This thesis is composed of two parts. The first part presents a new turbulence generation method based on stochastic wavelets and tests various properties of the generated turbulence field in both the homogeneous and inhomogeneous cases. Numerical results indicate that turbulence fields can be generated with much smaller bases in comparison to synthetic Fourier methods while maintaining comparable accuracy. Adaptive generation of inhomogeneous turbulence is achieved by a scale reduction algorithm, which greatly reduces the computational cost and practically introduces no error. The generating formula proposed in this research could be adjusted to generate fully inhomogeneous and anisotropic turbulence with given RANS data under a divergence-free constraint, which was not achieved previously in similar research. Numerical examples show that the generated homogeneous and inhomoge- neous turbulence are in good agreement with the input data and theoretical results.

The second part presents a framework of solving turbulence deconvolution problems using optimization techniques on Riemannian manifolds. A filtered velocity field was deconvoluted without any information of the filter. The deconvolution results shows high accuracy compared with the original velocity field. The computational cost of the optimization problem was largely reduced using wavelet representation while still maintaining high accuracy. Utilization of divergence-free wavelets ensures the incompressible property of deconvolution results, which was barely achieved in previous research.