| Turbulent flows are non-linear and involve a wide spectrum of wave-numbers. However most turbulent flows are governed by coherent structures which are localized events that occur at a narrow bandwidth of flow scales. Evolving these alone can reduce the cost of direct numerical simulations. Wavelets seem to be suitable for the representation of coherent structures due to their multiscale and compact-support properties. To model incompressible turbulence, one can combine incompressibility with a wavelet representation, which offers additional reduction in the number of degrees of freedom. With these motivations, our study has been three-fold.; First, a divergence-free wavelet analysis tool has been developed and used for the analysis of turbulence data. It has been shown that small artificial compressibilities do not influence the shape of the coherent structures significantly. In addition, wavelet compression of flow data has revealed that adaptive wavelet simulation of turbulence can lead to significant reduction in the degrees of freedom.; Second, divergence-free wavelets appropriate for the simulation of wall bounded turbulence have been developed. For this, we have used tensor products of wavelets composed of boundary adapted and boundary-free functions. The divergence-free projector for the scaling function coefficients has been obtained from the kernel of the divergence operator computed with an SVD algorithm.; Finally, a wavelet/Galerkin code has been developed as a test bed for the new basis. Our numerical experiments involving the Stokes problem indicate boundary adaptation imposes limitations on the lowest level one can go down to and the condition number of the operator matrices. The bound on the lowest level keeps the size of the SVD problem big and few of the implemented and tested preconditioners have demonstrated improvement in the condition numbers. As these problems are less severe in 2D, we were able to treat the time dependent Navier-Stokes equations.; For 3D simulations, we propose the use of the “fictitious domain” approach which uses Lagrange multipliers to impose wall boundary conditions in the fully periodic setting. This approach has potential in resolving the ill-conditioning problem and relaxing the limitation on the choice of the coarsest scale. |
