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.

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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

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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

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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

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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.

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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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[4] R.J. Duro and J. Santos, “Discrete Time Backpropagation for Training

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