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CIMSA 2005 – IEEE International Conference on
Computational Intelligence for Measurement Systems and Applications
Giardini Naxos, Italy, 20-22 July 2005
On the use of Delay Based Networks in the Analysis of Turbulent Signals
F. Lopez Peña, F. Bellas, R. J. Duro, & M. L. Sanchez Simon
Integrated Group for Engineering, University of Corunna,
Mendizabal s.n., 15403 Ferrol, Spain
Phone: +34-981337400, Fax: +34-981337410, Email: flop@udc.es
Abstract – The signals obtained by a hot wire anemometer when
measuring at various points within two different and well known
turbulent flow fields are used as test cases to asses an analysis
system including delay based neural networks. The analysis method
thus developed proves to be suitable for reconstructing the turbulent
signals and, additionally, for extracting from them some of their
main dynamic features corresponding to the large structures
embedded in the turbulence.
Keywords – Turbulence, Signal Analysis, Neural Networks,
Coherent Structures
I. INTRODUCTION
Since the XIX Century, turbulence in fluids is considered
a very interesting and challenging matter by physicists,
mathematicians and engineers alike. For physicists and
mathematicians fluid turbulence represents the challenge of a
major unsolved problem having some interesting aspects
such as the fact that it displays some universal characteristics
or that it involves processes combining ordered and
disordered phenomena thus merging both random and
deterministic dynamics. For the engineer, controlling
turbulence and its effects is of paramount importance in
many practical processes. These can be greatly improved in
some cases by diminishing turbulence levels -as for instance
when trying to reduce loses induced by drag forces- or in
some others by increasing it -as in the case of industrial
mixing processes-.
Any turbulent fluid flow motion contains a broad range of
spatial scales and frequencies corresponding to the many
different sizes of the eddies conforming the turbulent flow.
The so called Kolmogorov scale is the smallest one and
corresponds to the size of the tiny eddies where dissipation
takes place. On the other hand, most of the turbulent kinetic
energy resides in some of the larger scales called the integral
scales. These large scales usually exhibit quite evident
structures, namely coherent structures, whereas the small
scales are relatively isotropic [1]. The relative importance of
the coherent structures in the turbulence dynamics changes
very much from case to case depending on the flow geometry
and conditions. In many cases they have a relatively large
influence on the statistic values of flow magnitudes when
applying the classical approach of carrying out the analysis
after statistically averaging the fluctuations of turbulence
within the flow field. Thus, in order to improve the results of
calculations, they should be treated separately within the
analysis. Following this broad picture, turbulence can be
considered a chaotic phenomenon having many degrees of
freedom both in the time and space domains [2].
Nevertheless, the investigation of turbulent flows by
approaches other than purely statistical are not common and
have been attempted only during the last two or three
decades.
Synaptic delay based artificial networks are applied in the
present investigation to analyze the signals coming from a
hot wire anemometer when used to perform measurements in
two different and very well known turbulent flow fields,
employed here as workbenches. The first of these test cases is
the wake of a round cylinder and the second one is an air jet
exhausted into the atmosphere. These two cases belong to the
turbulent free shear layer family where the turbulence can
develop freely without any interference from any nearby
boundary. The fundamental parameter governing these or any
other turbulent flow is the so called Reynolds number,
defined as de ratio between inertial and viscous forces. For
both cases under analysis, the Reynolds number has a value:
Re= VD/ , (1)
being the density of the fluid and its viscosity, D the jet
or cylinder diameter, and V the mean flow velocity. The
cylinder wake presents strong embedded coherent structures
for values of the Reynolds number in the few thousands
range. The turbulent jet case is much noisier as it has a
surrounding shear layer acting as a strong source of
turbulence and the coherent structures present are not as
marked as in the other case. In both cases turbulence decays
when moving far downstream from the cylinder or the jet
exit.
A hot wire anemometer is used to perform the
measurements. This instrument is a basic fluid dynamics
research tool that is very suitable for the experimental
characterization of turbulent flows due to its ability to
measure flow fluctuations up to the hundred thousand Hertz
range [3]. The use of a single hot wire probe to perform the
experiments provides a time dependent signal corresponding
to the flow velocity fluctuations at the point where the probe
is set. For every test case, the analysis is performed
exclusively on the data array of the corresponding signal
without introducing any other information. This represents a
typical situation where the analysis of a high dimensional
complex phenomenon should be performed on a very limited
set of data having a much lower dimensionality than the
phenomenon under measurement.
0-7803-9025-3/05/$20.00 ©2005 IEEE 341
II. EXPERIMENTAL SET-UP
The experimental set-up for the round cylinder wake test
case is shown in Figure 1 where a round cylindrical bar with
a diameter of 8 mm is placed transversally in the test section
of an open jet wind tunnel. The density of air is calculated
applying the perfect gas equation by using the values of the
temperature in the incoming flow and the ambient pressure.
In order to be able to measure these magnitudes, a barometric
pressure sensor and a solid state temperature sensor are used.
The calculated value of the density of air under the test
conditions is then applied to calculate the value of the air
speed after measuring the difference in pressure in a static-
Pitot probe by means of a differential pressure transducer. All
the above mentioned sensors, as well as the hot wire
anemometer processor, are connected to a data acquisition
board installed in a personal computer trough an interface
box. This box contains all the filters and amplifiers needed to
condition the signals coming from the different transducers
and sensors. A RS232 communication cable is used to
communicate the hot wire anemometer processor from the
computer. A specific virtual instrument has been developed
under the Labview environment to perform all the acquisition
and control the wind tunnel in an appropriate way. This
instrument calculates the Reynolds number in real time,
displays it on the screen and uses it as control parameter in a
PID control subsystem.
Fig. 1. Experimental Set-up for the round cylinder wake case.
Figure 2 represents the experimental set-up for the
turbulent jet test case. The air jet under analysis is coming
out of a device taking air from the pressurized air supply of
the lab after passing through a settling chamber followed by a
contraction. In this case the air flow is controlled manually.
Most of the instrumentation is the same as in the previous
case except for a thermocouple that is used to measure the jet
temperature. This thermocouple is also used by the hot-wire
anemometer to perform corrections to errors in the velocity
measurements due to temperature fluctuations. In addition, in
this case the differential pressure transducer is connected
between the settling chamber and the ambient.
Fig. 2. Experimental Set-up for the turbulent jet case.
It is necessary to extract from the signals produced by the
sensors those features that are relevant to the classification
problem, thus making the classifier design simpler and
improving classification accuracy. Here the problem is
understanding which features describe the process in a way
almost as accurate as the full signals, and which constitute a
minimal set, avoiding both the “curse of dimensionality” and
incomplete process description.
III. TURBULENT SIGNAL PREDICTION AND
ANALYSIS
Predicting the evolution of a complex chaotic signal is a
problem of dynamic reconstruction which involves obtaining
some type of description of a given chaotic time series
obviating the need for detailed mathematical knowledge of
the underlying processes that conform its dynamics.
Basically, what is required is some type of function that can
autonomously adapt to the signal under study to the level of
detail required so that after some type of training or
parameter adjustment, it can generate it. This is the approach
followed here in order to study the turbulent signals
experimentally obtained in the different test cases. For each
of these cases, a sample of the corresponding signal is used at
first as training set for a neural network that learns to predict
it and which afterwards is quite efficient when performing
342
multi-step prediction processes of the given chaotic time
series. After the trained network proves to be suitable for
reconstructing the turbulent signal, its input is disconnected
from the real signal and is fed back instead from its previous
output. Thus, in this configuration, the network becomes a
signal generator. The signal generated this way is, in the first
steps, very similar to the original turbulent one, but soon it
starts a rapid decay by first loosing the fine details
corresponding to the smaller scales and then those of the
larger ones until it becomes flat or sinusoidal. In this work,
we show that these types of artificial neural networks learn
the regularities in these signals and thus reproduce them in
the remaining sinusoidal signal or during a transient time that
many times is large enough to allow obtaining the most
significant spectral peaks; these are shadows of the original
signal main features, especially those related with the
coherent structures present in the larger scales. The
application of the networks configured in this way as an
analysis tool was show by the authors in a previous work but
only applied to the cylinder wake case [6] and mainly to
detect the presence of coherent structures in situations were
standard tools did not perform adequately. The current
investigation presents some more test cases of this type of
flows and introduces this application to the analysis of
turbulent jet flow field which has quite different turbulent
characteristics when compared to the other test case. For
every test case the analysis procedure proves to be very good
when used in its signal prediction mode. In the generator
mode it is much more efficient than traditional frequency
analysis methods in detecting the presence of peaks
embedded in the jumbled spectrum characterizing some of
these signals. These peaks are an evidence of the presence of
organized structures in the flow and their frequencies
correspond to the ones obtained by traditional linear methods
when moving the probe to a position where they are strong
enough to be detected by these means.
IV. THE NEURAL NETWORK
The artificial neural network we consider for training
consists of several layers of neurons connected as a Multiple
Layer Perceptron (MLP); it is represented schematically in
Figure 3. There are two differences with respect to traditional
MLPs. The first one is that the synapses include a trainable
delay term in addition to the classical weight term [4]. That
is, now the synaptic connections between neurons are
described by a pair of values, (W, �), where W is the weight,
representing the ability of the synapse to transmit
information, and � is a delay, which in a certain sense
provides an indication of the length of the synapses. The
longer it is it will take more time for information to traverse
it and reach the target neuron. The second one is that some of
the nodes implement a product combination function instead
of the traditional sum.
We have developed an extension of the backpropagation
algorithm for training both parameters of the connections,
(W, �), and have called it Pi Discrete Time Backpropagation
( -DTB) [5]. This algorithm permits training the network
through variations of synaptic delays and weights, in effect
changing the length of the synapses and their transmission
capacity in order to adapt to the problem in hand.
In addition, through the appropriate determination of the
delay terms in the synapses the -DTB algorithm performs
an automatic selection of the signal points to be correlated.
Consequently, when speaking in the language of the dynamic
reconstruction of signals, the network automatically obtains
the embedding delay and embedding dimension.
If we take into account the description of the network in
terms of synaptic weights and synaptic delays, the main
assumption during training is that each neuron in a given
layer can choose which of the previous outputs of the
neurons in the previous layer it wishes to input in a given
instant of time. Time is discretized into instants, each one of
which corresponds to the period of time between an input to
the network and the next input. During this instant of time,
each of the neurons of the network computes an output,
working its way from the first to the last layer. Thus, for each
input, there is an output assigned to it.
In order to choose from the possible inputs to a neuron the
ones we are actually going to input in a given instant of time,
we add a selection function to the processing of the neuron.
This selection function could be something as simple as:
������
ji
jiij
0
1 (2)
so that the output of a traditional (3) and a product (4) node k
in an instant of time t is given by:
����
���
N
i
t
j
ijiktjkt hwFOik
0 0
)( (3)
����
������
N
i
t
j
ijiktjkt hwFOik
0 0
)(� (4)
where F is the activation function of the neuron, hij is the
output of neuron i of the previous layer in instant j and wik is
the weight of the synapse between neuron i and neuron k. The
first sum (or product) is over all the neurons that reach
neuron k (those of the previous layer) and the second one is
over all the instants of time we are considering (let´s say
since the beginning of time, although in practical applications
the necessary time is finite).
The result of this function is the sum or product of the
outputs of the hidden neurons in times t-�ik (where �ik is the
delay in the corresponding connection) weighed by the
corresponding weight values.
Now that we know what the output of each neuron is as a
function of the outputs of the neurons in the previous layer
343
and the weights and delays in the synapses that connect these
neurons to it, what we need is an algorithm that allows us to
modify these weights and delays so that the network may
learn to associate a set of inputs to a set of outputs. The basic
gradient descent algorithm employed in traditional
backpropagation may be used, but we must now take into
account the delay terms when computing the gradients of the
error with respect to weights and delays.
As shown in [5], these gradient terms are:
)( jktjk
jk
total hw
E ���
(5)
)1()( jkjk tjtjjkk
jk
total hhwE �����
(6)
being Etotal the total squared error for all the training vectors
and:
)()(2 kkk
k
k
k
totalk ONetFTO
ONet
O
O
E �����
(7)
where Tk is the desired output, Ok the one really obtained and
ONetk is the combination of inputs to neuron k, when we
consider output neurons. For hidden neurons connected to the
input layer, and defining as before:
kr
r
rk
k
totalk whNetF
hNet
E �� )(��
(8)
where index r represents the neuron of the next layer,
whether output or hidden, we have the following derivatives
for the weights and connections:
)( jktjk
jk
k
k
total
jk
total Iw
hNet
hNet
E
w
E �������
(9)
)1()( jkjk tjtjjkk
jk
k
k
total
jk
total
IIw
hNet
hNet
EE
!"!!!!"!
(10)
where the second derivative in (9) is the result of:
jk
N
i
t
n
iniktn
jk
k
IwhNet ik
!"!!"! #$
%&'())
0 0
)(
(11)
when we consider neurons in a hidden layer. It may be
observed that the derivative in hNet of equation (11) has been
discretized in order to obtain (10), implicitly assuming there
is a certain continuity in the temporal variation of the outputs
of the neurons, which in practice turns out to be a valid
assumption.
Regarding the product units in the first hidden layer, we
have:
)(
0
rktrrk
N
r
k IwhNet *+ (12)
and the derivatives are:
jk
trrk
N
r
jk
k
w
Iw
w
hNet rk )(
0
*!!,
(13)
)1()(
)(
)(
0
,jkjk
jk
rk
tjtj
tj
trrk
N
r
jk
k III
IwhNet ****
!"!
(14)
where index r denotes the input nodes connected to the
corresponding product hidden unit. The importance of the
procedure is that now the appropriate delays for these
connections, and consequently the temporal values that make
up the product terms in the product units are obtained
automatically, and are not imposed beforehand.
As we show, by discretizing the time derivative we obtain
simple expressions for the modification of the weights and
delays of the synapses, in an algorithm that is basically a
back-propagation algorithm where we have modified the
transfer function of the neuron to permit a choice of inputs
between all of the previous outputs of the neurons of the
previous layer.
Fig. 3. The network used.
V. RESULTS
As indicated above, neural networks have been applied to
analyze hot wire signal in two very different ways. As a first
approach, they were applied to predict a few time steps in
advance the measured signals in both the wake and the jet
344
cases. Our results show that the prediction obtained by the
networks for all cases tested appears always to be very good.
An example of this multi-step prediction can be seen in
Figure 4 representing the signal obtained by the hot wire
anemometer in the cylinder wake at a position 20 diameters
downstream of the cylinder when the Reynolds number value
is 2000. Figure 5 presents the case of a turbulent jet with a
Reynolds number of 5000 and the probe placed at 5
diameters from the jet exit and aligned to its edge. It can be
seen that the performance achieved by the multi-step
prediction process is adequate in both examples. As
mentioned earlier, this has been always like this in all cases.
Fig. 4. Measured (solid line) and predicted (dotted line) signal in a cylinder
wake at 20 diameters downstream of the cylinder and with Re=2000.
Fig. 5. Measured (solid line) and predicted (dotted line) signal in an air jet at
5 diameters downstream of the jet exit and with Re=5000.
When the ANN is used as a signal generator, that is, as an
analysis tool, the behavior can be different in both types of
flows. In the wake case it is able to detect a main peak of the
signal in conditions where a standard FFT is incapable of
doing it. For instance, figure 6 presents the power spectra of
the signal presented in figure 4 and of the signal generated by
the ANN during a time of 0.2 s starting 0,1 s after the real
signal input has been shut off. The spectrum of the real signal
presents a maximum at a frequency of 96 Hz and so does the
generated signal spectrum. However, the latter has lost a lot
of the details. If the FFT is performed with a later time
window, everything but the main peak will vanish. Figure 7
presents the power spectra of these signals taken at the same
position but after doubling de air speed value leading to a
Reynolds number of 4000. We can see that no peak can be
distinguished in the original signal spectrum because of the
increase in turbulence. Nevertheless, the spectrum of the
generated signal presents a peak at twice the frequency of the
previous case which is in fact the frequency corresponding to
the coherent structures under these flow conditions.
Fig. 6. Power spectra of the measured (solid line) and self-generated (dotted
line) signal in a cylinder wake at 20 diameters downstream of the cylinder
and with Re=2000.
Fig. 7. Power spectra of the measured (solid line) and self-generated (dotted
line) signal in a cylinder wake at 20 diameters downstream of the cylinder
and with Re=4000.
The spectra presented in figure 8 correspond to the
turbulent jet signal of figure 5 and its decaying generation by
the ANN in analysis mode. The real signal spectrum (solid
line) presents a maximum that is not well marked because the
random component of the turbulent signal is more intense in
this type of flow. The spectrum taken in a time window
during the decay of the self generated signal presents a better
discernible maximum at the same frequency.
Fig. 8. Power spectra of the measured (solid line) and self-generated (dotted
line) signal in a turbulent jet at 5 diameters downstream of the jet exit with
Re=5000.
345
The analysis of the jet signals by this system have proven
to be more difficult that the wake ones as in this type of flows
the self-generated signal decays much more rapidly than in
the other case and it always goes to a flat signal. Still, if a
maximum exist its decay is always slower than the rest of the
spectrum. In many cases the system can also detect a
maximum where no peak is perceptible in the real signal
spectrum. Therefore, this analysis mode of the network
appears to be a useful tool for detecting organized structures
within the turbulent flow by looking exclusively at the signal
generated in a single point and without having any other
spatial information of the flow-field.
The complexity of turbulence in the jet test case
sometimes produces situations were the analysis system may
present two peaks in the spectrum. As an example, figure 9
shows the power spectrum of the original signal (solid line)
and the self-generated one obtained by the network (dotted
line) for the case presented in figures 5 and 8 but placing the
probe at half of the distance from the jet exit. In this case, the
network provides a successful result presenting two main
frequencies at 64 Hz and 117 Hz while the original ones are
in 67.6 Hz and 114 Hz. As we can see, the higher frequency
is not clearly defined in the original signal, but in the
prediction there is a clear peak. For this case, the network
used consisted in two hidden layers each one of them with 30
neurons.
Fig. 9. Power spectra of the measured (solid line) and self-generated (dotted
line) signal in a turbulent jet at 2.5 diameters downstream of the jet exit with
Re=5000.
VI. CONCLUSIONS
In this paper we present a new approach for the dynamic
reconstruction through artificial neural networks of turbulent
signals taken in a single point within the flow field. Two
different well known turbulent flows have been used for
testing; a round cylinder turbulent wake and a turbulent jet
exhausted into the atmosphere. After training on the real
signal on line, this network has been able to make a multi-
step prediction of the signal in every case tested. The network
is then used in an analysis mode in which, and after some
time of predicting, the input signal is shut off and the
network generates the learned signal by using as inputs its
own predicted outputs. In this mode the generated signal
represents a decay of the original one towards a sinusoidal or
a flat signal. In many cases this signal presents during its
decay clear peaks in its spectrum corresponding to the main
characteristic frequencies of the original signal, even in cases
where these peaks cannot be seen in the original power
spectrum. Therefore, the signals thus generated reproduce the
basic spectral features of the flows under consideration and
facilitate their analysis.
ACKNOWLEDGMENT
This work was funded by the MCYT of Spain through
project VEM2003-20088-C04-01 and Xunta de Galicia
through projects referenced PGIDIT03TIC16601PR and
PGIDIT04DPI166004PR.
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Structures, Dynamical Systems and Symmetry”. Cambridge University
Press, 1998.
[2] M. Lesieur. “Turbulence in Fluids”, Third Revised and Enlarged
Edition. Kluwer Academic Publishers. Dordrecht, 1997.
[3] H. H. Bruun. “Hot-wire anemometry: principles and signal analysis”.
Oxford University Press. Oxford, 1995.
[4] R.J. Duro and J. Santos, “Discrete Time Backpropagation for Training
Synaptic Delay Based Artificial Neural Networks”, IEEE Transactions
on Neural Networks, Vol.10, No. 4, pp. 779-789, 1999.
[5] J. Santos and R.J. Duro, “Pi-DTB Discrete Time Backpropagation with
Product Units”, Connectionist Models of Neurons, Learning Processes
and Artificial Intelligence, Springer Verlag, pp 207-214, 2001.
[6] F. López Peña, R. J. Duro, and M. l. Sánchez Simón. “Detecting
Coherent Structures in a Turbulent Wake by Using Delay Based
Networks”. Computer Standards and Interfaces, 24 pp 171–184 (2002)
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