Gradual wavelet reconstruction of the velocity increments for turbulent wakes
This work explores the properties of the velocity increment distributions for wakes of contrasting local Reynolds number and nature of generation (a cylinder wake and a multiscale-forced case, respectively). It makes use of a technique called gradual wavelet reconstruction (GWR) to generate constrained randomizations of the original data, the nature of which is a function of a parameter, ϑ. This controls the proportion of the energy between the Markov-Einstein length (∼ 0.8 Taylor scales) and integral scale that is fixed in place in the synthetic data. The properties of the increments for these synthetic data are then compared to the original data as a function of ϑ. We write a Fokker-Planck equation for the evolution of the velocity increments as a function of spatial scale, r, and, in line with previous work, expand the drift and diffusion terms in terms up to fourth order in the increments and find no terms are relevant beyond the quadratic terms. Only the linear contribution to the expansion of the drift coefficient is non-zero and it exhibits a consistent scaling with ϑ for different flows above a low threshold. For the diffusion coefficient, we find a local Reynolds number independence in the relation between the constant term and ϑ for the multiscale-forced wakes. This term characterizes small scale structure and can be contrasted with the results for the Kolmogorov capacity of the zero-crossings of the velocity signals, which measures structure over all scales and clearly distinguishes between the types of forcing. Using GWR shows that results for the linear and quadratic terms in the expansion of the diffusion coefficient are significant, providing a new means for identifying intermittency and anomalous scaling in turbulence datasets. All our data showed a similar scaling behavior for these parameters irrespective of forcing type or Reynolds number, indicating a degree of universality to the anomalous scaling of turbulence. Hence, these terms are a useful metric for testing the efficacy of synthetic turbulence generation schemes used in large eddy simulation, and we also discuss the implications of our approach for reduced order modeling of the Navier-Stokes equations.