Leonardo Maria Lalicata
email: leonardo.lalicata@city.ac.uk

Luc Thorel

Augusto Desideri

G M Rotisciani

Francesca Casini

Model preparation for unsaturated soil testing in a centrifuge environment

Keywords: Unsaturated Soils, Static Compaction, Water Retention Curve, Centrifuge Modelling, Hydro-Mechanical Stress Paths

This paper presents selected aspects of the model preparation and the experimental procedures used for testing the unsaturated soil behaviour in centrifuge environment. The study is based on the results of centrifuge tests carried out to investigate the influence of partial saturation on the behaviour of laterally loaded piles. The soil used in the experimentation is a high permeability silty clay soil, named B-grade kaolin. The models were statically compacted at two different densities with the same water content. In order to reduce the after compaction suction, the samples were subjected to an imbibition process at 1×g connecting the bottom of the model with the water reservoir. The influence of the compaction process and the imbibition stage on the subsequent inflight soil conditions are discussed and analysed. The centrifuge tests were realised at the centrifuge centre of the University Gustave Eiffel, in Nantes, France, in the framework of the GEOTRANSALP project.

INTRODUCTION

Centrifuge modelling represents a valid methodology to investigate the performance of different types of geotechnical works such us shallow and deep foundations, tunnels, slopes and embankments. Traditionally, studies have been conducted on fully saturated or completely dry soil models. However, in some cases the variation of the water content in the soil due to the water table oscillation, rainfall events, interaction with the atmosphere and exceptional events (i.e. pipe rupture etc…) may be crucial for the geotechnical works behaviour. These and other problems have been recently investigated by centrifuge modelling [START_REF] Casini | Centrifuge modelling of an unsaturated collapsible soil[END_REF][START_REF] Soranzo | Face stability of shallow tunnels in partially saturated soil: centrifuge testing and numerical analysis[END_REF][START_REF] Thorel | Physical modelling of wetting-induced collapse in embankment base[END_REF]. Scaling laws for unsaturated soils have been experimentally investigated, among others, by [START_REF] Depountis | Centrifuge modelling of capillary rise[END_REF] and [START_REF] Caicedo | Validation of time scale factor of expansive soils in centrifuge modeling[END_REF] under different soil condition and g-levels. The authors found that capillary rise and diffusion time in the centrifuge could be scaled by 1/N and 1/N 2 , respectively. Details of the analytical formulations can be found in [START_REF] Caicedo | Centrifuge Modelling of Unsaturated soils. Advances in the Mechanics of Unsaturated soils[END_REF].

Compaction is the most common technique to prepare unsaturated soil samples [START_REF] Caicedo | Centrifuge Modelling of Unsaturated soils. Advances in the Mechanics of Unsaturated soils[END_REF]. Dynamic compaction allows to control the density but not the compaction stress. On the other hand, with the uniaxial static compaction procedure both density and compaction stress can be measured although the horizontal stresses are in general unknown.

This paper presents the details of the model preparation of the experimental investigation conducted by [START_REF] Lalicata | Experimental observation on a laterally loaded pile in unsaturated silty soil[END_REF]. The study aimed at assessing the influence of partial saturation on the response of laterally loaded piles installed in fine-grained soils. The centrifuge tests (N = 100×g) were carried out using the centrifuge facilities of the University Gustave Eiffel, located close to Nantes (France) in the framework of the GEOTRANSALP-PILE-UNSAT agreement.

MATERIALS

The soil used

The material used is a commercial soil named B-Grade kaolin (90% fine silt and 10% clay). This soil has a low plasticity index (IP=14%) and a low liquid limits (wL=42%), the plastic limit is wP=28%. A complete hydro-mechanical characterisation was conducted in the geotechnical laboratory using standard apparatus for unsaturated and saturated soils. Results are reported in [START_REF] Lalicata | Effect of saturation degree on the mechanical behaviour of a single pile subjected to lateral forces[END_REF] and [START_REF] Lalicata | Numerical Study of Laterally Loaded Pile in Unsaturated Soils[END_REF]. For the sake of brevity, only the flooding oedometric tests and the suction controlled oedometer tests are presented here. The hydro-mechanical parameters of the soil are listed in Table 1. Several imbibition tests in the oedometer apparatus were undertaken to define the after compaction conditions (w, e) for centrifuge models. The samples (18 mm thick and 70 mm in diameter) were prepared with different compaction features and subjected to different total vertical stresses v applied before the wetting stage. In the main series of the tests, the vertical stress was 150 kPa corresponding to the vertical stress active at half of the pile embedded length (z=8m). Samples were left equalise under the external load before filling the oedometer cell with water.

The soil powder was dried at 105 °C for 24 hours before adding the demineralized water to reach the target water content. The moisture was manually mixed and then it was kept in sealed bags for 24-48 hours to homogenise in a temperature-controlled environment. The specimens were statically compacted (v=1.5 mm/min) directly in the oedometric ring.

Three voids ratio are taken into account (e0 = 0.77, 0.92 and 1.12), for water content ranging from 10% to 26%, with steps of 4%. For each grid point, at least two samples were tested.

Figure 1a shows the wetting induced deformation under v =150 kPa in function of the compaction void ratios. Water content measurements at the end of the test have shown that all the samples reached degrees of saturation closed to one (Srmin=0.95).

Results are grouped for the initial water content. For the material, the latter does not have a significant influence on the wetting deformation while the results strongly depends on the initial void ratios.

The deformations reduce approximately linearly as the voids ratio decrease. Little swelling was observed for the more compacted samples (e0=0.77).

The influence of the applied external load during samples inundation is depicted in Figure 1b, for stresses ranging from 0 to 600 kPa. Samples were prepared at e0=1.12 and w=14%. Soil swelling occurs only in absence of external load applied, while since from v= 50 kPa the soil experience positive deformations that increase up to v= 300 kPa and then they slightly reduce in magnitude. Thus, in the centrifuge, this soil would have experienced modest swelling in the shallower layer and volume reductions increasing with depth. In order to explore the influence of the compaction conditions of soil in the centrifuge tests, two void ratios are selected (0.93 and 0.75) prepared with the same water content w=15%. The two densities are considered representative, respectively, of a collapsible soil and a swelling soil under wetting processes. The higher void ratios of 1.12 was discarded as showed deformation too elevated, Figure 1b. The soil water retention curve (SWRC) was obtained by means of the suction controlled oedometer [START_REF] Romero | Development of a new suction temperature controlled oedometer cell[END_REF]. The data corresponding to the main wetting curve a two different void ratios, 0.93 and 0.75, are presented in Figure 2 in terms of suction s (in log scale) versus the degree of saturation Sr. In the suction range experimentally studied, the data exhibit a marked influence of porosity in the retention behaviour. In fact, as porosity decreases, air entry value increases and the slope of the transition zone reduces in the Sr-s plane. The Van Genuchten (1980) equation was used to fit the data:

  , , , 1 () 1 m r r res r sat r res n S S S S s           (1)
Where Sr,res and Sr,sat are respectively the residual and saturated degree of saturation;  is the inverse of the air entry value, n governs the slope of the transition zone and m controls the inflection point at the residual values of suction. The values of the parameters are listed in Table 2. Neglecting the hysteresis of the SWRC, eq (1) can be used to estimate the as-compacted suction and thus the initial effective stress in the model according to the Bishop effective stresses definition: ' r Ss     [START_REF] Bishop | Some aspects of effective stress in saturated and partly saturated soils[END_REF]. As the total stress is zero after the compaction the effective stresses in the soil reduces to the product r Ss  , and are 387 kPa for e0=0.93 (Sr0=0.43) and 1768 kPa for e0=0.75 (Sr0=0.52) respectively.

The model pile

The model pile (scale 1/100) is a close-ended aluminium tube (external diameter D=12mm, thickness t=1mm) embedded into the soil for 150 mm. The pile was instrumented with 10 pairs of strain gauges arranged every 15 mm, providing the bending moment profile along the pile length. The strain gauges were calibrated in the laboratory adopting the cantilever scheme with point load. The experimental arrangement for the calibration is presented in Figure 3.

The experimental apparatus

The axial symmetric models 180 mm deep were prepared in 300 mm diameter rigid tub. As shown in Figure 4, two twin containers were spin up at the same time: one was used for CPT measurement before, during and after the test. The other was used for the load test on the pile that was bored in the centre of the model, at 1×g, by means of a manual screwing system. The screw was mounted into a rigid frame to maintain position and verticality.

The model was extensively instrumented in order to follow both the equalisation phases and pile loading (Figure 5).

Model preparation

Samples were prepared by static compaction at two different voids ratios (e0=0.93 and 0.75) and the same water content (w=15%).

The moisture was prepared following the same procedure used for the flooding tests. After the moisture homogenisation, the soil was then passed in a mechanical mixer to size down the aggregates and stored again in sealed plastic bags.

Compaction

The soil models were statically compacted, in a rigid tub of 300 mm in diameter. An extension is required to compact the soil to the target density. A 10 mm thick sand layer, surrounded by geotextile, allowed the drainage at the bottom of the model. A 2 mm thick plastic sheet coated the inner side to reduce skin friction during model preparation and during the test.

The soil is compacted in six layers of final height 30 mm with a displacement rate of 1.5 mm/min. The compaction is a displacement-controlled process: the press stops when the target displacement was reached. In Figure 6 typical results of the compaction process are presented for one loose soil sample (e0=0.93).

After the compaction of each layer, its surface was scarified to achieve a good connection with the sequent one. The soil surface was covered with a plastic film to prevent water evaporation in the upper part of the model. The reasonable homogeneity of the distribution of vertical compaction stresses, Figure 7, and the good repeatability of the results in the different tests confirm the validity of the procedure used. It has to be noticed, however, that the dispersion increases with the soil density. The compaction stress increases with dry density from a mean value of 500 kPa for the loose soil (e0 = 0.93) to 1400 kPa for the dense soil (e0 = 0.75). The bottom of the model was connected to a water reservoir, the level of which was governed by an electro-pneumatic valve system remotely controlled by the operator from the centrifuge control room. A laser sensor measured the water height in the tank. Five LVDT sensors measured soil settlements, three tensiometers, installed on the side of the container, measured the pore water pressure (negative and positive) in the soil. The lateral load was applied at 35 mm from ground level by a hydraulic actuator. The loading phase was displacement-controlled (v=0.003 mm/s at model scale), and a load cell measured the lateral load. One LVDT built in on a rotational joint gave the pile's vertical displacement and rotation.

CENTRIFUGE TEST PROCEDURE

In total four centrifuge tests were undertaken, varying both the elevation of the water table, zw, and the initial void ratio of the soil, e0 (Table 3). In the two tests where the water table was 70 mm from the ground level the pile was loaded until a lateral displacement, y, of 30-40% of the diameter, D, was reached. In the following step, the water table level was raised to ground level keeping the load constant. For the sake of comparison, two additional tests were conducted up to soil failure in fully saturated condition (zw=0) on soil samples prepared at the same voids ratio of the main tests. Measurements of pile displacements, lateral load and bending moments were recorded.

In order to reduce the as compacted suction, the samples were subjected to an imbibition process at 1g, connecting the bottom of the model with the water reservoir.

Before pile loading, equalisation of pore pressures in the soil was allowed. The load was applied with a very slow displacement rate to avoid the creation of excess pore pressures. At the end of each test, undisturbed specimen were sampled to obtain the water content and voids ratio distribution with depth.

ANALYSIS OF THE RESULTS

In this section, selected results are presented to illustrate first the influence of the centrifuge test procedure on the soil behaviour. The complete results are presented in [START_REF] Lalicata | Experimental observation on a laterally loaded pile in unsaturated silty soil[END_REF] and [START_REF] Lalicata | Effect of saturation degree on the mechanical behaviour of a single pile subjected to lateral forces[END_REF].

1g imbibition

The absorbed water volume in this stage is presented in Figure 8a. Figure 8b shows the temporal evolution of the vertical displacements at the soil surface for the two densities considered. The wetting process induces the development of vertical displacements directed upwards attaining a final values of approximately 3 mm for the loose soil and ~6 mm for the dense soil. The water infiltration into the soil layer entails an increase in water volume by around 2 lt in both cases.

The vertical displacements increase as the wetting front advances. Despite the absorbed water volume is approximately coincident in the two cases, the wetting-induced swelling in the dense soil is roughly the double of that found in the loose soil. This outcome reflects the relevant difference in the as-compacted suction and the degree of saturation (see Table 8). Because of the different shape of the SWRC, (see Figure 2), the same approximate variation in Sr leads to a suction decrease and, thus to a stress unloading, greater in the dense than in the loose soils.

The overall imbibition process induced by capillary rise at 1g with zero pore pressure at base had not still reached stationary conditions, because the main aim of this phase was to increase the suction to speed-up the equilibrium in the following in-flight phase.

In flight phases

Figure 9 shows the temporal evolution of the vertical displacements at the ground surface during the in-flight phase for T06 (e0=0.93) and T08 (e0=0.75). The increase in the gravity loading, together with the hydraulic boundary condition imposed at the base, leads to an instantaneous settlement of about 3 mm and 7 mm for the dense and the loose soil respectively. The vertical displacements increase in time up to 3.5 mm and 10 mm in the two cases.

After the pile loading, the water table was raised up of 70 mm saturating the whole model. The loose sample globally settles as the volumetric collapse of the unsaturated part is greater than the volumetric swelling of the deeper soil layer. On the other hand, the LVDTs measured very little increase in soil volume in the dense sample. The instantaneous soil volume reduction observed as g increases is possible because the soil is not fully saturated. Indeed, the increment of total stresses is taken all by the effective stresses, as the presence of air in pores allows for a soil skeleton compression. According to the previous imbibition stage, the soil degree of saturation is not uniform in the soil but reduces moving towards the ground surface (far from the bottom boundary condition). Thus, the initial settlement is likely to be related to the compression of the upper part of the model. Conversely, the consolidation settlement has to be attributed to the excess pore pressures dissipation in the lower part of the model.

The pore pressure evolution for the test T06 shown in Figure 10 qualitatively confirms these findings. In fact, the in lower part of the model (z=140mm), the pore pressure rises as g increases and then reduces with time. In the upper part (z=40mm), it reduces slightly at the beginnings. After four hours of consolidation the trends seems to be stabilised but the absolute values are different from those expected. The tensiometers correctly capture the pore pressure variation during the rising of the water table; they measured approximately +70 kPa at the end the test. Since the instruments worked well at 1g environment both before and after each test, further investigations are needed to understand the validity of this tools in high gravitational fields. The pile head displacement and force evolution versus time are presented in Figure 11 a-b for the two soil densities. During the saturation at constant load, the piles head displacements increased more for the sample with e0=0.93 than the one with e0=0.75. In both cases, the inundation event provoked the collapse of the pile. In fact, the displacements keeps growing even if the pore pressures tend to the equilibrium (Figure 10).

End of test measurements

The voids ratio profiles with depth, at the end of the tests, in Figure 12 allows drawing same general considerations about the soil state at the end of the tests. The reference test on saturated soil conditions for the dense soil, T09 (e0=0.75), shows a profile of voids ratio approximately constant with depth to a value of ~0.75. For the loose soil (T05) the voids ratio decreases with depth from a value of ~0.9 at the ground surface to a value of 0.76 for a depth of 170 mm. The latter distribution well fit the oedometric normal consolidation line, NCL, below 40 mm of depth, while the denser ones intercept it at z=120 mm. The models were subjected to the same experimental procedure (1g imbibition, increase of total stress and in-flight equalisation). Thus, these differences are related both to the different initial void ratio and the different shape of the SWRC that controls the variation of mean effective stress and the preconsolidation pressure during hydro-mechanical stress paths.

55

In test T06, after the water table rising, below z=90 mm, the voids ratio is the same of T05 (saturated reference test). From 40 to 90 mm of depth the effects of the volumetric collapse upon wetting are very significant while this has a small influence in the shallower part of the model (z<40 mm).

CONCLUDING REMARKS

The paper presents the experimental details of the centrifuge testing on an unsaturated fine graded soil focusing on the soil behaviour from the model preparation to the end of the test.

The study demonstrate the feasibility of centrifuge modelling in unsaturated soils highlighting the influence of the initial density on the soil response during the flight:  Low compacted soils experience significant positive deformations during both flight and the rising of the water table.  Dense soils show a moderate compression during the g-level increase and the application of the hydraulic boundary condition at the bottom. Moderate swelling was observed during the water table rising. Useful insights on the soil response in flight can be collected from simple flooding oedometer tests. On the other hand, the knowledge of the SWRC, and its dependence from void ratios, is crucial to analyse the hydraulic behaviour of the soil during the centrifuge tests.

The 1g imbibition phase prior testing efficiently reduces the soil suction and speed up the in-flight consolidation time. Some attention is required to not over saturate the material before the centrifuge test beginning.

The tensiometers correctly reproduced the pore pressures variations in-flight but more work must be done to investigate the shift observed from the expected values.

To date, the procedure for the model preparation requires a significant effort from the researcher that limits the applicability to large experimental campaigns. Further reflections on possible improvements of the experimental procedures are needed.

However, the findings of this study encourage the use of centrifuge modelling to investigate the soil structure interaction in unsaturated soils.

Figure 1 .

 1 Figure 1. Wetting induced deformation versus: a) initial void ratios, b) total vertical stress.

Figure 2 .

 2 Figure 2. Soil water retention curves, SWRC, for the two densities used in the centrifuge tests.

Figure 3 .

 3 Figure 3. Model pile and calibration set-up.

Figure 4 .

 4 Figure 4. Model assembled in the strongbox Figure 5. Model instrumentation.

Figure 6 .

 6 Figure 6. Typical results of the compaction process for one of the low density models.

Figure 7 .

 7 Figure 7. Compaction stress profiles in the tests.

Figure 8 .

 8 Figure 8. 1g imbibition: a) adsorbed volume, b) vertical displacement of soil.

Figure 9 .

 9 Figure 9. Soil vertical displacements versus time in test T06 (e0=0.93) and T08 (e0=0.75).

Figure 10 .

 10 Figure 10. Pore pressures versus time in T06 (e0=0.93).

Figure 11 .

 11 Figure 11. Pile loading and rising of the water table stages: a) lateral displacement versus time, b) latera load versus time.

Figure 12 .

 12 Figure 12. Void ratios profiles with depth at the end of the tests.

Table 1 .

 1 Hydro-mechanical properties of B-Grade kaolin.

	s	CC	CS	N0 *	'	cv
	(g/cm 3 )				(°)	(mm 2 /s)
	2.66	0.26	0.078	1.36	22	1.0

*N0 is the voids ratio at 'v=1 kPa under oedometric conditions.

Table 2 .

 2 Van Genuchten parameters for different void ratios.

Table 3 .

 3 Initial conditions of the centrifuge tests.

	Test	e0	zw	w0	vc	Sr0
	ID		(mm)		(kPa)
	T 05	0.93	0	15.03	580	42.02
	T 06	0.93	70	14.67	559	42.01
	T 08	0.75	70	14.72	1395	51.03
	T 09	0.75	0	14.72	1395	51.03


Luc Thorel
email: luc.thorel@univ-eiffel.fr

Semaan Maatouk
email: semaan.matouk@univ-eiffel.fr

Jose Antonio Schiavon
email: schiavon@ita.br

Isfog2020

Austin

CYCLIC LOADING OF HELICAL PILE AS ANCHOR FOR FLOATING WINDTURBINES: CENTRIFUGE TESTS

Keywords: helical pile, mono-helix, tension loading, cyclic loading NOTATIONS

Helical piles are currently used in several countries for onshore applications. In the perspective of the development of floating wind turbines, helical pile should be used as an anchoring system. The understanding of the behavior of this anchoring system is investigated under cyclic loading. Installation phase in a saturated sand, monotonic tension and cyclic behavior are analyzed, based on small scale (1/10) centrifuge models. After a presentation of the experimental device, some parameters are investigated, such as the installation torque for two rotation rates and the displacements under cyclic loading.

INTRODUCTION

Offshore Wind Turbines (OWT) follow an exponential development since 1991 [START_REF] Orsted | The world's first offshore wind farm is retiring[END_REF] as a new source of "green and blue" energy. If today the market is devoted to OWTs fixed on the underwater ground, the trends of the future concern deeper water and floating solutions [START_REF] Butterfield | Engineering Challenges for Floating Offshore Wind Turbines[END_REF], such as the prototypes recently developed in Europe (e.g. Floatgen, Hywind/Equinor, Windfloat/EDP, Eolink), North America (e.g. Aqua ventus) or Japan (e.g. Hibiki, NEDO Channel 2019).

Catenary mooring lines are used for the existing Floating Wind Turbines (FWT). This system is based on the use of free hanging lines fixed at the seabed with the use of drag anchors or suction anchors. The mass of the catenary line is the key aspect to the stability of this system. Some studies have treated the development and optimization of the use of such type of mooring system [START_REF] Goldschmidt | Coupled mooring systems for floating wind farms[END_REF]Muskulus 2015, Brommundt et al. 2012). As an alternative of the use of the catenary lines, Tension Leg Platform (TLP) system has been proposed for deeper water. This system is based on the use of tension steel leg fixed by piles or anchors at the seabed. A perspective use of helical pile for the fixation of these types of structures presents many advantages in comparison with the conventional driven piles: (1) Reduced noise pollution in comparison with piledriving method, (2) Easy unscrewing which will aid the decommissioning of the structure at the end of its service [START_REF] Byrne | Helical piles: an innovative foundation design option for offshore wind turbines[END_REF]. A major practical advantage also of the use of helical pile is that the capacity of these piles can be predicted from the torque measurement during installation [START_REF] Tsuha | Relation between installation torque and uplift capacity of deep helical piles in sand[END_REF].

The tension and compression static behavior of helical piles has been well treated in the literature [START_REF] Ghaly | Installation torque of screw anchors in dry sand[END_REF][START_REF] Gavin | Field investigation of the axial resistance of helical piles in dense sand[END_REF][START_REF] Merifield | Ultimate Uplift Capacity of Multiplate Helical Type Anchors in Clay[END_REF][START_REF] Elsherbiny | Axial compressive capacity of helical piles from field tests and numerical study[END_REF][START_REF] Ghaly | Pullout performance of inclined helical screw anchors in sand[END_REF][START_REF] George | Helical Piles Installed in Cohesionless Soil by Displacement Method[END_REF][START_REF] Hao | Tensile Monotonic capacity of helical anchors in sand: interaction between helices[END_REF]. On the other hand, fewer studies exist on the cyclic behavior of helical piles [START_REF] Schiavon | Behaviour of helical anchors subjected to cyclic loadings[END_REF][START_REF] Schiavon | Cyclic and post-cyclic monotonic response of a single-helix anchor in sand[END_REF][START_REF] El Sharnouby | Field investigation of axial monotonic and cyclic performances of reinforced helical pulldown micropiles[END_REF][START_REF] Wada | Effects of cyclic vertical loading on bearing and pullout capacities of piles with continuous helix wing[END_REF][START_REF] Newgard | Cyclic response of shallow helical anchors in a medium dense sand[END_REF]. The focus of previous research on helical pile was oriented to onshore application and the saturation of the soil was not a priority. Here, centrifuge tests in saturated Hostun HN38 sand, have been performed on single-helix helical model pile in order to study the cyclic behavior.

METHODOLOGY

Centrifuge modeling

Centrifuge modeling is used here on small scale models installed in a macro gravity field allowing for the replication of stress state occurring within prototype soil. The Ifsttar's geotechnical centrifuge (Fig. 1) is a 200×g.ton machine with a radius of 5.5 m that allows to carry 2000 kg at a g-level of 100. The tests presented are carried out on 1/10 scale piles scaled at 10 times the Earth's gravity (10×g). In this case, the main scaling laws [START_REF] Garnier | Catalogue of scaling laws and similitude questions in geotechnical centrifuge modelling[END_REF]) used to link the prototype (full scale) to the (small-scale) centrifuge model are listed in Table 1: for instance x*=x model /x prototype .

Distance x* L 1/N Stress σ* M/LT 2 1 Density ρ* M/L 3 1 Gravity g* L/T 2 N Displacement u* L 1/N Strain ε* - 1 Velocity v* L/T 1 Acceleration a* L/T 2 N Frequency f* 1/T N Force F* ML/T 2 1/N 2 Unit weight γ* M/L 2 T 2 1 Mass m* M 1/N 3 Torque T* ML 2 /T 2 1/N 3

Model soil

The model soil consists of Hostun HN38 sand (Table 2) with a relative density of 89% obtained by air pluviation into a rectangular strongbox of internal dimensions 1200 mm × 800 mm × 360 mm (length × width × depth). Once the pluviation finished, the strongbox is connected to a water tank by the underside to progressively saturate the sand mass. The final water table elevation is 4 cm above the level of the sand surface.

Model pile and experimental campaign

The model pile is a single helix rigid steel pile with a shaft diameter of d = 10 mm, helical plate diameter of D = 33 mm, a helix pitch of 9.7 mm and helix thickness of 1.5 mm. The corresponding prototype pile is then 100 mm in shaft diameter, 330 mm of helix diameter, 97 mm of helix pitch and helix thickness of 15 mm. The experimental set-up presented in Fig. 2 permits in-flight installation and loading of the pile. The pile is installed with a controlled rotation rate up to a helix embedment of 7.4D. The vertical jacking rate is also controlled in order to ensure a penetration of one pitch per revolution of the pile. A load-torque sensor is placed at the head of the pile and permits the measurement of the force and the torque applied at the pile during the installation and the following loading tests. The cyclic testing from installation, loading, and pull-out followed the typical procedure outlined in [START_REF] Schiavon | Centrifuge modelling of a helical anchor under different cyclic loading conditions in sand[END_REF]. Each cyclic loading was conducted up to a vertical accumulated displacement of 10%D, or at least for a minimum number of cycles N = 1000. The cyclic loading was controlled in force performing a constant-amplitude sine function. The characterizing parameters of cyclic loading Qm (mean axial cyclic load) and Qc (axial cyclic load amplitude) are defined by:

/2 [1] /2 [2]
in which, Qmax and Qmin are the maximum and minimum axial load values applied to the model pile head during cycling (Fig. 3).

Fig. 3: Load controlled cyclic loading

A set of 18 cyclic tests with variable amplitudes has been performed in this study [START_REF] Maatouk | Behavior of helical piles used as a floating wind turbine anchor: physical modelling in a centrifuge[END_REF], using a frequency of 0.3 or 0.4 Hz. The initial Qm is reached under a displacement-controlled movement of the helical pile, and then the cyclic loading is force-controlled.

RESULTS

Effect of installation rotation speed in saturated sand

In the field (at full scale), classical rotation rate during helical pile installation is generally lower than 20 rpm. Here the question of quasi-static or dynamic loading, locally on the helix, is raised as the scaling laws (e.g. [START_REF] Garnier | Catalogue of scaling laws and similitude questions in geotechnical centrifuge modelling[END_REF]) suggest that in case of non-quasi-static loading, the viscosity on the pore fluid should be increased by a factor 10. Another point is about the pore pressure dissipation: is the behavior of the soil in the vicinity of the helix under drained or undrained conditions? In order to estimate the potential effect of the rotation rate of the helical pile during the installation, two rotation rates of installation have been tested: 1 and 13 rpm. Taking into account the geometry of the pile (d = 10 mm and D = 33 mm) and the rotation rates, the relative helix/sand grain local shear velocity varies from the range 0.52-1.7 mm/s to the range 6.8-22 mm/s.The installation phase shows for the torque calculated over the deepest 3D (T3D) a variation of 15% between the 2 tests (Fig. 4), while the difference between the maximum tensile loads is less than 4% (Fig. 5). Regarding the dimensionless torque factor (QT.D/T3D) suggested by [START_REF] Byrne | Helical piles: an innovative foundation design option for offshore wind turbines[END_REF], it increases from 4.80 to 5.45 (13% of variation) when increasing the rotation rate by 13. It can be concluded that, even if a difference is observed on the torque profiles, the whole behavior in tension is not affected so much. Of course, additional tests with different rotation rates should be useful for a better understanding of the phenomena, including pore pressure measurements.

Stability diagram

The study of foundation systems subjected to cyclic loads must include not only its capacity but also its displacements over time. [START_REF] Poulos | Cyclic stability diagram for axially loaded piles[END_REF] proposed a stability diagram to evaluate the response of regular piles under cyclic loading: the axes are ⁄ versus ⁄ , where is the monotonic ultimate resistance in tension (Fig. 5). [START_REF] Tsuha | Behaviour of displacement piles in sand under cyclic axial loading[END_REF] recommend a definition of the stability zones according to the displacement rate. They are classified into three zones: stable, unstable and metastable.

• A Stable (S) zone, where axial displacements stabilize or accumulate very slowly for a considerable number of cycles greater than 1000.

• An Unstable (U) zone, where displacements accumulate rapidly until reaching the failure for a small number of cycles less than 100.

• An intermediate Meta-Stable (M) zone, where displacements accumulate at moderate rates until reaching the failure between 100 and 1000 cycles.

In this cyclic diagram (Fig. 6), the theoretically allowable loading is inside the hatched triangle below the diagonal corresponding to the safety factor equal to 1. All the performed tests are in the one-way tensile cyclic loading zone. Based on the anchor head displacement, there are ten cyclic tests where the displacement of the anchor head accumulates slowly and does not reach the displacement limit of 10% of the helix diameter (33 mm in prototype scale) during the 1000 cycles and can be classified as stable (S). Three cyclic tests are classified as Unstable (U) in which the displacement increases rapidly reaching a limiting value of 10%D during the first 100 cycles, and five cyclic tests are classified as Meta-Stable (M) in which the anchor head reaches a displacement of 10%D between the 100 and 1000 cycles. The stable, Meta-Stable and Unstable zones cannot be precisely delimited due to the low number of tests performed but a safety factor of 1.6 can be defined during the design of the studied pile (d = 100 mm and D = 330 mm in prototype scale) to insure a stable response of the pile. Additional tests are necessary to more accurately investigate the boundaries between the 3 zones.

CONCLUSIONS

Centrifuge tests on 1/10 scale models of single-helix anchor have been performed, including in-flight installation phase and tension loading. If the installation torque appears 15% smaller for rotation rate 13 times higher the ultimate monotonic resistance is only 4% smaller, which is in the range of classical natural variability, showing almost no effect of the installation rotation rate on the tension resistance. A set of 18 cyclic tests has permitted to identify the limits between the stable, metastable and unstable zones in the stability diagram, suggesting a possible safety factor of 1.6.

Fig. 1

 1 Fig.1 University Gustave Eiffel's geotechnical centrifuge

Fig. 2 :

 2 Fig. 2: Experimental set-up (inner dimensions)

Fig. 4 :Fig. 5 :

 45 Fig. 4: Installation torque against depth for rotation rate of 1 and 13 rpm.

Fig. 6 :

 6 Fig. 6: Cyclic interaction diagram.

Table 1 . Scaling law for different parameters (with N = 10) Parameter Notation Unit Scaling factor

 1

Table 2 : Characteristics of Hostun sand HN38 (Schiavon et al. 2017)

 2

	Sand	d60/d10	d50 (µm)	min (kg/m 3 )	max (kg/m 3 )
	Hostun HN38	1.97	120	1223	1565

© 2020 Deep Foundations Institute

ACKNOWLEDGEMENTS

The authors wish to thank WEAMEC (Projects REDENV-EOL and JE-CORECT), the Region Pays de Loire, Nantes Metropoles and CARENE for their financial support to the project grants, within the context of which this study has been conducted. Special thanks to the Univ. Gustave Eiffel Centrifuge team for its technical support and assistance during the centrifuge experimental campaign.


David Cathie
email: david.cathie@cathiegroup.com

Cathie

Khalida Sassi
email: khalida.sassi@sbmoffshore.com

Matthieu Blanc
email: matthieu.blanc@univ-eiffel.fr

Luc

HOLDING CAPACITY OF SUCTION ANCHORS WITH TRENCH -CENTRIFUGE TEST RESULTS AND INTERPRETATION

Keywords: holding capacity, anchor, trench, centrifuge modelling, interpretation

As part of the research initiated by SBMO on the chain trenching phenomena, a series for centrifuge tests (mainly at 75g) were performed on model anchors embedded in reconsolidated and intact Gulf of Guinea clay. Vertical and inclined pull-out tests were performed both with and without a trench. Anchor movement and pore pressures (suction) at different positions in the soil below the anchor were monitored during loading to assess reverse end bearing mobilisation. The results demonstrate that the trench does not significantly affect reverse end bearing but the holding capacity for typical inclined loading is reduced by approximately 20% for this specific anchor and trench geometry. 2D and 3D total and effective stress finite element analyses were performed as part of the interpretation work and some results are briefly presented in this paper.

INTRODUCTION

In-service oscillations of the anchor chain in semi-taut mooring systems have been observed to create a trench in front of the anchor in some soils [START_REF] Bhattacharjee | Serpentina FPSO Mooring Integrity Issues and System Replacement: Unique Fast Track Approach[END_REF][START_REF] Arslan | Remaining Capacity of the Suction Pile due to Seabed Trenching[END_REF][START_REF] Colliat | Formation and development of seabed trenching from subsea inspection data of deepwater gulf of guinea moorings[END_REF][START_REF] Sassi | Effect of Seabed Trenching on the Holding Capacity of Suction Anchors in Soft Deepwater Gulf of Guinea Clays[END_REF]. This potentially affects the ultimate holding capacity of the anchor.

As part of the re-assessment of the holding capacity of anchor piles with a trench, SBM Offshore (SBMO) in collaboration with Total commissioned a series of centrifuge tests at the University Gustave Eiffel (Nantes Campus). SBMO commissioned Cathie Associates (Cathie) to provide advice on the program and provide interpretative reports following the tests.

The scope of the centrifuge testing included a series of vertical and inclined holding capacity tests on pile anchors without a trench (to provide reference test cases) and with a trench. A maximum trench depth equal to the padeye depth (9.5m below mudline or about 2/3 of the pile penetration) was considered. Centrifuge tests included measurement of pore pressures at different levels below the anchor tip and measurements of pile head settlements and rotations to provide insights into the failure mechanism and mobilisation of reverse end bearing (REB). In total, 10 series of centrifuge tests were performed in 10 containers with different types of sample reconstitution (CT1 -CT10). These included vertical pull-out test series (CT2, CT7, CT8) and inclined load test series (CT6, CT9 and CT10) which are the subject of this paper.

Finite element (FE) simulations of the tests were performed but only selected results are provided in this paper. Further details of the FE work are available in [START_REF] Sassi | Effect of Seabed Trenching on the Holding Capacity of Suction Anchors in Soft Deepwater Gulf of Guinea Clays[END_REF].

CENTRIFUGE MODELLING

General

All reported centrifuge tests were performed in the Univ. Gustave Eiffel (formerly Ifsttar) fixed beam centrifuge operating at 75 or 80g. Further details are provided in [START_REF] Sassi | Effect of Seabed Trenching on the Holding Capacity of Suction Anchors in Soft Deepwater Gulf of Guinea Clays[END_REF]. Two slightly different model piles were used so that the prototype anchor pile was 4.5m OD and 15m penetration. For series CT2, the pile was made of aluminium with a machined surface and 56mm wall thickness. For all other tests, the piles were of stainless steel, sandblasted with a wall thickness of 37.5mm. The padeye was attached at 9.5m below seabed level. In the first tests, installation was by push-in. In subsequent tests, a combination of self-weight penetration and suction installation was used.

Soil model

Gulf of Guinea (GoG) clay was used for these centrifuge tests, either in reconsolidated slurry form or as intact blocks cut directly from undisturbed large diameter samples. Intact blocks were used over the whole model for CT2. All other tests reported here used reconsolidated GoG slurry to form the main soil body, in order to mobilize the REB, which was not developed in blocky reconstituted soil mass (probably due to preferential drainage at the blocks' interfaces). A zone was then cut out (at 1g) in front of the pile and replaced with undisturbed blocks into which a trench was cut. These model soils are referred to as the "hybrid" model. The undisturbed soil allowed a realistic and stable trench to be formed. However, the use of blocks creates interfaces between blocks which were approximately normally consolidated after the final centrifuge consolidation stage. As the particle size of GoG clay is very small in comparison to the size of the suction caisson model or of the T-bar, no soil material scale effect needs to be considered (e.g. [START_REF] Garnier | Catalogue of scaling laws and similitude questions in geotechnical centrifuge modelling[END_REF].

Water content profiles were taken at the end of most of the tests and several miniature T-bar tests (5mm OD) were performed at different stages of testing in both reconstituted clay and in blocks (at 3mm/sec to ensure undrained conditions). The results are interpreted using N T=10.5 (e.g. [START_REF] Gaudin | Un aperçu de l'utilisation du T-Bar miniature en centrifugeuse Journées Nationales de Géotechnique et de Géologie de l'Ingénieur[END_REF]) in order to estimate the undrained shear strength (Figure 1). The differences in shear strength profile are a result of slight changes between the test series affecting consolidation procedures, consolidation stresses, layering or incomplete consolidation at 1g due to scheduling limits. Field target strength profiles are enveloped by the range achieved. The lower water content and thus higher unit weight of the reconstituted clay is clearly apparent in Figure 1 (right-hand panel). Cyclic T-bar tests were also performed to characterize the sensitivity. A simplified set of undrained strength profiles (Table 1) are considered representative of the test conditions.

Trench

Tests CT2 -CT9 were performed with a trench geometry which had a width of 2D at the surface reducing to 1D at a depth of 1D, and then constant width of 1D down to the padeye level. This was selected at the start of the program before actual field data from surveys was available [START_REF] Sassi | Effect of Seabed Trenching on the Holding Capacity of Suction Anchors in Soft Deepwater Gulf of Guinea Clays[END_REF][START_REF] Colliat | Formation and development of seabed trenching from subsea inspection data of deepwater gulf of guinea moorings[END_REF]. For the final test set (CT10), a realistic geometry with a side slope of approximately 60° was adopted (based on observed stable slopes in the field).

Loading

Tests reported herein were performed for vertical and inclined loading (between 30° and 40° representative of extreme conditions for semi-taut deepwater mooring systems). Table 2 summarises the loading and trench conditions for the tests presented in this paper. All test series except CT2 were performed at a loading rate of 25mm/min. This may be compared to the T-bar penetration rate of 180mm/min (3mm/sec). No rate correction has been applied in this report to the inclined load tests since all were performed at the same rate. However, a rate correction in the form of a strength reduction factor is applicable to matching FE modelling based on the T-bar measured shear strength with centrifuge model results.

VERTICAL LOADING TEST RESULTS (CT2, CT7, CT8)

Test series CT2 was performed in the blocky clay model. The net pull-out load-deflection response of the CT2 series of tests are presented in Figure 2, where the net load is the measured pull-out load less the anchor weight.

The presence of the trench has negligible influence on the ultimate vertical pull-out capacity when full REB is available (tests C2P1 and C2P2) as the trench does not affect the mobilized end bearing resistance. A slight reduction in stiffness is identified which may be due to the reduced volume of soil around the anchor due to the trench.

For the two tests C2P3 without REB (the anchor tip was in contact with a sand layer at the bottom of the container which was drained), the consistency is very good. Furthermore, a markedly different behaviour is obtained for these tests without REB, showing that the REB capacity represents about 50% of the ultimate pull-out capacity, as noted by [START_REF] Andersen | Suction anchors for deepwater applications: Keynote lecture[END_REF]. Peak-residual response is apparent for shaft resistance only, as observed also for pile shaft resistance in sensitive soft clays. The REB is mobilized at higher displacements (10%) than shaft resistance (1.4%). This is in good agreement with previously published centrifuge tests results, e.g. [START_REF] Raines | Centrifuge modelling of suction piles in clay[END_REF]. An average interface factor α of 0.34 can be back calculated for the inside and outside shaft resistance from tests C2P3.

Test series CT7 was designed to replicate CT2, but with the model soil primarily consolidated from clay slurry and including measurement of the pore pressures. A thicker model soil was used to reduce the base boundary effects, and the anchor was penetrated by suction. In the area of the trench, the consolidated slurry was replaced by intact clay blocks ("hybrid" model soil). Results for the series CT7 and CT8 are presented in Figure 3. In the reference test C8P1, the pile had to be removed and re-installed which probably explains the lower pull-out capacity.

A larger set of experiments (more than 2) is ideally required to confirm these results.

A summary of the results of all vertical pull-out tests is provided in Table 3. Comparing CT7 and CT8, the uplift resistance in both reference cases is lower than the resistance with a trench. Considering the installation issues with both tests for the reference pile, it is not prudent to draw conclusions from these results except that the pull-out capacity does not appear to be significantly affected by the trench. This is probably due to the large contribution of the REB as revealed by the suction pressures.

Note: P1 -reference tests without trench, P2 -tests with trench Suction pore water pressures at the pile tip are slightly lower for the trench case in both tests although the effect of disturbance in the reference tests on these pressures is unknown. This may imply a reduction in REB due to the trench but if there is any reduction it is very limited. Pore pressure dissipation was negligible based on a reasonable estimate of the coefficient of consolidation and drainage path length between pile tip and the trench surface.

The lower uplift capacities in series CT2 (Figure 2) when compared to CT7 and 8 (Figure 3) may be due to slower pull-out rate, partial dissipation of suction pressures on block interfaces, and the proximity of the bottom sand boundary at 1.3D below the anchor tip. The machined aluminium pile material used for CT2, may also have contributed to the lower uplift resistance compared to the CT7 and 8 tests.

The effective REB bearing capacity factor Nc (defined here as ∆u/su_tip, where ∆u is the suction pressure at the pile tip) is shown in

INCLINED LOADING TEST RESULTS (CT6, CT9, CT10)

Results for the inclined loading tests are summarised in Table 4 and the load-deflection curves in Figure 4 (continuous line for reference tests, dotted lines for with trench tests). ). The lower capacity exhibited in C6P1 is probably due to the lower shear strength profile (on the order of 70% -see Figure 1). The presence of the trench causes a reduction in holding capacity of between 15 -21% although the effect of the different loading angles may also play a part. In the case of CT10, in which the loading angle was the same, the capacity reduction was 15%.

In all the tests the maximum capacity is mobilized between 14% and 22%D; at these ratios, the suction pressure measurements at the tip were typically in the range 80 to 100kPa which is much less than the fully mobilised suction pressure observed in the vertical loading tests (150 -200kPa, see Table 3). This limited mobilisation of suction pressure is due to the translational and rotational failure mechanism of the pile. Measured pore pressures broadly follow the trend of the force-displacement response suggesting a direct relationship.

Finite element (FE) modelling of the tests was performed; further details are provided in [START_REF] Sassi | Effect of Seabed Trenching on the Holding Capacity of Suction Anchors in Soft Deepwater Gulf of Guinea Clays[END_REF]. The ability of a simple Mohr Coulomb/Tresca model, and an effective stress model to predict the holding capacity of the anchor with and without trench is summarised in Table 5. Soil parameters for the modelling were based on best estimates from the in-situ and laboratory testing. No attempt has been made to calibrate the finite element model to fit the centrifuge results.

Without a strain rate correction, the FE results at 0.1D tend to overshoot the measured capacities. If the strain rate correction is applied (0.84 -see above) the FE results for CT6 and CT9 match quite well but undershoot significantly for CT10.

The failure mechanism of piles with and without a trench for the inclined load tests is by backwards rotation and forward translation. The failure plane is visible from a section which was excavated after the test (Figure 5). This figure also reveals full attachment of the soil and pile in the passive zone. No gap developed despite quite large rotations and translation. Concentric shear failure planes were observed in the soil behind the pile after large deformations (0.5D). These are also consistent with undrained shear failure rather than tensile cracking. The failure mechanism of a caisson anchor in soft clay is dominated by the suction pressures at the rear (which do not allow development of a gap) and below the caisson (i.e. REB). Suction pressures are the undrained response of the soil to the applied total stress changes and can be seen in the FE simulation shown in Figure 6.

Comparison with other published results

Failure envelopes for similar pile dimensions and in similar soil conditions with and without a trench have been published by [START_REF] Arslan | Remaining Capacity of the Suction Pile due to Seabed Trenching[END_REF] and [START_REF] Hernandez-Martinez | Trenching effects on holding capacity for in-service suction anchors in high plasticity clays[END_REF] based on finite element analyses. For comparison with the centrifuge data presented in this report, the calculated envelopes were normalised with their respective values of Vu (Nc*Su where Su is the undrained strength at tip level and Nc = 10 based on [START_REF] Martin | Vertical bearing capacity of skirted circular foundations on Tresca soil[END_REF] as shown in Figure 7.

The present centrifuge results are broadly consistent with the Hernandez findings although the centrifuge tests suggest a slightly higher holding capacity for the inclined load tests in which 2 of the 3 tests lie outside the respective envelopes.

CONCLUSIONS

The centrifuge model testing on caisson anchors in clay with a trench has led to the following conclusions:

1. For the anchor and trench geometry studied in this work, the trench in front of the anchor pile leads to a reduction in holding capacity of 15 -21% for load inclinations between 30° and 40°. Passive suction behind the anchor pile and reverse end bearing (provided by suction) ensures that the loss of capacity is limited.

2. Under vertical load, the trench has very little impact on the holding capacity because the main component of pull-out resistance is the REB. This was confirmed by the suction pressure measurements below the base of the anchor.

3. The failure mechanism for the pile tested was backward rotation combined with forward translation. This mechanism helps to ensure that the suction pressures at the pile top are limited.

4. Detachment was not observed at small and medium pull-out displacements (less than 50% of the diameter). At this deformation, concentric shear failure planes were observed in the soil behind the pile which is consistent with undrained shear failure rather than tensile cracking.

5.

Although not presented in detail in this paper, finite element modelling using the undrained shear strength measured in the model soil, with a strain rate correction applied can be used to predict the holding capacity of a pile anchor with a trench. Mohr Coulomb/Tresca criteria with φ=0 or an effective stress model with undrained conditions and c / φ' derived to match the average shear strength profile with depth may be used

  Note: Only simplified strength profiles from T-bar tests are shown for CT2, 6 and 7 for legibility

Figure 1 :

 1 Figure 1: Static T-bar results and water content profiles

Figure 2 :

 2 Figure 2: Vertical pull-out test series CT2 using intact clay blocks showing influence of REB and trench

Figure 3 :

 3 Figure 3: Vertical pull-out tests CT7 and CT8

Figure 4 :

 4 Figure 4: Comparison of results of inclined pull-out tests CT6, CT9, CT10

Figure 5 :

 5 Figure 5: Backward rotation and failure plane (C9P2)

Figure 6 :Figure 7 :

 67 Figure 6: Excess pore water pressures

Table 1 : Simplified soil profiles

 1

	Parameter	Unit	CT2	CT6	CT7	CT8	CT9	CT10
	Submerged unit weight	kN/m 3	3.5	6	6	6	6	6
	su0	kPa	1	1	2	3	2	2
	k	kPa/m	1.4	1.4	1.35	1.9	2.1	1.8
	su at pile tip	kPa	22	22	22	32	34	29

Notes: Su based on T-bar test with NT=10.5 Higher unit weight in CT6-10 corresponds to reconstituted slurry with lower water content than natural Angola clay

Table 2 : Summary of centrifuge tests

 2

Table 3

 3

	. Values of Nc between 6 and 8 agree with the

Table 3 : Results of vertical pull-out test series -CT2, CT7 and CT8

 3

	Test	Installation method	Trench	Peak pull-out resistance [MN]	∆u peak [kPa]	[%] δ/D at peak resistance	REB Nc ∆u/su (3)
	C2P1	Push in	No	3.1	-	9
	C2P2	Push in	Yes	3.0	-	12
	C2P3 (1)	Push in	No	1.6	-	1.4
	C7P1	Self-weight/suction	No	3.6 (2)	-172	12	7.8
	C7P2	Self-weight/suction	Yes	4.2	-148	5	6.7
	C8P1	Self-weight/suction	No	4.6 (2)	-206	20	6.4
	C8P2	Self-weight/suction	Yes	5.1	-185	20	5.9

Notes: (1) Coring and vented; (2) soil disturbed due to repeat installation probably reduced uplift resistance; (3) ignoring potential rate correction which would reduce su and increase Nc

Table 4 : Comparison of inclined pull-out tests CT6, CT9 and CT10

 4

	Test	Installation method	Trench	Load Inclination [°]	Maximum pull-out resistance [MN]	∆u 1 [kPa]	δ/D 2 [%]
	C6P1	Push-in	No	30	6.7	-80	22
	C6P2	Push-in	Yes	40	5.4	-76	16
	C9P1	Self-weight/suction	No	30	8.0	-97	19
	C9P2	Self-weight/suction	Yes	40	6.3	-	16
	C10P1	Self-weight/suction	No	35	8.2	-95	14
	C10P2	Self-weight/suction	Yes	35	7.0	-91	14
	δ = anchor displacement at load application point, D = pile diameter, ∆u = tip pore pressure 1 at peak pullout resistance 2 at peak pullout resistance -displacements corrected to compensate for ovalization (plastic deformation) of the
	pile
	C9P1 and C10P1 reference cases indicate very similar load/deflection behaviour and ultimate
	capacity (~8MN

Table 5 : Comparison of measured and predicted holding capacity for inclined tests

 5

					Predicted	Predicted
	Test	Trench	Load Inclination [°]	Measured holding Rm [MN] capacity,	holding capacity [MN] FEA 1 , from	capacity corrected effect (2) , Rp rate for strain	Ratio Rm/Rp
						[MN]
	C6P1	No	30	6.7	7.7	6.5	1.04
	C6P2	Yes	40	5.4	6.2	5.2	1.04
	C9P1	No	30	8.0	9.2	7.7	1.04
	C9P2	Yes	40	6.3	6.9 (3)	5.8	1.09
	C10P1	No	35	8.2	6.6	5.5	1.48
	C10P2	Yes	35	7.0	6.9	5.8	1.21

Notes: (1) FE holding capacity taken at 0.1D; (2) strain rate correction 0.84; (3) Effective stress soil model

ACKNOWLEDGEMENTS

The authors acknowledge the contributions of colleagues to this work: Matteo Castelletti, Fabrizio Calidonna, Chiara Prearo and Emilio Nicolini of Cathie, and Fabrizio Ardoino and Robin Givet of SBMO. Alain Puech of Fugro assisted with the initial planning, and Jean-Louis Colliat of Total has followed and guided the project throughout.


A coupled LDVM / IBM investigation of incompressible flows around moving profiles at low Reynolds number

A numerical solver based on the immersed boundary method has been recently developed on OpenFoam [3], a free software.

In this study, we investigate the Immersed Boundary Method (IBM) at low Reynolds number (Re=1000) for incompressible flow around NACA 0012 airfoil and then around a flat plate and a NACA 23012 airfoil in rotation (from 0° to 90°). The Leading-edge suction parameter Discrete Vortex Method (LDVM) is also investigated in the models using a rotation of the airfoils.

INTRODUCTION

Deep stall is a specific dangerous form of airplane stall. It occurs when there is a stall on the main wing and the horizontal tail is inside the detached wake of the main wing. The tail loses its efficiency, leading to a stable pitching equilibrium without any easy recovery method. This phenomenon affects certain aircraft designs, most notably those with a T-tail configuration. It is so of primary importance to investigate in details the flow features as well as the physical mechanisms leading to this phenomenon in order to develop efficient control strategies as well as reliable predictions methods during the conceptual and preliminary design phase of T-tail aircraft.

In this study, we aimed at investigating the flow developing over two-NACA 23 012-airfoils as proposed in Hétru (2015) using a high-fidelity model of DNS implemented in OpenFOAM and based on a recent and efficient Immersed Boundary Method [START_REF] Constant | An immersed boundary method in OpenFOAM: verification and validation[END_REF]. The IBMpisoFoam solver was validated with a cylinder geometry [START_REF] Constant | An immersed boundary method in OpenFOAM: verification and validation[END_REF] and for thin airfoils [START_REF] Riahi | A pressure-corrected Immersed Boundary Method for the numerical simulation of compressible flows[END_REF] at low Reynolds numbers (Re<200). However, some preliminary tests show that the mass error (Eq14) seems to increase with the value of the Reynolds numbers. For a first validation, simulation for thin airfoil profile at higher Reynolds (Re=1000) is necessary.

The validation of the method carried out for flows over bluff bodies [START_REF] Constant | An immersed boundary method in OpenFOAM: verification and validation[END_REF] is extended here for flows over airfoils at low Reynolds number (Re = 1000) and at low angle of attack (0° and 10°) [10]. The flow over a moving flat plate with a rotation from 0° to 90° is then considered. The Lagrangian mesh used to model the airfoil in the Immersed Boundary Method is adapted here to avoid mass leakage when moving over the Eulerian fixed mesh where fluid equations are resolved [START_REF] Riahi | A pressure-corrected Immersed Boundary Method for the numerical simulation of compressible flows[END_REF]. Despite spurious high frequencies oscillations due to the spatial uncertainty of the moving solid frontier, the aerodynamics coefficients of drag and lift are accurately recovered with respect to the literature. In addition, a Leading-edge-suction-parameter modulated Discrete Vortex Method [START_REF] Ramesh | Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding[END_REF] is calibrated from the DNS to be used as a reduced model in future investigations.

RESEARCH METHOD

Immersed Boundary method (IBM)

The incompressible flow is described by the unsteady and dimensionless Navier-Stockes equations:

∇. 𝑢 = 0 (1) ∂𝑢 𝜕𝑡 + ∇. (𝑢𝑢) = -∇𝑝 + 1 𝑅 𝑒 ∆𝑢 (2)
The geometry is defined by the use of Lagrangian points in the Eulerian mesh (Fig. 1). A penalization method is used to impose a non slip condition in the cells where the Lagrangian markers are located.

Figure 1: Definition of the geometry

The predictor-corrector solver pisoFOAM (used in body-fitted method) has been modified [START_REF] Constant | An immersed boundary method in OpenFOAM: verification and validation[END_REF] to inclined the definition of a force term f (Fig. 2). 𝑥 𝑗 is the position of the centre of the cell j, 𝑋 𝑠 is the position of the Lagrangian marker s and ∆𝑣 refers to the Eulerian quadrature (∆𝑣 = ∆𝑥∆𝑦∆𝑧) for the case of a Cartesian uniform mesh 𝛿 ℎ is the discretized delta function used in [START_REF] Peskin | An adaptative version of the immersed boundary method[END_REF].

𝛿 ℎ = { 1 3 (1 + √-3𝑟 2 + 1) 0 ≤ 𝑟 ≤ 0.5 1 6
(5 -3𝑟 -√-3(1 -𝑟) 2 + 1) 0.5 ≤ 𝑟 ≤ 1.5

0 1.5 < 𝑟 } (6) 𝑓 𝑛+1 (𝑥 𝑗 ) = ∑ 𝐹 𝑠 𝑛+1 𝛿 ℎ (𝑥 𝑗 -𝑋 𝑘 )𝜀 𝑗∈𝐷 𝑠 𝑘 (7)
The k-index refers to a loop over the lagrangian markers whose support contains the Eulerian node j. 𝜀 𝑘 is the Lagrangian quadrature, which is calculated by solving a linear system: 𝐴𝜀 = 1 (8) Where the vectors 𝜀 = (𝜀 1 , … , 𝜀 𝑁𝑠 ) 𝑇 and 1 = (1, … ,1) 𝑇 have a dimension Ns corresponding to the number of lagrangian markers. A is the matrix defined by the product between the 𝑘 𝑡ℎ and the 𝑚 𝑡ℎ interpolation kernels such that:

𝐴 𝑘𝑚 = ∑ 𝛿 ℎ (𝑥 𝑗 -𝑋 𝑘 )𝛿 ℎ (𝑥 𝑗 -𝑋 𝑚 ) 𝑗∈𝐷 𝑚 (9

Correction of the force term

The main problem with a penalization method in IBM is the precision in the non-slipping condition. In fact, for the incompressible case, the force term is not corrected during the correction loop. This error can be estimated with me maximum residual velocity value (Vres) or the error mass value.

𝑉 𝑟𝑒𝑠 = max(𝑢 𝑘 )/𝑈 ∞ [START_REF] Riahi | A pressure-corrected Immersed Boundary Method for the numerical simulation of compressible flows[END_REF] 𝑒𝑟𝑟𝑜𝑟 𝑚𝑎𝑠𝑠 = max(𝜌 𝑖𝑗 𝑑𝑖𝑣(𝑢 𝑖𝑗 )𝑆 𝑖𝑗 )

With 𝑆 𝑖𝑗 the area of the cell (i,j) and 𝑢 𝑘 the value of the residual velocity at the 𝑘 𝑡ℎ lagrangian marker.

Mesh

The present study performs simulations at Re=1000, the flow is laminar, therefore a large domain is needed (Fig. 3). The structured grid is used and since this is a DNS simulation, the cell size shall be dx=dy=1/Re=0.001c near the geometry (Fig. 4,5,6).

LDVM method

The Leading edge suction parameter Discrete Vortex Method (LDVM) is based on the potential thin airfoil theory in unsteady flows, applicable for large values of the angle of attack [START_REF] Ramesh | Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding[END_REF]. It is built on the time-stepping approach of [START_REF] Katz | Low-Speed Aerodynamics[END_REF] with the addition of a criterion for the leading edge detachment. In Fig. 7, the aerodynamic frame (X,Z) is fixed and the airfoil frame (x,z) sees an upstream velocity U aligned on the X axis with a time-dependant angle of attack (t).

A vertical motion h(t) is considered along the Z axis. The fluid velocity components are respectively U and W in the aerodynamic frame and w is the velocity along z in the airfoil frame. The local circulation distribution over the airfoil is obtained from a Fourier series:

( ) ( ) ( )       + + =   =  1 0 sin sin cos 1 2 , n n n t A t A U t      (16)
with A0(t), … , An(t) the time-dependant Fourier coefficients, c the airfoil chord and with:

( )  cos 1 2 - = c x ( 17
)
The Kutta condition is enforced through the form of the Fourier series. The Fourier coefficients are determined from the instantaneous local downwash w(,t) by enforcing the potential flow boundary condition that the velocity is tangential to the airfoil surface:

( ) ( )   - =     0 0 , 1 d U t w t A (18) ( ) ( )   =      0 cos , 2 d n U t w t A n ( 19
)
where w(,t) is the velocity normal to the airfoil calculated from the motion kinematics:

( ) ( )

x x h x x U x x h U x t w LEV TEV p LEV TEV   -   - - - - -         +   + +   =              cos sin sin cos ,    (20)
with TEV, LEV the velocity potentials associated with the leading edge and trailing edge vortices,  the airfoil mean camber line and xp the pivot position. At each time step, if the flow around the airfoil is attached, a trailing edge vortex (TEV) is released and advected by the flow at the following time step. However, a separation model is needed for large values of the angle of attack. The separation on the airfoil is obtained with an inviscid parameter developed by [START_REF] Ramesh | Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding[END_REF]. The Leading Edge Suction Parameter (LESP) is a non-dimensional measure of the suction at the leading edge, and equates the first Fourier coefficient:

( ) ( ) t A t 0 LESP = (21)
The critical value LESPcrit corresponds to the A0 value associated with the angle of attack for which spikes appear in the negative part of the friction coefficient, near the leading edge. It is a measure of the maximum suction that a given airfoil can bear before separation and is independent of the motion. Beyond that LESPcrit value, the airfoil suction side boundary layer separates from the leading edge, which corresponds to the release of a leading edge vortex (LEV). In that case, there is shedding of both a TEV and a LEV. These vortices, at every time step, must enforce Kelvin's circulation theorem:

( ) 0 1 , 1 , = + +   = = i l l LEV i k k TEV B t    (22)
with the airfoil bound circulation:

( ) ( ) ( ) ( )       + = =   2 , 1 0 0 t A t A c U d t t B       (23)
The previous variables are written in a non-dimensional form for the fluid mechanics problem as:

c U c Z Z c X X U W W U w U U w w           = = = = = =   (24)
The velocity induced by a given vortex is described with the model of Vatistas which incorporates a finite core radius rc [START_REF] Vatistas | A simpler model for concentrated vortices[END_REF].

( ) ( )   ( ) ( )   4 2 2 2 4 2 2 2 2 2                   + - + - - - = + - + - - = c k k k k k c k k k k k r Z Z X X X X W r Z Z X X Z Z U     (25)
The non-dimensional time step is:

015 . 0 = =   c tU t   (26)
The vortex core radius is taken to be 1.3 times the average spacing between the vortices [START_REF] Faure | Simulation of the aeroelastic behavior of a possibly detached flow airfoil by a discrete vortex method[END_REF]:

  = = t c r r c c  3 . 1 (27)
The last shed vortex is placed at one third of the distance from the shedding edge to the previously shed vortex [START_REF] Katz | Low-Speed Aerodynamics[END_REF]: For each time-step, the unknown parameters are the circulations corresponding to the newly shed vortices, advected by the velocity field. Firstly, if there is only a TEV shed at the iteration i, equation ( 20) is reduced to:

( ) ( ) ( ) ( )   -     -     -     -   - + = - + = - + = - + = LE l LEV LE l LEV LE l LEV LE l LEV TE k TEV TE k TEV TE k TEV TE k TEV Z Z Z Z X X X X Z Z Z Z X X X X 1 , , 1 , , 1 , , 1 , , 3 1 3
( ) 2 , 1 , T T t w i TEV i    + =   (29)
where T1 and T2 are terms depending on the angle of attack, the vertical displacement and the summation of the influence of the previously shed vortices. The only unknown parameter is the circulation of the TEV shed at iteration i. The airfoil bound circulation is obtained from (8) substituting the Fourier coefficients: Secondly, if both a TEV and LEV are shed at iteration i, then equation ( 5) is reduced to:

( ) 3 , 2 , 1 , T T T t w i LEV i TEV i     + + =    (32)
where T1, T2 and T3 are terms depending on the angle of attack, the vertical displacement and the summation of the influence of the previously shed vortices. There is two unknown parameters which are the circulations of the TEV and LEV shed at iteration i, requiring two equations. The airfoil bound circulation is obtained from (8) substituting the Fourier coefficients:

   + + = i LEV i TEV B I I I , 3 , 2 1    (33)
where I1, I2 and I3 are terms resulting of the integrals of T1, T2 and T3. Kelvin's theorem (22) and the criterion on the critical LESP provide:

0 LESP 0 0 1 1 , 1 1 , , , = - = + + + +   - =  - =     crit i l l LEV i k k TEV i LEV i TEV B A      (34)
Substituting the bound circulation and the value of A0, equation ( 19) is:

( ) ( ) 0 LESP 0 1 1 , 3 , 2 1 1 1 , 1 1 , 3 , 2 , 1 = - + + = + + + + + +   - =  - =      crit i LEV i TEV i l l LEV i k k TEV i LEV i TEV J J J I I I       (35)
Note that this is a linear system, and no iteration scheme is required as previously mentioned in [START_REF] Ramesh | Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding[END_REF] and [START_REF] Katz | Low-Speed Aerodynamics[END_REF]. Therefore, a significant gain in calculation time is expected [START_REF] Faure | A modified discretevortex method with shedding criterion for unsteady or constant angle of attack flow prediction[END_REF]. As only the shed LEV and TEV are computed with the method, it is worth noticing that the simulation time is increasing with the number of vortices. In order to reduce that number, the vortices located four chords downstream of the leading edge can be amalgamated into larger structures [START_REF] Sarpkaya | An inviscid model of twodimensional vortex shedding for transient and asymptotically steady separated flow over an inclined plate[END_REF]. That clustering can be realized with a multidimensional binary search tree or k-d tree [2], for all the vortices downstream of four chords of the airfoil. The unsteady form of the Bernoulli theorem is used to calculate the pressure distribution along the airfoil:

( )           -   + - = - t t V V p p
V         +         +         + + =       sin cos ,  (38) ps LEV ps TEV ps B ps t x x x h U V         +         +         + + =       sin cos ,  (39)
From the thin airfoil theory:

( ) ( ) 2 , 2 , t x x t x x ps B ss B     - =         =         (40) Then:  ( ) t x x x h U V V LEV TEV ps t ss t , sin cos 2 2 , 2 ,           +   + + = -   (41)
The potential functions time derivatives are:

( )    =   -   x ps d t t t t 0 ,      (42) Hence (21) becomes: ( ) ( )      +      +         + + = -   x LEV TEV ss ps d t t x x x h U p p 0 , sin cos           (43)
The normal force on the airfoil is obtained:

( ) ( )      +       +          + + =     c x LEV c TEV N dx d t t d t x x x h U F 0 0 0 , , sin cos            (44)
Using the Fourier coefficients, it is reduced to:

( ) ( )            +   +         + + +          + + =    c LEV TEV N dx t x x x A A A c A A h U c U F 0 2 1 0 1 0 , 8 4 4 3 2 sin cos             (45)
The axial force is given by the Blasius formula [START_REF] Katz | Low-Speed Aerodynamics[END_REF]:

2 0 2 A c U F A    = (46)
Similarly, the moment about the position xref on the airfoil is:

( )  ( )            +   -         - + + +    - +    + - =   
             (47)

RESULTS

Validation of the model

The Immersed boundary method is compared with the body-fitted method and the numerical simulations from Kurtulus [10] for 2D NACA 0012 wings at 0° and 10° (Fig. 8).

As expected (Tab.1), the simulations with the body-fitted method (pisoFoam solver) correspond to the other CFD models (1% error for 𝐶 𝐷 and 𝐶 𝐿 ) [10] and despite a one order of magnitude higher of difference for the error mass between the IBM solver and the body-fitted method, the solver seems quite accurate (11% difference for the value of 𝐶 𝐿 and 4% difference with the 𝐶 𝐷 value). (Fig. 9,10) The highest errors on the penalization force term are concentrated at the leading edge. In fact, the maximum residual velocity is at this location and his value is approximately five times the mean value of residual velocities.

The correction of the force term with the pressure gradient do not succeed to increase the accuracy of the penalisation method but increase the error on the residual velocity value.

A high correlation seems to exist (R²>0.95) between the value of Vres and the error obtained for the 𝐶 𝐷 and 𝐶 𝐿 value (considering to the body-fitted method as a reference).

Computation of rotational movement of a flat plate

Following the objective of studying the deep stall phenomenon, the displacement of airfoils must be implemented and validated for the solver. In this study, we use the rotation of a flat plate from 0° to 90° angle as described in the case 5A in Ramesh (2014) [START_REF] Ramesh | Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding[END_REF].

The function of the movement is smoothed [4] [12] to avoid a virtual mass phenomenon. ). (Fig. 12)

A high frequency oscillation can be observed during the movement on the lift and drag coefficient value. This perturbation is due to the change of state of the cells when the Lagrangian markers move. Moreover, even if a time step of 𝑑𝑡 = 5 × 10 -4 is sufficient to keep the courant number < 1, the lift and drag coefficient became independent of the value of the time step for 𝑑𝑡 ≤ 5 × 10 -5 . The values of lift and drag coefficient obtained from the Immersed boundary method follows the one obtained from other CFD method [START_REF] Katz | Low-Speed Aerodynamics[END_REF] (with a relative difference <10%). (Fig. 12,13).

Computation of a NACA 23012 rotation motion

The next step of this study is to apply a rotation motion on an airfoil. The same rotation is applied on a NACA23012 airfoil (incompressible flow, Re=1000).

The vorticity distribution around the airfoil is used [START_REF] Ramesh | Effect of airfoil shape and Reynolds number on leading edge vortex shedding in unsteady flows[END_REF] to define the angle corresponding to the leading-edge separation which is 21.4°. This allow to define a LESPcrit =0. However, there is a great disparity in the lift and drag coefficient between the IBM and the LDVM. The gap between the values of 𝐶 𝐿 is twice the one observed for the flat plane (Fig 16). This result is due to the geometry of the NACA23012. In fact, this geometry is out of the domain of validity of the LDVM method: in one hand, the LEV does not appear at the exact position of the leading edge, and in other hand a detached flow occurs on the trailing edge at low angles (α<10°) ( Fig 15). None of these phenomena are included in the LDVM solver. A modification of the method to determine the exact location of the generated vortex may increase the domain of validity of the LDVM method to thick airfoils.

CONCLUSION

The Immersed Boundary incompressible solver developed on OpenFoam is quite accurate for the simulation of flows around airfoils at low Reynolds number (Re=1000). The main flaw of the solver is the accuracy of penalization force which directly depends on the time step since the force term is not modified during the correction loop (Eq 11 and Eq12).

The LDVM method actually developed by Ramesh is only adapted for thin airfoils and a way to determine the location of the vortex generated may increase the domain of validity for NACA 23012 airfoil.

That is why the Immersed Boundary Method will be prioritized over the LDVM for our study in vortex forcing resulting from the aerodynamic interaction between two NACA 23012 airfoils.

  𝑑 the target velocity and 𝐼[û] 𝑠 the interpolation of the fluid velocity at the Lagrangian marker s.𝐼[û] 𝑠 = ∑𝑢 𝑗 𝑛 𝛿 ℎ (𝑥 𝑗 -𝑋 𝑠 )∆𝑣 𝑗∈𝐷 𝑠[START_REF] Faure | Simulation of the aeroelastic behavior of a possibly detached flow airfoil by a discrete vortex method[END_REF]

-

  𝑛+1,𝑞+1 = -∇. (𝑢 𝑛+1,𝑞 ∇𝑢 𝑛+1,𝑞 ) + ∇. 𝑓(û) (11) ∂𝑢 𝑛+1,𝑞+1 𝜕𝑡 + ∇. (𝑢 𝑛+1,𝑞+1 𝑢 𝑛+1,𝑞+1 ) =

Figure 2 : 4 ) 3 ) 3 Following

 2433 Figure 2: IBMpisoFOAM solver at each time step (n+1)

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Numerical domain

Figure 6 :

 6 Figure 6: Body fitted mesh

Figure 7 .

 7 Figure 7. Airfoil motion and frames of reference.

  where I1 and I2 are terms resulting of the integrals of T1 and T2. Substituting (15) into Kelvin's theorem (22):

  ps and ss respectively for the pressure side and suction side and Vt the tangential velocity. As the flow potential function is the sum of the potential functions of the bound circulation, TEV and LEV:

Figure 8 :

 8 Figure 8: Streamlines of mean velocity field for 0° and 10° angle of attack (NACA0012 Re=1000)

Figure 9 :Figure 11 :

 911 Figure 10: Drag coefficient NACA0012 Re=1000

Figure 13 :Figure 12 :

 1312 Figure13: Drag coefficient of a flat plate in rotation (Re=1000) and comparison with Ramesh (2014)[START_REF] Katz | Low-Speed Aerodynamics[END_REF]

Figure 14 :

 14 Figure 14: Vorticity around a NACA23012 airfoil in rotation (Re=1000). Start of the leading-edge separation.

Figure 15 :Figure 17 :

 1517 Figure 15: Vorticity around a NACA23012 airfoil in rotation (Re=1000)

Figure 16 :

 16 Figure 16: Lift coefficient of an airfoil in rotation (Re=1000)

Table 1 :

 1 NACA0012 Re=1000 α=10°

		𝐶 𝐿	𝐶 𝐷 Strouhal 𝐶 𝐿 /𝐶 𝐷	Error mass	Vres
	Dilek Funda Kurtulus [10]	0,425 0,165 0,876 2,58
	Body fitted method	0,420 0,164 0,8	2,56 3,5E-06
	IBM	0,479 0,172 0,8	2,78 2,5E-05 2,5%
	IBM and
	correction	0,501 0,176 0,7	2,85 2,5E-05 5,4%
	force term