. At large wave-number and large ti, we show that the t 2 i dependence in the leading order contribution crosses over to a |ti| dependence. The expression of the correlation functions in this regime was not derived before, even for the 2-point function. Both predictions can be tested in direct numerical simulations and in experiments.

Stationary, isotropic and homogeneous two-dimensional turbulence: a first non-perturbative renormalization group approach

“…The BMW strategy has turned out to be very successful in the context of turbulence, since the expanded flow equations can be closed at zero fields thanks to the Ward identities, whereas it generically requires to keep a whole dependence in background fields. This was first noticed in [24] for the two-point function, and generalized in [21] where the exact leading order term in the large wave-number expansion of the flow equation of an arbitrary n-point correlation function was obtained in 3D. The striking feature of these flow equations is that they do not exhibit the decoupling property usually expected for flow equations, e.g.…”

“…This equation can be simplified in both the limits of small and large time delays, as [21] ∂ κ G (n)…”

“…The set of Ward identities related to the time-gauged Galilean symmetry for generic vertex functions are derived in [21,24] in the velocity formulation. We present in this appendix their derivation in the stream formulation.…”

“…II, where its extended symmetries are analyzed in details. The NPRG formalism to study this field theory, developed in [21,23,24], is briefly presented in Sec. III, and the Ward identities related to the extended symmetries are derived in this framework.…”

“…This framework allows one to compute statistical properties of turbulence from "first principles", in the sense that it is based on the forced NS equation and does not require phenomenological inputs [22]. It was exploited in 3D to obtain the exact time dependence of n-point generalized velocity correlation functions at leading order at large wave-numbers at non-equal times [21]. The case of equal times is much more involved and its complete analysis in 3D is still lacking.The purpose of this paper is to use the NPRG formalism to investigate 2D turbulence.The outcome is three-fold.…”