0 The subject of this talk is ``wavelet analysis on the interval and application to data from turbulent channel flow''. Let me first motivate this study.

1 Near-wall turbulence dynamics is an active area of research---as we can witness at the present conference---and still not fully understood. Particularly, this goes for the regeneration cycle of turbulent structures and the details of the interaction between outer and inner scales of motion. But we do know that this is a flow which involves important coherent structures at various scales. Therefore, it is potentially a good candidate to be subjected to wavelet analysis which offers just that: access to information about position {\it and} scales simultaneously. However, in near-wall turbulence we have by definition at least one bounded coordinate direction which in turn poses problems in traditional wavelet analysis. When a construction for the real line or periodic functions are used, artifacts are obtained near the boundary. This is the reason why in practice a wavelet analysis is often carried out in wall-parallel planes only (like in a previous talk in this session), thereby not giving any information about the wall-normal scales. This point is addressed in the present talk where we present a new wavelet basis for analysis of data on the interval and will apply it to data from turbulent plane channel flow.

2 A quick recap of discrete-orthogonal wavelet analysis, where the basis functions are---in the classical case---images of a single ``mother function'', but rescaled and translated by {\it discrete} amounts. If the basis is well-chosen, one can attribute a meaningful position in space and a characteristic (length, time,...) scale to each member of the basis. The functions are orthogonal to each other and---the transform being linear---one can write a Parseval relation for the total energy of the signal (equal to the sum of the squares of the coefficients) thus allowing to interpret coefficient values in terms of their contribution to the total energy. Please note that the transform also involves the scaling functions which we will completely skip in the present considerations since they act merely as a low-pass filter. Finally, one desires a basic symmetry of the functions for the purpose of interpretation, a feature which we have to relax somewhat in the case of the interval. More specifically, we will only have symmetry in the center of the channel and approximately for wavelets far away from the boundaries.

3 Let us move on to the proposed basis. The idea is to simply perform a linear combination of Legendre polynomials in a sharply cut-off band of polynomial orders $k$ with shift coefficients ensuring the overall orthogonality of the resulting basis. Due to the choice of the polynomial type (Legendre), the orthogonality is w.r.t.\ a scalar product with a weight of unity as opposed to earlier constructions in the literature. This feature is important since otherwise an interpretation of coefficient values is extremely difficult and can be misleading.

4 Here we see the shape of the basis functions for a given scale index $j$ and at various shifts $i$. The functions are visibly self-similar near the center of the interval and get distorted when moving closer to the boundary. Therefore, we do not have translational invariance, and need to define a scale parameter $s_{ij}$ which is a function of both indices $i$ and $j$. Also, the shifts are not uniform, but rather more like Chebyshev collocation points, i.e.\ accumulated near the wall. The spatial localization is approximately proportional to $x^{-1}$ and the functions are visibly not entirely decayed at the opposite boundary, giving rise to these small ``tails'', which contain, however, very little energy.

5 So, bearing these special features of the basis in mind, let's look at the results when transforming some synthetic signals. The coefficients are obtained by simply evaluating the scalar product with the respective function and are visualized in form of the so-called ``scalogram'' which shows absolute value of coefficients (in dark shading) plotted as a function of position (horizontally) and negative of the logarithm of scale (vertically), i.e.\ small scales are located above. For a pure sine wave, this leads to a more or less horizontal band of response. Notice that the ``holes'' in the response are not due to the character of the present analysis, but are a consequence of the real-valued nature of the wavelets and is a common feature of discrete-orthogonal bases. Also, I would like to point out that the signal is {\it not} periodical over the length of the interval. The second signal is localized, i.e.\ a simple Gauss bump. Here we can read position and characteristic scale from the peak of the response in the scalogram.

6 In order to perform a quantitative analysis local spectra can be computed from the coefficients. We define a spectral energy density analog to the Fourier case, including a wave number or ``scale number'', which is just the inverse of the characteristic scale. Then we obtain a quantity which gives us for a given position in the interval the contribution to the total energy by all the scales which are active at the chosen point, i.e.\ corresponding to all basis functions which have their support there. We apply this analysis to data from direct simulation of plane channel flow at a Reynolds number $Re_\tau=590$ which was done at extremely high spectral resolution in a reasonable box. Statistics were accumulated over one ``wash-out'' cycle.

7 Here are the wall-normal spectra, in form of pre-multiplied spectra, for the streamwise velocity and the wall-normal velocity separately. The flow scales are upper-bounded by the channel height and the range of excited scales grows when approaching the wall. What is interesting here is the comparison between the two velocity components: the energy of the wall-normal velocity is visibly constrained by the presence of the wall since it has its peak at small to medium sized scales, growing with distance from the wall. The streamwise velocity always has the peak at the largest possible size. This finding is consistent with Townsend's attached eddy hypothesis, basically stating that the impermeability constrained is felt even relatively far into the flow, whereas the no-slip condition is not.

8 For the analysis of multi-dimensional data, the method of tensor products offers a rather simple way of combining one-dimensional bases. In this two-dimensional example we obtain wavelet functions with scale and position indices for the two directions separately. There is a tilde marked on one of the components because the two elementary bases do not need to be identical. Since we are interested in slices from channel flow in the following, we use our present Legendre wavelets for the wall-normal direction and well-known spline wavelets (of 4th order) in the wall-parallel direction. The quantity which is of interest here is the so-called ``intermittency index'', defined as the ratio between the coefficient energy at a given pair of scales and the spatial mean of the energy at those scales. In other words, the intermittency index shows regions of locally high energy, i.e.\ activity, for pre-selected scales.

9 The data is from the same simulation as seen before, i.e.\ at $Re_\tau=590$. The top images show the signal, but only the streamwise component of velocity, whereas the intermittency coefficient is computed from the sum of all three of the components. The left shows the streamwise cut, on the right is the spanwise slice. At small scales $s_x^+=s_y^+=15$, the intermittency index picks up buffer layer streaks in the spanwise cut and break-up regions of those streaks in the streamwise cut. At the medium scale $s_x^+=s_y^+=60$, we pick up events in the logarithmic region, like this strong shear layer here, and more rounded bulges in the streamwise analysis. The exact correspondence to hairpin structures in the latter case is not known at the moment. The maximum of intermittency increases about tenfold between the medium and the small scales and becomes more spatially localized.

10 As a conclusion I would like to summarize as follows. We have seen that the present Legendre-polynomial-based wavelet basis is orthogonal w.r.t.\ unity and therefore allows to analyze signals including a bounded coordinate direction to be analyzed with the help of such quantities as local spectra. As a perspective I would like to come back to the localization of the basis functions which was seen to be proportional to $x^{-1}$ only. It is our intention to improve this property in the following way.

11 Instead of having a sharp cut-off in polynomial space, i.e.\ the bounds of the sum over polynomials do not overlap for neighboring scale indices $j$, we can introduce smooth spectral window functions with the sum then going over all orders $k$ and the window selecting the appropriate combination. With some additional conditions, overall orthogonality can be maintained and on the lower figure we can see a plot of a localized version of the present wavelets which decays as $x^{-5}$. This is still work in progress, but I myself am interested in seeing the results (and differences if any) of applying the localized basis to the same data.