Petr Dobiáš
email: petr.dobias@irisa.fr

Emmanuel Casseau

Oliver Sinnen

Restricted Scheduling Windows for Dynamic Fault-Tolerant Primary/Backup Approach-Based Scheduling on Embedded Systems

Keywords: Dynamic scheduling, Fault-tolerance, Homogeneous faulty processors, Low complexity, Multiprocessors, Primary/backup approach

This paper is aimed at studying fault-tolerant design of the real-time multi-processor systems and is in particular concerned with the dynamic mapping and scheduling of tasks on embedded systems. The effort is concentrated on scheduling strategy having reduced complexity and guaranteeing that, when a task is input into the system and accepted, then it is correctly executed prior to the task deadline. The chosen method makes use of the primary/backup approach and this paper describes its refinement based on reduction of windows within which the primary and the backup copies can be scheduled. The results show that the use of restricted scheduling windows reduces the algorithm complexity by up to 15%.

I. INTRODUCTION

At the present time, requirements for higher performance and lower energy consumption to meet the demands of more and more complex computations are increasing. To fulfil these goals, it is necessary to ensure system functionality and therefore design fault-tolerant systems. One of the possibilities to provide such systems is to make use of redundancy in space and time on multi-core platforms. Actually, they are less vulnerable when one processor is faulty because other processors can take over its scheduled tasks. Nevertheless, there exist several issues when conceiving a solution. It is mainly concerned with a trade-off between the parallel computing accounting for system performance and the spatial redundancy arranging for reliability, and with the dynamic mapping and scheduling control, which is in general an NP problem.

The main objective of this paper is to investigate how to dynamically schedule tasks onto homogeneous faulty processors with reduced complexity to execute the algorithm at run-time onto an energy efficient embedded system. The principal idea is to make use of the multi-processor system on chip and take advantage of its inherent properties. We developed a run-time algorithm based on the primary/backup (PB) approach reducing the scheduling window when searching for free slots for primary and backup copies and we evaluated its performances. The primary/backup approach is a commonly used technique to provide fault-tolerance, for example it can be adapted for satellites [START_REF] Zhu | Analysis and Design of Fault-Tolerant Scheduling for Real-Time Tasks on Earth-Observation Satellites[END_REF] or employed in virtualised clouds [START_REF] Zhu | Fault-Tolerant Scheduling for Real-Time Scientific Workflows with Elastic Resource Provisioning in Virtualized Clouds[END_REF].

The rest of this paper is organised as follows. Section II sums up the related work. Our assumptions and our scheduling model are presented in Sect. III. Section IV describes and then analyses the experiments. Section V concludes the paper.

II. RELATED WORK

Manimaran et al. [START_REF] Manimaran | A Fault-Tolerant Dynamic Scheduling Algorithm for Multiprocessor Real-Time Systems and its Analysis[END_REF] assumed that resources can be reclaimed not only after backup deallocation (if this technique is employed) but also when a copy finishes its execution earlier than scheduled.

Other authors study more elaborated techniques for overloading of task copies. Al-Omari et al. [START_REF] Al-Omari | Efficient Overloading Techniques for Primary-Backup Scheduling in Real-Time Systems[END_REF] introduced the primarybackup overloading allowing primary and backup copies of two different tasks to overlap each other on the same processor. It was found that this technique improves schedulability comparing to the basic version of backup-backup overloading but its reliability, measured by means of time to next fault (TTNF), can be diminished. Tsuchiya et al. [START_REF] Tsuchiya | A New Fault-Tolerant Scheduling Technique for Real-Time Multiprocessor Systems[END_REF] proposed the active primary/backup approach in order to schedule tasks with tight deadlines. This method, which is mainly interesting for real-time systems, enables primary and backup copies of the same task to overload each other on two different processors. However, there are system overheads owing to the overlap and the simultaneous execution of task copies.

Naedele [START_REF] Naedele | Fault-Tolerant Real-Time Scheduling under Execution Time Constraints[END_REF] tried to improve the schedulability using the decision deadline, which authorised a task to be scheduled on probation if there is no free slots to place a new arriving task. In other words, when a new task T i arrives on the system and if it is not possible to immediately upon arrival determine schedules for primary and backup copies, this technique allows the algorithm to wait a certain amount of time, until the decision deadline, before it makes a decision whether the task T i is rejected or not.

Several authors consider that the system can tolerate more than one fault at the same time. Manimaran et al. [START_REF] Manimaran | A Fault-Tolerant Dynamic Scheduling Algorithm for Multiprocessor Real-Time Systems and its Analysis[END_REF] proposed the algorithm, which divides the total number of processors into groups. There are as many groups as the maximum possible number of faults in the system all at once. Regarding that each task, i.e. its primary and backup copies, must be scheduled within one group, the more groups are considered, the less possibilities of available schedules and thus the higher rejection rate. Al-Omari et al. [START_REF] Al-Omari | Efficient Overloading Techniques for Primary-Backup Scheduling in Real-Time Systems[END_REF] improved the method allowing several faults in the system at one time by introducing the dynamic grouping. It means that groups are dynamically formed and dissolved depending on the task arrival and execution. It was shown that this method performs better but with system overheads than the one proposed by [START_REF] Manimaran | A Fault-Tolerant Dynamic Scheduling Algorithm for Multiprocessor Real-Time Systems and its Analysis[END_REF] or the conventional primary/backup approach.

Zheng et al. [START_REF] Zheng | On the Design of Fault-Tolerant Scheduling Strategies Using Primary-Backup Approach for Computational Grids with Low Replication Costs[END_REF] noted that mapping and scheduling of backup copies have significant influence on the system schedulability. Consequently, they presented the algorithm, which computes the replication cost for each possible backup copy schedule and chooses the schedule having the lowest replication cost and, in case of tie, situated as soon as possible. In addition, they considered boundary schedules of backup copies defined as schedules having its start time and/or finish time at the same moment as boundaries of free slot or boundaries of overloadable backup copies. Such schedules enable the algorithm not to evaluate replication cost for all schedules on each processor and therefore lessen the complexity.

This section summarised the related work concerning the primary/backup approach. Most enhancements were proposed to improve the system schedulability or to diminish the rejection rate but without analysing and reducing the algorithm complexity.

III. ASSUMPTIONS AND SCHEDULING STRATEGY

The system consists of P homogeneous processors1 accommodating aperiodic tasks, which are dynamically mapped and scheduled on the system without preemption. The system is equipped with a fault detection mechanism promptly informing about a fault occurrence. We consider that only one permanent or transient fault can occur at any instant of time.

Regarding our task model, each task has three attributes: arrival time a, computation time c and deadline d. The task window tw, defined as da, is a multiple α of c, i.e. tw = α • c.

The studied algorithm is based on the primary/backup (PB) approach [START_REF] Ghosh | Fault-Tolerance Through Scheduling Of Aperiodic Tasks in Hard Real-Time Multiprocessor Systems[END_REF], which is known for its minimal resources utilisation and high reliability. Its principal rule is that, when a task arrives, the system creates two identical copies: the primary copy P C and the backup copy BC. These copies are then scheduled on two different processors. Both copies must be scheduled within the task window and they are not permitted overlapping each other on different processors to prevent system overheads. The primary copy is scheduled as soon as possible and the backup one as late as possible in order to avoid idle processors just after the task arrival time and possible high processor load later.

In order to improve the schedulability and to minimise resources usage, we consider the backup deallocation and the backup overloading [START_REF] Ghosh | Fault-Tolerance Through Scheduling Of Aperiodic Tasks in Hard Real-Time Multiprocessor Systems[END_REF]. The former technique consists in deallocating the backup copy BC i , if the primary copy P C i was correctly executed. The latter one allows overloading of several backup copies, except ones having their primary copies scheduled on the same processor.

When implementing the backup overloading, we employed the method from [START_REF] Zheng | On the Design of Fault-Tolerant Scheduling Strategies Using Primary-Backup Approach for Computational Grids with Low Replication Costs[END_REF], which maximises the backup overloading and makes use of boundary schedules. Thus, the algorithm computes replication cost to evaluate overlap percentage of arriving backup copy. If two possible schedules have the same replication cost, the one having the latter start time is chosen. for all (P ) processors do 3:

Algorithm 1 Studied algorithm

Map and schedule P Ci (as soon as possible) within PC window 4:

end for 5:

for (P -1) processors do 6:

Map and schedule BCi (with minimum replication cost and as late as possible) within BC window 7:

end for 8:

if PC and BC schedules exist then 9:

Commit the task Ti end if 13: end if

The algorithm, presented in Algorithm 1, carries out the exhaustive search on all processors. Therefore, it finds the best solution, i.e. the one having the primary copy scheduled as soon as possible and the backup one maximising the backup overloading and placed as late as possible. On the one hand, this method is known to be the best for the PB approach regarding the rejection rate and processor load. On the other hand, the algorithm needs to test all boundary schedules within the scheduling window, which requires a non negligible number of comparisons. The aim of this paper is to show that the number of comparisons can be reduced.

A. Restricted Scheduling Windows

To lower the complexity when searching for a schedule, we define scheduling windows for primary and backup copies within which respective copy can be scheduled. Restricting the scheduling windows for both copies reduces the mutual scheduling interference and the complexity (measured by means of number of comparisons of schedules delimited by boundaries), and encourages to place the primary copies as soon as possible and the backup ones as late as possible, thus improving the system schedulability.

Figure 1: Primary/backup approach with restricted scheduling windows (p = 1/3)

The size of scheduling window is defined by a parameter p representing the percentage of task window. The primary scheduling window is thereby delimited by a and a + p • tw and the backup one by dp • tw and d. Thus, for our algorithm the percentage is within 0 < p 1, while p = 1.0 for conventional algorithm. Figure 1 depicts an example of restricted scheduling windows.

B. Evaluation of the Complexity

In this section, we theoretically focus on the value of the complexity when placing tasks using the restricted scheduling windows.

Inspired by [START_REF] Zheng | On the Design of Fault-Tolerant Scheduling Strategies Using Primary-Backup Approach for Computational Grids with Low Replication Costs[END_REF], we define N ps the number of all possible schedules where a copy can be placed. This number2 is not easy to estimate in advance and that is why experimental results are essential to observe the trend. Regarding that, when searching for a backup copy schedule, there are more possibilities to examine (due to possible overloading), we denote N ps (P C) and N ps (BC) the number of all possibilities within the scheduling window when placing a primary copy or a backup copy, respectively. We remind the reader that we consider P processors and that a backup copy cannot be scheduled on the same processor as the primary copy. Thereby, the complexity comp is defined as follows:

comp(P C) = P • N ps (P C) • max 1 α ; min 1 - 1 α ; p (1) comp(BC) = (P -1) • N ps (BC) • max 1 α ; min 1 - 1 α ; p (2) comp = comp(P C) + comp(BC) (3)
An example is depicted in Fig. 2. It can be stated that, when the percentage of task window decreases, the complexity is reduced since there are less possible schedules to test. Nonetheless, we cannot theoretically evaluate system performances, such as rejection rate, and experiments are consequently necessary.

A. Experiment Set-Up

Simulations were carried out to study the algorithm behaviour. Their parameters are summed up in Table I. For each simulation scenario, 100 simulations of 10 000 tasks were treated and the obtained values were averaged. We consider that no fault occurs and therefore all backup copies can be deallocated once their respective primary copies finish.

The Targeted Processor Load (TPL) is a parameter necessary to generate task arrival times. If T P L = 1.0, task attributes a and c are generated so that every processor is considered to be working all the time at 100%. The arrival time is defined using the Poisson distribution with parameter λ depending on computation time. In order to compare our refined algorithm to the basic one, the results of different percentage p are analysed, including p = 1.0, i.e. the case without the restricted windows as in the conventional primary/backup approach presented in [START_REF] Ghosh | Fault-Tolerance Through Scheduling Of Aperiodic Tasks in Hard Real-Time Multiprocessor Systems[END_REF]. In this paper, most of the results are described for 14-processor system having the same input task set generated assuming T P L = 0.5 and T P L = 1.0, respectively. Actually, as depicted in Fig. 3c, it can be found that, when the number of processors is greater than 12, observed metrics have already smooth trend and analogous results are obtained. The same conclusions were noted for example in [START_REF] Zheng | On the Design of Fault-Tolerant Scheduling Strategies Using Primary-Backup Approach for Computational Grids with Low Replication Costs[END_REF].

Table I: Simulation parameters

B. Results

Figure 3c, plotting the processor load for p = 0.5 as a function of the number of processors, illustrates that for each task set both methods, i.e. the PB approaches with and without overloading, overlap each other. Similar results are obtained for p = 1.0.

Figures 3a and 3b depict the studied metrics as a function of the percentage p of task window. These figures show that the represented curves remain constant from p = 0.1 to p = 0.2 and from p = 0.8 to p = 1.0, which is due to minimal considered ratio of computation times to task window in our experiment set-up (2c d 5c and thus c/d max = 1/5).

The rejection rate, represented in Fig. 3a, is the ratio of rejected tasks to all arriving tasks. First of all, it can be noticed that, if the percentage p drops below 0.5, the rejection rate climbs for all methods because of the rejection of tasks with hard time constraints. Then, we focus on p between 0.5 and 1.0 and it can be seen that, when T P L = 0.5, the rejection rate is the same for both methods, almost constant and close to 0%. For T P L = 1.0, the PB approach with deallocation slightly achieves better result than the PB approach with deallocation and overloading. When p decreases, the rejection rate increases because there are less possible schedules. The higher the T P L, the more noticeable this phenomenon. Nonetheless, in the worst case (T P L = 1.0), the difference between p = 0.5 and p = 1.0 does not exceed 2%.

The algorithm complexity is evaluated by number of comparisons accounting for the number of tested schedules delimited by boundaries. All tasks are taken into account, no matter whether they are finally accepted or rejected. This metric is essential since it is related to energy consumption and rapidity of mapping and scheduling. The evolution of the maximum value of this number is depicted in Fig. 3b. The figure points out that the PB approach with deallocation has lower number of comparisons, which is due to less possible schedules to test (no backup overloading). Moreover, if we focus on p between 0.5 and 1.0, the higher value of p, the higher number of comparisons.

To sum up performances of our proposed refinement of the PB algorithm, it can be stated that the number of comparisons arranging for the algorithm complexity presents the most notable change. In order to accept tasks having deadline between 2c and 5c and arriving on the system, we consider the restricted scheduling window fixed at p = 0.5. This value is thus a reasonable trade-off between the complexity and the rejection rate. In this case, the number of comparisons is reduced by 10% and 15% for PB approach with deallocation and PB approach with deallocation and overloading, respectively.

V. CONCLUSION

This paper is concerned with the dynamic mapping and scheduling of tasks on multi-core systems which are currently challenging problems of fault-tolerant design. We refined the algorithm based on the PB approach using restricted scheduling windows for primary and backup copies and aiming at reducing its complexity. It was shown that our algorithm has the complexity reduced by up to 15%, the processor utilisation stays the same and the rejection rate rises by at most 2% (compared to the conventional PB approach).

Taking into account the trade-off between the complexity and the rejection rate, the best results are obtained for the percentage of task window close to 0.5, i.e. when the first half of the task window is reserved for primary copies and the second half for backup ones.

Input:

  Task Ti, Mapping and scheduling M S of already scheduled tasks Output: Updated M S 1: if new task Ti arrives then 2:

Figure 2 :

 2 Figure 2: Example of theoretical complexity

  2c) -(a + 5c)

Figure 3 :

 3 Figure 3: Experimental results comparing primary/backup approaches based on the exhaustive search and on the boundary schedule technique with different percentage of task window and T P L

This work was funded by the Reliasic project granted by the CominLabs Excellence Laboratory (ANR-10-LABX-07-01), French "Investissements d'Avenir" program.

In this paper for the sake of simplicity, a system with only homogeneous processors is considered. Nevertheless, the studied model can be easily extended to a system with heterogeneous processors, such as in[START_REF] Zheng | On the Design of Fault-Tolerant Scheduling Strategies Using Primary-Backup Approach for Computational Grids with Low Replication Costs[END_REF].

To simplify, we consider that Nps is uniformly distributed within the task window and has the same value on all processors.


Petr Dobiáš
email: petr.dobias@irisa.fr

Emmanuel Casseau

Oliver Sinnen

Comparison of Different Methods Making Use of Backup Copies for Fault-Tolerant Scheduling on Embedded Multiprocessor Systems

Keywords: Embedded real-time systems, Fault tolerance, Multiprocessor systems, On-line scheduling, Primary/backup approach

As transistors scale down, systems are more vulnerable to faults. Their reliability consequently becomes the main concern, especially in safety-critical applications such as automotive sector, aeronautics or nuclear plants. Many methods have already been introduced to conceive fault-tolerant systems and therefore improve the reliability. Nevertheless, several of them are not suitable for real-time embedded systems since they incur significant overheads, other methods may be less intrusive but at the cost of being too specific to a dedicated system. The aim of this paper is to analyse a method making use of two task copies when on-line scheduling tasks on multiprocessor systems. This method can guarantee the system reliability without causing too much overhead and requiring any special hardware components. In addition, it remains general and thus applicable to large amount of systems. Last but not least, this paper studies two techniques of processor allocation policies: the exhaustive search and the first found solution search. It is shown that the exhaustive search is not necessary for efficient fault-tolerant scheduling and that the latter search significantly reduces the computation complexity, which is interesting for embedded systems.

I. INTRODUCTION

Each system component is liable to fail and it will cease running correctly sooner or later. As a result, the system can exhibit a malfunction. There are applications where a system failure can have catastrophic consequences like advanced driver-assistance systems, air traffic control or medical equipment. In order to solve this problem, systems should be fault-tolerant, i.e. more robust to be able to properly work even if faults occur. The fault-tolerant design is thus a topical issue, especially at the present time when transistors scale down to fulfil stern constraints on energy-consumption and area, which goes hand in hand with higher vulnerability to fault occurrence. It is therefore necessary to ensure the system reliability, in particular for safety-critical applications.

The main aim of this paper is to compare different techniques intended for the primary/backup approach, which is an approach to fault-tolerant scheduling on embedded multiprocessor systems making use of two tasks copies: primary and backup ones [START_REF] Ghosh | Fault-Tolerance Through Scheduling Of Aperiodic Tasks in Hard Real-Time Multiprocessor Systems[END_REF]. The primary/backup approach is a commonly used technique when conceiving fault-tolerant systems owing to its easy application and minimal system overheads. Several additional enhancements [START_REF] Ghosh | Fault-Tolerance Through Scheduling Of Aperiodic Tasks in Hard Real-Time Multiprocessor Systems[END_REF]- [START_REF] Naedele | Fault-Tolerant Real-Time Scheduling under Execution Time Constraints[END_REF] to this approach have been already presented but few studies dealing with overall comparisons have been carried out.

The remainder of this paper is organised as follows. The related work is summarised in Section II. Section III presents our assumptions and it describes the scheduling model and tested techniques. The experiment framework is introduced in Section IV and results are then analysed in Section V.

II. RELATED WORK

Since systems are more vulnerable to faults and the reliability becomes the main concern, Naithani et al. [START_REF] Naithani | Reliability-Aware Scheduling on Heterogeneous Multicore Processors[END_REF] emphasized the necessity to consider the reliability during scheduling. They showed that it is better to make use of reliability-aware scheduling rather than performance-optimised scheduling. On average, although the reliability-aware scheduling degrades performance by 6%, it improves the system reliability by 25.4% compared to the performance-optimised scheduling.

The next question is which possibilities are available to conceive fault-tolerant systems. The first solution is to make the system more robust by adding additional components but, regarding that extra logic requires changes in hardware, this possibility is not applicable to already existing systems.

The second solution is to modify the code in order to verify its correct execution. Some error detection code can be inserted in high level code [START_REF] Tang | A Fault Detection Approach for MPSoC[END_REF] or in instruction-level one [START_REF] Reis | Automatic Instruction-Level Software-Only Recovery[END_REF]. Goloubeva et al. [START_REF] Goloubeva | Soft-error Detection Using Control Flow Assertions[END_REF] introduced additional executable assertions to check the correct execution of the program control flow. Their technique mainly aims at safety-critical applications. However, depending on the application and program, the obtained overheads are considerable: memory ones (minimum: 124%, average: 283%, maximum: 630%) and performance ones (minimum: 107%, average: 185%, maximum: 426%).

A hybrid solution putting the first and second solutions together was proposed by Bernardi et al. [START_REF] Bernardi | A New Hybrid Fault Detection Technique for Systemson-a-Chip[END_REF]. Their approach combined software-based techniques with an Infrastructure IP to detect transient faults in processor-based SoCs and it was tested using several benchmarks. The results showed significant overheads in execution time (minimum: 78%, average: 126%, maximum: 209%), in code size (minimum: 68%, average: 162%, maximum: 270%) and in data size (minimum: 102%, average: 107%, maximum: 113%).

The third possibility is to take advantage of the redundancy on multiprocessor systems, which can be effectively used to design fault-tolerant systems. While the redundancy in space, such as N -modular redundancy [START_REF] Dubrova | Fault-Tolerant Design[END_REF] incurs system overheads even during normal operation, the redundancy in time does not cause any of them in fault-free environment. An example of such redundancy is a primary/backup approach [START_REF] Ghosh | Fault-Tolerance Through Scheduling Of Aperiodic Tasks in Hard Real-Time Multiprocessor Systems[END_REF], which makes use of two task copies: primary and backup ones and which is commonly used for its high-reliability. It can be found in satellites [START_REF] Zhu | Analysis and Design of Fault-Tolerant Scheduling for Real-Time Tasks on Earth-Observation Satellites[END_REF] or employed in the virtualised cloud [START_REF] Zhu | Fault-Tolerant Scheduling for Real-Time Scientific Workflows with Elastic Resource Provisioning in Virtualized Clouds[END_REF]. Furthermore, its uncomplicated principle makes it compatible with different implementations, such as the genetic algorithm [START_REF] Kumar | Aco and Ga Based Fault-Tolerant Scheduling of Real-Time Tasks on Multiprocessor Systems -A Comparative Study[END_REF], [START_REF] Samal | Bio-Inspired Approach to Fault-Tolerant Scheduling of Real-Time Tasks on Multiprocessor -A Study[END_REF].

III. ASSUMPTIONS AND SCHEDULING MODEL

The studied system consists of P identical1 processors sharing the same memory2 . All tasks, characterised by their arrival time a, computation time c and deadline d, are considered to be aperiodic and independent. They are on-line and immediately mapped and scheduled on the system without preemption. We assume that only one processor failure (transient or permanent) can occur at any instant of time and that a fault detection mechanism3 exists to promptly inform about a fault occurrence. The principle of the primary/backup (PB) approach [START_REF] Ghosh | Fault-Tolerance Through Scheduling Of Aperiodic Tasks in Hard Real-Time Multiprocessor Systems[END_REF] is that, when a task arrives, the system creates two identical task copies. A time interval on which a task copy is placed is called a slot. The first task copy is called a primary copy and denoted P C, the second one is a backup copy labelled BC. These copies are then scheduled on two different processors within the task window 4 without overlapping each other, as depicted in Fig. 1. The aim is to keep system overheads down when no fault occurs. Thus, if the primary copy of a task cannot provide correct results due to a fault, the scheduled backup copy is run and its results are taken into account. In general, when scheduling a task, the algorithm first places its primary copy as soon as possible and then its backup copy as late as possible, as summarised in Algorithm 1. This strategy avoids idle processors just after the task arrival and possible high processor load later.

In order to increase the system schedulability and minimise resources utilisation, i.e. system overheads, two enhancing techniques exist: the backup deallocation and the backup overloading [START_REF] Ghosh | Fault-Tolerance Through Scheduling Of Aperiodic Tasks in Hard Real-Time Multiprocessor Systems[END_REF]. The former technique deallocates the backup copy BC i , if the primary copy P C i was correctly executed. The advantage of this technique is not to unnecessarily occupy resources and thus free the occupied slot for new arriving tasks. The latter technique overloads several backup copies, except ones having their primary copies placed on the same processor, and therefore saves up processor resources.

A. Processor Allocation Policies

The subsequent question is how a search for PC and BC slots are carried out. In this paper, we analyse two possibilities for processor allocation policies: the exhaustive search (ES) and the first found solution search (FFSS). The exhaustive search checks all processors for available slots and chooses the best solution, i.e. the one having its primary copy scheduled as soon as possible and the backup copy as late as possible. In contrast, the first found solution search assesses processors in turn and Algorithm 1 Primary/backup approach Input: Task T i Mapping and scheduling of already scheduled tasks Output: Updated mapping and scheduling 1: if new task T i arrives then 2:

Map and schedule P C i (ES or FFSS policy)

3:

Map and schedule BC i (ES or FFSS policy) 4:

if P C i and BC i slots exist then 5:

Commit the task T i 6:

else 7:
Reject the task T i 8:

end if 9: end if 10: if fault detection mechanism reports a fault then 11:

Do not take into consideration the result from P C i 12:

Wait for the result from BC i 13: else 14:

Consider the result from P C i 15: end if it looks only for the first suitable slot for PC and then for the first BC one no matter their positions within the task window. In this case, the best solution can be consequently missed.

Moreover, in order to avoid that the first processor has higher charge than other system processors, the FFSS starts its search for a PC slot on the processor following the processor on which the primary copy of previous task was successfully scheduled. The search then continues in ascending order until a free slot is found or no more processor is available [START_REF] Naedele | Fault-Tolerant Real-Time Scheduling under Execution Time Constraints[END_REF]. Then, if the primary copy of the new task is found on processor P i , a search for a BC slot can be conducted. It starts on processor P i-1 and it continues in decreasing order.

B. Active Primary/Backup Approach

Until now, the passive primary/backup approach was considered, i.e. that primary and backup copies of the same task cannot overlap each other on two different processors. Nevertheless, this approach may be too restrictive for some real-time systems since the deadline may be earlier than two times the computation time and, therefore, the active primary/backup approach should be considered [START_REF] Tsuchiya | A New Fault-Tolerant Scheduling Technique for Real-Time Multiprocessor Systems[END_REF]. On the one hand, it allows the primary and backup copies to overlap each other in space and thus helps to schedule tasks with tight deadlines. On the other hand, this approach gives rise to system overheads because the system entirely or partially executes the backup copy (during the execution of the corresponding primary copy). Besides, the active approach adds more schedulability constraints: backup copies scheduled by means of this method cannot overload other backup copies and cannot be overloaded as they always need to be executed (in total or in part).

IV. EXPERIMENTS FRAMEWORK

To evaluate the system performance using aforementioned techniques, simulations were carried out based on the parameters summarised in Table I. Each simulation contains 10 000 tasks and the results present the average of 100 simulations. We assume that there is no fault occurrence and all backup copies can be consequently deallocated once their respective primary copies finish.

In order to generate task arrival times, we define the Targeted Processor Load (TPL). If this parameter is equal to 1.0, task attribute a is generated so that every processor is theoretically charged all the time at 100%. In practice, the processor load can be less than 100% when tasks are rejected due to missing free slots. The performance of realised simulations were then assessed by means of the following criteria. The rejection rate represents the ratio of rejected tasks to all arriving tasks. The processor load is the effective system load taking into account the backup deallocation and rejected tasks. This metric allows us to evaluate the resources utilisation: when it equals 100%, the scheduling efficiency is maximal. The number of comparisons, which is necessary to carry out before finally accepting or rejecting a task, accounts for the algorithm complexity. One comparison corresponds to evaluation whether a free slot is large enough to accommodate a task copy on a given processor.

V. RESULTS

In this section, we first present the baseline results in order to observe the improvement of the backup deallocation technique presented in the subsequent section. Afterwards, we compare two processor allocation policies and evaluate the use of the active PB approach.

A. Baseline Results

Fig. 2 represent the rejection rate and the processor load as a function of the number of processors for the passive PB approach without any additional technique and the passive PB approach with BC overloading. The targeted processor load equals 0.5 and 1.0, respectively, and the chosen processor allocation policy is the first found solution search.

First of all, we note that the results of the rejection rate for the PB approach with BC overloading are better than the ones for the PB approach alone (up to 13%) since the implemented technique allows backup copies to overload each other, unless their primary copies are on the same processor, which saves up free slots that can be used for new arriving tasks. Regarding the processor load, both approaches reach similar values.

Then, it can be seen in Fig. 2a that the rejection rate decreases with the increasing number of processors. We remind the reader that the targeted processor load is set as a constant. Thus, according to the definition of the Poisson distribution parameter λ, when the number of processors increases, the parameter λ decreases, which implies that tasks have shorter interarrival time and arrive more often. Actually, the addition of processors brings more possibilities to find a suitable slot so that the processor load eventually decreases. Besides, the higher the targeted processor load, the higher the rejection rate because the system becomes more charged and consequently rejects more tasks.

Finally, the processor load is shown in Fig. 2b. The continuous lines account for the workload of the whole system, i.e. when the primary and backup copies are taken into account. The processor load increases when the number of processors rises. It can be noticed that, since the BC deallocation is not used, the system performs the same computation twice even though only one execution is necessary in the fault-free environment. The dashed lines in Fig. 2b stand for the processor load when backup copies are not considered. Actually, it is the effective processor load from the user's point of view. Even when the BC overloading is put into practice, the effective processor load is about 50%. Consequently, to improve the system performances, the technique of the BC deallocation should be introduced and analysed, which is the aim of the next section.

B. Merit of Backup Deallocation

To obtain comparable results to the previous ones, the algorithm makes use of the first found solution search and the T P L is fixed at 0.5 and 1.0, respectively. Fig. 3 represent the studied metrics as a function of the number of processors for the passive PB approach with BC deallocation only and for the passive PB approach with BC deallocation and BC overloading.

Foremost, it can be noticed that the PB approach with BC deallocation and BC overloading achieves better results than the PB approach with BC deallocation only, which means that the BC deallocation and the BC overloading can be used fruitfully together. Fig. 3a depicts that the rejection rate is significantly improved compared to Fig. 2 thanks to the BC deallocation. For example for 20-processor system and T P L = 1.0, the gain is about 75% no matter whether the technique of BC overloading is implemented or not.

The curves of processor load depicted in Fig. 3b account for the workload of the whole system and the workload of the primary copies only as well because all backup copies are deallocated and no fault occurs. It means that when the BC deallocation is put into practice, the system can accept twice as more tasks compared to the system without this technique. Moreover, when a system has higher number of processors, curves tend to the value of the targeted processor load (0.5 or 1.0) showing the effectiveness of the BC deallocation technique. In this section, we compare two processor allocation policies described in Section III-A: the exhaustive search (ES) and the first found solution search (FFSS). The T P L is still respectively set at 0.5 and 1.0 and the passive PB approach with BC deallocation and BC overloading is considered. Fig. 4 respectively depict the rejection rate, the processor load and the mean and maximum number of comparisons as a function of the number of processors.

C. Comparison of Two Processor Allocation Policies

Fig. 4a illustrates that the results of the rejection rate for two processor allocation policies are almost the same when T P L = 0.5 and that the ES achieves slightly better results than the FFSS when T P L = 1.0. Nevertheless, the difference between two techniques diminishes with the increasing number of processors and becomes negligible for 20-processor system.

The processor load represented in Fig. 4b shows that, when T P L = 1.0, the FFSS reaches lower values than the ES (approximatively about 10%). Taking into account that, the FFSS and the ES have similar rejection rate, the FFSS manages better the system workload, which can put off the ageing of processors.

The results of the mean number of comparisons are significantly improved and the ones of the maximum number of comparisons are also ameliorated. For instance for a 20-processor system, the mean number of comparisons drops by 94% and 74% when T P L = 0.5 and T P L = 1.0, respectively, and the maximum number of comparisons decreases by 58% and 14% when T P L = 0.5 and T P L = 1.0, respectively. We notice that the reduction in number of comparisons is more notable for the mean value than for the maximum one. Actually, when a solution using the FFSS is found, a search terminates. Nonetheless, in the worst case, i.e. when a solution is not found, all processors are tested.

Fig. 4c also depicts standard deviations for each value of the number of comparisons. It can be seen that the more processors in the system and thus the higher value of the mean number of comparisons, the larger the standard deviation. Furthermore, when the value of TPL increases, the standard deviation becomes larger because the processor load is higher and there are more comparisons to carry out and therefore higher chance to have larger standard deviation.

Similar results were obtained for the PB approach with BC deallocation only (and without BC overloading).

To sum up, we can state that, even though the FFSS can miss the best solution, i.e. the one where the PC is scheduled as soon as possible and the BC as late as possible, the FFSS achieves similar performances as the ES, especially when the number of processors is high. Moreover, the FFSS has lower number of comparisons, which makes this technique more suitable for real-time embedded systems and can reduce the energy consumption.

D. Evaluation of the Active Primary/Backup Approach

To evaluate the merit of the active PB approach, we make use of the standard simulation parameters as summarised in Table I but instead of the task window between 2c and 5c we consider its size between c and 5c. Regarding that the passive PB approach requires the size of at least 2c, this scenario allows us to assess the active approach.

The algorithm is based on the first found solution search and the T P L is fixed at 1.0. Fig. 5 depicts the rejection rate as a function of the number of processors for the PB approach with BC deallocation only and for the PB approach with BC deallocation and BC overloading and for their respective versions using the active approach.

As it was mentioned in Section III-B, the active PB approach induces system overheads. Consequently, when we employ this approach, we limit its application for tasks with tight deadline. We therefore introduce a threshold A, which is defined as tw = da < A • c and which decides whether the active approach is used or not. The value of A in Fig. 5 respectively equals to 1.5, 2.0 and 2.5.

Since the simulation scenario remains the same and only the value of A changes, the results of passive PB approaches are identical. In general, it can be observed that the active PB approach helps to reduce the rejection rate regardless the value of A. The lowest rejection rate is obtained when A = 2.0 because this value is located at the transition from the passive PB approach to the active one. For example, the active approach for a 20-processor system reduces the rejection rate by 21% for the PB approach with BC deallocation and by 22% for the PB approach with BC deallocation and BC overloading.

In addition, if A is less than 2.0, some tasks are automatically rejected due to the tight deadline, which is the reason why the rejection rate for A = 1.5 (Fig. 5a) is higher than the one for A = 2.0 (Fig. 5b). If A is larger than 2.0, no task is automatically rejected but the merit of active approach diminishes. The results (not presented in this paper) show that the higher the value of A, the higher the rejection rate of active approach and therefore the smaller the difference between the passive and active approaches.

VI. CONCLUSION

This paper analyses a method using two task copies for fault-tolerant on-line scheduling on multiprocessor embedded systems. It first presents the results of the primary/backup approach alone and the ones with the backup overloading. It can be seen that the technique of backup overloading helps to reduce the rejection rate by up to 13%. When the technique of the backup deallocation is then added, the improvement is even more noteworthy. For instance for 20-processor system and T P L = 1.0, the gain is about 75% compared to the baseline PB approach and regardless whether the technique of backup overloading is implemented or not.

Afterwards, two processor allocation policies are studied: the exhaustive search and the first found solution search. It was observed that for a multiprocessor system the both techniques achieve similar rejection rate. Even though the exhaustive search has slightly lower rejection rate, the first found solution search surpasses it in the processor load (about 10%) and especially in the number of comparisons (up to 74% for the mean value of this number and 14% for the maximum one). Such a significant gain can be very useful when conceiving fault-tolerant embedded systems subject to energy constraints.

Finally, the active approach is evaluated. It was demonstrated that it helps systems dealing with tasks with tight deadline and it therefore reduces the rejection rate. For 20-processor system, there is a drop of 21% for the PB approach with BC deallocation and of 22% for the PB approach with BC deallocation and BC overloading.

There exists also another interesting metric to evaluate system performances. It is the Time To Next Fault (TTNF) [START_REF] Ghosh | Fault-Tolerance Through Scheduling Of Aperiodic Tasks in Hard Real-Time Multiprocessor Systems[END_REF] defined as the time elapsed between a chosen time instant and the time when a new fault may occur without violating the assumption of only one fault in the system all at once (Section III). The lower the TTNF value, the better. Our results were not presented in this paper for lack of space but it was found that all obtained values are of the same magnitude and that the exhaustive search achieves slightly better results than the first found solution search.

Our future work deals with an enhancement for the primary/backup approach, which further decreases the number of comparisons compared to the first found solution search but without worsening system performances, especially the rejection rate. The aim is to improve the approach so that it can be efficiently implemented onto embedded systems with restricted power consumption budget. Moreover, we are about to carry out simulations with fault injection to asses performances of different approaches in a faulty environment not satisfying the TTNF.

Figure 1 :

 1 Figure 1: Principle of the primary/backup approach

Figure 2 :

 2 Figure 2: Rejection rate and processor load for the passive PB approach alone and for the one with BC overloading as a function of the number of processors and T P L (first found solution search)

Figure 3 :

 3 Figure 3: Rejection rate and processor load for the passive PB approach with BC deallocation and for the one with BC deallocation and BC overloading as a function of the number of processors and T P L (first found solution search)

Figure 4 :

 4 Figure 4: Experimental results for the passive PB approach with BC deallocation and BC overloading comparing two processor allocation policies as a function of the number of processors and T P L

Figure 5 :

 5 Figure 5: Rejection rate of the active PB approach as a function of the number of processors for different values of A (T P L = 1.0)

Table I :

 I Standard simulation parameters

	Parameter	Symbol	Distribution	Value(s)
	Number of processors	P	-	2 -25
	Targeted processor load	T P L	-	0.5 -1.0
	Computation time	c	Uniform	1 -20
	Arrival time Deadline	a d	Poisson Uniform	λ = average execution time T P L•P (a + 2c) -(a + 5c)

This work was funded by the Reliasic project granted by the CominLabs Excellence Laboratory (ANR-10-LABX-07-01), French "Investissements d'Avenir" program.

To simplify, a system with homogeneous processors is considered in this paper. Nonetheless, the studied model can be easily extended to a system consisting of heterogeneous processors, such as in[START_REF] Zheng | On the Design of Fault-Tolerant Scheduling Strategies Using Primary-Backup Approach for Computational Grids with Low Replication Costs[END_REF].

In the case of distributed memory, it is necessary to take delays of data transfers into account but the principle of the method remains the same.

A fault can be detected by acceptance tests, such as timing, coding, reasonableness or structural checks[START_REF] Dubrova | Fault-Tolerant Design[END_REF].

A task window tw is defined as the difference between the deadline d and the arrival time a.


Emmanuel Guilmineau
email: emmanuel.guilmineau@ec-nantes.fr

& F Chometon

Effect of Side Wind on a Simplified Car Model: Experimental and Numerical Analysis

Keywords: Generic Car Body, Numerical Simulation, Experimental Data, Crosswind Effect, Vehicle Aerodynamics. Guilmineau

A prior analysis of the effect of steady cross wind on full size cars or models must be conducted when dealing with transient cross wind gusts effects on automobiles. The experimental and numerical tests presented in this paper are performed on the Willy square-back test model. This model is realistic compared to a van-type vehicle; its plane under-body surface is parallel to the ground and separations are limited to the base for moderated yaw angles. Experiments were carried out in the semi-open test section at the Conservatoire National des Arts et Métiers (CNAM) and computations were performed at the Ecole Centrale de Nantes (ECN). The ISIS-CFD flow solver, developed by the CFD Department of the Fluid Mechanics Laboratory of ECN, used the incompressible unsteady Reynolds-Averaged Navier-Stokes equations. In this paper, the results of experiments obtained at a Reynolds number of 0.9 10 6 are compared with numerical data at the same Reynolds number for steady flows. In both the experiments and numerical results, the yaw angle varies from 0 • to 30 • . The comparison between experimental and numerical results obtained for aerodynamic forces, wall pressures and total pressure maps shows that the unsteady ISIS-CFD solver correctly reflects the physics of steady three-dimensional separated flows around bluff bodies. This encouraging result allows us to move to a second step dealing with the analysis of unsteady separated flows around the Willy model.

Introduction

As passenger vehicles become lighter to reduce fuel consumption, the stability and handling characteristics become more sensitive to aerodynamic forces. Moreover, a vehicle on the road is submitted to a non uniform turbulent atmospheric flow and transient cross winds. These phenomena, such as passing or crossing maneuvers, are factors in the vehicle's dynamic stability [START_REF] Hucho | Aerodynamics of Road Vehicles[END_REF].

From an experimental point of view, the cross wind effect has already been studied and several techniques allow the reproduction of a side gust of wind in a wind tunnel. For Baker and Humphreys [START_REF] Baker | Assessment of the Adequacy of Various Wind Tunnel Techniques to Obtain Aerodynamic Data for Ground Vehicles in Cross Winds[END_REF],

Macklin et al [START_REF] Macklin | Comparing Static and Dynamic Testing Techniques for the Crosswind Sensitivity of Road Vehicles[END_REF], this reproduction was obtained by propelling the model on a rail crossing the test section of the wind tunnel. Ryan and Dominy [START_REF] Ryan | The Aerodynamic Forces Induced on a Passenger Vehicle in Response to a Transient Cross-Wind Gust at a Relative Incidence of 3 • 0[END_REF] experimented on a generic hatchback model and used a technique where the side wind was produced by a cross jet. Another technique consists of using oscillating airfoil sections positioned upstream of a stationary model [START_REF] Passmore | An Experimental Study of Unsteady Vehicle Aerodynamics[END_REF]. Many experimental studies of cross wind were carried out on car models [START_REF] Ryan | The Aerodynamic Forces Induced on a Passenger Vehicle in Response to a Transient Cross-Wind Gust at a Relative Incidence of 3 • 0[END_REF][START_REF] Passmore | An Experimental Study of Unsteady Vehicle Aerodynamics[END_REF][START_REF] Baker | Ground Vehicles in High Cross Winds, Part 1: Steady Aerodynamic Forces[END_REF][START_REF] Ryan | Wake Surveys Behind a Passager Car Subjected to a Transient Cross-Wind Gust[END_REF], and lorry models [START_REF] Baker | Ground Vehicles in High Cross Winds, Part 1: Steady Aerodynamic Forces[END_REF][START_REF] Garry | Some Effects of Ground Clearance and Ground Plane Boundary Layer Thickness on the Mean Base Pressure of a Bluff Vehicle Type Body[END_REF], and real cars were also used in several studies [START_REF] Howell | The Side Load Distribution on a Rover 800 Saloon Car under Crosswind Conditions[END_REF][START_REF] Özdemir | Turbulent Structure of Three-Dimensional Flow Behind a Model Car: 2. Exposed to Crosswind[END_REF]. The approach retained here consists in submitting the model to a periodic movement in a steady wind [START_REF] Garry | Comparison of Quasi-Static and Dynamic Wind Tunnel Measurements on Simplified Tractor-Trailer Models[END_REF]. This approach does not directly simulate the side gust of wind but it permits the analysis of the phenomena of phase shifting and hysteresis associated with an unsteady wind [START_REF] Chometon | Experimental Study of Unsteady Wakes Behind an Oscillating Car Model[END_REF].

Numerical simulation is well integrated in the automotive industry, and is now an engineering tool used in parallel with experiments performed during the design process of road vehicles. Much fundamental research is performed on the Ahmed body which includes most of the aerodynamic features found on a real car. This simplified vehicle geometry has been widely used in a large number of experiments [START_REF] Ahmed | Some Salient Features of the Time-Averaged Ground Vehicle Wake[END_REF][START_REF] Lienhart | Flow and Turbulence in the Wake of a Simplified Car Model[END_REF] and many numerical simulations have been performed at zero yaw angle to evaluate CFD tools and their ability to predict drag and lift for several rear slant angles [START_REF] Han | Computational Analysis of Three-Dimensional Turbulent Flow around a Bluff Body in Ground Proximity[END_REF][START_REF] Bayraktar | Experimental and Computation Investigation of Ahmed Body for Ground Vehicle Aerodynamics[END_REF][START_REF] Sims-Williams | The Ahmed Model Unsteady Wake: Experimental and Computational Analyses[END_REF][START_REF] Kapadia | Detached Eddy Simulation Over a Reference Ahmed Car Model[END_REF][START_REF] Krajnovic | Flow Around a Simplified Car -part 1: Large Eddy Simulation[END_REF][START_REF] Fares | Unsteady Flow Simulation of The Ahmed Reference Body using a Lattice Boltzmann Approach[END_REF]. To date, most of the numerical simulation of the side wind effects on road vehicles are performed on trucks [START_REF] Maddox | Detached-eddy simulation of the flow around the ground transportation system[END_REF][START_REF] Mccallen | Aerodynamic Drag of Heavy Vehicles (Calls 7-8): Simulation and Benchmarking[END_REF] and on trains [START_REF] Khier | Flow Structure around Trains under Side Wind Conditions: a Numerical Study[END_REF][START_REF] Hemida | Large-Eddy Simulation of the Flow around Simplified High-Speed Trains under Side Wind Conditions[END_REF]. In all these numerical studies, several methods were used to model turbulence: classic turbulence models such as k-ε, the Detached Eddy Simulation (DES) or Large Eddy Simulation (LES) and even the Lattice Boltzmann approach. In this study, we will attempt to show that CFD can be used for industrial applications, such as complex geometries, while not being too high in term of CPU time. We use Reynolds-Averaged Navier-Stokes (RANS) simulations instead of LES, since the CPU time is between four and five time higher for Guilmineau 3 FE-07-1184 used to obtain design quantities, such as drag, in the external aerodynamics of ground vehicles [START_REF] Krajnovic | Flow Around a Simplified Car -part 1: Large Eddy Simulation[END_REF] since such techniques require a fine resolution in the near-wall regions that leads to grids with a large number of points. Several approaches that reduce resolution requirements by modeling the near-wall region, such as the hybrid methods, so-called hybrid LES-RANS or DES, are still not fully developed and several problems remain to be solved, such as the choice of a turbulence model, and the question of where to put the matching line between the RANS and LES regions. Once these problems have been solved, these techniques will be available for use in vehicle aerodynamics.

The research presented in this paper is the first step toward a more general goal which is the analysis of numerical simulations, based on RANS equations approaches, to calculate the evolution of aerodynamic characteristics of passenger vehicles as a function of unsteady situations such as passing, crossing, or the presence of an unsteady gust of wind. The approach is based on the comparison of experimental data with numerical data obtained on a new car model (Willy car model) which was tested for unsteady conditions [START_REF] Chometon | Experimental Study of Unsteady Wakes Behind an Oscillating Car Model[END_REF].

The goal of the current project is to understand the physics of separated flow around bluff bodies with application to side gust wind effects on automobiles. We found that a generic model with no sharp corners on the fore body and a square base was more convenient for analysing unsteady separations limited to its leeward side and base. An alternative would be to use the generic Ahmed body which is built around a rectangular box with a vertical front end, parallel sides, roof and underbody, and a rear slant. This model is very useful for analysing the total drag by its segmentation into the contribution of each part of the body [START_REF] Hucho | Aerodynamics of Road Vehicles[END_REF] and is commonly used as a basic body for the comparison of CFD approaches. Many results show that there are large separations at the level of the parallel sides and roof [START_REF] Ahmed | Some Salient Features of the Time-Averaged Ground Vehicle Wake[END_REF]. These separations interact with the base flow and make the analysis of effects of the base slant angle variations more difficult. For these reasons, we did not choose the Ahmed body, preferring a model with no sharp corners.

In this paper, we study the aerodynamic characteristics of the Willy model in a steady cross wind. The

Test Model

Experimental and numerical tests are performed on the square-back Willy test model which was designed to satisfy the following criteria:

• The geometry is realistic, compared to a real vehicle,

• The model's plane underbody surface is parallel to the ground,

• The separations are limited to the region of the base for a moderated yaw angle, i.e. β = 10 • .

The digital definition of the model for the axis (X, Y, Z) is analytical and obtained by combining the following elements, see Figure 1: 1. A NACA mean line (A) of chord Ch and geometric factor a = 1. The maximum camber of this mean line is located at mid chord, X/Ch = 0.5. Its generic equation is, see von Doenhoff [START_REF] Abbott | Theory of Wing Sections[END_REF]:

Y = 0 Z = - 1 4π 1 - X Ch ln 1 - X Ch + X Ch ln X Ch (1)
In Figure 1(a), which describes the side view of the model, the origin of the mean line (A) is Om, and point M is the location of its maximum camber, at a distance of Lm = Ch/2 from Om. The value of the camber is Ca. The digital definition of the model allowed the modification of the shape through 4 parameters. These parameters are p 1 = L/φ, where L is the length of the model, p 2 = Lm/L with the condition 0.5 < p 2 < 1, p 3 = Ca/φ with the condition 0 < p 3 < 0.5, and p 4 which is the ratio between the axes of the ellipses of the upper surface. The value of the radius R is defined by:

R L = 1 -2p 3 2p 1 ( 2
)
For the model analysed in this work, p 1 = 3.3750, p 2 = 0.7037, p 3 = 0.3050, and p 4 = 1.2.

The overall length of the model is L = 675 mm, the width is W = 240 mm, the maximum height is H = 192 mm, and its surface reference is the maximum cross section S re f = 41791 mm 2 . The ground clearance is G = 29 mm and the diameter of the four feet (f) which are used to secure the model to the floor of the wind tunnel is φ1 = 20 mm. A cylinder (c) with a diameter φ2 = 40 mm is used to protect the pressure tubes passing from the pressure taps to the multi-manometer. Other dimensions are defined in Table 1.

Experimental Set-Up

The description of the model mounted in the Prandtl-type wind tunnel of the CNAM is given in Figures 2.

The The value of the yaw angle β is positive when the right side of the car model is windward. The turbulence level at the center of the test section is 2%. The results of the experiments given in this paper are obtained at a Reynolds number of 0.9 10 6 based on the velocity V 0 and the length L of the model. Force and moment coefficients are non-dimensionalized by the maximum cross section S re f of the model, its length L and the free stream dynamic pressure q 0 = 1 2 ρV 2 0 . The drag coefficient Cx is given by:

Cx = Fx q 0 S re f (3) Guilmineau 6 FE-07-1184
where Fx is the drag force. The side force coefficient is given by:

Cy = Fy q 0 S re f ( 4
)
where Fy is the side force. The moment coefficient is defined by:

CN = N q 0 S re f L ( 5
)
where N is the moment along the Z axis. Uncertainty in the drag Cx is ± 1.7% and uncertainty in the side force Cy and in the yawing moment CN is ± 1.9%. Tomographies of total pressure are measured using a Kiel probe with an external diameter of 6 mm and yaw and pitch ranges of about 40 • . The non-dimensional total pressure coefficient, Cp i , is given by:

Cp i = p i0 -p i q 0 ( 6
)
where p i0 and p i are respectively the upstream and the local total pressure. For technical reasons, experimental total pressure tomographies are measured in (Yo, Zo) Eiffel planes. Non-dimensional locations of the tomographies are given in the reference axis Xo and the reference length is the length L of the model.

Wall pressures are measured along the curve (Pt) of the model, see Figures 1(a) and 1(c). The internal diameter of the pressure taps is 1.5 mm. Pressure coefficients Cp are defined by:

Cp = p -p 0 q 0 ( 7
)
where p 0 and p are respectively the upstream static pressure and the local pressure. The uncertainty in the pressure coefficients is estimated to ± 2.2%.

Numerics

Flow Solver

The ISIS-CFD flow solver, developed by the EMN (Equipe Modélisation Numérique) of the Fluid Mechanics Laboratory of the Ecole Centrale de Nantes, uses the incompressible unsteady Reynolds-Averaged Navier-Stokes equations. This solver is based on the finite volume method to build a spatial discretization of the transport equations. Guilmineau 7 FE-07-1184

The incompressible unsteady Reynolds-Averaged Navier-Stokes equations can be written (using the generalized form of Gauss' theorem) as:

∂ ∂t Z V ρdV + Z S ρ( - → U - - → U d ) • - → n dS = 0 (8a) ∂ ∂t Z V ρU i dV + Z S ρU i ( - → U - - → U d ) • - → n dS = Z S (τ i j I j -pI i ) • - → n dS (8b)
where V is the domain of interest, or control volume, bounded by a closed surface S moving at a velocity -→ U d with a unit outward normal vector -→ n .

-→ U and p are respectively the velocity and pressure fields. τ i j are the components of the viscous stress tensor whereas I j is the identity vector whose components are null, except for the component j, which is equal to unity.

All flow variables are stored at the geometric center of arbitrary shaped cells. Volume and surface integrals are evaluated with second-order accurate approximations. The face-based method is generalized to two-dimensional or three-dimensional unstructured meshes for which non-overlapping control volumes are bounded by an arbitrary number of constitutive faces. Numerical fluxes are reconstructed on mesh faces, by linear extrapolation of integrand from the neighboring cell centers. A centered scheme is used for the diffusion terms, whereas for the convective fluxes, the scheme implemented in the ISIS-CFD code is the Gamma Differencing Scheme (GDS) [START_REF] Jasak | High Resolution NVD Differencing Scheme for Arbitrarily Unstructured Meshes[END_REF]. Through a Normalized Variable Diagram (NVD) analysis, this scheme enforces local monotonicity and convection boundedness criterion.

The velocity field is obtained from the momentum conservation equations, and the pressure field is extracted from the mass conservation constraint, or continuity equation, transformed into a pressure-equation.

The pressure equation is obtained in the spirit of Rhie and Chow [START_REF] Rhie | A Numerical Study of the Turbulent Flow past an Isolated Aerofoil with Trailing Edge Separation[END_REF]. Momentum and pressure equations are solved in an segregated manner as in the SIMPLE coupling procedure [START_REF] Issa | Solution of the Implicitly Discretized Fluid Flow Equations by Operator-Splitting[END_REF].

A second-order backward difference scheme is used to discretize time. All spatial terms appearing in equations (8a) and (8b) are treated in a fully implicit manner. In this paper, we only use the steady equations.

Therefore, the velocity -→ U d and the first term in the equations (8a) and (8b) are null.

The numerical implementation is detailed by Queutey & Visonneau [START_REF] Queutey | An Interface Capturing Method for Free-Surface Hydrodynamic Flows[END_REF], other researchers of the EMN group of the laboratory.

Guilmineau 8 FE-07-1184

Turbulence Modeling

Several turbulence closures are included in the flow solver, ranging from linear eddy-viscosity based models to full second-order closures [START_REF] Duvigneau | On the Role Played by Turbulence Closures in Hull Ship Optimization at Model and Full Scale[END_REF]. For these studies, the Explicit Algebraic Stress Model (EASM) model has been chosen. Previous studies of the EMN group [START_REF] Duvigneau | On the Role Played by Turbulence Closures in Hull Ship Optimization at Model and Full Scale[END_REF][START_REF] Deng | Comparison of Explicit Algebraic Stress Models and Second-Order Turbulence Closures for Steady Flow around Ships[END_REF][START_REF] Deng | Three-Dimensional Flow Computation with Reynolds Stress and Algebraic Stress Models[END_REF] have shown that anisotropic turbulence closures (such as Reynolds Stress Transport Models (RSTM) or Explicit Algebraic Stress Models (EASM)) are the most reliable statistical turbulence modeling for predicting three-dimensional flows such as ship flows, when intense longitudinal vortices spread out in the aft part of the ship.

In the EASM approach, the Reynolds stress tensor is given by:

τ i j = 2 3 kδ i j -2ν t S i j + a 2 a 4 S ik W k j -W ik S k j -2a 3 a 4 S ik S k j - 1 3 S mn S mn δ i j (9)
S i j and W i j are the strain rate and rotation rate tensors defined respectively as:

S i j = 1 2 ∂u i ∂x j + ∂u j ∂x i W i j = 1 2 ∂u i ∂x j - ∂u j ∂x i (10)
The turbulent eddy viscosity is determined from:

ν t = max -kα 1 , 0.0005 k 2 ε ( 11
)
where α 1 is obtained from the solution to the following cubic equation:

α 1 τ 3 + t α 1 τ 2 + s α 1 τ + r = 0 ( 12
)
where τ = k/ε is the turbulence time scale, and

t = - γ 1 η 2 τ 2 γ 0 , s = 1 (2η 2 τ 2 γ 0 ) 2 γ 2 1 -2η 2 τ 2 γ 0 a 1 - 2 3 η 2 τ 2 a 2 3 + 2R 2 η 2 τ 2 a 2 2 , r = γ 1 a 1 (2η 2 τ 2 γ 0 ) 2
The root of equation ( 12) may be real or complex. The correct root is the root with the lowest real part [START_REF] Jongen | A Unified Analysis of Planar Homogeneous Turbulence using Single-Point Closure Equations[END_REF].

The parameters are given by

η 2 = S i j S i j , {W 2 } = -W i j W i j , R 2 = -{W 2 }/η 2 , a 1 = 1 2 4 3 -C 2 , a 2 = 1 2 (2 -C 4 ) , a 3 = 1 2 (2 -C 3 ) , a 4 = γ 1 -2γ 0 (α 1 /τ)η 2 τ 2 -1 τ, γ 0 = C 1 1 /2, γ 1 = C 0 1 /2 + (C ε2 -C ε1 ) / (C ε2 -1) , C ε1 = 1.44 C ε2 = 1.83 C 0 1 = 3.4 C 1 1 = 1.8 C 2 = 0.36 C 3 = 1.25 C 4 = 0.4 (13)
Guilmineau

FE-07-1184

To implement the EASM model, the turbulent velocity and length scales are determined by using two transport equations, the k-ω BSL model proposed by Menter [START_REF] Menter | Zonal Two-Equation kω Turbulence Models for Aerodynamic Flows[END_REF]. Compared to the original model, only two natural modifications are introduced. Firstly, the turbulent eddy viscosity ν t is replaced by the value given by [START_REF] Garry | Comparison of Quasi-Static and Dynamic Wind Tunnel Measurements on Simplified Tractor-Trailer Models[END_REF]. Secondly, the contribution of the nonlinear part of the Reynolds stress used to turbulence production is taken into account in three-dimensional flow computation. The EASM model implemented can be integrated down to the wall and gives correct log law behaviors without any recalibration. Details of this model can be found in [START_REF] Deng | Three-Dimensional Flow Computation with Reynolds Stress and Algebraic Stress Models[END_REF].

The CPU cost of the EASM model is 30% higher than the k-ω of Menter.

Computational Details

The computational domain starts at 4.4 × L from the origin 0 in front the model and extends to 5.9 × L behind the model. The width of the domain is 3 × L and its height is 1.6 × L, see Figure 2. The mesh is generated using HEXPRESS T M , an automatic unstructured mesh generator. This software generates meshes containing only hexaedrals. For the surface of the model, we use a no-slip boundary condition, and the wall normal resolution is set to 0.006 mm, i.e. y+ = 0.5. For the wind tunnel floor, we use a wall function, and the distance between the floor and the first fluid points is set to 0.6 mm. This approach, used to take the walls into account, is the same as the approach proposed by Krajnović and Davidson [START_REF] Krajnovic | Flow Around a Simplified Car -part 1: Large Eddy Simulation[END_REF], who use the LES approach around a generic ground model. At the outlet of the computational domain, the pressure is prescribed, while for the other boundaries we use a Dirichlet condition for velocity and turbulence. In order to establish a grid-independent solution, computations have been performed at the yaw angle β = 10 • for four meshes with approximately 1.6 × 10 6 points, 6.6 × 10 6 points, 10.8 × 10 6 points, and 19.3 × 10 6 points, which are named G0, G1, G2, and G3, respectively. Close to the wall of the model, the fine computational grid has ∆x = 1.6 mm in the streamwise direction and 0.7 mm ≤ ∆y ≤ 1.7 mm in the direction parallel to the surface and normal to the streamwise direction. The maximum size, ∆x = ∆y = ∆z = 0.1 m, are located at the top of the computational domain. The number of points of the mesh in the boundary layer is approximately 25 for the fine computational grid. The grid G3 is distributed on 60 blocks, the grid G2 on 34 blocks, the grid G1 on 20 blocks and the grid G0 on 8 blocks. The characteristics of these meshes are detailed in Table 2.

In this study, the computations are steady simulations. Consequently, any time step is used. To obtain converged results, 7000 non-linear iterations are necessary. For the fine mesh, the CPU time is 70 hours on an IBM Power 4.

Figure 4 presents the pressure coefficient along the curve (Pt) of the model (see Figure 1) versus the mesh for β = 10 • . We note that the prediction of the pressure along the curve (Pt) is independent of the mesh used. We only have a small difference near the base of the model with the coarse mesh.

Table 3 presents the drag coefficient versus the mesh. We note that the solution converges when the grid is increasingly fine. According to these results, all the numerical results presented thereafter were obtained on grids similar to the grid G3, i.e. with approximately 20 million points.

6 Results

Flow Structure

An experimental and a numerical reconstruction of the separated flow and of the wake obtained at β = 30 • is performed from tomographies of total pressure coefficients Cpi drawn in Eiffel planes (Yo, Zo), see Figures 5. The experimental tomographies are post-processed in order to suppress the noise measured outside the body, and experimental values are set to zero when the total pressure coefficients are in the range ∆Cpi = ±0.04. This general view shows that the numerical model reproduces the physics of the experimental three-dimensional turbulent wake. We note that the direction of the wake is strongly deviated by the Guilmineau 11 FE-07-1184 upstream velocity. The diffusion of the numerical model is lower than the diffusion observed experimentally. This result must be considered as a quality of the solver even if the turbulence level of the wind tunnel produces more mixing of turbulence scales.

The details of the formation of the leeward side wake and of the vortices shedding are described in The results obtained at the yaw angle β = 30 10. The cross flows drawn in Figure 6 show that the intensities of the vortices continue to increase, and that the main vortex (V2) remains attached to the wall. The vortex (V3) which appears at X/L = 0.30 in Figure 6 is diffused at X/L = 0.45 and is visible in the numerical and experimental tomographies of total pressure drawn in Figure 9. The vortex (V1) which is visible in Figure 9(a) is diffused in the experimental results, see Figure 9(b).

Secondary vortices (V4) and (V5) which are linked to vortex (V2), are visible at X/L = 0.30 and at X/L = 0.45, see Figure 6. These cross flows are linked to the wall friction lines drawn in Figure 10 The results obtained at β = 30 • are completed by the tomographies drawn in Figure 11, obtained in the wake at X/L = 0.65. During its transport, the numerical vortex (V2) keeps its own individuality and is not mixed with the two counter-rotating vortices emitted from the base, see Figure 11(a). In a similar manner, these two counter-rotating vortices also remain separated, as confirmed by the attachment line (A3) shown in Figure 10. In our experiments, the vortex system emitted from the base and the vortex (V2) approach an isotropic state more rapidly [START_REF] Bernard | Turbulent flow properties of large-scale vortex systems[END_REF]. Finally, all the vortices emitted in the wake gather in a unique structure linked to the floor boundary layer as shown in the three-dimensional views in Figure 5.

The numerical and experimental tomographies obtained in a horizontal plane at Zo = -14.5 mm, at midpoint of the ground clearance, are drawn in Figure 12. The position of the cross section Xo/L = 0.55 is given in order to locate the vortices (V1), (V3), and (V6), see Figure 9. Again, if the directions of the wakes of the feet (f) and of the cylinder (c) are correct, the numerical diffusion is lower than the diffusion revealed in the experiments. This fact, which is common to the entire computation domain, is overstated here by the Guilmineau 13 FE-07-1184 loss of velocity, and in turn by the decrease of the local Reynolds number, in the boundary layer developed at the level of the floor, see Figure 3.

Wall Pressure and Forces

The evolution of the pressure coefficient Cp along the curve (Pt), which is the projection of the curve (A)

on the side of the model, is drawn in Figure 13 the drag decreases. This behavior is typical of a square-back model [START_REF] Hucho | Aerodynamics of Road Vehicles[END_REF]. For the small yaw angles and up to β = 20 • , the numerical prediction under-predicts the drag and over-predicts this force for a larger yaw angle. We can connect this behavior with the prediction of production and transport of vorticity when the yaw angle increases, see Figure 6. In fact, for a yaw angle less than β = 10 • , only the vortex (V3) appears on the leeward side of the model and separations are limited to the base. For β = 15 • , the vortex (V2) remains weak at X/L = 0.30 and X/L = 0.45. The drag is then under-estimated compared to the experimental data.

When the yaw angle is greater than β = 20 • , the intensity of the vortex (V2) increases and a larger amount of vorticity is shed and transported in the wake. This fact leads to an overestimation of the drag. This result is a classic defect of RANS simulations, see Maddox et al [START_REF] Maddox | Detached-eddy simulation of the flow around the ground transportation system[END_REF]. The side force, see Figure 14 The friction lines obtained at β = 30 • on the underbody of the model, see Figure 10, shows that the wake of the cylinder (c) used to protect the pressure tubes interacts strongly with the wakes of the feet (f1) and (f2), which are upstream the cylinder, and that the cylinder itself interacts with the foot (f3) which is downstream.

In future experiments dealing with the analysis of unsteady side gusts of wind these interactions could deform conclusions concerning the unsteadiness of the wake. This fact is confirmed by the reading of the numerical friction lines obtained without the cylinder (c) at β = 30 10 shows that the friction lines on the underbody are changed, the consequences being the formation of a separation line (S3) starting at the foot (f1) and finishing at the foot (f3). Moreover, the change of the relative positions of the singular friction lines (A2), (S2), (A1) and (S1) indicates a modification of the structure of the vortex system composed of the main vortex (V2) and of the secondary vortices (V5) and (V6), see Figure 6 and the frames obtained at β = 30 • .

Comparing Figure 9 and Figure 17 where the numerical tomographies obtained at Xo/L = 0.55 are drawn, just downstream the base, shows the changes of the wake when cylinder (c) is removed. When the cylinder is present, the vortex (V3) which amalgamates the wakes of the cylinder and of the feet (f1) et (f2)

is intense and the loss of total pressure is Cpi = 1.2 at the level of its center C, see Figure 9(a) and Figure 12.

Without the cylinder, this vortex, named (V3) in Figure 17, is weak and amalgamates the wakes of the feet (f1) and (f2) only. The pressure coefficient in its center is close to Cpi = 0. Finally, the trajectory of vortex (V1) is not deviated by the wake of the cylinder and moves from about Y = + 15 mm, see Figure 9(a), to Y = -10 mm, see Figure 17. The trajectory of the wake (V6) of the foot (f4) is not changed. Future experiments analysing the physics of unsteady side gusts will be performed on the Willy model and it appears that the presence of the cylinder will deform the analysis concerning the effects of phase shift or hysteresis linked to the unsteadiness of the wake. To conclude this point, the cylinder will be removed, the wall pressure measurements will be performed by dynamic pressure transducers placed inside the model, and electric wires will pass through the feet of the model.

Conclusion

This paper presents numerical simulations and experiments on a square-back model which has been de-

  technique used to simulate this condition is to yaw the model relative to the freestream flow above a fixed ground plane. The results presented here deal with the analysis of steady flows for various yaw angles β Guilmineau 4 FE-07-1184 from β = 0 • to β = 30 • . The value β = 30 • corresponds to a side wind equal to half the value of the vehicle velocity.

2 .

 2 A Rankine half-body with a downstream diameter φ. The curve (B), which is in the symmetry plane of the body, is obtained by distributing the thickness of the Rankine body on the mean line (A) in direction Z. A cross section of the body, obtained at the point M, is given in Figure1(c). Point b is the intersection of the curve (B) with the cross section. The curve (abc) is an ellipse, the sides (ga) and (cd) are vertical and are connected to the horizontal underbody (ef) by quarter circles of radius R. The points a, M, and c are on the same horizontal line. Guilmineau 5 FE-07-1184

  semi-open test section of the wind tunnel has a cross section of 1.45 m by 1.45 m and the ground is fixed. Forces and moments are measured with a six-component balance. The reference axis (Xo, Yo, Zo) are the Eiffel axis where Xo is parallel to the upstream velocity V 0 . The Lilienthal axis (X, Y, Z) are linked to the model. The origin of the axis lies at the point O located on the floor of the model. This point O is the center of rotation of the model, see Figure 1.

  The normalized experimental and numerical velocity profiles of the boundary layer on the floor of the wind tunnel, at Xo = -670 mm, and Yo = 10 mm, are given in Figure3. The results are obtained for an upstream velocity Vo = 20 m/s and the model at the yaw angle β = 30 • . The flat plate 1/7th power law boundary layer profile is also given. The numerical result is typical of a turbulent boundary layer submitted to a weak positive pressure gradient, with a modification of the velocity gradient and an increase of the shape factor, see Cousteix[START_REF] Cousteix | Turbulence et Couche Limite[END_REF]. The position of the underbody of the model given in Figure3shows that the consequences of the discrepancies between the experiments and the present simulation should be limited to the underbody flow and should have only a weak influence on the rest of the model. Guilmineau 10 FE-07-1184

Figure 6

 6 Figure6where pseudo-streamlines are drawn in Lilienthal (Y, Z) planes at X/L = 0 which is at mid-length

  where (S1) and (S2) are separation lines and (A1) and (A2) are attachment lines. The main counter-clockwise vortex (V2), the secondary vortices (V4), which is counter-clockwise, and (V5), which is clockwise, are shed from the separation lines (S1) and (S2). The saddle point noted (Sa) noted in Figure 6 at X/L = 0.45 guarantees the topology of the cross flow.

  for different yaw angles β. Error bars are added on measured values. The results show that the pressure at the stagnation point is correctly predicted for all the angles, and that wall pressure is correctly simulated on windward and leeward sides up to β = 15 • . As the yaw angle increases, the stagnation point moves windward. The negative pressure gradient on the windward side decreases along the body. Up to β = 15 • , the pressure becomes relatively constant and tends towards the upstream static pressure p 0 . The flow is attached on the windward side and the prediction aligns with the results of the experiments up to β = 30 • . On the leeward side, the flow separates when the yaw angle is greater than 10 • , as shown by the crossflow obtained at X/L = 0.30, see Figure 6. The negative pressure gradient increases, and simulations and experiments align until the change of the sign of the pressure gradient. At approximately X = -200 mm, the rise of adverse pressure is the indication of a flow separation. The difference between experiments and simulations which is particularly visible at β = 30 • shows the tendency of the ISIS-CFD flow solver to over-predict vortex intensity, see Figure 9. This behavior is also observed by Maddox et al [21]. These authors simulate the flow around a simplified geometry of a ground transport system (GTS), at β = 0 • and β = 10 • . Calculations were conducted at a Reynolds number of 2 × 10 6 , based on upstream velocity and model width. The grid used was made up of 6 × 10 6 cells. Pressure coefficients are well-predicted at β = 0 • but at β = 10 • the DES and RANS simulation techniques calculate a overly large separated region near the front corner of the model, and consequently negative values of the pressure coefficients greater than those shown in the experiments. It must be noted that these discrepancies on the leeward side of the GTS are increased by the fact that the front of the model used by Maddox et al [21] is not streamlined compared to the Willy model.The drag coefficient and the side force coefficient, in the Lilienthal axis, are given in Figures14, where

  error bars are included. The drag, see Figure14(a), increases up to the yaw angle β = 20 • . After this angle, Guilmineau 14 FE-07-1184

6 . 3

 63 (b), shows a linear evolution. We note that the experimental value of the slope is (dCy/dβ) exp ∼ -2.36 rad -1 , and the numerical value of the slope is (dCy/dβ) num ∼ -2.94 rad -1 . The yawing moment versus the yaw angle is presented in Figure15. This coefficient rises linearly with β up to β = 25 • . The linearity of the side force and the yawing moment are typical characteristics of real vehicules[START_REF] Hucho | Aerodynamics of Road Vehicles[END_REF]. The slope of the experimental curve is (dCN/dβ) exp ∼ 0.40 rad -1 and (dCN/dβ) num ∼ 0.43 rad -1 for the numerical curve. The values of the slopes measured for the Willy model align with the values commonly obtained on real vehicles. For the yaw angle β = 30 • , the flow solver tends to under-predict the value of the yawing moment. One hypothesis, in accordance with Figure13(f), is that the intensity of the computed vortex (V2), see Figure10, on the leeward side of the model is too large compared to the experimental vortex. On real cars, this loss of CN is also observed when additional vortices are shed from separations on sharp edges at the back of a fastback vehicule. Influence of the Cylinder (c) for β = 30 •

Figure 1

 1 Figure 1: Model definition Guilmineau 23 FE-07-1184

Figure 2 :Figure 3 :

 23 Figure 2: Wind tunnel and model

Figure 4 : 26 FE

 426 Figure 4: β = 10 • : Pressure coefficient along the curve (Pt) versus the mesh

Figure 5 : 27 FE

 527 Figure 5: β = 30 • : 3D view of the wake

Figure 6 : 28 FE

 628 Figure 6: Crossflows for several yaw angles

Figure 7 :

 7 Figure 7: β = 10 • : Tomography at Xo/L = 0.60

Figure 8 :

 8 Figure 8: β = 20 • : Tomography at Xo/L = 0.60

Figure 9 :Figure 10 : 32 FE

 91032 Figure 9: β = 30 • : Tomography at Xo/L = 0.55

Figure 11 :

 11 Figure 11: β = 30 • : Tomography at Xo/L = 0.65

Figure 12 :

 12 Figure 12: β = 30 • : Tomography at Zo = -14.5 mm

Figure 13 :

 13 Figure 13: Pressure coefficient along the curve (Pt)

Figure 14 :Figure 15 :Figure 16 :Figure 17 :

 14151617 Figure 14: Force coefficient versus the yaw angle

  • are described through cross flows in Figures 6, tomographies drawn in Figures 9, 11, 12, and wall friction lines in Figure

  • , see Figure 16. Comparing of this

	result with Figure
	Guilmineau	15	FE-07-1184

Table 1 :

 1 Dimensions of the test model (data in mm) . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2 Characteristics of the four meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3 β = 10 • : Drag coefficient versus the mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Dimensions of the test model (data in mm)

	Guilmineau	39	FE-07-1184

Table 2 :

 2 Characteristics of the four meshes

		G0	G1	G2	G3
	Number of points	1,609,155 6,651,159 10,808,408 19,273,500
	Number of points on the model	30,905	102,522	205,366	342,432
	Number of points on the plate	20,848	49,857	61,313	98,664
	Number of cells	1,549,482 6,478,457 10,503,589 19,240,045

Table 3 :

 3 β = 10 • : Drag coefficient versus the mesh

	G0	G1	G2	G3
	Cx 0.434 0.397 0.384 0.378

Acknowledgments

Experiments were performed by Mr. Didier Legrand and Mr. E. Aïssaoui in the framework of preparation for an engineering degree from the CNAM. We would like to thank them for their help and cooperation. The authors gratefully acknowledge the Scientific Committee of IDRIS (project 0129) and of CINES (project dmn2049) for the attribution of CPU time.