Using Singular Systems Analysis to Characterise the Flow in the Wake of a Model Passenger Vehicle

*Corresponding author. Tel.: #44-1203-523-141; fax: #44-1203-418-922.E-mail address: [email protected] (C.T. Shaw)

Journal of Wind Engineeringand Industrial Aerodynamics 85 (2000) 1}30

Using singular systems analysis to characterise the#ow in the wake of a model passenger vehicle

C.T. Shaw!,*, K.P. Garry", T. Gress"!School of Engineering, University of Warwick, Coventry CV4 7AL, UK

"College of Aeronautics, Cranxeld University, Cranxeld, Bedford, MK43 0AL, UK

Received 23 June 1998; received in revised form 4 August 1999

Abstract

As the time-dependent #uid dynamics of wakes becomes important in industrial applicationssuch as vehicle design, so techniques need to be found that enable these dynamics to becharacterised. Whilst laser Doppler anemometry and particle image velocimetry are becomingwidespread in their application, they are not necessarily suitable for this application due to theirlow rate of data capture when air is the working #uid. In this paper, a methodology that hasalready been applied successfully to low Reynolds number #ows is applied to a turbulent wake.This involves the use of hot-wire anemometry to capture a large number of time series of velocitythroughout the wake of a model road passenger vehicle. These time series are then analysed bya mathematical analysis tool known as singular systems analysis, which enables the low-frequency components of a noisy signal to be determined. This is done in the framework of non-linear dynamical systems theory so that the underlying dynamics of the wake can be determined.From this it is possible to characterise those areas of the wake where coherent dynamicalstructures are present and to explore the mechanism responsible for the oscillation of the wake.The paper reviews the background to singular systems analysis systems analysis and describesthe application of the technique to the characterisation of the dynamics of the wake of a modelvehicle placed in an open jet wind tunnel. Results are presented for three cross-#ow planes inthe wake where the structure of the wake is revealed in a new light. In particular, it is clear that thetraditional picture of the vortex core appear to be present around the periphery of the vortex andin other areas where shear is apparent in the mean #ow. The analysis technique allows the motionof these to be tracked downstream through the wake, whereas simpler analysis techniques do notallow such tracking to be carried out. ( 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Wakes; Hot-wire anemometry; Vehicle aerodynamics; Unsteady #ow; Non-linear dynamics

0167-6105/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 1 6 7 - 6 1 0 5 ( 9 9 ) 0 0 1 0 4 - X

1. Introduction

1.1. Background

At present there is a trend for road passenger vehicles to become lighter andmore streamlined in an attempt to reduce fuel consumption and so increase thee$ciency of such vehicles, as well as to assist in materials recycling [1}3]. Thesechanges in vehicle design have a negative consequence in that future vehicles mightwell be more susceptible to any aerodynamic forcing of the vehicle body due tooscillations of the vehicle wake. Because of this, there is now an increased interest inthe prediction of the dynamic stability of a vehicle at an early stage in the designprocess.

As the low-frequency oscillation of the vehicle wake can a!ect the stability of thedriver}vehicle combination, a more detailed understanding of the oscillatory nature ofa wake is necessary, if appropriate design decisions are to be made. Traditionally,information on the #ow behaviour in the wake has been gathered using some form ofanemometry, usually using either laser Doppler anemometry (LDA) [4] or hot-wireanemometry (HWA) [5], to obtain time series of the #ow velocity in the wake.Recently, particle image velocimetry (PIV) [6] has also become available and has beenused to look at vehicle wakes [7].

Unfortunately, previous work has often assumed that the #ow in the wake behinda vehicle is steady or quasi-steady. It is from this assumption that the traditionalpicture of a vehicle wake with two contra-rotating vortices emerges [8]. Followingclose behind a road vehicle in the rain enables the wake motion to be made visible, asspray coming o! the vehicle moves through the wake. In these circumstances the wakeis seen to have a large time-dependent component demonstrating that the ideal-ised model of two vortices which are steady in time is not true. Recent PIV studies [7]have con"rmed this by showing large-scale vortical structures distributed throughoutthe wake that not only move with time but are also created and destroyed as timeevolves.

Determining the dynamics of the wake using either LDA or PIV methods is di$cultat present, however. For example, LDA methods measure the velocity of particles inthe #ow as they pass in a random way through the measurement volume. Thismeans that the time series generated is not evenly spaced in time and can also havea poor frequency resolution due to the low sampling rate, typically around 200 Hz for#ows involving air. Equally the time resolution of PIV is normally very poor, around15 Hz. Hence, there is still a place for HWA techniques, which have frequencyresolution in the kilohertz range, to provide a means of determining the dynamics ofthe wake.

Work has been going on for some time using singular systems analysis (SSA) toanalyse the wake of the #ow behind a cylinder at low Reynolds numbers, both beforeand after transition of the wake from laminar to turbulent #ow. This work [9}11] hasshown that the #ow structure of the wake is quite clearly made visible from the dataobtained with SSA, and also that the dynamics of the #ow can be determined bycareful analysis of the time-series data.

2 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30

1.2. Objectives

Given that the application of SSA and associated techniques in determining thedynamics of a wake has been successful at low Reynolds numbers, this work attemptsto extend the use of these techniques to higher Reynolds numbers where a fullyturbulent wake is present. The rationale for making this extension of the procedureis two-fold. First, in the high Reynolds number limit vortices are known to interact ina quasi-inviscid way [12], and so variation of the #ow dynamics with Reynoldsnumber should be small. Second, SSA acts like a low-pass "lter and so it should bepossible to remove the turbulence noise of the system from the signals, leaving behindthe dynamical signature of the #ow. In this work, the aim is to demonstrate that thesetechniques can be used: (1) to determine the dynamical structure of the #ow ina turbulent vehicle wake and (2) to enable some aspects of the dynamics of a wake tobe determined.

1.3. Structure of the paper

In the next section, the mathematics behind SSA will be explained. The authorsacknowledge that many will be unfamiliar with this mathematical techniques and thephilosophy that lies behind it. As a consequence of this, Section 2 will focus on thepractical implementation of the method as applied to the analysis of signals takenfrom a HWA in a wind tunnel. Section 3 will then describe the model set-up in thewind tunnel and the data capture procedure. Analysis of the captured data byconventional techniques and SSA follows in Section 4. The results are then discussedin Section 5 with the conclusions of the study being given in Section 6.

2. Singular systems analysis

2.1. Some concepts of dynamical systems theory

As is well known, if a #uid is assumed to be a continuum and not a discrete set ofparticles, then #ow problems are governed by a set of partial di!erential equations, theNavier}Strokes equations. These relate the conservation of momentum and mass tovelocity components and pressure of a #uid at every position in the #ow. Hence, ifa solution is to be found to a #ow problem, i.e., the #ow is to be known everywhere,then the velocity components and pressure need to be found at all points in the #owfor all time. E!ectively this means "nding velocity and pressure at an in"nite set ofpoints in space and in time. Clearly, this is an impossible task for most #ows, andbecause an in"nite amount of information is needed to de"ne the #ow the problem issaid, in the mathematical sense, to be of in"nite dimension.

For most situations, some discretisation process restricts the number of points intime and space, and the dimensions of the problem are reduced to some "nite number.For example, in a computational procedure, the mesh of points analysed will be "nite,if large. The mathematical dimension of a #ow can be seen to be restricted by other

C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30 3

means too. In certain #ows, such as the periodic vortex shedding in the wake ofa circular cylinder of in"nite length, the mathematical dimension is very low. Lookingat the wake of a cylinder, vortices are shed from each side of the cylinder at regularintervals. Measurement of the #ow velocity at a point in the wake would reveala simple repetitive pattern as the velocity changes with time. This variation with timewill be similar to that described by simple harmonic motion, and could well bepredicted by a small number of ordinary di!erential equations. In the jargon ofdynamical systems, the number of equations used in the prediction is said to be thedimension of the dynamical problem. By attempting to "nd the underlying mathemat-ical dimensions of a #ow situation, it might be possible to produce simple models thatdescribe quite complex phenomena. SSA is a technique that provides a means ofestimating the dimensionality of the dynamics of a #ow. It has been successfullyapplied to both the laminar and transitional wake of a "nite length cylinder [10,11],with the structure of these laminar and transitional #ows being exposed in consider-able detail. In this work the intension is to see if SSA can illuminate the #ow structurein a turbulent wake, that behind a model road vehicle.

2.2. Investigating a time series of velocity components

Using a single hot-wire probe, the variation of the #ow in the wake with respect totime can be found at a given point. Samples of the combination of velocity compo-nents perpendicular to the wire are captured for a number of discrete points in time.To capture the #ow variation in full at a point a three-wire system would be requiredsuch that the individual Cartesian components can be extracted from the captureddata and then plotted against time. With the single wire system all of the informationis not captured. However, the missing information can be constructed in a pseudo-form by using the method of delays, as developed by Takens [13]. For time seriessuch as those captured by a single-wire system, the method was "rst explained byBroomhead and King [14]. They show examples where time series are generatedfor a dynamical system determined by several ordinary di!erential equations. Takingonly one time series, they recreate the dynamics of the problem by creating pseudo-vectors of data. These are then analysed to see if the underlying dimensions ofthe problem can be found. Once this is done the dynamics of the full system canbe recreated. Essentially, the method involves reconstructing the phase space ofthe dynamics of the #ow as the physical point.

2.3. Using the method of delays

Imagine that a HWA system with a single wire captures a signal from the wireagainst time. The time series can be said to be a collection of real numbers l(t). Ifsay 1000 values are captured then l(t) will be a single vector of 1000 values:l1, l

2l3*l

1000. The method of delays is then used to convert this single vector into

a multi-dimensional set of values at a number of discrete points in time. The numberof values created at each time is called the embedding dimension. For example, if theembedding dimension is three then the set of values used at the "rst time point will be


Ml1, l

2, l

3N, at the second time point it will be Ml

2, l

3, l

4N and the nth time step it will

be Mln, l

n`1, l

n`2N.

In the language of dynamical systems the time series l(t) is transformed into anm-dimensional phase space position Y(t) using the transformation:

Y(t)"[l(t), l(t#q),2,l(t#(m!1)q)], (1)

where q is the time interval between samples and m is the embedding dimension. If theembedding dimension m is su$ciently large, then Takens' embedding theorem ensuresthat the reconstructed phase space is simply a smooth non-linear transform of the truephase space. This means that if su$cient values are used at each time point, therecreated dynamics will re#ect the underlying dynamics of the real system that is beingreconstructed. Once the values of Y(t) are known, they can be combined for all times asa matrix X which has as many rows as time points and has a number of columns equalto the embedding dimension. For the example used here, with an embedding dimen-sion of three, the matrix X is

X"Cl1

l2

l3

l2

l3

l4

l3

l4

l5

2l998

l999

l1000

D.Note that the number of rows shown, 998, is the maximum possible for the method

of creation described here. Also the columns of the matrix are simply the original timeseries shifted up by one row each time.

2.4. Using SSA to xnd the embedding dimension and other parameters

To determine the embedding dimension, the ideas of Broomhead and King [14] areused. Here the matrix X, the so-called trajectory matrix, formed from data at the set ofpoints Y(t) is used. Singular value decomposition of X is performed to produce a set ofsingular values, the eigenvalues of the problem, and singular vectors, the associatedeigenvectors. To do this, the covariance matrix XTX is formed, which is a squarematrix of dimension m]m. Then the eigenvalues and eigenvectors associated with thecovariance matrix are found. The eigenvectors are the signi"cant directions throughthe data and the eigenvalues are the variance of the data described by each eigenvec-tor. If two dimensions are used, i.e., m"2, then the process is directly equivalent toleast-squares "tting of a line to the data. The eigenvector associated with the largesteigenvalue would be the equation of the line of best "t to the data and the othereigenvector would be a line orthogonal to it.

In the jargon of dynamical systems analysis, the singular vectors are an optimalbasis set for the reconstructed phase space, being in e!ect the dominant directions ofthe m-dimensional phase space, and the singular values are a measure of the varianceaccounted for by the corresponding singular vectors. Hence, not only does the method


yield the dominant directions, but it also gives a measure of the relative importance ofeach of these directions. Note that in many cases the singular values do not reduce tozero as might be expected, but that rather an ordered set of the singular values showsthat a &noise #oor' exists where "nite but small values are recorded for the singularvalues.

Once the data has been reduced in this way for a number of embedding dimensions,an estimate of the actual dimensionality of the dynamics can be determined. This isdone by looking at the relationship between the eigenvalues. As the embeddingdimension is increased so the relative size of the "rst few eigenvalues changes until anembedding dimension is reached where the relationship stays relatively constant.Increasing the embedding dimension beyond this has little e!ect. Once some conver-gence in the eigenvalue relationship is found, the sum of the "rst few eigenvaluesdivided by the sum of all the eigenvalues gives the variance of the data for thoseeigenvalues. This means that if the eigenvalue sum of say "ve eigenvalues is 70% thenwe can say that 70% of the variance of the data is described by the "rst "veeigenvectors, and the dimensionality of the problems is 5 for 70% of the data.

In the analysis of the wake #ow presented later, contours of the sum of the "rst feweigenvalues are presented as a way of showing the structure of the wake. If, say with9 eigenvalues, the variance is high then the #ow is relatively simple and of lowdimension around 9. If, on the other hand, the variance is low then the #ow isrelatively complex and has a dimension much greater than 9.

Finally, by projecting the trajectory matrix X onto one of the singular vectorsa reconstructed time series is produced. These can then be analysed alone or incombination with projections onto other singular vectors to determine the dynamicsof the #ow.

3. Use of hot-wire anemometry

In this example the #ow in the wake of a model vehicle is to be analysed. A suitablemodel for this is shown in Fig. 1 where three views of the model are shown. The frontof the model has rounded edges with a radius large enough to ensure that the #owdoes not separate. At the rear the slant base is at an angle of 263 to the horizontal. Thisensures that a vortical #ow is produced in the wake, without separation at the start ofthe slanting plane.

The model has been placed is an open return blower tunnel that has a closedworking section of dimensions 460 mm wide by 456 mm high and is powered bya 15 kW motor. For the tests carried out here the #ow speed was set to 17 m/s givinga Reynolds number of 2.2]105. Behind the model, at the exit of the tunnel a traverseis placed and the wake has been surveyed using HWA for three cross-#ow (x}y)planes, 50, 100 and 200 mm behind the rear of the model. A 20]20 grid with a spacingof 5 mm has been used when capturing the data. The extent of the measurement areain x is from 20 to 115 mm and in y from 43 to 138 mm. As Takens theorem is to beused to recreate the dynamics at each sample point as described in Section 2, onlya single-wire system need be used as has been done here.


Fig. 1. The wind tunnel model and co-ordinate systems.

A TSI constant temperature anemometry system has been used which gives alinearised output. This output has been digitised with 12-bit precision at a sampl-ing rate of 2 kHz. This rate was chosen as dominant frequencies for this modelat this Reynolds number are known to be of the order of 200 Hz or less. Eachtime series was recorded for a time of 6 s and so at each point the time seriescontains 12 000 values. Full details of the experimental set-up are given by Gress[15].

4. Analysis of the time series

4.1. Mean yow structure

By taking the mean value of the velocity recorded by the single hot-wire andproducing a contour plot of the results, some idea as to the structure of the mean #owcan be obtained. Gress [15] con"rmed that this was the case as he compared the meanvalues to contours of total pressure taken at the same locations. Fig. 2 shows thecontours of the mean velocity recorded for the three measurement planes, with themean taken over 4096 points of a time series in each case. Note that the vortex core isdisplaced down and to the left as it moves downstream, and that it di!uses down-stream too.


Fig. 2. Contours of mean velocity (m/s) in the wake (a) 50 mm downstream (b) 100 mm downstream(c) 200 mm downstream.


Fig. 2. Continued.

Looking at Fig. 2(a), taken at 50 mm downstream, the vortex is seen to coexist witha shear layer which is horizontal from the vehicle centreline and runs below the vortexbefore turning vertically down outboard of the vortex centre. Also there is anisland-like structure to the right of the vortex. Moving to Fig. 2(b), taken at 100 mmdownstream, the shear layer has disappeared from view but the island-like structurenow appears to the left of the vortex as if it has been convected around the vortex fromthe position it held at 50 mm downstream. Similarly, in Fig. 2(c) at 200 mm down-stream, the shear layer has also disappeared but the island-like structure is now abovethe vortex. Measuring the angular position of the centre of the island structure at eachplane shows that from 50 to 100 mm it has moved by approximately 1353 and thatfrom 100 to 200 mm it has moved approximately 2603. This is consistent witha constant angular rotation with distance downstream.

To provide some physical understanding of the dynamical structure obtained,a detailed #ow visualisation experiment has also been carried out. This has beenachieved using a laser light sheet placed at the same measurement planes behind themodel. The #ow has been made visible using smoke particles injected into the vortexcore. Photographs of the vortex con"rm the mean #ow structure as shown in Fig. 2,but give no information about the time dependence of the #ow. The only informationabout the motion of the vortex available from the smoke #ow is that the vortexoscillates laterally rather than vertically.


Fig. 3. Contours of the root mean square of the #uctuating velocity (m/s) in the wake (a) 50 mmdownstream (b) 100 mm downstream (c) 200 mm downstream.


Fig. 3. Continued.

4.2. Fluctuating yow structure

The root mean square (RMS) of the #uctuating component of the velocitytime series, i.e., the time series with the mean value of the signal subtracted, canalso be computed. Contours of this are shown in Fig. 3. Note that by comparingFig. 3(a) with Fig. 2(a), the high values of the RMS value occur where there ishigh shear around the vortex and in the shear layer. Also comparing the equivalent"gures for the 100 and 200 mm planes, some correspondence can also be seen, inparticular in those areas which have the island-like structure. At 50 mm thiscorrespondence is con"rmed by "nding that the correlation coe$cient for thegradient of velocity and RMS values over the plane is 0.51. Better correlation, equalto 0.75, is achieved by using a non-dimensionalised shear where the shear valueis divided by the local mean velocity. At the 100 mm plane the correlation coe$cientsare lower at 0.11 and 0.13, respectively, as they are at the 200 mm plane wherethe values are 0.16 and 0.30, respectively. Hence, the unsteadiness is generated inthe areas of high shear in the wake but the unsteadiness is di!used down-stream.

4.3. Dominant frequencies

Contours of the dominant frequency can also be produced and these are shown inFig. 4. Now the picture obtained is much less clear, but the higher frequencies can be


Fig. 4. Contours of dominant frequency (Hz) in the wake (a) 50 mm downstream (b) 100 mm downstream(c) 200 mm downstream.


Fig. 4. Continued.

seen to be scattered around the periphery of the vortex. Plotting the dominantfrequency against the non-dimensionalised shear, for say the plane 50 mmdownstream, produces the scatter plot shown in Fig. 5. In this "gure, a largenumber of points occur with very low frequency for a range of shear values. It canbe presumed that these points are where the #ow is predominantly turbulent withthe dominant frequency being characteristically low. However, there is also a two-lobed structure evident, leading to the conclusion that high shear rates are relatedto low, but "nite, dominant frequencies and that high dominant frequencies arerelated to low shear rates. Similar scatter plots are obtained at the other twomeasurements planes.

4.4. SSA of the time series

By looking at the overall #ow characteristics in terms of the mean velocity recordedby the hot-wire, the RMS values of the velocity #uctuations and the dominantfrequency of the #ow a broad picture of the #ow structure can be produced. However,the expectation is that the use of SSA will provide more information on the dynamicsof the #ow and its structure.

As a "rst step in this procedure the embedding dimension m must be chosen.Various means of doing this are available. For example, one method is to simply take


Fig. 5. Scatter plot of non-dimensionalised shear (y-axis) against dominant frequency (x-axis) at 50 mmdownstream.

the ratio of the sampling frequency to the maximum frequency of interest that in thiscase gives a value of 8}10 for frequencies of interest in the range 200}250 Hz. Equally,Takens [13] suggests that the embedding dimensions m is given by

m2n#1, (2)

where n is the dimension of the attractor that describes the dynamics. Gress [15] hasshown that a reasonable estimate for the value of the attractor dimension n is around10. To do this Gress looked at the structure of the eigenvalues returned by the singularvalue decomposition and determined that the relative magnitudes of the "rst 10 valueswere much the same regardless of the embedding dimension when this was greaterthan 20. Using the value of 20 for the embedding dimension m in Eq. (2) predicts thatthe attractor dimension n is close to 10. As a result of this the embedding dimensionused for the analysis has been set to the slightly more conservative value of 30. Also,the length of each time series is 4096 points and 3000 trajectory points have been usedto create the trajectory matrix.

In the initial exploration of the data, SSA was used to produce the singular valuesfor the system and these were normalised by the sum of the singular values at eachpoint, before computing the contours for the largest singular value alone and for thesum of the largest three singular values. Clearly, di!erent contours were produced in


each case. Also, it was noted that there was little correlation between any of thevalues of shear, RMS #uctuation and dominant frequency and the largest singularvalue. However, the correlation between the RMS value and the sum of the largestthree singular values was high for all three measurement planes, being 0.70 at 50 mm,0.66 at 100 mm and 0.61 at 200 mm. Clearly, the dynamics of the #ow is beingdescribed by both of these parameters, with large #uctuating velocity values beingcorrelated to high values of the sum of the three singular values, i.e., a low-dimensional#ow.

To determine the structure of the dynamics in a systematic way, contours of the sumof the largest two, four, six, eight and ten singular values are plotted in Figs. 6}8 forthe three measurement planes. Note that the "gures show that as more singular valuesare summed so a convergence in the dynamical structure is achieved. Clearly, if thesummation is carried out over all of the singular values then the sum will be unity, andso there must be some optimum number over which to carry out the summation. Inthe cases shown here there is some evidence that this optimum is eight values, as thecontours are less detailed when ten values are summed.

Concentrating on Fig. 6, and comparing this to the contours of mean value andRMS value shown in Figs. 2(a) and 3(a), the SSA produces a di!erent structure to bothof the previous, more simplistic, analyses. Now, the island-structure is much morewide ranging around the vortex, and also more complex. To interpret these pictures itshould be remembered that where the sum of singular values is high, the dynamics isdescribed by very few singular vectors and so is relatively simple. Conversely, wherethe sum of singular values is low, the dynamics is described by many singular valuesand so is relatively complex. Looking at Fig. 6(e), it can be seen that the islands to theleft of the vortex are denoted by lower numbers than those to the right of the vortex.Hence, the islands to the left have more complex dynamics than those on the right.Similar structures can also be seen in Figs. 7 and 8.

Note that the contours derived from the SSA shown in Figs. 6}8 illustrate a muchmore detailed #ow structure than those shown in Figs. 2}4 derived from a simpleranalysis. This is especially true for the planes at 100 and 200 mm downstream. Hence,it is clear that the SSA provides a more discriminating technique.

4.5. Using SSA to determine frequency content

Carrying out a detailed frequency content analysis of the raw data shows that the#ow can be said to be turbulent everywhere. At all points the spectrum is broadband.However, the SSA can be used to extract a spectrum with the noise reduced and thesignature of the #ow dynamics remaining. This has been done at all points in the 12thhorizontal line through the measurement planes (where y is 98 mm), as this corres-ponds to a line through the centre of the vortex 50 mm downstream. The sum of thefrequency content from the "rst "ve singular vectors has been taken. Again thefrequency content is very complicated, with a rich variation in the frequency spectraobtained. One thing is noticeable, however that the spectra contain only frequenciesbelow 200 Hz, as the SSA e!ectively acts as a noise "lter, removing the signals ofgreater frequency.


Fig. 6. Contours of the normalised sum of the dominant singular values (eigenvalues) at 50 mm down-stream (a) "rst two eigenvalues (b) "rst four eigenvalues (c) "rst six eigenvalues (d) "rst eight eigenvalues(e) "rst ten eigenvalues.


Fig. 6. Continued.


Fig. 6. Continued.

In previous work [11] it has been found useful to compare the energy contentof spectra by summing the contents of frequency bins over several bins. Here thishas been done by adding the content of the frequency bins from 0 to 50 Hz, thenfrom 50 to 100 Hz, 100 to 150 Hz and from 150 to 200 Hz. Fig. 9 shows theenergy content across the line without normalisation and then Fig. 10 shows theenergy content normalised by the total energy in the four frequency ranges at a givenpoint.

To develop understanding of the energy transfer that takes place along the line, it isuseful to compare the contours in Fig. 6 with the energy shown in Figs. 9 and 10.Along the 12th horizontal line of data the sum of the singular values falls steadily fromleft to right with a minimum at the eighth position. Then the sum rises through thevortex core to a local maximum at the centre of the core (positions 10}12) beforefalling to position 13, rising to position 14, falling to position 15, rising to position17}18 and "nally falling. Figs. 9(a) and (b) also show exactly this trend. Consequently,the SSA has created contours which correspond to the sub-100 Hz energy content ofthe #ow dynamics. This means that where the dynamics of the #ow is complex (lowersums of singular values) the energy content is smaller in the sub-100 Hz region.Equally, areas where the #ow us simpler (higher sums of singular values) there isa larger energy content in the sub-100 Hz region. Fig. 9(c) shows broadly similartrends for the energy content in the 100}150 Hz region. However, Fig. 9(d) shows thatthe energy content in the 150}200 Hz region is somewhat di!erent. In particular, the


Fig. 7. Continued.


Fig. 7. Continued.

energy content in this band is higher where the sum of singular values is botha maximum and a minimum.

Looking at the variation of the normalised energy content along this horizontalline, as shown in Fig. 10, di!erent trends cans be seen. Where the dynamics arecomplex, the lowest amounts of energy are held in the sub-100 Hz bands andthe highest amounts in the bands over 100 Hz. Equally, where the dynamics are rela-tively simple, the percentages of energy in the bands over 100 Hz are dramaticallyreduced (positions 10}12) compared to an increase in the energy content at lowerfrequencies.

5. Discussion

As was stated in Section 1, the aim of this work has been to demonstrate the use ofSSA in (1) determining the dynamical structure of the #ow in a turbulent vehicle wakeand (2) determining some aspects of the dynamics of a wake. Before SSA has beencarried out the raw data gathered using the HWA has been analysed in terms ofmean and RMS values, together with the calculation of dominant frequencies andmean shear. The contours of mean #ow velocity show the traditional picture ofa vortex behind a vehicle with island-like structures that are convected around the vortex.High RMS values are also found to be generated in areas of high shear close to the


Fig. 8. Continued.


Fig. 8. Continued.

vehicle. Similarly, the highest dominant frequencies are generated around the vortexcore, particularly in areas where the mean shear is low. This pre-analysis is useful inits' own right and sheds some light on the dynamics of the wake, in particular thegeneration of unsteadiness.

Moving to the SSA, the #ow is seen to have a rich structure as determined by thecontours of sums of singular values, which are measures of the complexity of the #ow.However, whilst the contours of the sums of singular values show those areas of the#ow where the dynamics are relatively simple or relatively complex, they do notgive any more information about the #ow dynamics. To understand the dynamics inmore detail the energy content of the #ow between 0 and 200 Hz has been analysed infour frequency ranges each of 50 Hz width. This has been compared to the sums ofsingular values.

This analysis shows that the areas in the #ow, which the SSA identi"es as being ofsimpler dynamics, have a large energy content in all four frequency ranges. Equally,those area identi"ed as being of more complex dynamics have low-energy content inthe ranges 0}50, 50}100 and 100}150 Hz, but high-energy content in the 150}200 Hzrange. These "ndings come from an analysis of the magnitude of the #uctuatingenergy. However, it is also useful to consider the energy content in each frequencyrange as a percentage of the total #uctuating energy at a point. Such a comparisonshows that areas of #ow complexity are characterised by a high percentage content inthe 100}150 Hz and 150}200 Hz ranges and a much lower percentage content in the


Fig. 9. Energy content (y-axis, notional units) at 50 mm downstream for y"98 mm (a) Up to 50 Hz(b) From 50 to 100 Hz (c) From 100 to 150 Hz (d) From 150 to 200 Hz.


Fig. 9. Continued.


Fig. 10. Normalised energy content (y-axis) at 50 mm downstream for y"98 mm (a) Up to 50 Hz(b) From 50 to 100 Hz (c) From 100 to 150 Hz (d) From 150 to 200 Hz.


Fig. 10. Continued.


two lower-frequency ranges. Hence, as #ow complexity increases there is a transfer ofenergy from low frequencies to higher frequencies.

From this energy analysis, it can be seen that SSA has identi"ed the complexity ofthe #ow dynamics in the wake in a way which is consistent with the energy content ofthe #uctuations in the #ow. Areas of simple dynamics have the energy concentrated infrequencies below 100 Hz and area of complex dynamics have the energy concen-trated in frequencies above 100 Hz. The contours of sums of singular values inFigs. 6}8 show that the #ow has simple dynamics (high contour values) in the vortexcore, in horizontal and vertical shear layers inboard and below the vortex core andoutboard and below the core, respectively, and also in islands around the vortex core.This is true for all three planes analysed here. Also the motion of these areas can beseen as the wake develops downstream. As the vortex moves downwards so thehorizontal shear layer is distorted and the islands are convected around the core.

Hence, a model of the #ow structure can be postulated, despite the noise in thesystem. The basic #ow structure generated by the model consists of a vortex and twoshear layers. In these areas the #ow has simple dynamics, with the oscillations in thewake generated in these regions at low frequencies. Also islands of #ow are foundaround the vortex with simple dynamics. As the wake develops these areas arein#uenced by the overall mean #ow and the dynamical structure is convected aroundthe main vortex system.

It is the use of SSA that has enabled this picture of the #ow structure to bedetermined as the use of simpler measures such as mean and RMS velocitites, anddominant frequency, does not show this structure in the second and third planesdownstream where contours of these quantities become very blurred. It is as if the SSAis a more discriminating technique, capable of resolving the dynamics despite thelevels of noise present in the system.

6. Conclusions

Detailed hot-wire measurements are necessary to resolve the frequency variation inthe time-dependent wake of a model passenger vehicle. Whilst simple measures of thehot-wire data shed some light on the #ow structure near the vehicle, they do not inplanes further downstream. By using SSA to analyse the #ow, areas of simple #owdynamics and complex #ow dynamics are found for the three planes analysed here.This enables a model of the wake dynamics to be postulated, despite the high levels ofnoise in the turbulent wake. In this model areas of relatively simple dynamics aremoved under the in#uence of the main vortex.

References

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Using Singular Systems Analysis to Characterise the Flow in the Wake of a Model Passenger Vehicle