Modelling two-time velocity correlations for prediction of both Lagrangian and Eulerian statistics
Seminar Room 1, Newton Institute
AbstractMore information on two-point two-time velocity correlations are needed for a better prediction of turbulent dispersion as well as radiated noise using an acoustic analogy. Conceptual aspects will be emphasized and not applications. Only isotropic turbulence will be considered, although many applications are developped in our team towards strongly anisotropic turbulence, mainly in rotating, stably stratified and/or MHD flows.
A simple synthetic model of isotropic turbulence is firstly considered, using a random superposition of Fourier modes : This is the KS (Kinematic simulation) following Kraichnan and Fung et al. Unsteadiness of velocity field realizations is mimicked using temporal frequencies, which are expressed in term of a prescribed energy spectrum and the wavenumber. Even if the orientation of the wavevector is randomly chosen, the link of the temporal frequency to the wavenumber is deterministic in the simpler version of the KS model. Although such a model was relevant for several applications, it is dramatically questioned for the evaluation of two-time velocity correlations. It is shown that spurious oscillations are generated, and that it is needed to model the temporal frequencies as random Gaussian variables with a standard deviation of the same order of magnitude as their mean value. Further applications to noise radiation are touched upon, in order to illustrate dominant (Lagrangian) `straining' or dominant (Eulerian) `sweeping' effects, according to the scale under consideration.
The role of a typical time-scale for the decorrelation of triple velocity correlations is then recalled and discussed in the classical `triadic closures' from the Orszag and Kraichnan's legacy, such as EDQNM, DIA and semi-Lagrangian more sophisticated variants.
Finally, these different concepts (diffusive and/or dispersive eddy dampings, straining or sweeping processes) are applied to a recent closure theory of weakly compressible isotropic turbulence. A Gaussian kernel for the decorrelation of triple velocity correlations was shown to give much better results than the classical exponential kernel inherited from EDQNM in the incompressible case. A new explanation is given in accordance with the renormalization of the acoustic wave frequency by a pure random term with zero mean but with a standard deviation of the same order of the eddy damping term formerly used in EDQNM. This analysis can be related to the concept of Kraichnan's random oscillator, recently revisited by Kaneda (2007), with a connection to the much simpler KS problem firstly presented (see also the monograph `homogeneous turbulence dynamics' by Pierre Sagaut and Claude Cambon, just published in Cambridge University Press.)