Earth Surface Processes and Landforms Visualization of turbulence Earth Surf. Process. Landformsdata 32, 637–647 (2007) Published online 5 September 2006 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/esp.1423
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Technical communications
The visualization of turbulence data using a waveletbased method C. J. Keylock* Earth and Biosphere Institute and School of Geography, University of Leeds, UK
*Correspondence to: C. J. Keylock, Earth and Biosphere Institute and School of Geography, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK. E-mail:
[email protected] Received 17 March 2006; Revised 26 June 2006; Accepted 10 July 2006
Abstract This technical communication presents some MATLAB® routines for visualizing the structure in turbulent signals based on a standard conditional averaging and thresholding approach. Up to three velocity components may be analysed and the resulting output highlights the time at which important flow events occur as well as the frequency levels that contribute the greatest energy to this particular event. The latter information is derived from a wavelet decomposition of the signal and may assist in providing a process-based explanation of observed flow features. Copyright © 2006 John Wiley & Sons, Ltd. Keywords: turbulence; wavelets; quadrant analysis; ejection-sweep cycle
Introduction An analysis of coherent structures within a turbulent flow requires the application of appropriate algorithms for the identification of particular flow events as well as the calculation of flow properties such as the turbulent stresses and vorticity. In geomorphological studies this may well then involve linking sediment transport events to the instantaneous flow structure (Heathershaw and Thorne, 1985; Nelson et al., 1995). In fluvial and oceanographic research, recent advances in technology mean that it is now possible to go into the field and obtain high frequency information for three orthogonal velocity components at a point using an acoustic Doppler velocimeter (Kraus et al., 1994; Lane et al., 1998) or at a variety of depths using acoustic Doppler current profilers (Simpson et al., 1996; Kostaschuk et al., 2005; Parsons et al., 2005). In addition, the increasing use of sophisticated techniques such as large-eddy simulation (Lesieur and Métais, 1996; Keylock et al., 2005) permits explicit analyses of aspects of turbulence structure from numerical modelling output. The simplest way to highlight important flow events is to undertake a Reynolds decomposition of the velocity timeseries to obtain the fluctuating velocity signal: u′i = ui − ui
(1)
where ui is the instantaneous velocity measured in an orthogonal direction i = 1, 2, 3, u′i is the fluctuating velocity and ui is the mean velocity. In this paper, the directions 1, 2 and 3 correspond to x (streamwise), y (transverse) and z (vertical). Hence, the velocities u1, u2 and u3 could be written as u, v and w or ux, uy and uz in alternative notations. Using Equation (1), it is possible to analyse individual velocity components or the behaviour of their joint distribution. In addition, one can introduce a threshold size H (the hole) to consider only the extreme events. The decomposition given by (1) forms the basis of the most popular techniques for detecting flow events in turbulent flows (see, e.g., Bennett and Best, 1995; Sambrook Smith and Nicholas, 2005; Schindler and Robert, 2005) and it is these approaches that are considered in this paper. The u-level method (Bogard and Tiederman, 1986) just examines the downstream u = u1 = ux component. The justification for this simplification is that the downstream velocity component is usually dominant, that sediment entrainment is often correlated with fluctuations in this component (rather than with fluctuations in Reynolds stress, Heathershaw and Thorne, 1985), and that in a boundary-layer flow variation in w = u3 = uz is necessarily correlated Copyright © 2006 John Wiley & Sons, Ltd.
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Table I. Definition of flow quadrants in terms of downstream (u = ux = u1) and vertical (w = uw = u3) velocity components.
Quadrant name Outward interaction Ejection Inward interaction Sweep
Quadrant number
Sign of u1′
Sign of u3′
Contribution to Reynolds stress
1 2 3 4
+ − − +
+ + − −
Negative Positive Negative Positive
with ux in order to maintain a positive Reynolds stress (Hudgins and Kaspersen, 1999). A recent example of the use of this method is due to Roy et al. (2004), who developed their low and high speed wedge model for flow in gravel-bed rivers using this detector. However, studying two components gives improved results (Lu and Willmarth, 1973; Bogard and Tiederman, 1986) and is the most commonly used approach in the hydraulic (Biron et al., 1996; Ferguson et al., 1996) and atmospheric sciences (Katul et al., 1994). Based on the joint behaviour of the two components, one may define quadrant flow events as explained in Table I. Because Reynolds stresses are given by
τ ij = − ρui′u ′j
(2)
it follows that in a boundary-layer flow τij is dominated by the action of ejection and sweep events that make positive contributions to the shear stress with, typically, ejections correlated with entrainment of material into suspension (Cellino and Lemmin, 2004) and sweeps leading to the movement of bedload. However, as noted by Nelson et al. (1995), sweeps are not necessarily more effective than outward interactions at mobilizing bedload; it is just that events of the requisite size are more frequent. A deeper understanding of the behaviour of different quadrants in a shear flow was provided by Nakagawa and Nezu (1977), who gave results for the proportion of time occupied by and shear stress exerted by an open channel flow in each of the four quadrants by deriving the conditional probability density functions for each quadrant from a third-order Gram–Charlier distribution. Analysis of all three velocity components using an octant based approach (Table II) is less common but examples do exist in the literature. For example, Olçmen et al. (2006) use octants to study an extended version of a turbulence model proposed by Nagano and Tagawa (1990). While the transverse component is less significant in terms of boundary-layer theory, knowledge of its relation to the other velocity components may be important for eddy-structure analysis, validating large-eddy simulations or in the application of more sophisticated model validation criteria (Moeckel and Murray, 1997; Keylock, in press). Hence, whether one is analysing a signal consisting of one, two or three velocity components, there will often be a desire to visualize the data in a simple and effective manner. This aim of this technical communication is to introduce some computer routines that assist such visualization. I describe the underlying principle behind these algorithms and illustrate the type of output that is obtained below. The approach taken is to use a wavelet transform to decompose the velocity field so that intensive quantities can be identified by eye and correlated to particular flow events.
Table II. Definition of octants introduced in this paper. The sign of the lateral (v = uy = u2) velocity component is added to the quadrant number. Octant number −1 +1 −2 +2 −3 +3 −4 +4
Copyright © 2006 John Wiley & Sons, Ltd.
Sign of u1′
Sign of u3′
Sign of u2′
+ + – – – – + +
+ + + + – – – –
– + – + – + – +
Earth Surf. Process. Landforms 32, 637–647 (2007) DOI: 10.1002/esp
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Figure 1. A u-level decomposition of a time-series for thresholds of 0·5 σ1, 1·0 σ1 and 1·5 σ1. The normalized fluctuating velocity is shown in (d) as a solid black line, with dotted lines indicating the thresholds. The panels above give the events exceeding the three different thresholds with positive exceedances given in black and negative in white, and data that fall into the ‘hole’ are given in grey.
Signal Thresholding and Visualization Figure 1 shows an example normalized signal in the bottom panel together with six horizontal lines indicating the three chosen thresholds at both positive and negative positions. The number of exceedances clearly decreases as the threshold increases. Buffin-Bélanger et al. (2000) presented a method for visualizing the space–time correlation structure of simultaneous measurements made at various heights in a gravel-bed river. The approach taken here is inspired by the type of output that their work produced, but applies to time-series data from a single location, indicating different thresholds on the same plot rather than different heights. The method could be generalized to incorporate information gained from more than one sensor by converting the plots from two to three dimensions. An inspection of Figure 1 clearly shows that the exceedance of H = 1·5 at t = 30 is a different type of event to that at t = 82. The former arises due to relatively small high frequency fluctuations superimposed on a large, low frequency positive departure from the mean. The latter is a sudden and dramatic excursion from the mean that is not superimposed on an obvious larger scale fluctuation. It would be desirable to visualize this information in a clear manner so that the scales of turbulence that actively contribute to an event can be determined, which may in turn be linked to different scales of processes. The method chosen to accomplish this uses wavelet analysis to decompose the signal into different frequencies while retaining information in the time domain.
Visualizing the Active Scales of Turbulence Interest in using wavelet analysis in environmental turbulence studies has grown since the earliest applications (Farge, 1992; Hagelberg and Gamage, 1994; Smith et al., 1998). The method is a multiresolution method for analysing data based on translation and dilation of a compactly supported (i.e. finite length) function called a mother wavelet along a signal and the convolution of this function with the signal. It permits time and frequency information to be seen at the same time, a deficiency of spectral methods that only provide frequency information. The continuous wavelet transform (CWT) for a signal g that varies with time t based on a mother wavelet ψ is given by
CWT (Θ,s) =
Copyright © 2006 John Wiley & Sons, Ltd.
g (t )ψ |s|
1
t−Θ dt s
(3)
Earth Surf. Process. Landforms 32, 637–647 (2007) DOI: 10.1002/esp
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where s and Θ are the dilation and translation parameters of the mother wavelet ψ. If the length of the signal g(t) is N, the CWT produces N coefficients at every scale analysed. A whole range of different wavelet basis functions may be defined (Daubechies, 1988; Mallat, 1999) and these differ in terms of a variety of properties, including their time and frequency localization. Wavelets that have ‘sharp edges’ in the time domain such as the Haar (or Daubechies type 2) wavelet have excellent temporal localization (the properties of the signal at a point in time correspond closely to the values for the wavelet coefficients at that time, with little smearing of the coefficients across neighbouring time positions) but the frequency localization is poor. Wavelets with a much higher number of vanishing moments have the opposite tendency. In this study the Haar wavelet is adopted. For many forms of wavelet analysis this is a poor choice and a compromise is made regarding time and frequency localization. However, for the graphical tools developed here, a clear correspondence between the times at which particular thresholds are exceeded and the wavelet coefficients at these times is a desirable property. A more efficient way to undertake wavelet analysis is to replace the CWT by the discrete wavelet transform. In this case, the dilation is performed in powers of two (the mother wavelet starts at its minimum width and this is doubled at each dyadic scale j). This discrete or dyadic wavelet transform (Shensa, 1992) is calculated using a hierarchical cascade of filter banks, and is thus more efficient numerically, while dramatically reducing the number of wavelet coefficients produced. However, as the scale doubles, the number of wavelet coefficients halves, making comparative analysis between scales problematic (particularly for the types of visual analysis undertaken here. Hence, instead this study employs the stationary wavelet transform (SWT), also known as the maximal overlap discrete wavelet transform (MODWT) (Percival and Walden, 2000). This retains the efficiency of working just with dyadic scales but is an undecimated transform, which means that each point in time has a unique coefficient for each scale. The choice of the SWT also has the advantage that the results are not dependent upon where one breaks into the signal (Percival and Walden, 2000), which makes it useful for additional analysis purposes (Olhede and Walden, 2004; Keylock, 2006). The algorithm used to produce the figures is as follows: (a) perform the SWT using the Haar wavelet for 2 j−1 dyadic scales given a signal of length 2 j (extend the signal by zero-padding up to the next dyadic scale if required); (b) calculate the scale-by-scale and global variance of the wavelet coefficients for each velocity component individually or in combination (excluding coefficients that correspond to the extended portions of the signal); (c) set all wavelet coefficients to zero for locations in time where the velocity does not exceed the specified threshold H; (d) set all wavelet coefficients to zero where the sign of the coefficient at a point in time is opposite that of the sum of the coefficients at that point in time; (e) normalize all remaining coefficients by the global (Figures 2(a), 3(a) and 6) or scale-by-scale (Figures 2(b), 3(b) and 7) standard deviations for each component or the global and scale-by-scale values over all components (Figures 8 and 9); (f ) plot these wavelet decompositions of the signal. Compared with a standard SWT decomposition of the time-series (Figure 4), this new type of plot has a number of advantages. First, the amount of information that needs to be viewed is reduced from all the coefficients to just those for the detected positions. Second, if a flow event causes a significant excursion from the mean, only the wavelet coefficients that contribute are shaded, since coefficients with an opposite sign are in white. This makes it easier to focus upon the relevant scales for generating a particular flow event and hence the likely processes in operation. For example, Keylock et al. (2005) discussed three scales of turbulence behaviour in a parallel-channel confluence experiment. The highest frequency fluctuations were due to individual Kelvin–Helmholtz vortices and the intermediate scale was a consequence of the interaction of two shear layers, while the longer term response was due to shear layer flapping. The presence of dark grey or black shading at frequencies corresponding to the periodicities of these processes would permit a quick assessment of the importance of particular processes for generating particular flow events. Third, scaling with respect to a global variance indicates the scales that dominate the whole flow (in Figures 2(a), 3(a) and 4 this can be seen to be level 5), but makes it harder to determine the importance of a contribution at a particular scale to the detected flow event. Because the scale-by-scale variance of the wavelet coefficients can be used to produce a wavelet spectrum, analogous to the Fourier spectrum (Percival and Walden, 2000), normalizing with respect to such variances compensates the coefficients for differences in energy between scales (Figures 2(b) and 3(b)). From Figure 2(a) it is clear that the third to the sixth exceedances of the threshold resulted from a large amount of energy at level 5 (more than three times the global standard deviation of the wavelet coefficients). The ninth exceedance of the threshold appears to be relatively subdued in Figure 2(a), although the shift in energy towards higher frequencies is clear. However, in Figure 2(b) the difference in total energy between level 5 and level 1 is Copyright © 2006 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 32, 637–647 (2007) DOI: 10.1002/esp
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Figure 2. A wavelet decomposition of a time-series showing the results for a u-level threshold of 1·5 σ1. The normalized fluctuating velocity is shown in (c) as a solid line with the locations exceeding the threshold given by vertical dotted lines. The wavelet coefficients shown in the remaining panels are normalized by the global wavelet variance (a) and the scale-by-scale wavelet variance (b) with darker values indicating a higher value for the coefficients. Coefficients that are opposite in sign to the detected fluctuation are set to zero here.
Figure 3. A wavelet decomposition of a time-series showing the results for a u-level threshold of 1·0 σ1. See Figure 2 for additional information.
Copyright © 2006 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 32, 637–647 (2007) DOI: 10.1002/esp
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Figure 4. Wavelet coefficients resulting from an SWT decomposition of the time-series shown in the previous figures. The magnitude of the coefficients is shown with higher values shown in black (in accord with Figures 2 and 3).
accounted for and it can be seen that the high frequency coefficients that generated the ninth exceedance were, on their own terms, at least as unusual as the level 5 coefficients that drove the third to the sixth exceedances.
Results for Higher Dimensions Figures 5 to 9 give an example of a quadrant-based analysis (an examination of the longitudinal and vertical velocity components). For quadrant and octant analyses it is only possible to display the results from up to three choices of
Figure 5. A quadrant decomposition of a time-series for thresholds of 0·5 σ1σ3, 1·0 σ1σ3 and 1·5 σ1σ3. The normalized fluctuating components are shown in the bottom panel as a solid black line (u = u1) and a dotted line (w = u3). The second panel from the bottom shows the u1′u3′ product along with dotted lines indicating the thresholds. The panels above give the events exceeding the three different thresholds with sweeps given in black, outward interactions in dark grey, inward interactions in light grey and ejections in white. Copyright © 2006 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 32, 637–647 (2007) DOI: 10.1002/esp
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Figure 6. A wavelet decomposition of a time-series showing the results for a quadrant threshold of 1·5 σ1σ3. The normalized fluctuating velocities are shown in the bottom panel, with the type of threshold exceedance indicated in the panel above. The top two panels give the wavelet decompositions for both velocity components. In this case each velocity component is normalized separately by the variance of the coefficients at all scales.
Figure 7. A wavelet decomposition of a time-series showing the results for a quadrant threshold of 1·5 σ1σ3. The normalized fluctuating velocities are shown in the bottom panel, with the type of threshold exceedance indicated in the panel above. The top two panels give the wavelet decompositions for both velocity components. In this case each velocity component is normalized separately by the scale-to-scale variance of the coefficients.
threshold simultaneously. The reason for this is that, in addition to the two velocity time-series, their product is also displayed (Figure 5). Thresholding is then undertaken upon this product with the hole size H defined in terms of the product of the standard deviations for these two velocity components. An exceedance occurs if |u′1u′3| > Hσ1σ3 Copyright © 2006 John Wiley & Sons, Ltd.
(4) Earth Surf. Process. Landforms 32, 637–647 (2007) DOI: 10.1002/esp
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Figure 8. A wavelet decomposition of a time-series showing the results for a quadrant threshold of 1·5 σ1σ3. The normalized fluctuating velocities are shown in the bottom panel, with the type of threshold exceedance indicated in the panel above. The top two panels give the wavelet decompositions for both velocity components. In this case each velocity component is normalized by the variance of the coefficients at all scales over both components.
Figure 9. A wavelet decomposition of a time-series showing the results for a quadrant threshold of 1·5 σ1σ3. The normalized fluctuating velocities are shown in the bottom panel, with the type of threshold exceedance indicated in the panel above. The top two panels give the wavelet decompositions for both velocity components. In this case each velocity component is normalized by the scale-to-scale variance of the coefficients over both components.
Copyright © 2006 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 32, 637–647 (2007) DOI: 10.1002/esp
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Figure 10. An octant decomposition of a time-series for thresholds of 0·5 σ1σ2σ3, 1·0 σ1σ2σ3 and 1·5 σ1σ2σ3. The normalized fluctuating components are shown in the bottom panel as a solid black line (ux = u1), a dotted line (uz = u3) and a grey line (uy = u2). The second panel from the bottom shows the u 1′u 2′u 3′ product along with dotted lines indicating the thresholds. The panels above give the events exceeding the three different thresholds with sweeps given in black, outward interactions in dark grey, inward interactions in light grey and ejections in white. Shading is used to distinguish between positive and negative values for u 2′. The negative values for u 2′ are shaded, while positive values are solid blocks of colour.
The shading in Figure 5 respects that used in Figure 1 with positive excursions of u1′ shown in darker colours and negative in lighter colours. However, in this case, the events that make positive contributions to the Reynolds stresses and dominate in boundary-layer flow (sweeps and ejections) are allocated to black and white, while the dark and light grey colours are reserved for outward interactions (u1′ > 0, u3′ > 0) and inward interactions (u1′ < 0, u3′ < 0), respectively. There are now four wavelet decomposition plots for each threshold (Figure 6 to 9) instead of the one previously presented. These differ from each other in terms of the normalization applied to the variance of the wavelet coefficients. Now it is the case that the two wavelet decompositions in each plot correspond to the different velocity components. Figure 6 shows a normalization by the overall variance for each velocity component and Figure 7 is a normalization by the variance of each scale for each velocity component, similar to the approaches already presented. The new approaches are a normalization in terms of the variance across all scales for both components simultaneously (Figure 8) and in terms of the scale-by-scale variance for both components simultaneously (Figure 9). These additional normalizations are useful for seeing which velocity component dominates a particular event, particularly as the velocity time-series data are normalized in terms of their mean and standard deviation, which adds clarity in terms of structure identification (Figure 1 and Figure 5) but does not permit their relative energies to be discerned. The output for octants (Figure 10) is very similar to that for quadrants. Cross-hatching is used to distinguish the events where u2′ is negative from those where u2′ is positive, with the latter shown as solid colours. Hence, in Figure 10 for H = 1·5 it can be seen that all the sweeps are positive in u2′, while the two outward interactions are both negative in u2′. Equivalent methods for normalizing the variances are available for octants. The difference from the octant plots is that there are now three wavelet decompositions on each figure (one for each component). Although greyscale output has been presented here, colour versions of these routines are also available.
Conclusion The aim of this paper has been to demonstrate some tools for visualizing turbulent structures in velocity data. The method supplements the traditional conditional averaging approach to turbulence analysis by permitting the identification of the types of scale of behaviour that contribute to the exceedance of the thresholds at particular times. This can assist with a process-based explanation of experimental data and an assessment of the turbulence phenomena that need to be accurately represented in large eddy simulation modelling studies. Copyright © 2006 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 32, 637–647 (2007) DOI: 10.1002/esp
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In the algorithms presented, up to three components may be visualized and output produced in colour or black and white. Wavelet-based decompositions of the time-series aid interpretation by indicating the dominant periodicities of the turbulent structures that lead to the exceedance of the threshold. All these routines are written in MATLAB®, require the wavelet toolbox and can be downloaded from the author’s website at: http://www.geog.leeds.ac.uk/people/ c.keylock. They are run from the command line in MATLAB® by typing in the function name and any optional settings. For example, turbvis2bw(dataset) runs a quadrant analysis on the data contained in the variable dataset using all the default settings and produces plots in black and white. The defaults are thresholds of 1 σ1σ3 and 2 σ1σ3, the production of all four wavelet plots with the different normalizations and a scale for the time that equates to one unit per time-series datum. However, turbvis3(dataset,[1.5 3],[3 4],timescale) runs an octant analysis for thresholds of 1·5 σ1σ2σ3 and 3 σ1σ2σ3, only prints out wavelet plots that are normalized over all the velocity components (wavelet plot choices 3 and 4), uses units for the x-axis of the time-series given by the variable timescale and produces colour output. The comment lines at the top of each routine explain the optional settings in more detail.
Acknowledgements I am grateful to the editor and two referees for providing helpful comments that have led to improvements in this manuscript.
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Earth Surf. Process. Landforms 32, 637–647 (2007) DOI: 10.1002/esp
