Prediction of the dynamics of a backward-facing step flow using focused time-delay neural networks and particle image velocimetry data-sets
Graphical abstract
Introduction
Shear layer flows like boundary layers (BL), mixing layers, jets or Backward-Facing Step (BFS) flows (Beaudoin, Cadot, Aider, Wesfreid, 2004, Dergham, Sipp, Robinet, 2013) are ubiquitous in nature as well as in industrial flows. One of their most important properties is that they are convectively unstable (Huerre, Monkewitz, 1990, Chomaz, 2005) and then are very sensitive to background noise. They are also called “noise-amplifier flows” because of their ability to amplify some specific frequency ranges.
Noise-amplifier flows play an important role in many industrial flows, like separated flows around airfoils (Darabi and Wygnanski, 2004) and the complex 3D wakes of ground vehicles (Aider, Dubuc, Hulin, Elena, 2001, Beaudoin, Cadot, Aider, Gosse, Paranthoen, Hamelin, Tissier, Allano, Mutabazi, Gonzales, Wesfreid, 2004). Most of the time they are responsible of loss of efficiency (increase of aerodynamic drag, lift crisis) or nuisances like aeroacoustic noises or fluid-structure interactions. Controlling shear flows is then crucial for many industrial applications. For instance, controlling the shear layers to reduce the wake of a bluff-body or a ground vehicle has been proved to be a valuable strategy to reduce the aerodynamic drag of ground vehicles (Aider, Beaudoin, Wesfreid, 2009, Aider, Joseph, Ruiz, Gilotte, Eulalie, Edouard, Amandolese, 2014, Eulalie, Fournier, Gilotte, Holst, Johnson, Nayeri, Schutz, Wieser, 2018, Li, Barros, Borée, Cadot, Noack, Cordier, 2016, Grandemange, Ricot, Vartanian, Ruiz, Cadot, 2014).
Nevertheless, closed-loop flow control experiments are still challenges both for academic or industrial configurations. From a general point of view, the fist step is to choose sensors and actuators before searching for a proper control law allowing the modification of the targeted flow by the actuators based on information coming from the sensors. One can choose arbitrarily the sensors and actuators and propose a closed-loop law based on some physical properties of the targeted flow, like the Kelvin-Helmholtz frequency of a shear layer (Gautier and Aider, 2015b). One can also use a Machine Learning algorithm to search for a proper control law (Gautier, Aider, Duriez, Noack, M.Segond, M.Agel, 2015, Li, Barros, Borée, Cadot, Noack, Cordier, 2016, Debien, von Krbek, Mazellier, Duriez, Cordier, Noack, Abel, Kourta, 2016),Duriez et al.. Ultimately the problem of the relevance of the sensors and their ability to represent the global dynamics of the flow becomes crucial. One can use statistical data-based System Identification (SI) techniques to try to find a relation between one or multiple sensors and the dynamics of the flow as shown by (Guzmán, Sipp, Schmid, 2014, Varon, Guzman, Sipp, Schmid, Aider, 2015). Another option is to use machine learning techniques to find the best combinations of sensors and a proper Reduced-Order system to recover most of the dynamics of the flow. This is the objective of this study.
In modern experimental and testing / measuring techniques, data-driven methods are becoming of great interest, since they don’t require a priori knowledge of a model and the access to large data-sets is becoming easier. This is especially true when studying thoroughly non-stationary flows for different Reynolds numbers with various sensors such as Particle Image Velocimetry (PIV) or multiple Pitot / multiple hot-wire probes. Methods like statistical and machine learning algorithms, are becoming efficient and reliable for both academics or industrial applications. Neural networks (NNs) particularly are attracting attention in this “Big-Data” revolution.
In fluid systems, feed-forward artificial NNs have been used for data-driven reduced-order modelling (Müller, Milano, Koumoutsakos, 1999, Wang, Xiao, Fang, Govindan, Pain, Guo, 2018, Pan, Duraisamy, 2018) with many results showing better field reconstruction than traditional POD methods (San, Maulik, Ahmed, 2018, Erichson, Mathelin, Yao, Brunton, Mahoney, Kutz, 2019). They were also used by (Mi et al., 1998) for experimental flow regime identification in multiphase flows. We mention that it has also been proved by Baldi and Hornik (1989), that a linear NN can be equivalent to a Proper Orthogonal Decomposition (POD) basis structure. Convolutional NNs have also been used for the efficient real-time 2D and 3D inviscid simulations (Tompson et al., 2016) or along with pressure measurements for the velocity field prediction around a cylinder (Jin et al., 2018). Furthermore, if there are hints of deeper understanding of the underlying physics, simple shallow NNs can provide very good results in SI as well as for control laws creation (Lee, Kim, Babcock, Goodman, 1997, Herbert, Fan, Haritonidis, 1996, J.Rabault, Kuchta, Jensen, Reglade, Cerardi, 2018). Their potential has been demonstrated early for modelling surface pressure and aerodynamic coefficients of 3-dimensional unsteady cases of aircrafts flows (Faller and Schreck, 1996). Finally, deep NNs are increasingly more important in the fluid mechanics community, especially for the modelling of complex turbulent flows. Srinivasan et al. Srinivasan et al. (2019) compared deep feed-forward and recurrent Long-Short Term Memory (LSTM) networks for turbulent shear flows prediction. Recently Deng et al. (Deng et al., 2019), used LSTM networks to reconstruct the POD coefficient time series using sub-sampled distributed velocity sensors in an inverted flag flow PIV experiment. A short review of applications of deep NN to fluid mechanics can be found in Nathan Kutz (2017).
For complex flows, the number of degrees of freedom obtained from a 2D-2C (2-Components in a 2D velocity field) optical-flow Particle Image Velocimetry (PIV) measurement of a few millions pixels image is millions. Such a large system is impossible to handle and a reduced-order model (ROM) has to be identified. A dynamic observer can identify such a model based only on input-output measurements from measurable system quantities. As proposed firstly by Guzmán et al. (2014) and verified experimentally for PIV data by Varon et al. (2015) it is possible to predict the full dynamics of a transitional flat-plate BL flow in the form of POD coefficients using a few local upstream sensors. The first step in their method was to create a successful reduced-order system using POD. The second step was to identify an optimal state-space model using a statistical learning process (the so-called N4SID algorithm), in order to predict at any moment all the POD coefficients (outputs) by measuring one or two local variables upstream (sensors or inputs) in the flow. A similar approach was also presented in a paper from Beneddine et al. (2016), where the full frequency spectrum was obtained from local frequency information of the flow. Time-resolved field reconstruction was also successfully obtained for time-resolved PIV data of a round jet flow using point sensors and the mean flow Beneddine et al. (2017).
In the present study, we explore means of performing a local-to-global dynamics system identification (SI) using a Neural Network (NN) architecture. We show we can successfully apply a machine learning data-driven identification process in a complex experimental fluidic data-set, in order to learn the relation between local upstream sensors and the global fluctuation dynamics of the system. A different data-set is used to validate the learning-training step. The predicted dynamics then allow the reconstruction of the full flow field, which would help design realistic, experimental controllers targeting the kinetic energy of the full fluctuation field. We show the importance of the various choices (from sensors, to NN parameters) in a successful SI of an experimental, time-resolved, separated flow.
The paper is organised as follows: first, we present the various artificial NNs that can be found in the literature. Then we present the experimental setup used to study the BFS flow. The choice of the NN architecture is then discussed. Different NNs have been tested; only one NN lead to satisfactory results with the least number of parameters and a reasonable computational time. Then the choice of the sensors as well as of the NN parameters like the training data-set size or the value of the time-delay are discussed. The efficiency of the chosen shallow NN architecture for such a SI is then demonstrated through validation data-sets comparisons between real and estimated POD coefficients time-series. Finally, the reconstructed time-resolved velocity fields are successfully compared to measured velocity fields before turning to the conclusions.
Section snippets
Definitions
An artificial Neural Network (NN) can provide a non-linear mapping between a set of inputs and a set of corresponding outputs. Great progresses have been made lately due to the availability of large data-sets, the increasing number of optimised toolboxes and also the improvements of Graphics Processing Units (GPU) parallel programming. This is the reason why NNs are becoming more and more popular nowadays.
The key component of a NN is the neuron or “perceptron”. In general, one defines a weight w
Hydrodynamic channel
Experiments have been carried out in a hydrodynamic channel in which the flow is driven by gravity. The flow is stabilised by divergent and convergent sections separated by honeycombs, leading to a turbulence intensity of 0.8 %. A NACA 0020 profile is used to smoothly start the boundary layer. The test section is 80 cm long with a rectangular cross-section wide and high (see Fig. 4). The step height is . The vertical expansion ratio is and the spanwise
Characterisation of the BFS flow
The objective of this study is to evaluate the potential of a NN SI method on experimental data of a shear-layer flow. We focus on a BFS flow which is a typical case of noise-amplifiers. Upstream perturbations are amplified in the shear layer leading to significant downstream disturbances. Separation is imposed by a sharp edge creating a strong shear layer where Kelvin–Helmholtz instability leads to vortex shedding (Fig. 4). Another important feature is the creation of a large separation
System identification steps
In the present study we explore the potential of artificial NNs for local-to-global dynamics SI applied to a shear layer flow. A full scheme of the identification process is summarised in Fig. 8. First, in the full data-set of time-resolved PIV experiment is decomposed to identify the dynamics in the form of POD coefficients. Then, in this data-driven identification process, we just rely on the input (optical sensors) - output offline measurements for a period of time from the operation of the
Validation criterion
To evaluate the efficiency of the identification, one has to define a quantitative criterion to compare the POD coefficient time-series results obtained with the different NN architectures to the ones obtained experimentally. In the following, we compute the mean-squared error (MSE) at each time-step n for each POD coefficient am(n):
Then the averaged MSE for all the coefficients () time-series gives the final evaluation error for the specific NN architecture:
Choice of the input(s)
Our sensor consists of selected measurements from the time-resolved velocity field. Different inputs can be defined from these velocity measurements, or from velocity-derived variables like the vorticity or vortex identification criteria. First, it is necessary to choose the physical quantities measured by the sensor(s). As the optical sensors are the inputs in the identification process, their choice is a critical step. Our challenge is to identify vortices embedded in the shear layer, close
Optimal NN identification procedure
The goal of the NN identification method is to predict at each time step n the POD coefficients am(n) of the full field using local upstream sensors sj(n). The POD coefficients time-series have been calculated based on the PIV velocity fields. The sensors sj(n) were also monitored at the same time steps. The pairs [am(n), sj(n)] () is our identification data-set.
A FTDNN architecture has been chosen. Shallow and deeper LSTM architectures were also tested for this specific study without
Conclusions
A successful application of a NN System Identification method to a time-resolved PIV experiment of a typical noise-amplifier flow has been presented. We were able to predict with satisfying precision the global dynamics of the flow (in the form of POD coefficients), using visual sensors coming from local velocity measurements. A shallow FTDNN architecture was sufficient to recover the overall dynamics of the flow. There was no need for sophisticated LSTM gates or more than one hidden layers,
CRediT authorship contribution statement
Antonios Giannopoulos: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing - original draft, Writing - review & editing. Jean-Luc Aider: Conceptualization, Funding acquisition, Investigation, Methodology, Project administration, Resources, Supervision, Validation, Writing - review & editing.
Acknowledgements
The authors would also like to thank M. Aris Kanellopoulos from Georgia Institute of Technology for the valuable comments and discussions.
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