We decompose turbulent flows into two orthogonal parts: a coherent, inhomogeneous, non-Gaussian component and an incoherent, homogeneous, Gaussian component. The two components have different probability distributions and different correlations, hence different scaling laws. This separation into coherent vortices and incoherent background flow is done for each flow realization before averaging the results and calculating the next time step. To perform this decomposition we have developed a nonlinear scheme based on an objective threshold defined in terms of the wavelet coefficients of the vorticity. Results illustrate the efficiency of this coherent vortex extraction algorithm. As an example we show that in a 256 2 computation 0.7% of the modes correspond to the coherent vortices responsible for 99.2% of the energy and 94% of the enstrophy. We also present a detailed analysis of the nonlinear term, split into coherent and incoherent components, and compare it with the classical separation, e.g., used for large eddy simulation, into large scale and small scale components. We then propose a new method, called coherent vortex simulation ͑CVS͒, designed to compute and model two-dimensional turbulent flows using the previous wavelet decomposition at each time step. This method combines both deterministic and statistical approaches: ͑i͒ Since the coherent vortices are out of statistical equilibrium, they are computed deterministically in a wavelet basis which is remapped at each time step in order to follow their nonlinear motions. ͑ii͒ Since the incoherent background flow is homogeneous and in statistical equilibrium, the classical theory of homogeneous turbulence is valid there and we model statistically the effect of the incoherent background on the coherent vortices. To illustrate the CVS method we apply it to compute a two-dimensional turbulent mixing layer.

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An adaptive multilevel wavelet collocation method for solving multi-dimensional elliptic problems with localized structures is described. The method is based on multi-dimensional second generation wavelets, and is an extension of the dynamically adaptive second generation wavelet collocation method for evolution problems [Int. J. Comp. Fluid Dyn. 17 (2003) 151]. Wavelet decomposition is used for grid adaptation and interpolation, while a hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. The multilevel structure of the wavelet approximation provides a natural way to obtain the solution on a near optimal grid. In order to accelerate the convergence of the solver, an iterative procedure analogous to the multigrid algorithm is developed. The overall computational complexity of the solver is OðNÞ, where N is the number of adapted grid points. The accuracy and computational efficiency of the method are demonstrated for the solution of two-and three-dimensional elliptic test problems.

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Two mathematical approaches are combined to calculate high Reynolds number incompressible fluid-structure interaction: a wavelet method to dynamically adapt the computational grid to flow intermittency and obstacle motion, and Brinkman penalization to enforce solid boundaries of arbitrary complexity. We also implement a wavelet based multilevel solver for the Poisson problem for the pressure at each time step. The method is applied to two-dimensional flow around fixed and moving cylinders for Reynolds numbers in the range 3 × 10 1 ≤ Re ≤ 10 5. The compression ratios up to 1 000 are achieved. For the first time it is demonstrated in actual dynamic simulations that the compression scales like Re 1/2 over five orders of magnitude, while computational complexity scales like Re. This represents a significant improvement over the classical complexity estimate of Re 9/4 for two-dimensional turbulence.