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Reduced Order Modeling of a Turbulent Two Dimensional Cylinder Wake with Filtered POD and Artificial Neural

Networks

Akin Paksoy1, Buryan Apacoglu2 and Selin Aradag3

TOBB University of Economics and Technology, Ankara, 06560, Turkey

The aim of this research is to represent the flow behind a two-dimensional (2D) circular cylinder at a turbulent Reynolds number with the help of Artificial Neural Networks (ANN’s) in order to be able to control the vortex shedding formed in the wake region by training the network using Computational Fluid Dynamics (CFD) simulation results. The flow over a 2D circular cylinder is analyzed by CFD, and the results are validated with the experimental results given in literature. In order to observe turbulent flow structures and their effects in the wake region for control purposes with ANN, the order of the flow is reduced by application of the Proper Orthogonal Decomposition (POD) method to the data ensemble obtained from CFD analyses. A filtering technique based on Fast Fourier Transform (FFT) is added to the POD procedure, and it is observed that 99% of the total energy content is represented by using only the two most energetic modes where the effects of large scale turbulent structures are revealed. For further flow field observation with ANN estimation, six surface mounted pressure sensors are heuristically selected based on a regular snapshot-based POD application to the surface pressure data. Then, an ANN structure is designed to estimate the mode amplitudes of the two most energetic POD modes from the sensor pressure data.

Nomenclature CD = Drag coefficient

= Time step size = Spatial increment in the x-direction = Spatial increment in the y-direction

Re = Reynolds number based on diameter St = Strouhal number

i = Input to the ANN structure yi = Output from the ANN structure wi

k = Weight values assigned in the network bk = Bias value

= Activation function

I. Introduction HE phenomenon of vortex shedding behind bluff bodies has been a subject of extensive research. Many flows of engineering interest produce this phenomenon and the associated periodic lift and drag response1. External

flow over bluff bodies is an important research area because of its wide range of engineering applications. Although, the geometry of a bluff body can be simple, the flow behind it is chaotic and time-dependent after a certain value of Reynolds number. Forces acting on the body such as drag and lift also vary in time, and cause periodic loading on the body. These forces originate from momentum transfer from fluid to the body, where the magnitudes of them are strongly related to the shape of the body and properties of the flow.

1 Graduate Student, Department of Mechanical Engineering, Member AIAA. 2 Graduate Student, Department of Mechanical Engineering. 3 Assistant Professor, Department of Mechanical Engineering, Senior Member AIAA.

T

49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition4 - 7 January 2011, Orlando, Florida

AIAA 2011-58

Copyright © 2011 by Akin Paksoy, Buryan Apacoglu, Selin Aradag. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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Flow over a circular cylinder problem arises in diverse engineering applications such as hydrodynamic loading on marine pipelines, risers, offshore platform support legs, chemical mixing, lift enhancement etc.2,3 It is experimentally investigated by Norberg4 that when the Reynolds number of flow over a circular cylinder exceeds 48, vortices separate from the cylinder surface, and start to move downstream, where steady-state behavior of the flow turns into a time-dependent state. These periodically moving vortices at the downstream form self-excited oscillations called the von Karman vortex street3. Separation from the surface of the cylinder can either be laminar or turbulent according to the regime of the flow in the boundary layer. It is shown by Wisink and Rodi5 that flow with a Reynolds number between 1000 and 20,000 is called subcritical, and in this range, boundary layer on the cylinder is entirely laminar and transition from laminar to turbulent flow happens somewhere at the downstream. Although vortex street is fully turbulent after Re 20,000, laminar separation sustains up to a Reynolds number of 100,0006. There are several experimental and computational studies in literature that examine the flow over a circular cylinder at subcritical Reynolds numbers7-10. In order to observe a time-dependent system in detail, a practical procedure is needed to separate the space and time dependency. The Proper Orthogonal Decomposition (POD) is a reduced order modeling technique used to analyze experimental and computational data by identifying the most energetic modes in a sequence of snapshots from a time-dependent system11. It has been used in numerous applications to introduce low-dimensional descriptions of system dynamics by extracting dominant features and trends12. The POD technique was originally introduced by Karhunen and Loéve, which has several distinct names among different disciplines as Karhunen-Loéve Decomposition, Principal Component Analysis and Singular Systems Analysis13. The major theory of the POD technique offers infinite linear combinations of orthogonal functions to represent a stochastic process or a system14. The POD technique was originally developed in the context of pattern recognition, and it has been used successfully as a method for determining a low-dimensional description for human face, structural vibrations, damage detection and turbulent fluid flows15. In addition, the method has also been used for industrial and natural applications, such as supersonic jet modeling, thermal processing of foods, investigation of the dynamic wind pressures acting on buildings, weather forecasting and operational oceanography11. There are several studies in literature that utilize the POD method in fluid mechanics applications as a reduced order modeling tool16-20. Fluid mechanical systems that have turbulence show complex fluid motion and flow structures that display different ranges of length scales. Low-dimensional modeling by application of the POD method identifies orthogonal basis functions and time-dependent coefficients that reveal relevant modes of the flow field and enable representation of complex random flow fields by considering an optimal amount of the most energetic modes. The classical POD approach based on Snapshot Method is developed by Sirovich21, and it optimizes modes based on energy. In laminar flow data analyses, where energy drop-off for higher order modes is rather steep, the classical snapshot-based POD approach can successfully be used directly to analyze flow structures formed in the wake region22, such as vortices that persistently appear, disappear and then appear again continuously23. However, in energetic and randomly characterized turbulent flows, application of the snapshot-based POD method directly to the data ensemble obtained by CFD results poses problems24. Because turbulent flows contain the effects of both large and small-scale turbulent structures in the flow field, each resulting snapshot-based POD mode also contain structures of different sizes. This requires that many more modes have to be kept in the truncated model to represent a specific portion of the flow field. What is more, the classical snapshot-based POD procedure is not able to separate the flow structures according to their scales during configuration of the modes. This is an unfavorable result in many important engineering applications. For instance, if a control strategy is planned to be adapted to a system under investigation to control the vortices formed in the wake region, the control parameters are highly dependent on frequency and size of the structure formed behind the bluff body24. To overcome the difficulties of the snapshot-based POD approach in turbulent flow analyses, a couple of filtering techniques are offered in literature to separate the effects of small-scale turbulent fluctuations, and the remaining unwanted bifurcations observed in the wake region. In this study, one of the most promising filtering techniques for turbulent flow analyses, a Fast Fourier Transform (FFT)-based spatial filtering procedure offered by Aradag et al25 is applied to the data ensemble obtained from the CFD analyses of 2D flow over a circular cylinder. For practical applications, it is important to estimate the state of the flow, e.g. the relevant POD mode amplitudes, using body-surface mounted sensors. They are simple, relatively inexpensive and provide reliable data27. For real-time flow control applications, it is important to predict the flow based on surface sensors placed at a few discrete points. A study in which regular snapshot-based POD application is utilized for laminar flow conditions to the 2D cylinder surface pressure measurement data in order to evaluate optimal sensor locations for further ANN studies performed by Apacoglu et al27.

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Artificial Neural Networks (ANN’s) refer to computing systems the main idea of which is inspired from the analogy of information processing in biological nervous systems. A neural network works as a mathematical function which transforms a set of input variables into a set of output variables28,29. By using ANN’s, it is possible to obtain a solution for complex problems that do not have an analytical solution via application of conventional approaches. Currently, neural networks are used for the solution of problems in system identification, such as pattern recognition, data analysis, and control. Apart from these, ANN’s have also been applied in diverse problems in several fields such as insurance, medicine, economic predictions, speech recognition and image processing.30

Several notable features of ANN’s include relatively high processing speeds and the ability to learn the solution of a complex problem from a set of examples. For example, in a case studied by Zhang et al31, ANN’s are used to estimate flow characteristics from previous flow dynamics. They observed two-dimensional von Karman vortex structures in an elongated rectangular cross-sectional area of a static prism. They used previously obtained von Karman structural phases observed at certain Reynolds numbers to estimate phases for new Reynolds numbers.31

II. Objective In this study, an FFT-based Proper Orthogonal Decomposition (POD) procedure is applied to the computational

data obtained from the 2D CFD simulations performed for a circular cylinder for a Reynolds number of 20,000 to separate the undesired effects of small scale turbulent structures, and to reduce the order of the flow with the help of the POD technique to observe the effects of dominant large-scale turbulent structures in the wake region. Artificial Neural Networks (ANN’s) are used to predict the time coefficients obtained from the FFT-Filtered POD analysis by using only the sensor data from several locations on the cylinder surface. The training and validation data used for the neural nets are from several computational cases where the flow is either unforced or forced by air blowing using different blowing velocities.

III. Methodology

A. Computational Fluid Dynamics (CFD) Methodology For the CFD simulations, ANSYS Fluent® (v12) software is used. Pressure-based Navier-Stokes Equations are

solved, and incompressible turbulent flow is modeled in the simulations. Operating conditions and fluid properties are: density=0.01056 kg.m-3, velocity=34 m.s-1, viscosity=1.795x10-5 kg.(m.s)-1 and pressure=872.36 Pa. The dimensions of the computational domain are defined as 40 m x 40 m and the cylinder, the diameter of which is equal to 1 m, is placed at the center of this domain. For the computations, y+ value is smaller than 0.2. A time step of equals to 1x10-3 seconds is used for the computations. Spalart-Allmaras turbulence model is used to model turbulent structures. The details of the boundary consitions, grid refinement study and computations are provided by Apacoglu32. The drag coefficient (CD), Strouhal number (St), pressure coefficient distribution around the cylinder and the velocity profiles at the wake are validated using the experimental results of Lim and Lee9, Aradag et al10. Air blowing is applied from several slots located on the cylinder surface to force the flow as shown in Fig. 1. Slots are opened in different combinations for different blowing velocities as outlined below:

Blowing from all slots, u=0.1 U Blowing from slots numbered 1 and 4, u=0.1 U Blowing from slots numbered 1 and 2, u=0.1 U Blowing from slots numbered 2 and 3, u=0.1 U Blowing from slots numbered 1, u=0.1 U Blowing from all slots, u=0.5 U Blowing from slots numbered 1 and 4, u=0.5 U

where U=34 m.s-1. Futher details about the forced CFD simulations of turbulent flow are given in Apacoglu32. Velocity values obtained for the x-direction at the wake region are recorded in each time step during 10 periods of the flow time in order to obtain data ensemble required for the POD application and ANN estimations.

B. Reduced Order Modeling – Filtering and POD Methodology The snapshot-based POD method provides a basis for the modal decomposition of an ensemble of functions, containing the data obtained from experiments or computational simulations. The technique is used to analyze data with a view of extracting dominant features and trends named as coherent structures that are typically patterns of

Figure 1. Position of the slots located on the circumference of the cylinder

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space (basis functions) and time (mode amplitudes)33. Detailed mathematical procedure for this method is given in Paksoy et al22. In turbulent fluid flows, according to Kolmogoroff's hypothesis, firstly large-scale turbulent structures, and then, by further separation of them small-scale turbulent structures are generated in the wake region behind the bluff bodies. Since the small ones grow from large ones, if the large-scale turbulent structures are controlled, it may be possible to control the whole cylinder wake. This issue is important when a typical control strategy is needed to be applied in the wake region to control the vortex street targeting only the large-scale turbulent structures. However, turbulent flow has the characteristics of a more energetic and random behavior in which energy of the flow is transferred faster to the small-scale turbulent structures that hinders direct application of the classical snapshot-based POD approach to separate the structures according to their scales. Therefore, application of a filtering procedure integrated with POD method is inevitable to obtain necessary mode amplitudes in turbulent flows for a better observation of the von Karman vortex street formed in the wake region, and for further engineering flow control applications25.

There are a couple of filtering methods employed in literature including a variety of fields34-40. In this study, filtering the effects of small-scale turbulent structures from the flow field is achieved by application of a Fast Fourier Transform (FFT) method to the data ensemble obtained from CFD analysis. This procedure (FFT-Filtered POD method) was originally developed and used for three-dimensional flows by Aradag et al25.

C. Sensor Placement Methodology A one-dimensional regular snapshot-based POD analysis is carried out on cylinder surface pressure data to identify optimal sensor locations. In practical engineering applications, it is neither feasible to use sensors for measurements directly in the wake region nor possible to obtain accurate data, since it is not practical to place and fix sensors in the wake region. Besides, body-surface mounted sensors are simple, relatively inexpensive and provide reliable data for further analyses26. Pressure data obtained from sensors are essential in training of neurons in the ANN in order to help them to perform real-time estimations of the mode amplitudes. In the POD analysis of the cylinder surface pressure measurement, CFD data providing pressure signals on the cylinder surface at 360 locations with one-degree increments and corresponding 1337 snapshots are used. After evaluating spatial and temporal characteristics of the surface pressure data at 360x1337 points by the one-dimensional regular snapshot-based POD analysis, it is observed that considering only the most energetic three pressure-based POD modes contain more than 99% of the total energy content as shown in Table 1.

Table 1. Energy content change with respect to mode number including characteristic of the pressure data

Mode Number Energy Content (%) Mode Number Energy Content (%) 1 92.14 4 0.46 2 4.50 5 0.29 3 2.53 6 0.05

Total (3 Modes) 99.17 Total

(6 Modes) 99.97

Sensor placement methodology has a major challenge lying in the process of finding the appropriate number of sensors and determination of their locations that best enable the desired modal filtering which lead to the identification of the higher energy modes. In order to determine the optimum sensor locations, it is crucial to find the areas of most energetic activity that correspond to maxima and minima of the pressure-based POD modes, which enclose the characteristics of the surface pressure data26,41. For practical applications, it is desirable to reduce the amount of sensors required for the real-time estimation of the systems26. Since the contribution of the most energetic three pressure-based POD modes to the total energy content is greater than others, it is concluded that taking into account only the first three of them for identification of optimal sensor locations is enough. Sensor locations correlated to the energetic surface pressure maxima and minima of the first three pressure-based POD modes are given in Table 2. The locations of the sensors in Table 2 are referenced in terms of the circumferential angle measured from front stagnation point along the clockwise direction.

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Table 2. Sensor locations correlated to the energetic surface pressure maxima and minima

Mode Number

Related Sensor Location (degrees of angle)

1 (minima) 2 (maxima) 1 274 87 2 224 134 3 213 184

As it can be seen from Table 2, totally six sensors are placed onto the surface of the cylinder. Optimal sensor locations on the circumference of the cylinder are also shown in Fig. 2. The sensors corresponding to the first two modes target the periodic modes associated with the von Karman shedding frequency, whereas the sensors related with other modes target the non-periodic POD modes26.

Figure 2. Optimal sensor locations on the circumference of the cylinder

D. Artificial Neural Network (ANN) Methodology An Artificial Neural Network (ANN) is an interconnected assembly of simple processing elements, the functionality of which is loosely based on the biological neuron. The processing ability of the ANN is stored in the interunit connection strengths, or weights, obtained by a process of adaptation to, or learning from, a set of training patterns42. In ANN, a neuron is a processing element that takes number of inputs, weights them, sums them up, and uses the result as the argument for a singular valued function, which is called the activation function30. Figure 3 gives schematic representation of a single artificial neuron. It can be seen from Fig. 3 that change in every input causes also a change in the output, and the amplitude of this change is commonly dependent to the values of the weights and the type of the activation function. In Fig. 3, w1

k, w2

k, …, wmk represent weight values of the inputs that are shown with 1 2 m. Whereas, shows the bias value,

and corresponds to the activation function. The mathematical function of single neuron containing network structure is given by Eq. (1) and Eq. (2).

1

mk k

i ii

(1)

1 0

m mk k k k k k

i i i ii i

(2)

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Figure 3. Schematic representation of a single ANN neuron

Among a variety of network structures the most common one is the multilayer perceptron network (MLP) or also referred as the feedforward network that consists of two or more layers (Fig. 4)43.

Figure 4. Basic structure of a two-layer MLP network

In Fig. 4, the first layer is known as the hidden layer since it is in some sense hidden between the external inputs ( 1 to 4) and the output layer, which produces the output of the network (y1

k and y2k). W1 and W2 are the matrices

represent the weight values respectively connecting inputs to hidden layer neurons and correspondingly to output layer neurons. In order to determine the weight values included in W1 and W2, there has to be a set of examples of the outputs that are related to the inputs. The determination process of weights from the prior examples is known to be training or learning30,43. In this study, pressure data obtained from surface sensors and previously obtained POD mode amplitudes are used as inputs to the neural network structure. At the end of ANN study, it is needed to estimate mode amplitudes that are the same as the mode amplitudes obtained from the POD analysis of the CFD results, but without using further CFD simulations. Since the system consists of multi inputs (six sensor pressure data and sampling mode amplitudes), and requires multi outpus (each estimated mode amplitude will be an output), an MLP structure is used. The MLP network structure provides great harmony for discrete-time modeling of dynamic systems for which there is a nonlinear relationship between the system’s input and output. Assuming t counts of sampling period, while y(t) specifies the present output, y(t-1) signifies the output observed at the previous sampling instant. If it is assumed that the output of the dynamic system at discrete time instances can be described as a function of a number of past inputs and outputs as shown in Eq. (3), an MLP network can be used for approximation as given in Eq. (4) 30.

1 ,..., , 1 ,... ,y t S y t y t n u t u t m (3)

1ky

2ky

1

2

3

4

1kw

ky ku

kb 2kw

kmw

1

2

m

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, ,0 ,01 1

,hn n m

j j j p p j ij p

(4)

In this research, the ANN estimation method of choice including application of the MLP network structure is based on a nonlinear system identification approach described in Norgaard et al30, using ANN’s in collaboration with Auto-Regressive, eXternal input (ARX) model structure. This model includes nonlinear optimization techniques based on the Levenberg-Marquardt back propagation method. The Levenberg-Marquardt method minimizes the difference between the extracted POD mode amplitudes and the ANN estimations, while adjusting the weights of the model. The Levenberg-Marquardt method is a hybrid algorithm that combines the advantages of the steepest descent and Gauss-Newton methods to produce a more efficient method than either of these two methods44. The importance of the ARX engaged ANN dynamic network model structure is its strong stability capability even if the dynamic system under investigation is unstable. The stability task is at the highest level of importance when dealing with nonlinear systems of partial differential equations, such as the Navier-Stokes equations.30,45 Further mathematical definition of the ARX engaged ANN model is given in Norgaard et al30.

IV. Results

A. Reduced Order Modeling – Filtering and POD Results Codes are developed in MATLAB to perform the filtering procedure and the snapshot-based POD. The filtering

procedure starts with establishing a mesh structure in the x and y directions of the two-dimensional flow field where the x and y spatial domains change within 20.5 m – 25.5 m and 18 m – 22 m respectively with an increment of

obtained from the forced and unforced CFD analyses consists of x-direction velocities observed in the wake region. The velocity data is collected in a matrix with a size of 101x81. In this matrix, each row represents the increase in the x-direction, while each column shows the increase in the y-direction. The FFT technique is firstly introduced to the x-direction in the formed matrix. After this, the spatial x domain is converted to wave number domain. When the energy spectrum of the FFT analysis is investigated with respect to the x-direction wave number domain, it is observed that the first three values in each row includes effects of the large scale turbulent structures needed for further investigation. Later on, inverse FFT process is employed to the FFT-filtered data to reconstruct the x-direction velocity field containing only the effects of large-scale turbulent structures for further POD analysis. Finally, the steps performed for the x-direction is repeated for the data in the y-direction.

In the snapshot-based POD study, the FFT-filtered new data ensemble consisting of 1337 snapshots is analyzed. There are totally 10 periods and each period contains 137 time steps. Snapshots are equally spaced at 0.001 s apart from each other, in which they contain x-velocity data with respect to x and y coordinates. Table 3 shows the energy content variation with respect to mode number obtained after implementation of the snapshot-based POD technique to both unfiltered and FFT-filtered new data ensemble. According to Table 3, it is obvious that the FFT-filtered case reaches a greater amount of energy content by addition of fewer amounts of modes. For example, considering just the first two POD modes reveals approximately 99% of the total energy content for the FFT-filtered case, whereas, this value remains at 94% for the unfiltered case due to a more sluggish energy drop off trend.

Table 3. Energy contents of the first four most energetic POD modes

Mode Number

Energy Content (%) Unfiltered FFT-Filtered

1 49.95 54.68 2 44.08 43.96 3 2.78 0.77 4 2.28 0.58

Total (4 Modes) 99.09 99.99 To observe the effects of FFT-filtering and low-dimensional descriptions of the flow field, the most energetic four POD modes which contain information about the vortex formation in the wake region of the 2D cylinder for the unfiltered and FFT-filtered cases are shown in Fig. 5. In addition, Fig. 5 provides a comparison for the results of

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turbulent and laminar flow analyses for the 2D circular cylinder, where energy content for the modes diminishes from mode 1 to mode 4 in all cases22,27.

(a) (b) (c)

Figure 5. The most energetic first four POD modes a) for the unfiltered, b) for the FFT-filtered, and c) laminar flow snapshot-based POD analysis

In Fig. 5, since the effects of small-scale turbulent structures are separated from the data ensemble by FFT

filtering, application of the snapshot-based POD analysis to the FFT-filtered case results in smoother POD modes. In addition, the von Karman vortex street formed behind the cylinder and the velocity change in the wake region can be seen more clearly for the energetic turbulent fluid flow. Furthermore, after small-scale turbulent structures are

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filtered by FFT, because only von Karman vortex street remains in the flow field, turbulent FFT-filtered POD analysis results (Fig. 5b) resemble the results of laminar flow analysis (Fig. 5c). Figure 6 presents mode amplitudes with respect to snapshot number for both unfiltered and FFT-filtered turbulent fluid flow cases.

(a) (b)

Figure 6. Time coefficient vs. snapshot number change for the first four POD modes a) of the unfiltered, and b) of the FFT-filtered case

In Fig. 6, it can be observed that mode amplitudes for the FFT-filtered case are smaller than the ones obtained in

the unfiltered case. The major reason for this is that bifurcations caused from the effects of the small-scale turbulent structures are eliminated and only the larger-scaled ones remain which directly represent the effects of periodically moving vortices formed in the wake region. Furthermore, in both cases, due to rapid energy drop off after modes one and two, and reduced level of certain characteristics of the flow, such as velocity and vortex formation, mode amplitudes for the third and fourth modes are evaluated as significantly smaller than of the other modes. In order to test the accuracy of the results obtained by the POD technique, a reconstruction process is applied where the details of the process are explained in the study performed by Cohen et al41. Figure 7 represents the x- direction velocity contours of both original and reconstructed data in the FFT-filtered case for the snapshot number 1337, where use of only two POD modes for the reconstruction process gives reasonable results. However, if infinite number of modes can be used during the POD procedure, it is obvious that the original and reconstructed results will exactly be the same.

(a) (b) (c)

Figure 7. a) Original data, b) reconstructed data by considering the most energetic first two POD modes, and c) reconstructed data by considering the most energetic first four POD modes

of the 1337th snapshot in the FFT-filtered case

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B. Artificial Neural Network (ANN) Results The network is designed as a multilayer perceptron network (MLP) in order to enable observation of multi inputs and multi outputs. It consists of two layers (one hidden and one output layer) similar to the one shown in Fig. 4. The network uses a supervised learning (training) process with an adequate set of data that constitutes 669 time steps corresponding to approximately half of the 10 shedding cycles, and converged in approximately 90 iterations. The training process uses cylinder surface pressure data obtained from the six sensors being as one set of the inputs, and time histories of the mode amplitudes obtained from the POD method being as the other set of the inputs. Thus, the input section to the network comprises two different sets of data, one coming from the surface mounted pressure sensors and the other from the POD results in the form of mode amplitudes. The problem under investigation is a real-time system application, and the use of ANN connected with ARX provides harmony to observe such a dynamic real-time system via the parameter defined as time delay during the calculation procedure. Specification of this parameter to a certain value features the network architecture by saying that there is a continuous input that is delayed, and then it is sent as an input to the network. Taking this into consideration, time histories of the mode amplitudes evaluated from the POD method is fed into the network as a continuous input with a specified time delay value. The complexity and size of the network can be adjusted by varying both time delay and hidden layer neuron number parameters. There is only one hidden layer in the modeled neural network, and the activation function is based on the nonlinear tanh function. A single bias input has been added to the output from the hidden layer. Since one of the important network design parameters is the hidden neuron number (HNN) existing in the hidden layer, various HNN’s are used while time delay (TD) of 3 to determine a feasible HNN design parameter by making use of the order of the error signals obtained from the ANN estimation process as shown in Fig. 8.

(a) (b)

Figure 8. Order of the error signals for different hidden neuron numbers at TD=3 a) for mode 1 and b) for mode 2

According to FFT-filtered POD results, it is revealed that 99% of the total energy content is represented by using only the two most energetic modes where the effects of large scale turbulent structures are appropriately observed. Taking this into consideration, Fig. 8 represent time histories of the order of the error signals obtained while HNN equals to 10, 15 and 20 only for modes 1 and 2 after the ANN estimations. Especially in Fig. 8a, which shows mode 1 constituting the largest portion of the effects of large scale turbulent structures, it can be observed that the order of the error signals show no prominent difference for HNN’s 15 and 20, but provide better results than for HNN=10. However, it is crucial to keep in mind that increasing HNN increases network complexity, computational cost and effort for evaluation of the results. Therefore, from the analysis of the order of the error signals, it is determined to use 15 as the HNN design parameter for the further ANN analyses. Following specification of the HNN, the same analysis including the investigation of the order of the error signals is performed for determination of a feasible time delay (TD) parameter value (Fig. 9).

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(a) (b)

Figure 9. Order of the error signals for different time delays at HNN=15 a) for mode 1 and b) for mode 2 An increase in time delay value positively affects accuracy of the results, and relatively decreases the order of the error signals. For larger time delay values, there is more data available for the network to train itself by interconnecting the input sets via setting up larger weighing matrices, and hence weights, by making use of more known data coming from the past. From Fig. 9, it can clearly be seen that assigning TD to 6 and 9 provide much better results with relatively reduced levels of the order of the error signals with respect to TD=3. With this in mind, it is concluded to select TD=6, which provides 6 past time histories of the mode amplitudes obtained from the POD method together with the pressure data coming from six sensors as network input sets. The weighing matrices between the input layer and the hidden layer (W1, shown in Fig. 4) and between the hidden layer and the output layer (W2, shown in Fig. 4) depend on the number of available sensors on the surface. Since there are six cylinder surface mounted pressure sensors available, by specifying TD=6 and HNN=15, the maintained matrix dimensions for W1 is of the order of [15,37] and for W2 is of the order of [2,16]. They are initialized as “1” before starting the training process. The output layer has a linear activation function, and it consists of two outputs, namely mode amplitudes of the most energetic two POD modes. Following the training process, a validation step is performed to check efficiency of the design parameters. ANN estimations of the mode amplitudes and their comparison with the original data for mode 1 and mode 2 are shown respectively in Fig. 10.

(a) (b)

Figure 10. Comparison of the original and estimated mode amplitudes a) for mode 1 and b) for mode 2

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It can be observed from Fig. 10 that the resulting ANN estimations for the mode amplitudes of modes 1 and 2 show great coherency including only minor errors. In order to further check whether the specified network design parameters and the modeled network structure works for another real-time wake region observation problem, flow is forced by air blowing from the opening slots located on the surface of the cylinder as given in Fig. 1. The modeled ANN-ARX structure with TD=6 and HNN=15 is trained by using pressure data coming from six cylinder surface mounted sensors positioned as shown in Fig. 2, and mode amplitudes obtained from the POD method applied to a case by opening all slots and blowing air at 3.4 m/s. Another case is established by opening only slot 1 at 135° on the cylinder surface and blowing air at 3.4 m/s from this slot. Application of the POD method reveals a set of mode amplitudes for both of the cases. Since it is needed from the formed ANN-ARX structure to give predictions of mode amplitudes that give information about what is happening in the flow without further CFD simulations, the trained network structure with the mode amplitudes of the former case is used for estimation of the mode amplitudes of the latter just feeding the sensor pressure data this, slot 1-opened, case as an input set to the modeled network structure. Figure 11 provides comparison of the original and estimated mode amplitudes for the slot 1-opened and air blowing case.

(a) (b)

Figure 11. Comparison of the original and estimated mode amplitudes after addition of the opening slots on the surface of the cylinder and air blowing from slot 1 at 135° a) for mode 1 and b) for mode 2

As in the previously observed case, Fig. 11 shows that the results obtained from ANN estimation are in good agreement with the results obtained from the POD application. The low-level, acceptable ANN estimation errors are especially clustered at the lower end tips of the periodic curves. This shows that a robust and real-time estimator of mode amplitudes which is necessary for observation of the effects of dominant large-scale turbulent structures in the wake region can be evaluated effectively without requiring further CFD simulations by implementation of the ANN integrated with ARX.

V. Conclusions In this study, in order to observe the wake region in detail the Fast Fourier Transform (FFT)-filtered Proper

Orthogonal Decomposition (POD) method is applied to the velocity data obtained from the CFD analysis of the 2D simulations of a cylinder wake.

Turbulent flow fields contain the effects of both large and small-scale turbulent structures; hence, each resulting POD mode also contains the effects of those structures at different sizes. The classical snapshot-based POD procedure is not able to separate the flow structures according to their scales during the analysis. This is an undesired situation for many engineering applications. To overcome this drawback, the FFT-based spatial filtering procedure offered by Aradag et al25 is applied to the data ensemble containing x-direction velocities obtained from the CFD analyses. After taking the inverse FFT, a modified new data ensemble is formed to which the classical snapshot-based POD method is applied.

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FFT-filtering efficiently provides separation of the effects of small scale turbulent structures from the flow field, and it enables to analyze the vortices formed in the wake region more effectively for control purposes. Application of the FFT-filtered POD technique reveals that 99% of the total energy of the flow can be represented by using only two modes where the formed vortices alternate in time. This study shows that the FFT-filtered POD methodology given by Aradag et al25, which was originally developed for 3D flows, can also be used for 2D turbulent flows. By utilization of a one-dimensional regular snapshot-based POD analysis performed on cylinder surface pressure data obtained from CFD, six sensors are placed on the locations where there is the highest modal activity on the cylinder surface. Pressure data coming from six surface-mounted sensors and past mode amplitudes are used as inputs for ANN, while mode amplitudes for the remaining time steps are estimated using ANN. The modeled network structure is further used to estimate mode amplitudes of a forced flow case using air blowing on the cylinder surface, by training the network with sensor pressure data and prior mode amplitudes of another forced flow case. Following this, using only sensor pressure data of the former case, its mode amplitudes are successfully estimated from the previously trained network model. As a result, it is observed that real-time behaviour of the wake region can be observed accurately without requiring expensive CFD simulations by implementation of the ANN-ARX model structure for the estimation of the mode amplitudes.

Acknowledgments This research is supported by The Scientific and Technological Research Council of Turkey (TUBITAK) under

grant 108M549 and Turkish Academy of Sciences Distinguished Young Scientists Awards Programme (TUBA-GEBIP). The computations were performed at TOBB University of Economics and Technology CFD Laboratory. The authors would like to thank to Assoc. Prof. Kelly Cohen for his help, support and comments related to Artificial Neural Networks and POD analysis, and Asst. Prof. Cosku Kasnakoglu for his help regarding to POD analysis.

References 1 Cohen, K., Siegel, S., McLaughlin, T., Gillies, E., and Myatt, J., “Closed-Loop Approaches to Control of a Wake Flow

Modeled by Ginzburg-Landau Equation”, Computers and Fluids, Vol. 34, 2005, pp. 927-949. 2 Ong, M.C., Utnes, T., Holmedal, L.E., Myrhaug, D., and Pettersen, B., “Numerical Simulation of Flow around a Smooth

Circular Cylinder at Very High Reynolds Numbers”, Journal of Marine Structures, Vol. 22, No. 2, 2009, pp. 142-153. 3 Gillies, E. A., “Low-Dimensional Control of the Circular Cylinder Wake”, Journal of Fluid Mechanics, Vol. 371, 1998,

pp. 157-178. 4 Norberg, C., “Effects of Reynolds Number and a Low-Intensity Free Stream Turbulence on the Flow around a Circular

Cylinder”, Chalmers University of Technology, 1987, ISSN 02809265. 5 Wissink, J.G., Rodi, W., “Numerical Study of the Near Wake of a Circular Cylinder”, International Journal of Heat and

Fluid Flow, Vol. 29, 2008, pp. 1060-1070. 6 Travin, A., Shur, M., Strelets, M., and Spalart, P., “Detached-Eddy Simulations Past a Circular Cylinder Flow”,

Turbulence and Combustion, Vol. 63, 1999, pp. 293- 313. 7 Anderson, J.D., Fundamentals of Aerodynamics, 2nd Edition. McGraw Hill, 1991, New York. 8 Aradag, S., Cohen, K., Seaver, C., and McLaughlin. T, “Integrating CFD and Experiments for Undergraduate Research”,

Computer Applications in Engineering Education, 2009, DOI: 10.1002/cae.20278. 9 Lim, H., and Lee, S., “Flow Control of Circular Cylinders with Longitudinal Grooved Surfaces”, AIAA Journal, Vol. 40,

No. 10, 2002, pp. 2027-2036. 10 Aradag, S., “Unsteady Vortex Structure behind a Three Dimensional Turbulent Cylinder Wake”, Journal of Thermal

Science and Technology, Vol. 29, No. 1, 2009, pp. 91-98. 11 Cao, Y., Zhu, J., Luo, Z., and Navon, I., “Reduced Order Modeling of the Upper Tropical Pacific Ocean Model Using

Proper Orthogonal Decomposition”, Computer and Mathematics with Applications, Vol. 52, No. 8-9, 2006, pp. 1373-1386. 12 Lumley, J.L., “The Structure of Inhomogeneous Turbulent Flows”, Atm. Turb. and Wave Prop., 1967. 13 Holmes, P., Lumley, J.L., and Berkooz, G., “Turbulence, Coherent Structures, Dynamical Systems and Symmetry”,

Cambridge University Press, 1996, Cambridge, UK. 14 Newman, A.J., “Model Reduction via the Karhunen-Loéve Expansion Part 2: Some Elementary Examples”, Institute for

Systems Research, Technical Report No. 96-33, 1996. 15 Chatterjee, A., “An Introduction to the Proper Orthogonal Decomposition”, Computational Science Section Tutorial,

2000, Department of Engineering Science and Mechanics, Penn State University, Pennsylvania. 16 O'Donnell, B., and Helenbrook, B. “Proper Orthogonal Decomposition and Incompressible Flow: An Application to

Particle Modeling”, Computers and Fluids, Vol. 36, No. 7, 2007, pp. 1174-1186. 17 Sen., M, Bhaganagar, K., and Juttijudata, V., “Application of Proper Orthogonal Decomposition (POD) to Investigate a

Turbulent Boundary Layer in a Channel with Rough Walls”, Journal of Turbulence, Vol. 8, No. 41, 2007.

American Institute of Aeronautics and Astronautics

14

18 Connell, R., and Kulasiri, D., “Modeling Velocity Structures in Turbulent Floods Using Proper Orthogonal Decomposition”, International Congress on Modeling and Simulation, December 2005, Melbourne, Australia.

19 Lieu, T., Farhat, C., and Lesoinne, M., “Reduced-Order Fluid/Structure Modeling of a Complete Aircraft Configuration”, Computer Methods in Applied Mechanics and Engineering, Vol. 195, No. 41-43, 2006, pp. 5730-5742.

20 Unal, M., and Rockwell, D. “On Vortex Shedding from a Cylinder, Part 1: The Initial Instability”, Journal of Fluid Mechanics, Vol. 190, 2002, pp. 491-512.

21 Sirovich, L., “Turbulence and the Dynamics of Coherent Structures, Part I: Coherent Structures”, Quarterly Applied Mathematics, Vol. 45, No. 3, 1987, pp. 561-571.

22 Paksoy, A., Apacoglu, B., and Aradag, S. “Analysis of Flow over a Circular Cylinder by CFD and Reduced Order Modeling”, ASME 10th Biennial Conference on Engineering Systems Design and Analysis, 12-14 July 2010, Istanbul, Turkey.

23 Smith, T.R., Moehlis, J., Holmes, P., “Low Dimensional Models for Turbulent Plane Couette Flow in a Minimal Flow Unit”, Journal of Fluid Mechanics, Vol. 538, 2005, pp. 71-110.

24 Ma, X., and Karniadakis, G., “A Low-Dimensional Model for Simulating Three-Dimensional Cylinder Flow”, Journal of Fluid Mechanics, Vol. 458, 2002, pp. 181-190.

25 Aradag, S., Siegel, S., Seidel, J., Cohen, K., and McLaughlin, T., “Filtered POD-Based Low-Dimensional Modeling of the Three-Dimensional Turbulent Flow Behind a Circular Cylinder”, International Journal for Numerical Methods in Fluids, 2010, DOI: 10.1002/fld.2238.

26 Seidel, J., Cohen, K., Aradag, S., Siegel, S., and McLaughlin, T., “Reduced Order Modeling of a Turbulent Three Dimensional Cylinder Wake”, , 25-28 June 2007, Miami, Florida.

27 Apacoglu, B., Paksoy, A., and Aradag, S., “CFD Analysis and Reduced Order Modeling of Uncontrolled and Controlled Laminar Flow over a Circular Cylinder”, Engineering Applications of Computational Fluid Mechanics, in press.

28 Mehrotra, K., Chilukuri, K., M., and Ranka, S., “Elements of Artificial Neural Networks”, The MIT Press, 2000, Chap. 1. 29 Bishop, C., M., “Neural Networks and Their Applications, Review Article”, Review of Scientific Instruments, Vol. 65,

No. 6, 1994, pp. 1803-1832. 30 Norgaard, M., Ravn, O., Poulsen, N.K., and Hansen L.K., “Neural Networks for Modeling and Control of Dynamic

Systems”, cessing, Springer, 2003, UK. 31 Zhang, L., Akiyama, M., Huang, K., Sugiyama H., and Ninomiya, N., “Estimation of Flow Patterns by Applying

Artificial Neural Networks”, Information Intelligence and Systems, Vol. 1-4, 1996, pp. 1358-1363. 32 Apacoglu, B., “CFD Analyses of Uncontrolled and Controlled Laminar and Turbulent Flows around a Cylinder”, M.Sc.

Dissertation, Mechanical Engineering Dept., TOBB Univ. of Economics and Technology, Ankara, Turkey, 2010. 33 Berkooz, G., Holmes, P., and Lumley, J., “Turbulence, Coherent Structures, Dynamical Systems and Symmetry”,

Cambridge: Cambridge UP, 1996, pp. 86-93. 34 Prazenica, R., Kudila, A.J., and Vignola, J.F., “Spatial Filtering and Proper Orthogonal Decomposition of Scanning Laser

Doppler Vibrometry Data for the Nondestructive Evaluation of Frescoes”, Journal of Sound and Vibration, Vol. 304, No. 3-5, 2007, pp. 735-751.

35 Aubry, Holmes, P., Lumley, J., and Stone, E., “The Dynamics of Coherent Structures in the Wall Region of a Turbulent Boundary Layer”, Journal of Fluid Mechanics, Vol. 192, 1988, pp. 115-173.

36 Johansson, P., B., George, W., and Woodward, S.H., “Proper Orthogonal Decomposition of an Axisymmetric Turbulent Wake Behind a Disk”, Physics of Fluids, Vol. 14, No. 2508, 2002, DOI: 10.1063/1.1476301.

37 Gamard, S., and George, W., “Application of a ‘Slice’ Proper Orthogonal Decomposition to the Far Field of an Axisymmetric Turbulent Jet”, Physics of Fluids, Vol. 14, No. 2515, 2002, DOI: 10.1063/1.1471875.

38 Dowell, E.H., and Hall, K.C., “Modeling of Fluid Structure Interaction”, Annual Review of Fluid Mechanics, Vol. 33, 2001, pp. 445-490, DOI: 10.1146/annurev.fluid.33.1.445.

39 Attar, P., J., Dowell, E., H., White, J., R, and Thomas, J., P., “Reduced Order Nonlinear System Identification Methodology”, AIAA Journal, Vol. 44, No. 8, 2006, pp.1895-1904.

40 Tadmor, G., Bissex, D., Noack, B., R., Morzynsky, M., Colonius, T., and Taira, K., “Temporal-Harmonic Specific POD Mode Extraction”, 4th Flow Control Conference, Seattle, WA, AIAA paper 2008-4190, 2008.

41 Cohen, K., Siegel, S., and McLaughlin, T., “A Heuristic Approach to Effective Sensor Placement for Modeling of a Cylinder Wake”, Computers and Fluids, Vol. 35, 2006, pp. 103-120.

42 Gurney, K., “An Introduction to Neural Networks”, CRC Press, 1997, pp. 1-6, Chap. 1. 43 Samarasinghe, S., “Neural Networks for Applied Sciences and Engineering, From Fundamentals to Complex Pattern

Recognition”, Auerbach Publications, Taylor and Francis Group, 2006, pp. 11-68, Chap. 2. 44 Samarasinghe, S., “Neural Networks for Applied Sciences and Engineering, From Fundamentals to Complex Pattern

Recognition”, Auerbach Publications, Taylor and Francis Group, 2006, pp. 113-193, Chap. 4. 45 Siegel, S., Cohen, K., Seidel, J., Aradag, S., and McLaughlin, T., “Low Dimensional Model Development using Double

Proper Orthogonal Decomposition and System Identification”, 4th Flow Control Conference, 23-26 June 2008, Seattle, Washington, AIAA 2008-4193.