8.2 Spatial distribution of vorticity
8.2.3 Summary
−300 −295 −290 −285 −280 −275 −270
−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
Win11b x = −286 cm
−255 −250 −245 −240 −235 −230 −225 −220
−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
Win12a x = −238 cm
−210 −205 −200 −195 −190 −185 −180
−10
−8
−6
−4
−2 0 2 4 6 8 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
Win12b x = −193 cm
−160 −155 −150 −145 −140
−8
−6
−4
−2 0 2 4 6 8
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
Win13a x = −147 cm
−110 −108 −106 −104 −102 −100 −98 −96 −94 −92 −90
−6
−4
−2 0 2 4 6
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
Win13b x = −101 cm
(s−1)
−60 −40 −20 0 20 40 60 80
Figure 8.8: Contour plots of the vorticity of mean flow under the crests, measured at different cross-shore positions along the flume. The SWL is at elevationz= 0 cm in all the figures.
is dominated by three-dimensional structures. The vorticity fields, estimated using central difference method, revealed the circulation of the flow. Results have shown that the spatial distribution of instan- taneous vorticity is extremely patchy near the crest, with isolated pockets of high positive and negative values that have steep gradients. The axis of each vortex is titled in the direction of wave propagation.
In between the patches and below the SWL are large expanses with nearly zero vorticity. At the location where eddy interaction occurred, a zigzag pattern of the vorticity profile was observed due to the pres- ence of counter-rotating vorticity around the region. As flow progresses, observed eddies slow-down and vanish slowly when they go deeper, dissipated by bottom friction and slowed down by mean return flow.
Maximum intensity vorticity for the mean flow occurs near the shear layer, aroundz/h∼0, where the uprush opposes the undertow. At this point a large clockwise-rotating vorticity is formed underneath the wave crest. As the wave crest propagates along the flume, the width of the positive vorticity broadens as it sequentially interacts with downstream vorticity which are remnants of previous breaking waves. The principal feature of the phase-averaged vorticity is that a strong positive vortex appears in the flow near the shear boundary layer. This main vortex of the wave diffuses rapidly into the interior of the wave after breaking, as the eddies on the surface become more viscous. Advection and molecular diffusion play a part in stretching the vortex and redistributing the vorticity. Below the trough level, both instantaneous and mean flow vorticity are about two orders of magnitude smaller than the phase speed divided the local water depth,p
(gh)/h. For the same region,Chang&Liu[41] and [69] found that the vorticity generated by wave breaking, was of the same order of magnitude as the phase speed divided the local water depth.
Peak magnitudes of both instantaneous and mean flow vorticity generated by wave breaking, are one order of magnitude smaller than the phase speed divided by local depth,c/h.
Chapter 9: Wavelet analysis of the flow
”The wavelet transform is a tool that cuts up data, functions or operators into different frequency components, and then studies each component with a resolution matched to its scale.”
Ingrid Daubechies, Lucent, Princeton University
9.1 Introduction
Complex turbulent flow fields are described as the mixing of coherent vortical structures that are com- posed of different length scales of motion. In the study of turbulence, one commonly used technique for extraction and analysis of flow structures from instantaneous data, is the Fourier spectra which can exhibit a spectral density function distribution among various sizes of eddies (Liu et al.[205]). However, Fourier transforms can only provide spectrum information in the frequency domain, with the loss of the time domain information and which require the extra restriction on the stability of the flow. Therefore, as pointed out byLiu et al.[205], unconventional extraction and analysis techniques that involve continuous and discrete wavelet transforms for the PIV velocity fields are required to study the detailed local flow structures. The main advantage in using wavelets rather than the more classical Fourier decomposition, lies on the possibility of extracting localized properties such as local energy magnitudes by decomposing a given signal over basis functions. The present approach is based on this idea and is motivated by the successful results obtained by the application of wavelets to turbulent flow data byLongo [66].
One approach of finding strong events uses the wavelet decomposition technique (Farge [32]). This algorithm searches, at different scales, for similarities between the signal and a set of probing wavelet functions, which match the profile of a certain type of filamentary vortex. The wavelet analysis is a time- frequency analysis method, which plays an important role in processing non-stationary signals obtained by PIV to measure turbulent flow fields. Other applications of wavelet transforms for analyzing signals and turbulence can be found in Farge [32] and Mallat [91]. In the analysis, the signal is decomposed into different scales of frequencies (or wavenumber) using a continuous complex wavelet transform. The wavelets are generated based on a single basic wavelet,ψ(x), called themother wavelet, by scaling and translation. Farge [32] firstly introduced the wavelet analysis application to study turbulence.
Turbulent flows exhibit many different length and time scales. Hence, it is important to study energy transfer between the scales, in order to gain deeper insight into turbulence (Joshi & Rempfer [206]).
Wavelets offer potential for the analysis of the energy transfer in a turbulent flow. This is mainly due to their locality and scalability property. Locality here means that wavelets have compact support, which enables them to resolve local features in a flow. Wavelet representations thus have an ability to express and separate structures in a flow at different scales. Farge et al.[28] have performed substantial work in the field of coherent structure detection using orthonormal wavelets.
By decomposing the time series into time-frequency space, one is able to determine both the dominant modes of variability and how these modes vary in time. Application of wavelet analysis in flow structures, to detect, extract and analyze the characteristics of turbulence or eddy structures from our experimental data is presented. The main objective of this chapter is to use wavelets to decompose the velocity fluctuation signal and look at the percentage of signal energy in the various bands determined by the scales. This will enable the location of intervals where excess energy occurred in the breaker, in a certain frequency bands. In this application, the continuous wavelet transform (CWT) is employed to that effect. By analyzing the coefficients in a certain frequency band it is possible to locate the intervals where the energy was higher. A wavelet based algorithm developed by Torrence & Compo [82] was modified and employed to decompose the turbulence data into contributions at different frequencies and different locations from the still water line mark. The dominant frequencies and time period of the turbulent flow can be obtained from the temporal wavelet transform of the fluctuating velocity components. Furthermore, the energy contributions of structures at each scale were computed.
This chapter begins by exploring what is known about wavelet transforms and the characteristics of wavelet coefficients. The primary purpose of the discussion is to introduce notation and to provide a suitable background on the wavelet transform as a mathematical and microscopic tool for turbulent flow analysis. Reference to previous work on wavelets is made while the basic ingredients needed for the present analysis are given in sections below. The following description is limited to the needs of the present study, so readers are referred to other sources, such asFarge [32],Torrence &Compo [82], Mallat [91], Muller
&Vidakovic [207] andLabat et al.[208], for a more thorough description of wavelet analysis capabilities.
The main task of the present work, however, is to present and validate a technique for wavelet analysis of the fluctuating velocity vector fields. The wavelet based techniques that were applied to the turbulence flow are explained and the main results obtained are presented.