Jinjian Li
email: jinjian.li@imt-lille-douai.fr

Guillaume Lozenguez
email: guillaume.lozenguez@imt-nord-europe.fr

Jacques Boonaert

Arnaud Doniec

Multi-models machine learning methods for traffic flow estimation from Floating Car Data

Keywords: Estimation of traffic flows, simulation and modeling of transportation systems, Gaussian Process Regression (GPR), big data, machine learning, Floating Car Data (FCD)

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

With the rapid growth of urban centers during the last decades, the development of efficient urban transportation services has become a central issue to reduce the high wasted time during the daily commute. The resulting increasing demand in terms of transportation flows has to cope with the difficulty to adapt existing or create new transportation networks.

In this context, simulate daily transportation behaviors allows operators to experiment and to visualize decisions about infrastructure and regulation policies. One of the major basics on efficient simulation relies on the ability to produce models representing the way that transportation flows evolve with time, depending on the traffic demands and events that impact the transportation network. The estimation of traffic flow is one of the core requirements in those simulation. One of the costless solution would be to reconstruct the traffic flow from aggregated information (travel duration estimations), available on web services.

Previous work validates that machine learning based approach is a promising way to reconstruct "sensor like traffic flow data" from aggregated information like the ones proposed by Google services [START_REF] Li | Toward reliable estimations of urban traffic flows from machine learning and floating car data[END_REF][START_REF] Li | Traffic flow multi-model with machine learning method based on floating car data[END_REF]. Applied such an approach would permit, for instance, to infer on a realistic traffic demand at each entrance of a city (i.e. the flow of incoming vehicles).

Machine learning over aggregated information approach is based on accessible databases that can provide information regarding the transportation condition (travel duration) at a given location and at a given time. In 2007, the Google company has extended Google Maps by adding Google Live Traffic, the visualization of traffic information in real time [START_REF] Van Den Haak | Validation of google floating car data for applications in traffic management[END_REF] [START_REF] Jeske | Floating car data from smartphones: What google and waze know about you and how hackers can control traffic[END_REF]. Here, the notion of real time means the current state and is applied to qualify the service of FCD provided. In more detail, Google exploits users' position data of Android smart-phones, in order to get a significantly fast and accurate mapping of the traffic. This data is called Floating Car Data (FCD), which can also be collected by any localization system embedded in a car and sent to the service provider via a mobile connection. Generally, these raw Floating Car Data are aggregated to provide more intelligible and relevant information regarding traffic condition.

For example, in Google Maps, FCD are used to give a real-time traffic information using colored road section 1 which is determined by the navigation system or, as in the case of Google Live Traffic, by the smart phone and is sent to the service provider via a mobile phone connection. Therefore this allows the generation of real-time traffic information, which is visualized by the colors on Google Maps: red road points are related to a traffic jam or stop-and-go traffic, orange indicates heavy traffic and green points correspond to clear roads. However, those platforms generally provide only aggregated data like average travel duration more than the initial raw data.

Such an approach would permit operators to provide efficient global information while limiting the effort in continuously measuring road traffic flows based on physical sensors (radars, induction loops, etc.). The number of sensors, even in mid-sized cities, can increase very quickly. For example, to measure the input and output flows of a simple 4-points roundabout, at least 8 physical sensors are required. Therefore, such an expensive traffic flow measurement method makes the mentioned simulation process out of reach.

Preliminary result was published on the 15th World Conference on Transport Research 2019 [START_REF] Li | Toward reliable estimations of urban traffic flows from machine learning and floating car data[END_REF] and on the 6th International Conference on Control Decision and Information Technologies 2019 [START_REF] Li | Traffic flow multi-model with machine learning method based on floating car data[END_REF] where traffic flows is estimated according to Floating Car Data (FCD) from Google Maps only on the basis of regressors trained using machine learning techniques instead of using stationary physical equipment (such as loop detectors [START_REF] Cheung | Traffic measurement and vehicle classification with single magnetic sensor[END_REF] or video cameras [START_REF] Coifman | A real-time computer vision system for vehicle tracking and traffic surveillance[END_REF]). In this paper, we present an extension of the previous works with an increased experiment setup that permits to automatize the model definition. Firstly, we show experimentally that, among 19 types of regression methods (including Linear Regression Models, Regression Trees, Support Vector Machines and Ensemble of Trees), the Gaussian Process Regression (GPR) is the most suitable machine learning method to obtain the best fitting criterion with respect to our dataset. Secondly, a selection of the adequate regressor is computed from a set of regressors (multi-model) to estimate traffic flows from FCD, based on the different types of travel duration profiles. This multi-model approach can greatly reduce the estimation error, by precisely clustering days presenting different types of travel duration profiles. Experiments are conducted by comparing estimated flows with real ones provided from induction loop sensors. The results we obtain seem promising enough to say that correct transportation flows models could be obtained with a very light use of real traffic sensors. This paper is organized as follows: the next section describes related works and the different usages of aggregated FCD in the context of transportation networks modeling. The third section focuses on the problem we choose to address that is building sensor like data flow measurements from aggregated data. In this section, we also provide details regarding our problem formulation on the standpoint we took for solving it. The fourth section presents the single and multi-GPR machine learning method for the estimation of traffic volume. The fifth section deals with the experimental site, the results we obtained, the comparison between estimated traffic flow and real observed data prior to the discussion. The last section concludes this paper and presents some perspectives and further works based on it.

Related Work

The successful wide scale deployment of the Advanced Traveler Information Systems (ATIS) and Advanced Traffic Management Systems (ATMS) highly relies on the capability to perform accurate estimation of the real traffic states on road networks. Therefore, the use of real-time Floating Car Data (FCD), based on traces of Global Positioning System (GPS) positions of vehicles, is emerging as a reliable and cost-effective way to collect accurate traffic data for a wide area road network. Unlike other traffic data collection techniques (e.g. traffic cameras, induction loops embedded in the roadway, radar-based sensors), floating cars act as moving sensors traveling in a traffic stream and do not require additional instrumentation to be set up on the roadway.

The main communication architecture for FCD is based on the exchange of information between a fleet of floating cars traveling on a road network and a central data operation system. The floating cars periodically send their positions (latitude, longitude and altitude) and instantaneous velocity thanks to GPS receiver and Global System for Mobile communications (GSM) or General Packet Radio Service (GPRS) transmitter. While the central data operation system tracks the received FCD along the traveled path by matching the related trajectory data to the corresponding real road network. The frequency of sending/reporting is generally determined by the required resolution of the data and the performances of the available communication channels, for example, bandwidth. Therefore, FCD is an effective approach to determine the real traffic speed on the road network, based on gathering of data such as localization, speed, direction of travel and time information from the mobile phones of drivers and passengers of the vehicle. That is to say, every vehicle with an active mobile phone acts as a sensor for the network.

Using FCD for estimating travel duration and traffic states has received high attention over the years from the scientific community. The most common and useful information provided by FCD is travel-duration and speeds along road links or paths [START_REF] Miwa | En-route updating methodology oftravel time predictionusing accumulated probe-car data[END_REF][START_REF] Turksma | The various uses of floating car data[END_REF][START_REF] Yoon | Surface street traffic estimation[END_REF]. Works such as [START_REF] De Fabritiis | Traffic estimation and prediction based on real time floating car data[END_REF][START_REF] Nanthawichit | Application of probe-vehicle data for real-time traffic-state estimation and short-term travel-time prediction on a freeway[END_REF][START_REF] Erdelić | Estimating congestion zones and travel time indexes based on the floating car data[END_REF] calculate or forecast the travel duration on roadside according to the speed of the floating car on that road. Event detection, such as congestion and accidents has been discussed by using the trajectories or speed of floating cars [START_REF] Asakura | Incident detection methods using probe vehicles with on-board gps equipment[END_REF][START_REF] Chen | Spatial-temporal traffic congestion identification and correlation extraction using floating car data[END_REF][START_REF] Chen | Characteristics of urban road non-recurrent traffic congestion based on floating car data[END_REF]. The percentage of floating cars required for the estimation of travel duration is presented in the works like [START_REF] Dai | A simulation evaluation of a real-time traffic information system using probe vehicles[END_REF][START_REF] Hong | Spatial and temporal analysis of probe vehicle-based sampling for real-time traffic information system[END_REF][START_REF] Vázquez | A comparison of deep learning methods for urban traffic forecasting using floating car data[END_REF]. Reconstructing the traffic states from FCD has been previously addressed in papers such as [START_REF] Hong | Spatial and temporal analysis of probe vehicle-based sampling for real-time traffic information system[END_REF][START_REF] De Fabritiis | Traffic estimation and prediction based on real time floating car data[END_REF][START_REF] Kerner | Traffic state detection with floating car data in road networks[END_REF][START_REF] Sunderrajan | Traffic state estimation using floating car data[END_REF].

In [START_REF] De Fabritiis | Traffic estimation and prediction based on real time floating car data[END_REF], the authors attempt to estimate and predict the travel speed with the real-time FCD based on traces of GPS positions. Another work like [START_REF] Hong | Spatial and temporal analysis of probe vehicle-based sampling for real-time traffic information system[END_REF], both spatial and temporal characteristics to the domain of floating car sampling is introduced. And the analysis of this work can provide an insight for floating car based traffic state system designers on the transmitting period, sampling interval and penetration of floating car that are desirable in a traffic network in order to get certain coverage and accuracy in traffic state estimation. An interesting method is proposed in [START_REF] Kerner | Traffic state detection with floating car data in road networks[END_REF], whose purpose is to obtain a high quality on the reconstruction of travel times in the net with a smaller percentage of FCD vehicles and number of FCD messages. The most recent research about traffic state reconstruction using FCD is presented in work [START_REF] Sunderrajan | Traffic state estimation using floating car data[END_REF]. The speed is estimated based on FCD. Then the authors make use of the R 2 Statistic to build the function between average speed (V) and density (ρ). Finally, the traffic flow is calculated according to the

Fundamental diagram (Q = ρ • V).
In summary, the above works mainly deal with the following subjects: 1) strategy to collect FCD, 2) estimation of traffic state by building traditional function model. However, in this work, on the one hand, it is a pragmatic choice to apply the aggregated FCD in Google Maps, because it is the most widespread and best-known system from a more practical and industrial point of view, and it can provide lots of data in a complete transportation network considering scientific aspects. Other FCD providers (such as Uber) could have been employed, but the quality of the collected data is very similar while offering a much lower coverage than the Google-based services [START_REF] Wu | Comparing google maps and uber movement travel time data[END_REF]. Moreover, accessing the data sources (programming constraints) from the Google API is almost straight straightforward and free of charge (for a given number of requests per month). On the other hand, traffic flows are estimated according to FCD from Google Maps only, on the basis of regressors trained using machine learning techniques, instead of using traditional stationary devices (such as loop detectors [START_REF] Cheung | Traffic measurement and vehicle classification with single magnetic sensor[END_REF] or video cameras [START_REF] Coifman | A real-time computer vision system for vehicle tracking and traffic surveillance[END_REF]). To the best of our knowledge, this is an innovative approach that could help minimize the use of stationary sensors while providing a good enough estimation of traffic flows.

Problem Description and Proposed Mathematical Model

This section firstly describes the problem addressed in the work. Next step presents the proposed system structure, where two types of regressors based models are introduced, including single model and multi-model. Then the feature extraction method is shown. At last, the criteria used to evaluate the proposed system's performance are presented.

Problem description

The problem is the estimation of traffic flow on a lane based on travel duration estimated by Google Maps thanks to FCD, as illustrated in Fig. 1. We consider a road section of a certain length (1.2 km in the example of Fig. 1), we measure the traffic flow on this section thanks to a physical sensor (in the example of Fig. 1 data were collected using a Doppler radar (24.165 GHz / 100mw EIRP)) located in the middle of the section and in parallel we estimate the travel time on this section thanks to Google Maps. The objective is to find the relationship between this travel time and the vehicle flow measured on the ground.

According to the Fundamental diagram of traffic flow theory, travels duration and traffic flows are linked with each other, and their nonlinear relationship can be traditionally formulated with some special statistical model-based methods based on many assumptions, with lots of parameters to be tuned [START_REF] Wu | An algorithm for multi-class network equilibrium problem in pce of trucks: application to the scag travel demand model[END_REF] [START_REF] Lam | Calibration of the combined trip distribution and assignment model for multiple user classes[END_REF]. Several examples of such method are Van-Aerde-Function and exponential model [START_REF] Brilon | Speed-flow models for freeways[END_REF]. The parameters to be tuned can be mean travel time at zero and real flow, some other parameters to be estimated from empirical data, etc. However, it would be too complicated with too many parameters to be tuned to use a general statistical model-based method, with no guarantee of efficiency.

Therefore, this work tries to directly establish such a relationship on the basis of machine learning methods. Because machine-learning based approach can learn a general black model without the knowledge of a 'Physique Model' of the dynamic system, making it much more convenient to be propagated on all over 'similar' road sections.

Proposed system structure

The proposed system structure consists of three parts: data collection, machine learning model training and trained model testing, as illustrated in Fig. 2. For data collection, traffic flows are measured by a sensor over a specified and limited period of time , which is 26 weeks in this work. Over this same period of time, We retrieve directly travels durations, which are calculated by Google map servers with a complicated algorithm based on the collected floating car data. In other words, through the Google maps API, the travel duration can be obtained by giving the GPS for origin and destination of the chosen road segment, and the time-stamp, as shown in the Tab. 1. The GPS for origin and destination is got manually from the Google Map on the chosen road. The time-stamp is a serial of numbers to define a certain instantaneous time elapsed since 1970, January 1 st , 0 hours, 0 minutes, 0 seconds. The request can be done either on a Web browser or by Python programming. The travel duration data is captured each ten minutes from Google-map. In total, 26208 data are retrieved. Then the obtained traffic flows data and their corresponding travels durations are split into two sets: a training dataset with 50 percent for the machine learning model training, and a validation dataset with another 50 percent for the trained model testing. For the training of machine learning model, the estimation of traffic flows from travels durations is treated using regression models. Three types of models are proposed: single model, manual multi-model, and K-means multi-model (refers to 3.3). The machine learning algorithm applied is the training of a Gaussian Processes Regressors (GPR) (refers to 4.5). This corresponds to the red rectangle part in Fig. 2, that is presented more detailedly in the next section (refers to 3.3). At the end of the learning process, trained GPR models are obtained, which can take travels durations as input to estimate traffic flows as output, without requiring any additional "in site" flow sensors anymore. The estimated traffic flows are compared with the one measured by real sensors to evaluate the performance of trained GPR models.

Single model and multi-models proposed for traffic flows estimation

The traffic flows evolve over the days of a week, because of the difference of traffic demand and supply between working day and weekend. Therefore, using days' specific GPR models (multi-models approach) instead of a "generic" one (single model approach) may provide better estimations of the traffic flows along the week. In the manual multi-models, three GPR models are trained for weekdays, Saturday and Sunday in training data set.

As a prior step, we also have to verify that days of the week can be clustered on the basis of the travel durations profiles collected by Google Map. Here, the profile associated to a day is simply the sequence of travel durations measurements along this day. To do so, we apply a Principal Components Analysis (PCA), which is a method to reduce the dimension of a feature vector while keeping a significant part of its original information (refers to 4.2).

The classical k-means algorithm is further applied to this reduced feature space (refers to 4.3) to build clusters that are interpreted as different types of days. On the one hand, each of these clusters will, in turn, be used to train its own GPR model. On the other hand, the label results from k-means are applied to train a Support Vector Machine (SVM) model, which is used to cluster a new day (testing data sets) to choose the suitable trained GPR model from its corresponding FCD profile. Three provided strategies to build models are illustrated in Figs. 3 and4: 1. Single model: only one GPR model is trained using all the training data (variations of FCD profiles between the days are not taken into account during the GPR's training process, namely, all the training days are applied to the same regression model) [START_REF] Li | Toward reliable estimations of urban traffic flows from machine learning and floating car data[END_REF]. Sunday, one for Saturdays and one for Sundays, respectively), to model the fact that, generally speaking, the traffic flows seem to be similar during "regular" days of the week, and are different during Saturday and during Sunday. The term "manual selection" stands for the fact this partition is only based on the name of the day, without statistical analysis [START_REF] Li | Traffic flow multi-model with machine learning method based on floating car data[END_REF].

3. K-means multi-models: several different GPR models are trained based on the clustering results from k-means, in order to model the fact that, generally speaking, the FCD profiles clustered by k-means should have similar traffic flows. This K-means multi-models are new content compared with the above two types of models published on two International Conferences. Therefore, performing the clustering process using FCD or using flow data leads to two compatible partitions of the type of days.

Features extraction method

The feature extraction method is illustrated in Fig. 5. This paper address a regression problem but not a prediction problem. Then, for estimating traffic flow at time step k, "past" and "future" travel duration samples can be taken into account. Consequently, the feature extraction process is based on the assumption that the traffic flow f k at the given time step k can be approximated from a samples window with (2 • n + 1) width centered on travel durations d k at time step k, where, the notations are defined as follows: • Application of testing data set on the trained Machine Learning models to get estimated traffic flows;

• Comparison of the estimated traffic flow and the real one.

Main steps for traffic flow estimation with manual multi-models

• Request of FCD from Google Maps API and corresponding traffic flow from transportation management;

• Division of all FCD and traffic flow data into three groups of training data set and testing data set according to weekdays, Saturday and Sunday;

• Construction of input vector X k based on the selected half width of sample window n, as presented in the subsection 3.4, for training and testing data set respectively;

• Application of three groups of training data set to three different GPR models respectively, as shown in Fig. 6;

• Application of three groups of testing data set on the corresponding trained Machine Learning models according to weekdays, Saturday and Sunday, in order to get estimated traffic flows;

• Comparison of the estimated traffic flow and real one.

Main steps for traffic flow estimation with K-means multi-models

• Request of FCD from Google Maps API and corresponding traffic flow from transportation management;

• Division of all FCD and traffic flow data into training data set and testing data set with day as unit;

• Combination of all the FCD data within the same day in the training data set as input vector for PCA;

• Application of K-means methods to groups all days in the training data set into three different groups;

• Construction of input vector X k based on the selected half width of sample window n, as presented in the subsection 3.4, in the three different groups separately;

• Application of three groups of training data set to three different GPR models respectively, as shown in Fig. 6;

• Combination of all the FCD data within the same day in the testing data set as input vector for PCA;

• A classifier SVM trained based on the K-means results in the above step 6, in order to classify the testing data into suitable group;

• Application of corresponding trained Machine Learning models in each group on testing data set to get estimated traffic flow;

• Comparison of the estimated traffic flow and real one.

Principal Components Analysis (PCA) for reduction of the selected features space dimension

Principal Component Analysis (PCA) is the general name for a technique in multivariate data analysis aimed to reduce the number of dimensions, while keeping as much as possible of the data's variation [START_REF] Groth | Principal components analysis[END_REF][START_REF] Lim | Medical image compression using block-based pca algorithm[END_REF]. Instead of researching thousands of original variables, the first few components built from a linear combination of the original features and containing the majority of the data's variation are explored. The statistical analysis and visualization of these new variables, named the principal components, can assist to find similarities and differences between samples.

Important original variables that are the major contributors to the first few components can also be discovered. More precisely, PCA applies a vector space transformation to reduce the dimensionality of large data sets. By using mathematical projections, the original data set, which may have involved a great deal of variables, can often be interpreted in just a few variables ( named the principal components). Therefore, it is often the case that an examination of the reduced dimensional data set will allow the user to spot trends, patterns and outliers in the data, much more easily than without performing this principal component analysis. Therefore, in this paper, PCA is applied to reduce the features dimensions of the FCD profiles in order to facilitate the k-means clustering.

Clustering method K-means to dispatch all tested days into different clusters

The k-means is an unsupervised clustering algorithm applied to find groups within the data [START_REF] Sinaga | Unsupervised k-means clustering algorithm[END_REF][START_REF] Alsayat | Social media analysis using optimized k-means clustering[END_REF]. Given a set For a more detailed presentation about SVM, interested readers can refer to [START_REF] Vapnik | Statistical learning theory[END_REF] for more information.

Regression machine learning methods for traffic volume estimation based on the FCD

This section presents the description of regression machine learning methods for the estimation of traffic flow based on travel duration from Google Maps, as shown in Fig. 6.

Linear regression models

These models describe a linear relationship between an output and one or more input. Such model has the following characteristics: 1)the response (output) has a normal distribution with mean Y, for a set of predictors or inputs, which is named as X; 2)a coefficient vector b is defined and linearly combined with the predictors X; 3)the linear regression model is Y = Xb. The classical hyperparameters to tune when using linear regression models are the learning rate and the number of interactions. We can also choose to perform robust fitting using a set of weighting functions to cope with outliers or to consider non-Gaussian noise affecting the data.

Regression trees models

These models can give numeric responses based on input data. In order to predict a response, the decisions in the tree from the root node down to a leaf node should be followed, because each leaf node contains a response.

The trees applied are binary, which means that only one predictor (variable) is checked in each step of prediction.

The hyperparameters for regression trees allow to obtain different trees forms restricting the maximum depth of trees, defining the minimum data points to split a node, fixing the number of samples required to consider node as a leaf, etc.

Support Vector Machine (SVM) for regression

This model is identified by Vladimir Vapnik and his colleagues [START_REF] Vapnik | The nature of statistical learning theory[END_REF], is a very popular machine learning tool for regression. The SVM regression is a nonparametric technology, since it relies on kernel functions. In the Matlab Regression learner APP, the linear epsilon-insensitive SVM ( -SVM) regression is applied. In -SVM regression, predictor variables (X) and observed response values (Y) are included in the training data set, and the goal is to find a function f(X) that deviates from Y (observed response values) by a value no greater than (error or deviation) for each training point X, as shown in Eq.( 2). And the function f(X) should be as flat as possible.

| f (X) -Y| ≤ (2)
In addition to the choice of the kernel function (linear, quadratic, cubic, gaussian, etc.) two hyperparameters are mainly used to improve performance of SVM: parameter defining the width (margin) of the zone used to fit the training data and a factor, corresponding to a "box constraint" that tunes the cost of deviations larger than .

An ensemble of trees model

This model is a predictive model composed of a weighted combination of multiple regression trees. In general, combining multiple regression trees increases predictive performance. This means that results from many weak learners can be melded into one high-quality ensemble predictor. Here we find the same parameters as for the decision trees (maximum depth of each tree, the minimum data to split a node, etc) and some extra hyperparameters like the number of trees or the weighted combination of trees.

Gaussian Process Regression (GPR)

As we can see in the experimental results in the section 5 that Gaussian Process Regression (GPR) can help us to have good performance for the estimation of traffic flow. Therefore the GPR is introduced more detailedly as following. Firstly, the background of GPR machine learning method is introduced [START_REF] Rasmussen | Gaussian processes in machine learning[END_REF]. Then the algorithm of applying the GPR for the estimation of traffic flow is explained.

The GPR [START_REF] Rasmussen | Gaussian processes in machine learning[END_REF][35] is a supervised machine learning method that offers mapping function between input and (continuous) output data. The Gaussian Process framework is used in different areas extended from classification to regression problems, including speed estimation [START_REF] Hu | Short-term wind speed prediction using empirical wavelet transform and gaussian process regression[END_REF], travel duration estimation [START_REF] Idé | Travel-time prediction using gaussian process regression: A trajectory-based approach[END_REF], time-varying systems [START_REF] Hu | Online gaussian process regression for time-varying manufacturing systems[END_REF] etc.

In this work, a GPR model was adopted to model and to estimate traffic volume from the Google aggregated FCD.

Because the traffic volume is expected to have some complex relationship with the travel duration on the same road, simple parametric models such as linear or polynomial functions is inappropriate for this task [24][25]. Therefore, in this subsection, firstly, Gaussian Processes (GPs) formulation is presented. Then, based on the GPs formulation, the Gaussian Process Regression machine learning method is introduced.

Generally, GPs offer a Bayesian paradigm to learn an implicit functional relationship ŷ = f (x), according to a given training data set, D = {(X d i , y i )|i ∈ N}. The variable N is the size of the data set. The symbol X d i represents a vector for the i th observed input variable (also named predictor, regressor, control, or independent) in a d-dimensional feature space. And y i is a one-dimensional observed target value (also named predicted, regresse, response, or dependent), which is either continuous or discrete. However, unlike most classical Bayesian models [START_REF] Jensen | An introduction to Bayesian networks[END_REF], GPs directly infer a prior distribution on the whole function f (X). Thus, function f (X) is treated as a random field and is assumed to be a GP a prior, as the Eq.( 3) shows.

p( f (X)|θ) ∝ GP(m(X), k(X, X )) (3)
where, the prior GP is fully defined by a mean function m(X) and a covariance function k(X, X ). The notation θ means the prior's hyperparameters applied to parameterize the covariance function, as follows:

K(X, X ) = K(X, X ; θ) (4)
Strictly speaking, a GP model can also be treated as a probability distribution, which is defined over the following functions:

E[ f (X)] = m(X) (5) Cov[ f (X), f (X )] = k(X, X ) (6)
where f (X), f (X ) are random variables that are indexed by any pair of X and X . Then, a GP prior can be roughly considered as a probability distribution for an infinite number of random variables. Furthermore, a collection of function values, which are indexed by any finite number of X = [x 1 , x 2 , ..., x n ] T , e.g.,

F(X) = [ f (x 1 ), f (x 2 ), ..., f (x n )],
supposes a multivariate normal distribution in Eq.( 7)

p( f (X)|) = N(m(X), K(X, X )) (7)
where the average vector m(X) and covariance matrix K(X, X ) are determined directly based on m(•) and k(•, •), as following:

m(X) = [m(x 1 ), m(x 2 ), ..., m(x n )] T (8)
K i, j = k(x i , x j ) i, j = 1, ..., n (9)
For the sake of simplicity and without loss of generality, m = (x) = 0 is assumed, since the data can always be centered by the sample mean.

With the machine learning term, k(x, x ) is often named as a kernel function or simply a kernel instead of a covariance function. As detailed later, kernel functions generally take certain forms which are parameterized by one or several parameters θ. Therefore, a GP prior can be specified by determining a specific type of kernel function (also named covariance function) and the associated θ values. Eq.( 10)

p( f |D, θ) = p(y| f )p( f |X, θ) p(D|θ) (10)
where the input variables X should be made explicit in the prior and term p(D|θ) is called Marginal Likelihood, since it is a function of variable θ and given data set D. The noise model p(y| f ) is also a likelihood, for the reason that it is a function of f for a fixed set of observations y. Here, the p(y| f ) is introduced, since y i is a corrupted version of f (x i ). Therefore, the estimation distribution for a new input x new is achieved by using the Eq.( 11) with the posterior

p( f |D, θ) p( f new |x new , D, θ) = p( f new , f |D, θ)d f (11)
By the combination with Eq.( 11) and the noise model, the predictive distribution for y new is achieved in the Eq.( 12)

p(y new |x new , D, θ) = p(y new , f new |D, θ)d f new (12)
From the Eq.( 12), not only the estimated average values but also the associated uncertainty (error-bar) could be calculated. In the GP modeling, it is as collection of function values f (x) needed to be Gaussian instead of variables x itself, which are assumed to be distribution-free. Therefore, the GP model theoretically can handle data with any kinds of distributions. For a more detailed presentation, interested readers can refer to [34][40] [41] for more information.

Gaussian Process Regression machine learning method is introduced based on the GPs formulation. The GP model presented in the above subsection can solve non-linear regression problems, if the observed target value y i is continuous, and the noise model p(y| f ) is assumed as a normal distribution. Then the GPR model can be expressed in the Eq. [START_REF] Asakura | Incident detection methods using probe vehicles with on-board gps equipment[END_REF].

y i = f (x i ) + θ i θ i ∼ N(0, σ 2 ) ( 13
)
In this case, the inference of GPR model becomes analytically tractable, as a result of the Gaussianity of p(y| f ).

Accordingly, for a new input x new , the predictive mean and variance associated with fnew = f (x new ) = f new are defined in Eq.( 14)-( 15), respectively [START_REF] Rasmussen | Gaussian processes in machine learning[END_REF].

µ( f new ) = k(x new , X)[K(X, X ) + σ 2 I] -1 y (14)
and

Var( f new ) = k(x new , x new ) -K(x new , X)[K(X, X) + σ 2 I] -1 k(X, x new ) ( 15
)
where X and y mean the observed predictors and observe target value. I is defined as the identity matrix.

The main hyperparameter of GPR is σ the initial value for the noise standard deviation of the Gaussian process model.

Experiment and validation from the real data for single model

In this section, we conduct a series of experiments over two road segments to evaluate the proposed algorithm and compare the results with real data. Firstly, the performance is compared between estimated traffic flow and real data in the single model regarding data of a first road segment. Then the results between single model and multi-models are compared on another road segment.

Single model simulation case

The experiments are executed on a 1.2 km long road segment named "Boulevard de la République, Douai, 59500,

France" with GPS of origin (50.372329,3.070058) and destination (50.380205,3.082129) for a duration of 26 weeks.

Firstly, the protocol of our experiments are presented. Then, the comparison of RMSE among different machine learning regression methods is shown for n (half width of the sample window) from 2 to 24. Next, experimental performance during a whole week is presented in detail for the kernel function, named rational quadratic in GPR, because it can help us to achieve the lowest RMSE value, compared with others. Finally, an experiment with only weekdays, which performs better than the above case, is executed, for the reason that the profile of traffic flow exists big difference between weekdays and weekends.

Protocol and experiments description

The data of travel duration and real traffic flow on this road for a whole week from Monday to Sunday is respectively acquired from Google Maps, as shown in Fig. 7 and town council of Douai in France, as Fig. 8 shows. In total, 19 types of regression models are applied to find the best machine learning regression methods for such problem of estimating traffic flow from travel duration. The set of all the sample data is divided into two groups: 50 percent of sample data is used to train and validate the regression model; the remainder is extracted as new data to test the performance of the trained regression model. The 5-fold Cross Validation method is applied for the 19 types of regression machine learning algorithms to avoid overfitting problem. The n (half width of the sample window as shown in the Fig. 5) locates in the zone [START_REF] Li | Traffic flow multi-model with machine learning method based on floating car data[END_REF][START_REF] Wu | An algorithm for multi-class network equilibrium problem in pce of trucks: application to the scag travel demand model[END_REF]. Specifically, the sample time interval is from 20 to 240 minutes, since the sample step is 10 minutes. The unit for traffic flow is veh/h, travel duration s. Note that all the hyperparameters described in the presentation of each regression machine learning models have been automatically tuned by Matlab.

Evaluation criteria

The evaluation criteria, containing Root Mean Square Error (RMSE) and Root-Mean-Square Deviation (RMSD), are applied to evaluate and to compare the performances of the two proposed strategies (single model and multimodel), because they are the classical criteria and the most used evaluation criteria for the sequences data regression problem while some authors suggest the joint use of the MAE ('Mean absolute Error') [START_REF] Chai | Root mean square error (rmse) or mean absolute error (mae)?-arguments against avoiding rmse in the literature[END_REF]. In addition, the estimation error distribution looks near from being Gaussian (refer to histogram in pages [START_REF] Dai | A simulation evaluation of a real-time traffic information system using probe vehicles[END_REF][START_REF] Hong | Spatial and temporal analysis of probe vehicle-based sampling for real-time traffic information system[END_REF], which is a 'classical' use case for comparing models with MSE. Here, they are defined as follows: 16)

RMS E = 1 N N k=1 ( f (k) -f (k)) 2 (
AM = 1 N N k=1 ( f (k) -f (k)) ( 17
)
RMS D = 1 N -1 N k=1 (( f (k) -f (k)) -AM) 2 ( 18
)
where f (k) is the observed traffic flow at time step k and f (k) is the corresponding estimated traffic flow. AM is the arithmetic mean. N is the total number of samples.

Discussion about Machine Learning methods

This subsection presents the comparison of RMSE for all the 19 machine learning regression models, as shown in B. Regression trees. These models can achieve better performance than linear regression models with lower RMSE, for the regression trees can deal with nonliear system. The best result with lowest RMSE 70.206 happens when the n equals 24 under the Fine Tree (number 5 in the Tab. 2).

C. Support vector machines. The lowest RMSE for SVM is 59.811, which is obtained by the Fine Gaussian at n=24, and is 14.81 percent lower than that in the Fine Tree.

D. Ensemble of trees. Such models are better than the above regression trees by combining several trees together.

The lowest RMSE (57.69) is acquired by the bagged trees when n is 24, and is 17.83 percent lower than that in the Fine Tree.

E. Gaussian process regression models. The global best result is the lowest RMSE with value of 18.938, which is achieved by Rational quadratic kernel function with n=23, is 67.17 percent lower than that in the bagged trees. In summary, the estimation performance can be improved either by changing machine learner regression model or by increasing the value of n. Indeed, for most models, the RMSE decreases when n increases. The performance rank for different types of regression models for the problem of estimating traffic flow from travel duration is as following: Gaussian Process Regression models > Ensemble of Trees > Support Vector Machines > Regression Trees > Linear Regression Models. Therefore, this work chooses Gaussian Process Regression with Rational quadratic kernel function as regression model.

Discussion about half width of sample window n

Globally, for the GPR with kernel functions of rational quadratic, when n increases from 4 to 23, the RMSE value decreases accordingly, and the lowest one achieved at the point (n=23) is 18.9 veh/h. Then RMSE value lightly increases when n is 24, for the reason that when the n is too big, some data far away from the center point is not so related and give some extra noise influence to the estimation. However, the rate for reducing the RMSE is extremely different when n is augmented. For n from 4 to 16, the RMSE is reduced rapidly with an improvement of 65.3 veh/h. Then the RMSE is reduced slowly with an improvement of 10.3 veh/h for n from 16 to 24. Therefore, the best choice for n equals to 23 when the RMSE performance is the only criterion. Nevertheless n with 16 is the best choice if the RMSE performance and length of n should be considered together, because the bigger n is, the more data is needed to estimate the traffic flow, as shown in the Fig. 5.

Experimental performance for a whole week

The comparison between real and estimated traffic flow and the distribution of errors is shown in detail with n as:

4, 8, 16, 24, as shown in Fig. 1112131415161718. Firstly, with n=4, the trained GPR model can only capture the traffic flow tendency, because the n is too small and more information should be needed, as shown in the Fig. 11. As a result, for the corresponding distribution of error in the Fig. 12, only 45 percent of errors locates between the zone [-50, 50] (veh/h). The biggest error happens in 300 veh/h and the RMSE is 92 veh/h. However, when n is increased to 24, estimated traffic flow is more similar to the real one than that with n=4, as the Fig. 17 shows, which means that the increase of n can help to model the profile more exactly. Therefore, 86 percent of errors locates in the zone [-50, 50] (veh/h), which is almost twice bigger than that with n=4. The RMSE is 20.18 veh/h. Third times lower than the one with n=4. The biggest error is 170 veh/h. However, in Fig. 17, most of the obvious error happens in weekends, because the profile's shape of traffic flow is very different between workdays and the weekend, which motivates us to build a GPR model only for weekdays to improve the performance. This subject is discussed in detail in the following subsection. The performance is compared between GPR models for whole weeks and only weekdays. For n=4, compared with the GPR model for the whole week (refers to Fig. 11), the GPR model for only weekdays (refers to Fig. 19) can capture the real data more precisely, and can get a higher percentage of errors (62.8%) locating between the zone [-50,50]. Furthermore, when the n is increased to 24, as shown in the Fig. 26, 97 percent of errors in the zone [-50,50] , which is 11% higher than that in the GPR model for the whole week, as Fig. 18 shows. Therefore the performance can be improved by building a GPR model with smaller time zone, for example, only the weekdays instead of a whole week. Each day is expressed as a point, beside which the numbering schemes are defined as follows: 1) the first number represents the day within each week; 2) the last two numbers express the order of week. For example, the number "203" means the Tuesday in the third week.

Multi-models simulation case

Labeling results from PCA and k-means for training data

The clustering results are presented in Fig. 28 for days with different travel durations profiles. We can observe that PCA leads to a drastic reduction of the dimension of the features space from its original value to 2, significantly decreasing the complexity of the k-means based labeling process. At the end of the clustering, training days are divided into 4 clusters: cluster 1 and 2 include most of weekdays and a small part of weekends; cluster 3 consists of most of Saturday and some weekdays, which proves that the traffic flows during some weekdays are similar to Saturday. This is the main reason explaining the different performances between multi-model from k-means selection and multi-model from manual selection: in multi-model with manual selection, all weekdays belong to the same group (refers to Figs. 3 and4); cluster 4 is mainly made of Sunday. Therefore, PCA associated to k-means can cluster the travel durations profiles into different clusters more reasonably and precisely than the manual way. However, the multi-model with manual selection does not significantly improve the performance compared with single model, because such model cannot group the FCD flows profiles precisely. As we noticed before, all weekdays labeled using manual selection naturally belong to the same group. However, many FCD flows profiles during weekdays are similar to that in Saturday, as shown in Fig. 28.

Comparison of statistical results

The differences between the three models tend to narrow if we focus on peak hours. In the figures 29 to 40 we have considered the morning peak hours (6:00 -9:30 am) and the evening peak hours (4:00 -7:30 pm). Figures 29,[START_REF] Furey | Support vector machine classification and validation of cancer tissue samples using microarray expression data[END_REF][START_REF] Vapnik | The nature of statistical learning theory[END_REF][START_REF] Xie | Gaussian processes for short-term traffic volume forecasting[END_REF][START_REF] Xie | Gaussian processes for short-term traffic volume forecasting[END_REF][START_REF] Jensen | An introduction to Bayesian networks[END_REF] and 39 represent actual and estimated flows over a succession of peak hours. There is no time scale in these graphs, the peak hour periods succeed one another in chronological order. The corresponding error histograms are shown in Figures 30,[START_REF] Vapnik | Statistical learning theory[END_REF][START_REF] Hu | Short-term wind speed prediction using empirical wavelet transform and gaussian process regression[END_REF][START_REF] Hu | Short-term wind speed prediction using empirical wavelet transform and gaussian process regression[END_REF][START_REF] Hu | Short-term wind speed prediction using empirical wavelet transform and gaussian process regression[END_REF][START_REF] Mackay | Gaussian processes-a replacement for supervised neural networks?[END_REF] In a general way, we can see that by considering only the peak hours, the interest of the multi-models (manual or learned) is much more reduced. We can also see that the afternoon peak hours are better estimated than the morning peak hours (16% maximum error versus 24%). This is probably due to the fact that these peaks are much more variable over the week compared to the rest of the day. This leads to some noise in the estimation. FCD are sufficient for continuously providing a good enough estimation of the vehicles flow along that road, using our GPR regressor. Thus, as an alternative approach to investing in a costly static sensor systems for each of the roads requiring a flow estimation, we could rather use movable measurement units to acquire the data for training the GPR, and then move this system to another place and repeat the process until each model of these "strategic" roads is learned. Finaly, a GPR regressor combined with its corresponding FCD stream would act as a kind of virtual sensor, cost effective vehicles flow sensor.

These measurements, in turn, are used to build and update urban transportation networks' models, which are necessary components required for future mobility to better manage circulation flows within our cities and increase the safety of the users. In a near future, these users could be "human drivers" as well as autonomous cars. The latter may directly use the provided models while directly feeding them with their FCD for an even better quality of traffic estimation. This is exactly what we are currently developping in the scope of the ORIO project (refers to the Acknowledgment section). In fact, the proposed algorithm can be used for all traffic situation and not only urban traffic. The urban context is chosen in the work because the implementation of physical sensors (radar, induction loops) are more difficult than highway.

As a critical reflection, two main drawbacks can be highlighted. A first one concerns the use of non-sparse methods on input data which could be an issue with longer sequences to analyze, leading to obtaining a more complex learned model. A second one, classical in every machine learning solutions, relates to our dependence on the quality of input data. Let's remember that we need two types of input data: the actual flow measured on studied road section and travel time supplied by the FCD provider (e.g. Google in this article).

In a simulation perspective, our results are encouraging regarding the capacity of generating traffic flows from punctual real measurements and more global aggregated data. The virtual sensors reproduce flows according to the curve of the day. The aim is for these sensors to be distributed all over the network. The multi-model permits to gain in accuracy and to provide a collection of 'typical days' to use in simulations. Our perspective is first to generate virtual sensors in key road segments that capture the main entrance and way out of a studied area. Beyond the simulation perspective, questions remain on the possibility to take advantage of a model learned in a specific road segment for other 'similar' road segments in the network. The test of the proposed algorithms on other areas of the city and other transportation context will be the subject of our future work. To do so, the model can integrate topological information (dimension of the segment, relative position to the city center...) as well as temporal information in a multi-model learned from several road segments in order to help in producing generalizable models.

The input vector of the GPR regressor is a sequence of consecutive FCD values centered at the sample step for which the flow estimation has to be produced. To make it clear, the proposed GPR does not directly take into account the "time" (says hours, minutes and seconds), neither does it use the name of the day itself while it produces a rather good estimation of the flows. It is no secret saying that traffic flows evolve with the time of the day and the day of the week (the traffic during the weekend sometimes has nothing to do with what it is on business days!). Then, instead of feeding our GPR learning process with data including a complete week, we plan to build sets of "period of the day" related GPR regressors (for instance: night, morning, midday, afternoon and evening) in replacement of "weekly" GPR regressors computed from several road segment data.

Figure 1 :

 1 Figure 1: Estimation of traffic flow based on FCD (travel duration) from Google with machine learning method. It is assumed in this article that traffic flows should be uniform along the chosen road

Figure 2 :

 2 Figure 2: Proposed system structure

2 .Figure 3 :

 23 Figure 3: Flowchart for Single model and manual multi-models (an example for three clusters on training and testing data sets for a week, respectively)

Figure 4 :

 4 Figure 4: Flowchart for K-means multi-models (an example for three clusters on training and testing data sets for a week)

  Traffic flow (vehs/h)Traffic flow sample data for 20 steps n=4 as an example Machine Learning method n=4 as an example

Figure 5 :•

 5 Figure 5: Features extraction model

of observations (x 1 , 4 . 4 .

 144 x 2 , ..., x n ), where each observation is a d-dimensional vector, k-means clustering algorithm aims to divide the n observations into a set of k groups (k ≤ n), such as G = G 1 , G 2 , ..., G k , in order to minimize the within-group sum of distance squares, which is defined as the sum of distance functions of each point in the group to the corresponding center. The objective function of k-means is the following, where, c i means the centroid of points in group G i . Therefore, the k-means clustering algorithm can be used to cluster the days based on the FCD profiles.In this work, we choose the value of k based on the daily life's observation of our city in France. Typically, we have three types of day: working days, Saturdays and Sundays. During classical working days, people need to go to work, etc. Saturdays are characterized by less work and open shops. Sundays, generally, few people go to work and the possible activities are more restricted (most of the shops are closed for instance). Classification method SVM SVM is a typical machine learning algorithm for classification problem, which was originally introduced by Vapnik and co-workers[START_REF] Furey | Support vector machine classification and validation of cancer tissue samples using microarray expression data[END_REF][START_REF] Vapnik | Statistical learning theory[END_REF] and successively extended by plenty of other researchers. It can have a remarkably robust performance with respect to sparse and noisy data, which makes it useful in a good deal of applications from text categorization to protein function prediction. In particular, for the classification problem, it separates a given set of binary labeled training data with a hyperplane, which is maximally distant from training data sets (also known as 'the maximal margin hyperplane'). If no linear separation is possible, it also can work by combining with the technique of 'kernels' function, which can automatically realize a non-linear mapping to a feature space. In the end, the hyperplane found by the SVM in feature space corresponds to a non-linear decision boundary in the input space.

  Once a GP prior p( f |θ) and a "noise" model p(y| f ) are determined, p( f |D, θ) the posterior distribution of f can be easily obtained by updating the prior p( f |θ) based on the Bayes theorem with the training data set D, as shown in

Figure 7 :Figure 8 :

 78 Figure 7: Travels durations from Google FCD along a week

Fig. 10 and

 10 Fig.10 and Tab. 2-3, in order to find the best solution, which can estimate the traffic flow from travel duration with lowest RMSE. The 19 regression modules are grouped in the Tab. 2-3 into five families:A. Linear regression models. These models always get a very high RMSE, because the relationship between travel duration and traffic flow is nonlinear[START_REF] Wu | An algorithm for multi-class network equilibrium problem in pce of trucks: application to the scag travel demand model[END_REF][START_REF] Lam | Calibration of the combined trip distribution and assignment model for multiple user classes[END_REF].

Figure 9 :

 9 Figure 9: RMSE profile for regression models under n from 2 to 24

Figure 10 :

 10 Figure 10: RMSE profile for GPR under different n

Figure 11 :PercentageFigure 12 :Figure 13 :PercentageFigure 14 :Figure 15 :PercentageFigure 16 :Figure 17 :PercentageFigure 18 :Figure 19 :PercentageFigure 20 :

 11121314151617181920 Figure 11: Actual VS estimated traffic flow with n=4 for a week

Figure 21 :PercentageFigure 22 :Figure 23 :Figure 24 :Figure 25 :PercentageFigure 26 :

 212223242526 Figure 21: Actual VS estimated traffic flow with n=8 for weekdays only

Firstly, the

  experiments' parameters are presented. Secondly, FCD profiles on training data sets are clustered into k (here k = 4) clusters by k-means method. At last, the obtained results (estimated traffic flows) are compared with the real observed data (real traffic flows measured by sensors).Prior to the above clustering process, the dimension of FCD profiles is reduced by PCA (Principal Component Analysis) algorithm. Having a look at the eigen vectors' coefficients after the PCA has been applied to the initial representation shows us that the initial basis is not preserved. Indeed, most of the eigenvectors result from a combination of a large number of the initial base's vectors. As an illustration, figure27) represents the eigenvector basis as a 33x33 gray level image. The darker is the pixel, the closer to 0 is the magnitude of the corresponding eigenvector's coefficient. Rather than observing columns mainly made of dark pixels with a few bright pixels, we see a wide distribution of gray shaded pixels affecting a large part of the image.

Figure 27 :

 27 Figure 27: Representation of the eigenvector basis after PCA

4 Figure 28 :

 428 Figure28: Training data sets are clustered into four clusters by k-means algorithm, after their dimensions are reduced by PCA method. Each day is expressed as a point, beside which the numbering schemes are defined as follows: 1) the first number represents the day within each week; 2) the last two numbers express the order of week. For example, the number "203" means the Tuesday in the third week.

Figure 29 :Figure 30 :Figure 31 :Figure 32 :

 29303132 Figure 29: Actual (RED) VS estimated traffic flow (BLUE) with single model for morning peak hoursFigure 30: Normalized estimation error with single model for morning peak hours

Figure 33 :

 33 Figure 33: Actual (RED) VS estimated traffic flow (GREEN) with manual multi-models for morning peak hours

Figure 34 :

 34 Figure 34: Normalized estimation error with manual multi-models for morning peak hours

Figure 35 :

 35 Figure 35: Actual (RED) VS estimated traffic flow (GREEN) with manual multi-models for evening peak hours

Figure 36 :

 36 Figure 36: Normalized estimation error with manual multi-models for evening peak hours

Figure 37 :

 37 Figure 37: Actual (RED) VS estimated traffic flow (MAGENTA) with K-means multi-models for morning peak hours

Figure 38 :

 38 Figure 38: Normalized estimation error with K-means multi-models for morning peak hours

Figure 39 :

 39 Figure 39: Actual (RED) VS estimated traffic flow (MAGENTA) with K-means multi-models for evening peak hours

Figure 40 :

 40 Figure 40: Normalized estimation error with K-means multi-models for evening peak hours

Table 1 :

 1 Input and output data structure for requesting FCD with Google Maps API. For example, the 310 seconds in the output column means that it takes average 310 seconds for cars to pass from origin point GPS to the destination point GPS on the road at the given time stamp.

	Example	GPS of origin	Input GPS of destination	Time-stamp	Output Traffic duration (s)
	1	(50.372329,3.070058) Road name: Boulevard de la Republique,Douai,France (50.380205,3.082129)	1520677640 (10/03/2018;10:27:20)	310
	2	(50.380163,3.079186) Road name: Boulevard Lahure,Douai,France (50.390721,3.081124)	1521306610 (17/03/2018;17:10:10)	370

Table 2 :

 2 RMSE value for types 1-10 of machine learning regression models under n from 2 to 24 (PE is an abbreviation of Percentage Error)

	n	1	PE(%)	A. Linear Regression Models 2 PE(%) 3 PE(%)	4	PE(%)	5	PE(%)	B. Regression Trees 6 PE(%)	7	PE(%)	8	C. Support Vector Machines PE(%) 9 PE(%) 10	PE(%)
	2 139.34 57.89

Table 3 :

 3 RMSE value for types 11-19 of machine learning regression models under n from 2 to 24 (PE is an abbreviation of Percentage Error)

	n	11	C. Support Vector Machines PE(%) 12 PE(%) 13	PE(%)	14	D. Ensemble of Trees PE(%) 15	PE(%)	16	PE(%)	E. Gaussian Process Regression Models 17 PE(%) 18 PE(%)	19	PE(%)
	2 121.47 50.47	121.93 50.66	131.73 54.73	105.24 43.72	103.11 42.84	112.51 46.75	110.33 45.84	117.19 48.69	108.1 44.91
	3 117.17 48.65	117.32 48.71	129.65 53.83	100.68 41.80	97.405 40.44	108.85 45.19	106.05 44.03	108.45 45.03	102.96 42.75
	4 107.14 44.46	114.3 47.44	126.36 52.44	95.515 39.64	92.91 38.56	103.17 42.82	100.92 41.88	95.616 39.68	94.589 39.26
	5 98.045 40.69	110.38 45.81	124.06 51.48	90.612 37.60	87.285 36.22	101.27 42.03	93.715 38.89	85.182 35.35	85.625 35.53
	6 87.702 36.41	106.82 44.34	121.99 50.64	86.639 35.96	81.334 33.76	87.001 36.11	75.897 31.51	74.147 30.78	72.321 30.02
	7 82.244 34.16	104.93 43.59	120.37 50.00	83.458 34.67	77.281 32.10	74.919 31.12	63.406 26.34	66.032 27.43	61.221 25.43
	8 81.351 33.81

Table 4 :

 4 Comparison of statistical results (RMSE and RMSD are expressed in veh/h) The RMSE and RMSD for single model, manual multi-models and K-means multi-model are compared in the Tab.4. All types of models can estimate traffic flows with low RMSE and RMSD. Furthermore the K-means multimodels can reduce RMSE and RMSD by up to 36.2% and 35.9%, respectively, compared with single model. Because each day in testing data set is firstly input to the trained SVM model to choose the suitable trained GPR model.

	Criteria Single model Manual multi-models K-means multi-models
	RMSE	37.3434	37.1286	23.827
	RMSD	37.0281	36.9523	23.8084

4 colors are available: green for normal speed of traffic, orange for slower conditions, red for congestion and dark red for stopped traffic

Acknowledgment

This work was supported by the project ORIO (Observing the peRformances of urban Infrastructures and mobility / preventing collisions with vulnerable people using Opportunistic radar). The project ORIO is done within the framework of ELSAT2020, which is co-financed by the European Union with the European Regional Development Fund, the French state and the Hauts de France Region Council. The real traffic flow data from induction loops on the road "86 Boulevard de la République, Douai, France" was provided by the town hall of Douai (mairie de douai).


Charbel Habchi

Mohammad Oneissi

Serge Russeil
email: serge.russeil@mines-douai.fr

Daniel Bougeard

Thierry Lemenand

Streamwise

Comparison of eddy viscosity turbulence models and stereoscopic PIV measurements for a flow past rectangularwinglet pair vortex generator

Keywords: Rectangular winglet pair, vortex generator, longitudinal vortices, turbulence model, stereoscopic particle image velocimetry. Nomenclature

Vortex generators (VG) are widely used in enhancing the heat transfer coefficients in heat exchangers due to the development of longitudinal and transverse vortices. Therefore, understanding the development of these vortices has a high importance for the design and optimization of heat exchangers. When using numerical simulations, the choice of an appropriate turbulence model that can better predict the flow structure downstream a VG is fundamental. In the present study, three-dimensional numerical simulations, with two different commonly used eddy viscosity turbulence models, are performed for channel flow fitted with rectangular-winglet pairs (RWP) vortex generators. The numerical results are compared to experimental data obtained by stereoscopic particle image velocimetry (SPIV).

The shear-stress transport (SST) κ-ω model and the re-normalization-group (RNG) κ-ε model

Introduction

Heat exchangers are involved in various applications starting from small household radiators to large chillers and condensers in power plants. Several active, passive and compound techniques were used in order to increase the thermal performance of heat exchangers while keeping relatively low pumping power [START_REF] Aris | An experimental investigation into the deployment of 3-D, finned wing and shape memory alloy vortex generators in a forced air convection heat pipe fin stack[END_REF][START_REF] Li | Comparison and analysis of the arrangement of delta winglet pair vortex generators in a half coiled jacket for heat transfer enhancement[END_REF][START_REF] Ali | Three-dimensional numerical study of heat transfer and mixing enhancement in a circular pipe using selfsustained oscillating flexible vorticity generators[END_REF][START_REF] Zhang | Heat transfer enhancement by streamlined winglet pair vortex generators for helical channel with rectangular cross section[END_REF][START_REF] Karkaba | Multi objective optimization of vortex generators for heat transfer enhancement using large design space exploration[END_REF][START_REF] Habchi | Residence time distribution and heat transfer in circular pipe fitted with longitudinal rectangular wings[END_REF]. In particular, passive technique, and especially vortex generators (VG), is the most commonly studied method in enhancing heat exchangers due to many reasons such as their efficiency, easy manufacturing and maintenance [START_REF] Habchi | Residence time distribution and heat transfer in circular pipe fitted with longitudinal rectangular wings[END_REF][START_REF] Khoshvaght-Aliabadi | Performance of a plate-fin heat exchanger with vortex-generator channels: 3D-CFD simulation and experimental validation[END_REF]. Passive VG exist in various shapes [START_REF] Awais | Heat transfer enhancement using different types of vortex generators (VGs): A review on experimental and numerical activities[END_REF][START_REF] Kumar | A review of heat transfer and fluid flow mechanism in heat exchanger tube with inserts[END_REF], and they produce mainly two type of vortices, namely longitudinal vortices (LV) and transverse vortices (TV). TV are two-dimensional swirling flows with axes normal to the main flow direction, while LV rotate about an axis in the streamwise direction, implying a threedimensional helicoid swirling motion. LV are found to have an advantage over TV in terms of scalar mixing and heat transfer performances [START_REF] Fiebig | Embedded Vortices in Internal Flow: Heat Transfer and Pressure Loss Enhancement[END_REF][START_REF] Habchi | Mixing assessment by chemical probe[END_REF][START_REF] Oneissi | Heat transfer enhancement of inclined projected winglet pair vortex generators with protrusions[END_REF].

In order to investigate the flow structure induced by these vortex generators, three methods are commonly used in the open literature: experimental measurements, theoretical developments and numerical simulations. While theoretical analysis is limited to very specific academic flow configurations, experiments in fluid mechanics are considered a very important part for investigating flow structure characteristics by using for instance laser Doppler velocimetry and particle image velocimetry (PIV) [START_REF] Hamed | Vortical structures in the near wake of tabs with various geometries[END_REF][START_REF] Velte | Flow analysis of vortex generators on wing sections by stereoscopic particle image velocimetry measurements[END_REF][START_REF] Mokrani | Experimental study of the influence of the rows of vortex generators on turbulence structure in a tube[END_REF]. They can supplement and validate theoretical studies and numerical modelling of flow field downstream VG.

Computational fluid dynamics (CFD) simulation is widely used in studying the formation and development of TV and LV downstream VG. Since experimental methods are very expensive, time consuming and involve too much labor, numerical simulation has become more and more popular with the advances in computational power [START_REF] Lu | Effects of curved vortex generators on the air-side performance of fin-and-tube heat exchangers[END_REF][START_REF] Awais | Enhancement of thermal and hydraulic performance of compact finned-tube heat exchanger using vortex generators (VGs): A parametric study[END_REF][START_REF] Khanjian | Effect of the angle of attack of a rectangular wing on the heat transfer enhancement in channel flow at low Reynolds number[END_REF]. CFD simulations are flexible; since adjusting computerized model is much easier and cheaper than real prototypes.

Moreover, CFD simulations can provide profound analysis; because one can analyze physically unreachable zones, in addition to shape optimization for heat transfer and mixing enhancement [START_REF] Oneissi | Heat transfer enhancement of inclined projected winglet pair vortex generators with protrusions[END_REF][START_REF] Lemenand | Vorticity and convective heat transfer downstream a vortex generator[END_REF][START_REF] Garelli | Heat transfer enhancement in panel type radiators using delta-wing vortex generators[END_REF].

Consequently, numerous studies using numerical simulations have been carried in order to analyze the thermal and hydraulic performances of vortex generators in heat exchangers.

These studies have an important industrial impact since the intensification of heat transfer using VGs allows significant performance increases. But most often the choice of the turbulence model is made by comparison on the overall performance of the exchanger which does not allow to precisely quantify the capacity of the computer code to accurately predict detailed features of the turbulent wake downstream VGs [START_REF] Gupta | Numerical investigation towards implementation of punched winglet as vortex generator for performance improvement of a fin-and-tube heat exchanger[END_REF][START_REF] Li | Comparison and analysis of the arrangement of delta winglet pair vortex generators in a half coiled jacket for heat transfer enhancement[END_REF].

Various turbulence models were used by the authors to model transverse and longitudinal vortices in turbulent flows. Two of the most commonly used turbulence models are the re-normalization-group (RNG) κ-ε model [START_REF] Salviano | Optimization of winglet-type vortex generator positions and angles in plate-fin compact heat exchanger: Response Surface Methodology and Direct Optimization[END_REF][START_REF] Hatami | Experimental and numerical analysis of the optimized finned-tube heat exchanger for OM314 diesel exhaust exergy recovery[END_REF][START_REF] Habchi | Enhancing Heat Transfer in Vortex Generator-type Multifunctional Heat Exchangers[END_REF] and the shear-stress transport (SST) κ-ω model [START_REF] Tang | Flow structure and heat transfer in a narrow rectangular channel with different discrete rib arrays[END_REF][START_REF] Eiamsa-Ard | Numerical study on heat transfer of turbulent channel flow over periodic grooves[END_REF]. The RNG κ-ε model is well known to improve the accuracy of the result in rapidly strained and swirling or rotating flows. The SST κ-ω model is known to be accurate for wide range of flows such as transonic shock waves and adverse pressure gradients and it performs very well in the near-wall region as well as in the flow core [START_REF] Moukalled | The Finite Volume Method in Computational Fluid Dynamics[END_REF].

In simulating turbulent flows downstream VG, the selection of appropriate turbulence model plays a massive role in the prediction of the hydraulic performance and thus convective heat transfer. For this reason, and since the most used models for modeling turbulent flows downstream VG are the RNG κ-ε and SST κ-ω models, this paper focuses on comparing these two models with an experimental study using stereoscopic PIV (SPIV).

In open literature [START_REF] Oneissi | Novel design of delta winglet pair vortex generator for heat transfer enhancement[END_REF].

The numerical method, computational domain and mesh sensitivity analysis are presented in Section 2. Section 3 is devoted for the experimental setup. Comparison between experimental and CFD simulations of flow topologies in horizontal and vertical planes are presented in Section 4. Section 5 is devoted to the concluding remarks.

Numerical Procedure

Governing equations and turbulence models

The flow is governed by the 3D steady-state Reynolds-averaged Navier-Stokes (RANS) equations computed using ANSYS Fluent 15 which is based on a cell-centered finite volume discretization method [START_REF] Habchi | Enhancing Heat Transfer in Vortex Generator-type Multifunctional Heat Exchangers[END_REF]. RANS turbulence modelling approach allows the calculation of the mean flow without first calculating the full time-dependent flow field. Two turbulence models are particularly detailed and used in this study.

First, the re-normalization-group (RNG) κ-ε model was developed by Yakhot et al. [START_REF] Yakhot | Renormalization group analysis of turbulence:(i) basic theory[END_REF][START_REF] Yakhot | Development of turbulence models for shear flows by a double expansion technique[END_REF] using RNG method to renormalize the Navier-Stokes equations in order to capture the effects of small scale motion in a turbulent flow. In standard κ-ε model, the turbulent diffusion is accounted only at a specific scale, since eddy viscosity is resolved from a single length scale. This trend is not realistic since all turbulence scales will contribute to the turbulent diffusion. The modified κ-ε model solves this problem and involves different scales to the production term. The RNG κ-ε model shows fundamental improvements over the standard κ-ε model and enhances the final solution [START_REF] Launder | The numerical computation of turbulent flows[END_REF], where the effects of turbulence of strong streamline curvature, vortices and swirl effects are taken in account [START_REF] Choudhury | Introduction to the Renormalization Group Method and Turbulence Modeling[END_REF].

Second, the shear-stress transport (SST) κ-ω model developed by Menter [33] is also used in this study. This model solves two partial differential equations: one for the turbulence kinetic energy equation κ and the other for its specific dissipation rate ω. Also, the shearstress transport combines the use of κ-ω formulation in the inner parts of the boundary layer and the switching to a κ-ε behavior in the free-stream thus avoiding the κ-ω sensitivity to the inlet free-stream turbulence properties. In addition, it is characterized by its good behavior in adverse pressure gradients and separating flows while attaining accuracy and reliability [START_REF] Ansys | Standard and SST k-w Models[END_REF].

The preceding attributes give the SST κ-ω model additional accuracy and reliability thus providing it an advantage over the standard κ-ω model.The SST κ-ω model was used by many researchers in previous works proving a fair match with experimental results [START_REF] Yongsiri | Augmented heat transfer in a turbulent channel flow with inclined detached-ribs[END_REF][START_REF] Tang | Flow structure and heat transfer in a narrow rectangular channel with different discrete rib arrays[END_REF].

This approach necessitates assessment of the wall adjacent cell size analogous to the dimensionless wall distance y + lower than 4, ensuring that the viscous sublayer is meshed.

Low Reynolds correction and near-wall treatment are adopted for the SST κ-ω and RNG κ-ε turbulence models, respectively. Double precision and second order upwind numerical schemes are used for spatial discretization of the convective terms [START_REF] Warming | Upwind second-order difference schemes and applications in aerodynamic flows[END_REF]. Centraldifference and second order accuracy are used for the diffusion terms. The Coupled algorithm is used for the pressure-velocity coupling with the pseudo-transient option which is a form of implicit under-relaxation for steady-state cases. Pseudo-transient option allows the user to obtain solutions faster and more robustly. For detailed description of the different methods and model equations, the reader could refer to [START_REF] Ansys | Standard and SST k-w Models[END_REF].

Computational domain and boundary conditions

The test section consists of one row of rectangular winglet pair vortex generators that produce pairs of main longitudinal vortices (see next section). This configuration allows to assume two planes of symmetry; the first is at the channel midplane and the other at one side of the middle winglet pair. The channel streamwise direction is defined as x, spanwise direction as y and z for the vertical direction. The analytical velocity profile used to fit the experimental inlet velocity is based on the model developed by Štigler [START_REF] Štigler | Introduction of the analytical turbulent velocity profile between two parallel plates[END_REF] based on vorticity distribution between two parallel plates.

Štigler velocity profile fits realistic results better than the power law velocity profiles that was derived for pipe flows, since the derivative of the function does not tend to infinity at the boundaries, thus no infinite shear stress at the walls. In addition, Štigler analytical solution solves the Munson power law in which the derivative of the function at the middle of the channel does not tend to zero. Štigler's analytical profile is defined as [START_REF] Štigler | Introduction of the analytical turbulent velocity profile between two parallel plates[END_REF]:

= 1 - |2 -| (1)
= ( -) 4 -2 (2)
where is the longitudinal velocity (m.s -1 ), is the maximum velocity (m.s -1 ), is the channel height (m), is the vertical distance from the bottom wall (m), 0 ≤ z ≤ H, = 3.4 is a pressure drop depending constant, is the mean flow velocity (m.s -1 ), is the pressure drop (Pa) obtained from the friction factor inside an empty duct flow. This expression can be used for both laminar and turbulent velocity profiles.

Mesh sensitivity analysis

The mesh used in this work is similar to a previous study done by Oneissi et al. [START_REF] Oneissi | Novel design of delta winglet pair vortex generator for heat transfer enhancement[END_REF] in which a non-uniform mesh with polyhedral cells is adopted for the computational domain. All walls in the channel are treated with 10 inflation layers with 40 µm first layer thickness for attaining y + values less than 1. Table 1 presents the channel mesh characteristics used in this work. The mesh sensitivity analysis is done according to the method proposed by Celik et al. [START_REF] Celik | Procedure for Estimation and Reporting of Uncertainty Due to Discretization[END_REF]. According to this method a grid convergence index (#$%) is calculated for several flow parameters. In our study, we consider local velocity distribution in the wake of the VG as well as the pressure drop across the flow domain. Four mesh densities were studied as follows: 500,000 -860,000 -1,600,000 -2,500,000. It is found that for a mesh size of around 2.5 million cells, the #$% for local velocity did not exceed 1.2 % and the #$% for the pressure drop was lower than 0.7 %, for both eddy viscosity turbulence models. Thus, this mesh is adopted for the present numerical simulations.

To ensure good convergence of the results, the criteria are based on the continuity, linear momentum, and energy equations with residuals of at least 10 &' .

Table 1 Mesh characteristics

Maximum element size (mm) 1.30

Type of mesh Polyhedral

Number of elements 2.5x10 6

Inflation (μm) 40

Inflation layers 10

Maximum y + 0.40

Experimental Method

The experimental studies are performed in a close-loop air channel shown in Figure 2.

The same experimental setup has been used in a previous study where more details are given on the benchmark and measurement method in Ref. [START_REF] Oneissi | Experimental analysis by stereo-PIV of the development of streamwise vortices downstream of rectangular winglets[END_REF]. In this paper we give a brief description of the experimental benchmark and method adopted.

The test section is 1200 mm in length, 500 mm in width and of height equal to 37.8 mm.

The flow rate is controlled by varying the rotational speed of the fan. In the seeding room, seeding particles are generated through a controlled combustion process of which produces about 1 µm mean diameter smoke particles with a density around 1.06 kg/m 3 [START_REF] Wang | Chemical composition of fine particles from incense 29 burning in a large environmental chamber[END_REF][START_REF] Cheng | Incense Smoke: Characterization and Dynamics in Indoor Environments[END_REF] • xy measurement planes, where the laser sheet is placed horizontally above the winglets row parallel to the bottom wall.

• yz measurement planes, where the laser sheet is placed vertically in cross-sections normal to the main flow direction at distinct locations.

In each measurement plane, 3000 images were captured with a sampling frequency of 2

Hz.

Results and discussion

Streamwise velocity profile

Before comparing the experimental and numerical findings, the mean flow velocity profile at the inlet cross section is presented in along with the analytical velocity profile obtained from Eq. ( 2) for = 4400. The experimental and analytical velocity profiles show a great agreement and can be used as an inlet boundary conditions for the CFD simulations.

The average error between experimental and analytical velocities is less than 2%.

Horizontal velocity components

Flow structure in horizontal plane of both turbulence models are compared with SPIV experimental results in this section. In order to investigate the flow topology in the horizontal plane xy (z=24 mm from bottom wall), first, the three mean velocity components ) *, ,̅ and . / are compared with experimental results. Second, to investigate the turbulent characteristics of the flow, turbulent kinetic energy is also compared.

Comparison between SPIV and RNG κ-ε turbulence model

Results obtained by numerical simulation and experimental method are post-processed using Matlab adopting similar image filtering and colormaps.

Comparison between SPIV and SST κ-ω model

A similar strategy is followed in this section for comparing CFD using SST κ-ω turbulence model and experimental results. Figure 7 shows a comparison of the turbulent kinetic energy profiles, normalized with the square of mean flow velocity ( ), between experimental results (denoted by SPIV) and numerical simulation using SST κ-ω turbulence model at Re=4400. Both figures are not perfectly superposing, but they are in relatively good agreement. SST κ-ω turbulence model seems to have good agreement with experimental results for the turbulence kinetic energy of the primary vortex however, it does not reproduce fairly that for secondary vortex, especially from x/H higher than 3. Meanwhile, the effectiveness of the SST κ-ω is much better than the RNG κ-ε model as concluded from these two last sections.

Secondary flow structure

In addition to the flow field comparison in horizontal plane, flow topologies in vertical planes are extracted for further examination of the turbulence models prediction of longitudinal vortices at several locations in the wake region. In this section, Γ criterion proposed by Favelier et al. [START_REF] Favelier | Développement d'un critère d'identification de structures tourbillonnaires adapté aux mesures de vitesse par PIV[END_REF] and employed previously by Oneissi et al. [START_REF] Oneissi | Experimental analysis by stereo-PIV of the development of streamwise vortices downstream of rectangular winglets[END_REF] to identify and track the main and induced vortices in cross sections downstream the winglet pair, is also used here to validate turbulence model predictions. Comparing topologies of the Γ criterion with SST κ-ω turbulence model and RNG κ-ε turbulence model at different locations (x= H, 3H, 5H, 9H and 10.5H) are respectively illustrated in Figure 8 and Figure 9. The results are presented in a symmetrical manner for better visibility.

It is clearly shown from Figure 8 that the SST κ-ω turbulence model accurately predicts and identifies the generated vortices location at different vertical planes, while Figure 9 shows that the RNG κ-ε turbulence model could not predict the generation of these streamwise vortices which are the most responsible for macromixing and heat transfer enhancement [START_REF] Habchi | Residence time distribution and heat transfer in circular pipe fitted with longitudinal rectangular wings[END_REF][START_REF] Habchi | Mixing assessment by chemical probe[END_REF].The RNG κ-ε turbulence model only predicts the formation of one main vortex that develops along the channel, but it fails in predicting any induced vortices. Whereas, the SST κ-ω turbulence model predicts precisely the number of generated vortices and the relative position of these vortices. Figure 8 shows that SST κ-ω turbulence model predicts one main vortex in addition to two induced vortices near the bottom wall of the channel at a distance x=H. Also, for the other planes, the SST κ-ω turbulence model predicts accurately the main and induced vortices numbers and locations.

Conclusion

In this study, a comparison between CFD numerical simulation and SPIV experimental data of longitudinal vortices generated downstream rectangular winglet pairs in parallel platefin channel is investigated at a Reynolds number Re=4400. Two eddy viscosity models are adopted, namely the RNG 6 - This study highlights the dominance of the SST κ-ω turbulence model over the RNG κε turbulence model in predicting flow structure characteristics of longitudinal vortices in a parallel plate channel downstream vortex generator. Moreover, the SST κ-ω turbulence model can capture accurately the main and induced vortices and therefore it is preferable to be used in CFD studies dedicated for convective heat transfer enhancement using vortex generators.

  this study, we investigate longitudinal vortices generated by rectangular-winglet pairs in parallel plate turbulent channel flow to examine the flow structure characteristics. A row of rectangular winglet pairs is fixed in a parallel plate test bench channel. SPIV is used to harvest three dimensional (3D) instantaneous velocity fields in the flow configuration for a Reynolds number = 4000 (based on the channel's hydraulic diameter) which is a typical value in the range of those commonly studied in the open literature. The dimensions of the test section and VG as well as the Reynolds number used here are commonly used in studies from the

Figure 1

 1 Figure 1 presents an isometric view showing the computational domain and boundary conditions. The channel height is H=37.8 mm with breadth B=1.6H and length L=11H. Each rectangular winglet is 1.5H in length, H/2 in height and thickness equals to 0.05H. The winglets are set normal to the channel bottom wall with an angle of attack of 30° from the incident flow, and the distance separating the leading edges of VG is 0.2H.

Figure 1

 1 Figure 1 Computational domain and boundary conditions.

  used as flow tracers. A row of rectangular winglet pair VG is mounted on the bottom wall and used to generate longitudinal vortices in the test channel. The test section consists of four rectangular winglet pairs. Only one pair of rectangular winglets at the middle of the test section is investigated in this study due to symmetric boundary conditions as explained in the previous section for the CFD study. The VG are manufactured by means of an Objet Alaris 30 Desktop 3D Printer using photopolymer jetting technology. two digital cameras and image-based processing software. The three velocity field components are measured in two plane orientations:

Figure 2

 2 Figure 2 Close-loop air channel setup: (1) Seeding room; (2) Particle generator; (3) Honeycomb; (4) Convergent section; (5) Upstream channel; (6) Test section (lateral and upper glazed windows); (7) Convergent 90°-elbow; (8) Absolute pressure transducer; (9) Laminar flowmeter; (10) Temperature probe; (11) PVC suction pipe; (12) Centrifugal fan; (13) Wirings and cables; (14) Micro-manometer and temperature recorder ; (15) Normal camera; (16) Inclined camera.

Figure 3 Figure 3

 33 Figure 3 Mean flow velocity at the inlet cross section from experimental and analytical methods. In order to analyze the eddy viscosity turbulence models in predicting the generation and dissipation of longitudinal vortices generated by rectangular winglet pair, local comparison of the velocity field from CFD and SPIV measurements is conducted in this section. Measurements of longitudinal velocity profiles at a distance ( = downstream the VG trailing edge and vertical distance z=24 mm from the bottom wall are compared for instance in Figure 3. This figure shows a comparison between longitudinal mean velocity profiles () *) normalized by the bulk velocity ( ) extracted from experimental results with a

Figure 4 Figure 4 Figure 5

 445 Figure 4 Comparison of mean flow velocity components in the horizontal plane at Re=4400 and z=24 mm from bottom wall: (a) 0 /, (b) 1 /, (c) 2 / , upper section from experimental results and lower section from numerical simulations using RNG κ-ε turbulence model.

Figure 5

 5 Figure 5 Comparison of the normalized turbulent kinetic energy topologies at Re=4400 and z=24 mm from bottom wall, between experimental results (upper section) and numerical simulation using RNG κ-ε turbulence model (lower section).

Figure 6 Figure 6

 66 Figure 6 Comparison of mean flow velocity components in the horizontal plane at Re = 4400 and z=24 mm from bottom wall: (a) 0 /, (b) 1 /, (c) 2 / , upper section from experimental results and lower section from numerical simulations using SST κ-ω turbulence model.

Figure 7

 7 Figure 7 Comparison of the normalized turbulent kinetic energy topologies at Re=4400 and z=24 mm from bottom wall, between experimental results (upper section) and numerical simulation using SST κ-ω turbulence model (lower section).

Figure 8 Figure 9

 89 Figure 8 Comparison of Γ function topologies used for vortex identification downstream the winglet pair at Re=4400 between experimental results (left section) and numerical simulation using SST κ-ω turbulence model (right section).

7

 7 and SST 6 -8 turbulence models, which are the most widely used when dealing with flow and heat transfer downstream vortex generators. The aim of this study is to qualify turbulence model in predicting the flow structure of longitudinal vortices in a parallel plate channel, since these vortices are the most responsible for macromixing and heat transfer enhancement. The obtained results of the velocity profile from the simulations are compared locally with experimental data extracted using SPIV techniques. Local velocity profile in the wake of the VG obtained from the SST κ-ω turbulence model shows a good correspondence with experimental data, while RNG κ-ε turbulence model did not predict well the same behavior. Velocity fields on a horizontal flow sections in addition to the turbulent kinetic energy distribution are inspected. The SST κ-ω turbulence model shows fair agreement with experimental results. On the other hand, the RNG κ-ε turbulence model shows no consistency at all with experimental results. Meanwhile, the SST κ-ω turbulence model is found to not accurately predict the turbulent kinetic energy of the induced secondary vortex. The flow topology is also compared on several cross sections downstream the VG to show development and number of streamwise vortices by means of Γ criterion. The SST κ-ω turbulence model accurately predicts and identifies the generated vortices location at different planes, the number of generated vortices (main and induced) as well as their relative positions. The RNG κ-ε turbulence model could only predict the generation of main vortices, but it fails at predicting any induced vortices.