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We develop a stochastic model for the velocity gradients dynamics along aLagrangian trajectory. Comparing with different attempts proposed in theliterature, the present model, at the cost of introducing a free parameterknown in turbulence phenomenology as the intermittency coefficient, gives arealistic picture of velocity gradient statistics at any Reynolds number. Toachieve this level of accuracy, we use as a first modelling step a regularizedself-stretching term in the framework of the Recent Fluid Deformation (RFD)approximation that was shown to give a realistic picture of small scalesstatistics of turbulence only up to moderate Reynolds numbers. As a secondstep, we constrain the dynamics, in the spirit of Girimaji & Pope (1990), inorder to impose a peculiar statistical structure to the dissipation seen by theLagrangian particle. This probabilistic closure uses as a building block arandom field that fulfils the statistical description of the intermittency,i.e. multifractal, phenomenon. To do so, we define and generalize to astatistically stationary framework a proposition made by Schmitt (2003). Theseconsiderations lead us to propose a non-linear and non-Markovian closeddynamics for the elements of the velocity gradient tensor. We numericallyintegrate this dynamics and observe that a stationary regime is indeed reached,in which (i) the gradients variance is proportional to the Reynolds number,(ii) gradients are typically correlated over the (small) Kolmogorov time scaleand gradients norms over the (large) integral time scale (iii) the jointprobability distribution function of the two non vanishing invariants $Q$ and$R$ reproduces the characteristic teardrop shape, (iv) vorticity getspreferentially aligned with the intermediate eigendirection of the deformationtensor and (v) gradients are strongly non-Gaussian and intermittent.

A multifractal model for the velocity gradients dynamics in turbulent flows