Divergence-free Wavelet Analysis of Turbulent Flows

This project is funded by the U.S. National Science Foundation.




The study, analysis and simulation of turbulent flows is probably one of the most challenging problems in engineering, scientific computing and applied mathematics.
Wavelets offer some potential both for the analysis and the simulation of turbulent flows. This is mainly due to a good localization in frequency and physical domain which entails a good separation of features associated with different characteristic length scales. Localization in physical space can be enhanced by employing compactly supported wavelets which all subsequent investigations will be confined to.
A second important feature of wavelets is their compression property. By this we mean that often only relatively few but judiceously chosen terms in an infinite wavelet expansion suffice to recover the function within a desired tolerance. This relies on the so called cancellation properties of wavelets combined with the fact that certain weighted sequence norms of the expansion coefficients are equivalent to the norms for certain smoothness spaces such as Sobolev or Besov spaces (such estimates will be called norm equivalences). Such highly economical representations are therefore particularly promising for representing functions with a wide range of characteristic length scales such as those representing turbulent flow fields. A primary objective of this project is to demonstrate the usefulness of wavelet concepts and their key mechanisms for the analysis and compression of turbulent flows.
In particular, wavelet concepts are used to analyze the POD method which aims at approximating the underlying infinite-dimensional nonlinear dynamical system by some low-dimensional model. A whole variety of different approaches for this reduction can be found in the literature.
Wavelets have been already used for the analysis and simulation of turbulent flows. In most papers, the data analysis has been performed by means of orthogonal wavelets for the vorticity of periodic flows. For the simulation of periodic flows so far mainly exponentially decaying vaguelettes have been employed. Also interpolatory wavelets have been used for the simulation.
By contrast, we propose to utilize compactly supported divergence-free wavelets for the analysis of velocity fields. Since we can therefore work with the velocity/pressure formulation of the Navier Stokes equations these tools work naturally also for three-dimensional problems on bounded domains so that the influence of no slip boundary conditions can be taken into account. Since the incompressibility constraint is directly built into the basis functions, such systems are natural candidates for a quantitative analysis of the role of incompressibility and its perturbation. Specifically, we will be concerned with the following issues.
First, divergence-free wavelets a wavelet basis for an appropriate complement space give rise to a Helmholtz-type decomposition for spaces of vector fields. Using the divergence-free wavelet transform, we can split flow data obtained from experiments as well as direct numerical simulations into its solenoidal and compressible parts. In particular, this tool offers us insight into the stability of the {\em Proper Orthogonal Decomposition} (POD) method, which is a well-known technique for constructing a low-dimensional turbulence model. POD modes are given as the eigenfunctions of the autocorrelation tensor for an ensemble of flow realizations. These modes can be interpreted as the coherent structures in a turbulent flow. In order to produce incompressible flow model equations based on the POD-Galerkin method, the corresponding data fields have to be divergence-free. However, it turns out that all the considered flows do in fact contain some compressible components. By comparing the POD modes obtained from the available data and those from its divergence-free part we study the quantitative influence of this violation on the incompressible flow model.
We have investigated several snapshots of a two-dimensional velocity field obtained from the Particle Imaging Velocimetry (PIV) measurements of the turbulent wake behind an airfoil. A slice of a snapshot is shown in the above figure. This data has been kindly made accessible to us by M. Glauser and C.S. Yao from NASA Langley Research Center.
Second, the above mentioned ability of wavelet expansions to separate contributions from different characteristic lenght scales will be used to provide insight into the multiscale structure of the various flows and, in particular, of the coherent structures.
Finally, we explore the compression properties of divergence-free wavelet bases. Here the term compression refers to the reduction of data sets and should not be confused with the flow property compressibility. We compress several turbulent flow fields in the sense that we retain possibly few terms in their wavelet expansion while keeping most of the energy. Such a reduction without loosing essential information is closely related to the regularity of the underlying field in a certain scale of Besov spaces. Due to the above mentioned norm equivalences induced by wavelet expansions we are able to quantify the regularity of given data sets in the relevant scale and make conclusions about their compressibility or adaptive recovery. This latter aspect is particularly interesting with regard to adaptive direct numerical simulation.
In future research, we will be considered with the construction of an adaptive wavelet method for simulating wall-driven tubulent channel flow.