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.

We study the statistical properties of stationary, isotropic and homogeneous turbulence in twodimensional (2D) flows, focusing on the direct cascade, that is on wave-numbers large compared to the integral scale, where both energy and enstrophy are provided to the fluid. Our starting point is the 2D Navier-Stokes equation in the presence of a stochastic forcing, or more precisely the associated field theory. We unveil two extended symmetries of the Navier-Stokes (NS) action which were not identified yet, one related to time-dependent (or time-gauged) shifts of the response fields and existing in both 2D and 3D, and the other to time-gauged rotations and specific to 2D flows. We derive the corresponding Ward identities, and exploit them within the non-perturbative renormalization group formalism, and the large wave-number expansion scheme developed in [Phys.Fluids 30, 055102 (2018)]. We consider the flow equation for a generalized n-point correlation function, and calculate its leading order term in the large wave-number expansion. At this order, the resulting flow equation can be closed exactly. We solve the fixed point equation for the 2-point function, which yields its explicit time dependence, for both small and large time delays in the stationary turbulent state. On the other hand, at equal times, the leading order term vanishes, so we compute the next-to-leading order term. We find that the flow equations for simultaneous npoint correlation functions are not fully constrained by the set of extended symmetries, and discuss the consequences. * leonie.canet@grenoble.cnrs.fr forcing [16][17][18]. We refer the reader to reviews on 2D turbulence for a more exhaustive account [1, 2,8,19].However, the complete characterization of the statistical properties of 2D turbulence remains a fundamental quest. In particular, the previous results concentrate on the structure functions, which are equal-time quantities, but the time dependence of velocity or vorticity correlations is also of fundamental interest. In this respect, recent theoretical works have shown that the time dependence of correlation (and response) functions could be calculated within the Non-Perturbative (also named functional) Renormalisation Group (NPRG) formalism [20,21]. 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. We identify two new extended symmetries of the 2D NS action and derive the associated Ward identities. At leading order in wave-numbers, we show that the flow equation for a generic n-point correlation functi...

We consider the Diffusive Epidemic Process (DEP), a two-species reaction-diffusion process originally proposed to model disease spread within a population. This model exhibits a phase transition from an active epidemic to an absorbing state without sick individuals. Field-theoretic analyses suggest that this transition belongs to the universality class of Directed Percolation with a Conserved quantity (DP-C, not to be confused with conserved-directed percolation C-DP, appearing in the study of stochastic sandpiles). However, some exact predictions derived from the symmetries of DP-C seem to be in contradiction with lattice simulations. Here we revisit the field theory of both DP-C and DEP. We discuss in detail the symmetries present in the various formulations of both models. We then investigate the DP-C model using the derivative expansion of the nonperturbative renormalization group formalism. We recover previous results for DP-C near its upper critical dimension d_{c}=4, but show how the corresponding fixed point seems to no longer exist below d≲3. Consequences for the DEP universality class are considered.