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*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 -
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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)
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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
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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
4 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000)
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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
C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30
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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.
6 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000)
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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.
C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30
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Fig. 2. Contours of mean velocity (m/s) in the wake (a) 50 mm
downstream (b) 100 mm downstream(c) 200 mm downstream.
8 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000)
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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.
C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30
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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.
10 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000)
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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
C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30
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Fig. 4. Contours of dominant frequency (Hz) in the wake (a) 50
mm downstream (b) 100 mm downstream(c) 200 mm downstream.
12 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000)
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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
C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30
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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
14 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000)
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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.
C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30
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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.
16 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000)
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Fig. 6. Continued.
C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30
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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
18 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000)
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Fig. 7. Contours of the normalised sum of the dominant singular
values (eigenvalues) at 100 mm down-stream (a) "rst two eigenvalues
(b) "rst four eigenvalues (c) "rst six eigenvalues (d) "rst eight
eigenvalues(e) "rst ten eigenvalues.
C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30
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Fig. 7. Continued.
20 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000)
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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
C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30
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Fig. 8. Contours of the normalised sum of the dominant singular
values (eigenvalues) at 200 mm down-stream (a) "rst two eigenvalues
(b) "rst four eigenvalues (c) "rst six eigenvalues (d) "rst eight
eigenvalues(e) "rst ten eigenvalues.
22 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000)
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Fig. 8. Continued.
C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30
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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
24 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000)
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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.
C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30
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Fig. 9. Continued.
26 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000)
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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.
C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000) 1}30
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Fig. 10. Continued.
28 C.T. Shaw et al. / J. Wind Eng. Ind. Aerodyn. 85 (2000)
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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.
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