We perform numerical experiments of a dipole crashing into a wall, a generic event in two-dimensional incompressible flows with solid boundaries. The Reynolds number (Re) is varied from 985 to 7880, and no-slip boundary conditions are approximated by Navier boundary conditions with a slip length proportional to Re(-1). Energy dissipation is shown to first set up within a vorticity sheet of thickness proportional to Re(-1) in the neighborhood of the wall, and to continue as this sheet rolls up into a spiral and detaches from the wall. The energy dissipation rate integrated over these regions appears to converge towards Re-independent values, indicating the existence of energy dissipating structures that persist in the vanishing viscosity limit.

It is well known that solutions to the FourierGalerkin truncation of the inviscid Burgers equation (and other hyperbolic conservation laws) do not converge to the physically relevant entropy solution after the formation of the first shock. This loss of convergence was recently studied in detail in [S. S. Ray et al., Phys. Rev. E 84, 016301 (2011)], and traced back to the appearance of a spatially localized resonance phenomenon perturbing the solution. In this work, we propose a way to remove this resonance by filtering a wavelet representation of the Galerkin-truncated equations. A method previously developed with a complex-valued wavelet frame is applied and expanded to embrace the use of real-valued orthogonal wavelet basis, which we show to yield satisfactory results only under the condition of adding a safety zone in wavelet space. We also apply the complex-valued wavelet based method to the 2D Euler equation problem, showing that it is able to filter the resonances in this case as well.

Images acquired by cameras installed in tokamaks are difficult to interpret because the three-dimensional structure of the plasma is flattened in a non-trivial way. Nevertheless, taking advantage of the slow variation of the fluctuations along magnetic field lines, the optical transformation may be approximated by a generalized Abel transform, for which we propose an inversion technique based on the wavelet-vaguelette decomposition. After validation of the new method using an academic test case and numerical data obtained with the Tokam 2D code, we present an application to an experimental movie obtained in the tokamak Tore Supra. A comparison with a classical regularization technique for ill-posed inverse problems, the singular value decomposition, allows us to assess the efficiency. The superiority of the wavelet-vaguelette technique is reflected in preserving local features, such as blobs and fronts, in the denoised emissivity map.