Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
Numerical simulation data : from validation to physical analysis. P. Sagaut1 1
S. Deck2
L. Larchevêque3
Institut Jean Le Rond d’Alembert, Université Pierre et Marie Curie - Paris 6 2 ONERA, Département d’Aérodynamique Appliquée 3 IUSTI
[email protected] [email protected] [email protected]
1
Introduction
In the recent years, unsteady three-dimensional turbulent computations such as direct numerical, large-eddy or detached-eddy simulations have become widely used to analyse more and more complex flows. Beyond standard one-point statistics ore even budget analysis, such computations are able to give access to gigabytes of unsteady high-resolution data from which detailed information on the dynamics of the flows can be obtained. The purpose of this lecture is to described some statistical tools that can be used to characterise various physics using these often underexploited resources. The related signal processing can be carried out equivalently in either the physical or spectral spaces. However emphasis will mainly be put on analyses in the spectral space since they allow a natural scale separation that is almost mandatory in studying broadband data such as the ones related to turbulent flows. Scale separation can of course be achieved either in space or time but the latter choice is the most convenient because of the straightforward definition of Fourier transforms for evenly-spaced data such as time series. Moreover, information in the wavenumber space can often be partially recovered from the ones in the frequency space by means of some kind of Taylor hypothesis. However a genuine move to the wavenumber space is sometime desirable and some direct methods will also be described. All the methods described hereafter are based on the knowledge of some statistical quantities. However these statistical quantities are not computed exactly, they are evaluated by means of statistical estimators whose use induces some biases with respect to the true underlying values. These biases have to be taken into consideration when analysing data, especially those coming from numerical simulations because of the generally short duration of the time series compared to experimental ones. That the reason why the methods will first be applied to test signals of known characteristics that mimics in resolution and length series issued from turbulent computations in order to gain some knowledge of the limitation of each estimator. Moreover, simple thresholds of significance will be given for the basic estimators when possible. The estimators will then be applied to data issued from large-eddy simulations of turbulent cavity flows[19, 18]. Beyond turbulence, these flows often exhibit strong unsteadiness because of an aeroacoustic coupling that involves convected vortices in the mixing layer over the cavity and upstream propagating pressure waves. Therefore the combination of broadband turbulence, hydrodynamic instability and pressure waves of various length scales, time scales and convection/propagation velocities makes these flow attractive for a demonstration of the capabilities (and limitations) of the methods on real applications.
Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
2
Spectral Analysis
Spectral analysis has advanced in the last decades as a signal processing tool to extract significant information about certain properties of measured data. The objective of this section is to provide some basis on the most commonly used techniques in the field of spectral analysis. The minimum amount of theory that is necessary for our purpose will be provided and the emphasis will be put on the essential features that make the tools helpful for the purpose of characterizing turbulent flows data issued either from experiment or CFD. Non-parametric methods also referred to as classical techniques, to distinguish them from the more modern spectral estimation methods will be briefly presented and illustrated by some applications in the frame of fluid dynamics. The reader interested in more in-depth discussions of the subject is referred to several reference textbooks : Bendat and Piersol [2], Kay[3], Marple[4].
2.1
Classification of spectral analysis methods
The Power-Spectral-Density (PSD) function of a random sequence x, named Sxx (f ) describes how its mean squared value σ 2 is distributed in frequency since : Z +∞ Z +∞ 2 σx = Sxx (f )df = Gxx (f )df (1) 0
−∞
Sxx (f ) (respectively Gxx (f )) is often referred to as two-sided (respectively one-sided) power spectral density function. As an example, if x is a fluctuating pressure signal of time (respectively of distance) expressed in Pa, its PSD is expressed in P a2 /Hz (respectively in P a2 .m−1 ). Spectral analysis methods which aim to estimate Sˆxx (f ) can be, broadly speaking categorized into two major classes corresponding to non-parametric and parametric methods. • Non-parametric methods do not rely on assumption that the data are drawn from a model. Consequently they can be applied in situations where less is known about the application in questions. • Conversely, parametric methods assume that the underlying stochastic process is described parametrically. The task is here to estimate the parameters of the model that describe the signal. Especially Auto-regressive (AR) models are even more popular than the other linear parametric models like moving average (MA) and autoregressive moving average (ARMA) due to their inherent computational efficiency.
2.2
The smoothed periodogram of Welch
According to the Wiener-Kintchine theorem, the PSD for a sample record of limited length T is defined as the Fourier transform of its autocorrelation function or by the following nearly equivalent definition : 1 E |X(f, T )|2 T →∞ T
Sxx (f ) = lim
(2)
where X(f, T ) is the finite Fourier transform of x(t), X(f, T ) =
Z
T
x(t)e−2jπf t dt
(3)
0
per The estimate of Sxx (f ) by simply omitting the limiting and expectation operation in Eq.(2), i.e. Sˆxx (f ) = per 1 2 ˆ T |X(f, T )| is called periodogram. Unfortunately, it can be shown that Sxx (f ) is an inconsistant estimator since even if the data record length becomes large, the variance is a constant. To circumvent
Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
this problem, an averaged periodogram can be used[3]. Consider the signal x(t) and let xN [k] = x(kTe ) designate the sequence of N samples obtained by sampling this signal with a period Te . Let us now segment the data into L non-overlapping blocks of length M . In this manner, the data set becomes : (i)
xM [k] = x[iM + k],
0≤i≤L−1
0≤k ≤M −1
(4)
The averaged PSD estimate expressed in semi-discrete form is obtained from : L−1 (i) Te X |XM (f )|2 aper ˆ Sxx (f ) = L M
(5)
i=0
(i)
(i)
where XM (f) is the Fourier transform of xM . From a numerical point view, Eq (5) is computed at n frequencies fn = M Te . Besides, one can show that the variance of the periodogram is n=0...M/2
decreased by a factor of L.
A variation of the averaged periodogram has been proposed by Welch[5] in which each elementary sequence is multiplied by a weighting function wM : (i)
(i)
x ˜M [k] = xM [k].wM [k]
(6)
In addition, the elementary data sequence are overlapped by 50% thus allowing to average over more blocks and achieve additional some extra variance reduction. Similarly to Eq (5), the estimator of Welch is given by : L−1 ˜ (i) 2 Te X |X welch M (f )| Sˆxx (f ) = WL M
(7)
i=0
P −1 2 1 where W = (1/M ) M n=0 wM [n] accounts for the window function applied to the data since wM (i) modified the average power of the sequence x ˜M . One commonly uses the Hamming window defined Ham [k] = 0.54 − 0.46 cos by wM
2kπ M −1
k = 0...M − 1.
It is worth noting that increasing the number L of elementary data segments provides better estimates since the variance of the estimator is decreased. Nevertheless, increasing L leads inevitably to a lower frequency resolution (i.e. M decreases). A compromise must be reached between the resolution and the statistical stability of the PSD estimate. When dealing with short duration data as classically encountered in CFD, this compromise is far from being evident2 . If the resolution3 is not adequate for our given application, we must resort to other methods as discussed in the next section. 1
Most window functions afford more influence to the data at the center of the set than to data to the edges, which represent a loss of information. To compensate this loss, the individual data sets are commonly overlapped in time, usually by 50%. 2 In the case of non-overlapping segments, Kay[3] gives a (1 − α) × 100% confidence interval given by : "
aper aper 2L.Sˆxx (f ) 2L.Sˆxx (f ) ; 2 2 χ2L;1−α/2 χ2L;α/2
#
(8)
where L is the number segments, χ2n is the χ2 distribution with n degrees of freedom and α the confidence 2 of elementary 2 level such as P rob χn < χn;α = α . 3 An improved interpolation of the spectrum can be reached by expanding the input signal with zeros. This technique called “zero padding” does not change the spectrum (i.e. no extra resolution is afforded) but can improve its readability.
Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
2.3
The Autoregressive model
Parametric methods differ from non-parametric methods in that a model structure from data is specified a priori. Among this class of models, the auto-regressive model is probably the most popular. If x[k] is the k − th value of the time serie, the AR-model of order p is given by : x [k] =
p X
ai x (k − i) + e [k]
where p ≤ N
i=1
x [k]k=0...N −1 = x (kT e)
with
(9)
This equation tells us that each value of the serie x[k] depends only on a weighted sum of the previous values of a time series (i.e. a finite past) plus an error term e(k) (sometimes referred to as innovation since e(k) is independant of past samples). This error term is assumed to be a zero mean white noise with finite variance σ 2 . To estimate the p + 1 parameters (ai ,), Yule and Walker(YW) proposed a set of linear equations relating the parameters of the AR model with the autocorrelation sequence R :
R.
1 a
=
σ2 0
(10)
Several methods of autoregressive-parameter estimation from data samples can be considered : • Direct inversion like Gaussian elimination but a shortcoming of this approach lies in its huge computational time (the algorithm cost is O(K 3 ) if the autocorrelation matrix is of dimension K = p + 1). • Iterative methods are based on the concept of estimating the parameters of a model of order p from the parameters of a model of order p. The Levinson-Durbin algorithm recursively solves the symmetric Toeplitz4 system of linear equations given by Eq.(10). The algorithm requires O(K 2 ) flops and is thus much more efficient than the direct inversion. • Burg’s method[6] is preferred since it uses the data points directly unlike the latter method, which relies on the estimation of the autocorrelation function (generally a tricky task for small data segments issued from CFD). As one can surmise, the AR model can be of any order as desired. Nevertheless, it should be as accurate as possible in terms of signal representation. For example, a model with a too small order p will not represent the properties of the signal, whereas a too high model order will represent noise and will not be a reliable representation of the signal. After selecting the order of the model p, we can proceed with the estimation of the p + 1 parameters (ai , σ 2 ) using Burg’s algorithm. The PSD of the AR-p model is then given by
Sx (f ) =
Te σ 2 |1 −
−j2πif |2 i=1 ai e
Pp
(11)
Parametric methods can yield higher resolutions (and are thus sometimes refered to as highresolution methods) than nonparametric methods in cases where the signal length is short because they model the data. One has then to check the pertinence of this model (i.e. of this hypothesis). A frequency bias for estimates of sinusoids in noise is reported in the literrature. However, the authors have not experienced this in their CFD data and we recommend in any case to firstly perform a classical spectral analysis (such as Welch’s method) prior to using Burg’s method. 4
A Toeplitz matrix is a diagonnaly constant matrix in which each descending diagonal from left to right is constant
Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
2.4
Examples of application
This section focuses on some applications of spectral analysis of short data issued from CFD. Only the one-sided power spectral density (named for the sake of simplicity G(f )) will be considered. Several kind of representation will be used. As reminded previously, the Power Spectral Density (PSD) function of a random sequence, describes how the mean squared-value of this data is distributed in frequency[2] since : Z ∞ Z ∞ 2 σ = G(f )df = f.G(f )d [log (f )] (12) 0
0
Therefore, by plotting spectra as f.G(f )/σ 2 in linear/log axis, one can obtain directly the contribution to the total energy of the considered frequency band. When data (especially pressure signals) are known to consist of energetics sinusoids in white noise, it is common to express the Sound Pressure Level (noted SPL and misguidedly expressed in decibels per Hz (dB/Hz)) : ! p G(f ) SP L = 20 log 10 (13) 2.10−5 Two examples of application of spectral analysis method to process short data records are now presented. The first application (§2.4.1) involves a low-frequency periodic phenomenon in white noise while the second example (§2.4.2 and §??) concerns the comparison of CFD with experiment for broadband spectra. 2.4.1 Determination of a periodic motion in short data The transonic buffet is an aerodynamic phenomenon which results in a large scale self-sustained motion of the shock over the surface of an airfoil. While standard designs aim at minimising the intensity of the buffeting, assessment of buffet onset by means of relevant computational methods remains a challenging problem for aircraft industries. To get a better knowledge of this phenomenon, an experiment[7] has been conducted on the case of a two-dimensional rigid airfoil to serve as a reference test case for assessing current numerical tools. In parallel to the experimental effort, a Zonal-Detached Eddy Simulation (ZDES) of this flow has been conducted [8] (see figure 1). The Reynolds number, based on the free-stream velocity and the chord was 3.106 , whereas the free-stream Mach number was set to 0.73. The PSD of pressure fluctuation for a sensor located near the trailing edge is compared to experiment in figure 2. Strong harmonic peaks are present which clearly illustrate the periodic nature of the flow. The peak around 70Hz is the buffet main frequency of the oscillating shock motion. The duration of the wind tunnel signal (T=50s) allows for determining very accurately this low frequency. On the numerical side, the task is not so easy due to short data T=0.08s (imposed by high CPU cost of the RANS/LES calculation). To compare the simulation with experiment, one can firstly cut the experimental data into slices of same duration as for CFD. Note in figure 2 the spectral fluctuations at highest frequencies of the experimental slice of T=0.08s compared to the whole signal (T=50s). Indeed, because of numerical short data records, it is very important in interpreting spectral estimates to be able to determine whether a particular spectral detail is due to statistical fluctuation or is actually present. In the present case, let us be reminded that the spectra issued from the simulation have been obtained by using 3 overlapping segments leading to a frequency resolution of
Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
Figure 1 – Upper-part : Sonic surface (blue) and turbulent structures. Lower part : instantaneous dilatation field.
Figure 2 – PSD of pressure fluctuation (comparison CFD/Experiment).
Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
h i ˆ ); 7.2G(f ˆ ) 30Hz. Assuming these parameters, a 95% confidence interval may be given 5 by 0.47G(f h i ˆ ) − 3.3; G(f ˆ ) + 12 expressed in decibels, see figure 3). This interval appears rather large (or G(f and illustrates the main difficulty in estimating low-frequency phenomena with short data issued from CFD. Conversely, Welch’s method remains well-adapted to treat experimental data since in the present case, a frequency resolution F R = 2Hz hallows to average the i spectral content over 200 ˆ ˆ blocs yielding a 95% confidence interval given by 0.88G(f ); 1.16G(f ) (or expressed in decibels h i ˆ ) − 0.56; G(f ˆ ) + 0.65 ). G(f
Figure 3 – PSD of pressure fluctuation : 95% Confidence Interval. Subsequently, figure 4 shows that the “zero padding” technique enables a better interpolation of the spectrum than the classical method of Welch but does not afford any extra resolution. Conversely, the autoregressive model allows to accurately determine the frequency of the peak (and thus of the shock motion). In general, the autoregressive model method is particularly well adapted to study short data that are known to consist of sinusoids in white noise (see Refs [9, 8, 10] for examples of use of this method in the field of aerodynamics).
Here we estimate the χ2n distribution with a normal distribution (which is only correct for large values of n) and compute the (1 − α) × 100% confidence interval as follows : 5
"
q
where η =
2 L
ˆ ) G(f ˆ ) G(f ; 1+η 1−η
#
erf −1 (1 − α) and erf −1 is the inverse error function.
(14)
Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
Figure 4 – Upper-part : PSD of pressure fluctuation (comparison Welch/Burg) on signal issued from CFD (T=0.08s). Lower part : Comparison between short CFD data (Burg) and long-duration experiment data (Welch)
Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
2.4.2 Comparison Experiment/CFD of broad-band spectra Many applications involving high-Reynolds turbulent flows feature broad band spectra. As an example, the understanding of the structure of shear flows with separation is of major importance for the design and control of many engineering applications (structural loads, flight control among others). For instance, the flow downstream an axisymmetric afterbody (typical of launcher’s rearbodies) features large scale eddies. An hybrid RANS/LES simulation allows for investigating the spatial organization of the fluctuating pressure field in the separated flow behind an axisymmetric step at high subsonic regime. The body is immersed into a high subsonic flow with a free stream Mach number of 0.702 leading to a Reynolds number based on the forebody diameter D of about ReD ≈ 1.1 106 . Besides, an ideal truncated contoured nozzle (TIC) is integrated in the extension base center leading to an adapted supersonic jet. A detailed analysis of this kind of flows is given in reference [11] (see also Ref [12] for an application in the supersonic regime). The main characteristics of the experimental acquisition [13] and the numerical simulation are summarized in the following table : Parameters Experiment CF D
dutation (T) 1.6 s 0.2 s
sampling frequency (fe ) 10.240 kHz 100 kHz
∆tCF D / 2. 10−6 s
TAB. 1 – Sampling characteristics The main characteristics of the instantaneous and time-averaged flowfield are described in figure 5.
Figure 5 – Left part : Coherent structures downstream of the axisymmetric step flow educed using 2 ∞ iso-surface of QU = 70. Right part : time-averaged flow field. D2 The PSD of pressure fluctuation for a sensor located in the middle of the recirculating flow is given in figure 6. A frequency resolution FR=50 Hz is retained for both the experimental and numerical data together with the Welch non-parametric method and a Hamming window. These parameters allow to average the spectral content over 160 overlapping segments in the experimental case and roughly ten times less for the signal issued from CFD. One can notice that these spectra are dominated by a broadband high frequency contribution
Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
Figure 6 – Comparison of PSD of signals of different duration centered near 1400 Hz. Though this global shape as well as the averaged levels are well reproduced by the calculation, the spectrum issued from CFD present a too high level of fluctuations at highest frequencies. One could be inclined to say that the smallest eddies in the calculation are too energetic. As stressed earlier and from a statistical point of view, the statistical variance of the PSD estimate is very different when averaging over 15 blocks or 160 blocks. Let us now cut the experimental data into 8 non-overlapping slices of same duration as for CFD (i.e. 0.2s) so that the spectral content of each slice is also averaged 15 times. The 8 experimental spectra are plotted in figure 7 and compared to that obtained by CFD.
Figure 7 – Comparison of PSDs for signals (CFD/Exp) of same duration. One can notice that the CFD spectrum is in much better agreement with the experimental ones which also feature severe statistical fluctuations at high frequencies. This result shows firstly that computing accurate broadband spectra (in a statistical sense) with short data is a difficult task and se-
Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
condly that a conclusion issued from a comparison between CFD and experiment has to be conducted cautiously with full knowledge of the implication.
3 3.1
Linear coupling : characteristic scales Two-point/time correlation
A first step in the understanding of the dynamics of a given flow is to be able to characterise the length and time scale of the unsteady entities within it. A natural way to do so is to directly visualise them. However visualisations require at least two-dimensional data and therefore yield the storage of a very large amount of unsteady data. Moreover, for turbulent flows, visualisations can be of low efficiency because of the broadband range of scales that tends to hide the structures of interest. Another simple approach relies on the coherent nature of such structures : they are non-local (to some extent) in both time and space. Consequently their characteristic length (respectively time) -scales can be deduced from two-point (respectively two-time) cross-correlations. Since informations on the level of coupling give generally more insight than the total power involved in the coupling, it is often more convenient to consider the normalised cross-correlation coefficient which, given two space-time series a(x, t) and b(x, t), reads : E[a(x, t) b(x + ξ, t + τ )] − E[a(x, t)] E[b(x, t)] Cab (ξ, τ ) = rn h i o n h i o E a(x, t)2 − E[a(x, t)]2 E b(x, t)2 − E[b(x, t)]2
(15)
where E[·] denote the expected value operator. Series a and b being discrete and of finite length, one has to define a discrete estimator of this operator : nx −nξ nt −nτ X X 1 Γ(xi , tj ) E[Γ(x, t)] = (nx − nξ ) (nt − nτ ) i=1
(16)
j=1
where nt , nx nξ and nτ respectively stand for the number of samples in time, the number of samples in space (if there is an homogeneous direction), the number of sample corresponding to the separation distance ξ and the number of samples corresponding to the time delay τ . The injection of Eq. 16 in Eq. 15 yields the asymptotically unbiased estimator of the correlation coefficient. The correlation coefficient takes values in the range [−1, 1]. A large value of |Cab (ξ, τ )| is related to a linear correlation between a and b with a space shift ξ and a time delay τ . It is worth noting that under the null hypothesis of two uncorrelated series of large enough length (typically O(1000)), the probability for the correlation coefficient to be higher than a given value Cab is roughly equal p to erfc |Cab | N/2 with N the total number of averaged samples, i.e. N = (nx − nξ ) (nt − nτ ), according to Eq. 16. It implies that Cab has to be higher than the threshold r 2 Cab & 1.82 (17) 99% N for demonstrating a significant correlation between series a and b with a 99% confidence level. Lastly, positive values of Cab are associated with in-phase series whereas negative values denotes out-ofphase. For homogeneous turbulent flows, the correlation coefficient is vanishing at large separation distance and time delay. Therefore classical integral scales can be computed to obtain characteristic length and time scales. However, for flows exhibiting convection/propagation, the coefficients go alternatively in-phase and out-of-phase and integral scales can no longer be computed. The characteristic length scales (respectively time scales) are then obtained by seeking the first change of sign of
Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
(b) 0.0014
0.12
0.0012
0.1
0.001 0.0008
0.08
T
D
(a) 0.14
0.0006
0.06
0.0004 0.04 0.0002 0.02 0
0.2
0.4
0.6
X
0.8
1
0
0
0.2
0.4
0.6
0.8
1
X
Figure 8 – Characteristic (a) spatial and (b) temporal scales computed using two-time and two-point correlations from a test signal (see text for details) ; results with spline interpolation (−−−−−) and without any interpolation (−− −−) compared to (a) the imposed diameter of the blob and to (b) the diameter divided by the imposed convection velocity (◦). the ξ (resp. τ ) function Caa (ξ, τ = 0) (resp. Caa (ξ = 0, τ )) starting from a separation distance ξ = 0 (resp. from a time delay τ = 0). Note that for convection/propagation the characteristic time scale thus obtained is related to the travelling time of the structure. This method is applied to a test signal computed by convecting over a randomly-distributed mesh a Gaussian blob through a white-noise background with the size of the blob and the convection velocity being modulated over space according to known functions. The resulting estimates correspond to the dashed curves in Fig. 8 for (a) length and (b) time scales. This figure demonstrates that the estimates are able to recover the imposed values of the test signal (for Fig. 8 b the imposed time scale is obtained by dividing the imposed length scale by the imposed convection velocity). However the estimates also exhibit a high level of noise due to an under-sampling of the correlation coefficient. Since correlations are rather smooth functions both in ξ and τ provided that statistical convergence is achieved, they can be efficiently interpolated using spline functions to obtain more accurate values of the zero-crossing τ and ξ, as demonstrated by the solid curves in Fig.8. The correlation method is then used to analyse the size of vortices inside the mixing layer of the cavity flowfield described in Ref. [19]. These results are compared in Fig. 9 with the sizes estimated by visually educing vortical structures for every phase of the phase-averaged flowfield. Three vortices of individual sizes and trajectories where found per cycle using this latter method (see [19] for details). The correlation method has not such ability to discriminate among multiple vortices : the correlationestimated size is therefore related to some kind of weighted average of the individual sizes of each of the vortex. It is seen from Fig 9 that the size obtained by correlation (grey band) reasonably agrees with the size of the two largest structures (dotted and dashed plots) over the first half of the cavity. On the contrary, the size found using correlation tends to be close to the one of the smallest structures (solid plot) over the third quarter of the cavity length before further decreasing. Such evolution can be explained by noting that the smallest structure travels above the mesh line where data have been sampled for the first half of the cavity while the two largest ones respectively go up and down this line over the second half. This remark highlights the fact that structures whose characteristics are looked for must be convected along the line where data are sampled in order to obtain reliable estimates.
3.2
Coherence spectrum
As mentioned in the introduction, scale separation is desirable when dealing with multiple and/or broadband phenomena. Scale separation can be easily achieved by moving into the temporal spectral
Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
D/L
0.3
0.2
0.1
00
0.2
0.4
0.6
0.8
1
x/L
Figure 9 – Size of coherent structures inside the mixing layer developing over a deep cavity [19]. Vortex diameters computed from a vortex eduction scheme applied to phase-averaged flowfields for structures 1 (· · · · · · ), 2 (− − −) and 3 (−−−−−) and characteristic length scale computed using twopoint auto-correlation of the streamwise velocity (grey band). space. In this space, the image of the cross-correlation function is the cross-spectrum. Analogously to the definition of the auto-spectrum (or power spectrum, see Eq. 2 of the first part of this course), the cross-spectrum reads : 1 Sab (f ) = lim E[A(f, T ) B∗ (f, T )] (18) T →∞ T where A and B are the Fourier transform of time series a and b as defined by Eq. 3 of the first part of this course and ∗ denotes the complex conjugate. Various definitions of the expected value operator E[·] can be used, as for instance the one proposed by Welch (see Eqs. 5 and 7 of the first part of this course) relying on the average of Eq. 18 for half-overlapping windowed block of data. Contrary to the auto-spectrum, the cross-spectrum is complex-valued. Consequently it is sometime difficult do directly interpret and is easier analysed when split into two real-valued spectra. The first one is the (squared) coherence spectrum which is the analogous of the cross-correlation coefficient in the spectral space. It is defined as : γ 2 ab (f ) =
|Sab (f )|2 . Saa (f ) Sbb (f )
(19)
Squared coherence spectrum takes values in the range [0, 1]. These values correspond to the amount of linear correlation between the time series a and b at the frequency f . The threshold of significance of γ 2 for a non-zero coherence is given by :
γ 299%
& 1 − 10
with
−4 ndof
(see [17] for details)
(20)
18 nb (see [5] for details) (21) 11 for the Welch averaging method of nb blocks with 50% overlapping. It is worth noting that the average is mandatory since γ 2 (f ) ≡ 1 if computed from a single block, making the estimator useless. The coherence spectrum can be used similarly to the correlation coefficient to evaluated the characteristic length scale of a structure by computing the γ 2 values between a reference location and others locations with ξ spatial separation. However, contrary to the correlation coefficient, results are related to a lone frequency, therefore making possible a scale-by-scale analysis. The ability of this method to achieve scale separation is clearly demonstrated in Fig. 10 where the ξ evolution of the γ 2 spectrum for streamwise velocity at three frequencies respectively corresponding to the first aeroacoustic mode, the second aeroacoustic mode and the Kelvin-Helmholtz instability of the flow [19] are plotted. The reference location is taken at half the length of the cavity. It is seen that the two aeroacoustic modes induce large-distance correlations due to pressure whereas the size of ndof '
Congrès Francophone de Techniques Laser, CFTL 2008, Futuroscope, 16-19 septembre 2008
C(x=L/2,f,ξ)
1 0.8 0.6 0.4 0.2 0-0.2
-0.1
0
ξ/L
0.1
0.2
Figure 10 – Evolution with respect to the separation distance ξ of the squared coherence function of the streamwise velocity between location x = L/2 and x = L/2 + ξ inside the mixing layer of the cavity flowfield [19] for frequencies respectively related to the first aeroacoustic cavity mode (−−−−−), the second cavity aeroacoustic mode (· · · · · · ) and within the Kelvin-Helmholtz instability frequency band (− − −). the Kelvin-Helmholtz at the reference location can be estimated by integrating the dashed curve over the ξ regions where the coherence levels are significant. Note that the computation of γ 2 has been carried out by splitting the time series into 32 blocks following the Welch approach, therefore resulting in a significance threshold of 0.16 according to Eq. 20. The integration yields a length scale of 0.24 roughly similar to the one found by global correlation for X = L/2 in Fig 9.
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