A hyperbolic partial differential equation model for filtering turbulent flows

A two dimensional partial differential equation scheme that uses a second-order hyperbolic diffusion equation for image denoising is developed to a three dimensional model for filtering isotropic turbulence. The mathematical derivation of the model is introduced and a consistent finite difference numerical approximation scheme is proposed. The model is tested against the velocity fields of isotropic turbulence that are simulated by solving the lattice Boltzmann(LB) and the Navier–Stokes(NS) equations, respectively. The D3Q15 LB model and the Fourier spectral method are used to solve the LB equation and the Navier–Stokes vorticity equation at grid points of 256³, respectively. Also the filtering method with the choice of the same filtering parameters is applied against a synthetic turbulent field with the same number of points. The high rotation vortex identification method Q is used to visualize the total, coherent and incoherent fields in all cases of the study. Despite the different nature of the LB and NS direct numerical simulations and the synthetic turbulence, all input parameters for the hyperbolic filtering model are chosen the same in all cases. Comparisons between the LB and NS filtered energy spectra, coherent vortices, incoherent regions, skewness, flatness and the fourth order moments are also considered. The same features are calculated for the non-filtered and filtered synthetic turbulent fields. It is shown that the coherent part preserves all statistical features of turbulence. However, the isotropy is the only feature that has been preserved for the incoherent part and other statistical features are divergent in all cases of the study.