Search Results (3)

Homogeneous isotropic turbulence (HIT) has been a useful theoretical concept for more than fifty years of theory, modelling, and calculations. Some exact results are revisited in incompressible HIT, with special emphasis on the 4/5 Kolmogorov law. The finite Reynolds number effect (FRN), which yields corrections to that law, is investigated, using both Kármán–Howarth-type equations and a statistical spectral closure of the Eddy-Damped Quasi-Normal Markovian (EDQNM)-type. This discussion offers an opportunity to give an extended review of such spectral closures, from weak turbulence, as in wave turbulence theory, to a strong one. Extensions of the 4/5 or 4/3 Kolmogorov/Monin laws to anisotropic cases, such as stably stratified and MHD turbulence, are briefly touched on. Before addressing more recent work on compressible isotropic turbulence, the simplest case of quasi-incompressible turbulence subjected to externally imposed isotropic compression or dilatation is presented. Rapid distortion theory is found to be a poor model in this isotropic case, in contrast with its relevance in strongly anisotropic flow cases. Accordingly, a fully nonlinear approach based on a rescaling of all fluctuating variables is used, in order to show its interplay with the linear operator. This opens the discussion on the cases of homogeneous incompressible turbulence, where RDT and nonlinear models are relevant, provided that anisotropy is accounted for. Finally, isotropic compressible flows of increasing complexity are considered. Recent studies using weak turbulence theory, modelling, and DNS are discussed. A final unpublished study involves interactions between the solenoidal mode, inherited from incompressible turbulence, and the acoustic and entropic modes, which are specific to the compressible problem. An approach to acoustic wave turbulence, with resonant triads, is revisited on this occasion. Full article

In this paper, we focus on the existence of positive solutions for a singular tempered sub-diffusion fractional model involving a quasi-homogeneous nonlinear operator. By using the spectrum theory and computing the fixed point index, some new sufficient conditions for the existence of positive solutions are derived. It is worth pointing out that the nonlinearity of the equation contains a tempered fractional sub-diffusion term, and is allowed to possess strong singularities in time and space variables. In particular, the quasi-homogeneous operator is a nonlinear and non-symmetrical operator. Full article

In this paper, we analyze the problem of determining orbital hypernormal forms—that is, the simplest analytical expression that can be obtained for a given autonomous system around an isolated equilibrium point through time-reparametrizations and transformations in the state variables. We show that the computation of orbital hypernormal forms can be carried out degree by degree using quasi-homogeneous expansions of the vector field of the system by means of reduced time-reparametrizations and near-identity transformations, achieving an important reduction in the computational effort. Moreover, although the orbital hypernormal form procedure is essentially nonlinear in nature, our results show that orbital hypernormal forms are characterized by means of linear operators. Some applications are considered: the case of planar vector fields, with emphasis on a case of the Takens–Bogdanov singularity. Full article