Emilie Hachet

Hélène Van Den Berghe

Eric Bayma

Marc R Block

Rachel Auzély-Velty

Design of biomimetic cell-interactive substrates using hyaluronic acid hydrogels with tunable mechanical properties

Hyaluronic acid (HA) is a natural polysaccharide abundant in biological tissues with excellent potential for constructing synthetic extracellular matrix analogs. In this work, we established a simple and robust/reliable approach to prepare hyaluronic acid-based hydrogels with controlled stiffness and cell recognition properties for use as cell-interactive substrates. This approach relied on a new procedure for the synthesis of methacrylate-modified HA macromers (HA-MA) and, on photorheometry allowing to monitor gelation by photopolymerization. We showed in this way the ability to obtain gels that encompass the range of physiologically relevant moduli while maintaining the recognition properties of HA by specific cell surface receptors. These hydrogels were prepared from HA macromers having a degree of methacrylation in the range of 0.1-0.28, allowing to minimize compromising effects on the

INTRODUCTION

Matrix elasticity or stiffness has been recently shown to have a dramatic effect on many aspects of cell function including adhesion, migration, proliferation and differentiation. In many cases, cells appear to have a preferred substrate stiffness which correlates well with that of the tissue environment from which they are derived. [1][2][3][4][5] In order to mimic native tissue architecture for applications in biotechnology and tissue engineering, various polymer scaffolds have been developed as substitute of the body's natural extracellular matrix (ECM). [6][7] Among these, hydrogels are highly attractive for developing synthetic ECM analogs due to their ability to simulate the microenvironment of most soft tissues. These threedimensional networks made of cross-linked polymer chains possess high water contents allowing for easy solute transport. Furthermore, many hydrogels can be formed under mild, biocompatible conditions and can be straightforwardly modified to exhibit specific cell adhesion ligands, desired elasticity, and degradability. 8 As it is now well established that the cell microenvironment plays a crucial role in cell physiology and fate, the needs for hydrogels with tunable mechanical properties that can be optimally fitted for a wide variety of cell cultures are becoming more apparent, along with identifying the essential biochemical signals to incorporate into these synthetic ECM analogs. 9 . Hyaluronic acid (HA) is a natural glycosaminoglycan that is composed of the repeating disaccharide unit of D-glucuronic acid and N-acetyl-D-glucosamine. This polysaccharide exists in human and animal tissues as the major component of ECMs, particularly in synovial fluid and the vitreous body of the eye, inserm-00712208, version 1 -26 Jun 2012 and plays an important role in many biological processes including cell proliferation, cell differentiation, morphogenesis, inflammation, and wound repair. [10][11][12][13] Although not completely understood, the bioactive properties of HA originate from its capacity of specifically interacting with membrane receptors present on most mammalian cells (such as CD44 or RHAMM), [14][15][16] as well as with dedicated soluble ECM proteins (named hyaladherins). [17][18] Due to its unique viscoelastic and biological properties HA finds many biomedical applications such as ophthalmological surgery, treatment of osteoarthritis of the knee, prevention of post-surgical adhesion... [19][20] In addition, HA has excellent potential for constructing synthetic ECM analogs. 21 However, chemical modification of HA is required to prepare covalently cross-linked hydrogels and to modulate in vivo degradation rates and cell attachment abilities of the resulting biomaterials. An effective strategy for cross-linking HA under mild conditions is photopolymerization of pendent methacrylates, previously introduced through functionalization of the hydroxyl 11,[22][23][24][25][26][27][28][29][30][31] or carboxyl [32][33] groups of HA. This methodology provides some advantages such as increased spatial and temporal control over crosslinking and biocompatibility with in situ polymerization. [34][35] Among these photopolymerizable HA derivatives, most of them have resulted in hydrogels with low crosslinking density due to low degrees of methacrylation (DM  0.3). In order to extend applications of such HA hydrogels, Bencherif et al. recently proposed a new method to prepare HA-methacrylate conjugates with a broader range of DM (from 0.14 to 0.9). 22 In vitro studies of myoblast cell adhesion indicated that the resulting hydrogels possessing various crosslinking densities do not support alone cell spreading as previously reported for fibroblast and endothelial cells. 25,33,36 When functionalized with the GRGDS peptide, a consensus sequence found in various ECM proteins that trigger the interaction with integrin family cell surface receptors, the hydrogels promoted cell interactions but the degree of methacrylation had a minimal effect on myoblast cell attachment, spreading and proliferation. On the other hand, by using a HA macromer with ~ 100 % modification of the hydroxyl groups and varying the methacrylate consumption through a dual cross-linking method, Marklein et al. demonstrated the ability inserm-00712208, version 1 -26 Jun 2012 to control stem cell spreading and proliferation on HA hydrogels modified with RGD adhesion moieties. 37 These latter findings indicate that the mechanical properties of functionalized HA gels can impact some cell types in terms of morphology and functions. As a result, they underscore the importance of fine-tuning gel strength in a broad range of network cross-linking density in order to trigger specific changes in cellular responses. Yet, beside integrin engagement through RGD peptides, as the HA-cell receptor interaction is critically dependent on the molecular weight and conformational freedom of HA, care must be taken regarding the conditions for its chemical cross-linking. Indeed, if the aim is to additionally exploit its biological potential, it must be taken into account that HA interactions with cell surface receptors require minimum sequences of HA repeating units (REF Morra). Thus, HA 6 , which is three HA disaccharide units, is the minimum size of HA chain required to occupy the CD44 binding site, and HA 10 or greater is the optimal length (Ref Morra + tammi). TIndeed it has already reported that the degree of substitution of modified HA have an impact on its biological activity. Ref Oh

In this study, we aimed to establish a simple and reliable methodology for the synthesis of hyaluronic acid-based hydrogels that encompass the range of physiologically relevant moduli while maintaining binding affinity to HA receptors as biomimetic ECM platforms for cell culture and other tissue engineering applications.

To this end, we first developed new conditions for the methacrylation of HA based on its reaction with methacrylic anhydride, allowing us to prepare in a short time a series of macromer precursors with a large range of DM. Photo-rheometric measurements were then used to monitor in situ gelation of methacrylated HA derivatives under various photopolymerization conditions, including the concentration of the photoinitiator, light intensity, concentration of macromer and, degree of methacrylation. This in-depth rheological study allowed us to define the optimal conditions to prepare hydrogels with a broad range of elastic moduli. Lastly, we determined the effect of HA material rigidity on the attachment and spreading of different cell types. All the cell type tested had the ability to interact with these HA hydrogels. Surprisingly, cell spread on these substrates in a manner that was dependent inserm-00712208, version 1 -26 Jun 2012 on the hydrogel storage modulus. Using pre-osteoblasts, we determined that attachment and spreading were specifically triggered by both CD44 and its coreceptor RHAMM, the two major cell receptors of HA at the plasma membrane.

MATERIALS AND METHODS

Materials.

Hyaluronic acid (M w = 100000 g/mol) was purchased from Lifecore (USA). Methacrylic anhydride (AMA), N,N-dimethylformamide (DMF), phosphate buffer saline (PBS, pH ~ 7.4), (3aminopropyl)trimethoxysilane (APTS, ODTS), hyaluronidase (HAse type VIII, ∼300 units/mg, ref H3757) were purchased from Sigma-Aldrich-Fluka (L'isle d'Abeau, France). 2-hydroxy-1-[4-(2hydroxyethoxy)phenyl]-2-methyl-1-propanone (Irgacure 2959) was kindly provided by Ciba Specialty Chemicals (Basel, Switzerland). All chemicals were used without any further purification. The water used in all experiments was purified by a Elga Purelab purification system, with a resistivity of 18.2 MΩ cm.

Synthesis of methacrylated HA (HA-MA).

Hyaluronic acid was derivatized with methacrylic anhydride either in water as reported elsewhere 28 or, in a mixture of water/DMF (1/1, v/v). In the following, we describe the latter procedure which was optimized in this work and allowed us to obtain HA-MA derivatives with higher DM. HA (0.20 g, 0.50 mmol) was dissolved in ultrapure water (10 mL) at 4 °C and the resulting mixture was kept at 4 °C under continuous stirring overnight for complete dissolution. DMF (6.7 mL) was then added dropwise in order to have a water/DMF ratio of (3/2, v/v). AMA (0.5, 1, 2 or 3 molar equivalents with respect to the moles of repeating unit of HA, i.e. 0.039 g (0.25 mmol), 0.077 g (0.5 mmol), 0.154 g (1 mmol) or 0.231 g (1.5 mmol), respectively) was added while maintaining the pH between 8 and 9 (by adding 0.5 M NaOH) for 4 hours. The reaction was kept at 4 °C under continuous stirring for one night.

After this time, NaCl was added to the reaction mixture to have a NaCl concentration of 0.5 M. The polymer was precipitated by addition of ethanol (with a water/ EtOH (v/v) ratio of 2/3). After removal inserm-00712208, version 1 -26 Jun 2012 of the supernatant, the precipitate was successively washed with mixtures of water/EtOH (3/7, 1/4, 1/9, v/v) and finally dissolved in ultrapure water for a final purification by diafiltration (ultramembrane Amicon YM10) with ultrapure water. The purified product was recovered by freeze-drying and characterized by 1 H NMR spectroscopy (Supporting Information, Figure SI-1).

Characterization of methacrylated HA by 1 H NMR. 1 H NMR spectra of the HA-MA derivatives dissolved in deuterium oxide (6 mg/mL) were performed at 80 °C using a Bruker DRX400 spectrometer operating at 400 MHz. Deuterium oxide was obtained from SDS (Vitry, France). All spectra were recorded by applying a 45° tip angle for the excitation pulse, and a 10 s recycle delay for 128 scans. The DM of the samples was determined by digital integration of the anomeric protons signals or methyl protons signals of HA and of the methacrylate proton signals at ~ 6.1, ~ 5.7 and ~ 1.9 ppm.

Rheometry. An AR2000Ex rheometer (TA Instruments Inc.) fitted with a UV-curing cell ( = 365 nm) and an aluminum plate (diameter 20 mm) was used for the in situ measurement of the viscoelastic properties of the HA-methacrylate gels. Following deposition of 250 µL of HA-MA solution, the gap between the flat quartz plate and the aluminum plate was initially 0.7 mm (measuring ambient temperature). It was controlled during the experiments by maintaining the normal force at 0  0.1 N.

During oscillatory time sweep tests, drops of the G' modulus were observed a few min (~ 5-10 min) after its sharp increase (as a result of network formation) for the hydrogels with high stiffness. These were attributed to grip-slip caused by the release of water from the hydrogels. Therefore, the flat quartz plate was sanded, which allowed us to overcome this experimental problem (Supporting Information, Hydrogel immobilization. For the cell culture experiments, hydrogels were covalently linked to a coverslip during photopolymerization by methacrylation of the coverslip prior to use. The methacrylated coverslip was prepared by treating the slip previously modified with ODTS/APTS with methacrylic anhydride. The glass slip was modified with a mixture of ODTS and APTS as previously described. 38 Briefly, the modified slip with -NH 2 groups was immersed in PBS (5 mL) containing AMA (0.1 mL) and stirred at room temperature for half a day. The slip was then rinsed by EtOH/water (1/1, v/v) mixtures, water and dried in air before use. HA-MA solutions, deposited on the non-sanded quartz plate of the rheometer, were gelled in contact with the methacrylated glass coverslips. Under such conditions, hydrogels with different stiffness (G' modulus varying from 600 to 17000 Pa) were prepared. After preparation, hydrogels were transferred to cell culture media (Dulbecco's modified minimal essential medium, DMEM, Invitrogen) supplemented with 50 U/mL penicillin, and 50 mg/mL streptomycin and left for 12 h for equilibrium before cell culture studies. Media was refreshed after 12 h to remove any remaining monomer or initiator.

Cell culture and biological reagents. NIH-3T3L was a subclone of ATCC NIH3T3 selected for its inserm-00712208, version 1 -26 Jun 2012 high spreading ability and HeLa cells were from ATCC and grown inMEM and DMEM respectively, supplemented with 10% fetal calf serum and antibiotics in a humidified 5% CO 2 atmosphere. Primary osteoblasts (passage 2) were immortalized by transduction with a retrovirus expressing the large SV40 T antigen, 39 cloned, and tested for their ability to express alkaline phophatase upon differentiation, 40 as previously described 41 . They were cultivated under the standard conditions in DMEM supplemented in

RESULTS AND DISCUSSION

Synthesis of HA-methacrylate macromers. Several methods have been developed for synthesizing methacrylate-modified HA, some of them using methacrylic anhydride 23,[28][29][30] and others glycidyl methacrylate (GMA) 11, 24-27, 31, 42 . In both cases, the degree of methacrylation did not exceed 0.3 when methacrylation was performed in aqueous solution due in part to the low solubility and hydrolysis of AMA and GMA in water. Therefore, in order to enhance the DM, alternative methods consisting of the inserm-00712208, version 1 -26 Jun 2012 reaction of HA with GMA in organic solution or in a hydro-organic mixture were developed. 22,27 Bencherif et al. showed the ability to increase the DM up to 0.9 when the reaction of HA with GMA was performed in a water/DMF mixture at physiological pH but it required a long reaction time (10 days). 22 Different esters resulting from a reversible transesterification through the primary hydroxyl group and an irreversible ring-opening conjugation through the carboxylic acid group toward the highest substituted carbon of epoxide were assumed to be formed simultaneously. Their respective amounts were reported to be time-dependent, the concentration of ring-opening products thus increasing with time.

In this work, we synthesized methacrylated HA derivatives by reaction of methacrylic anhydride with the hydroxyl groups of HA, leading to only one type of ester (Figure SI-1).

As previously reported, 23,28,30 when the reaction of AMA with native HA was performed in aqueous solution, we obtained HA derivatives with low DM ( 0.20) although AMA was added in large excess (see Table 1). By comparing the molar ratio of AMA to HA in the reaction medium and the DM of modified HA, it can be noticed that for the two feed ratios, HA was derivatized with 1-5 % of the added AMA. Interestingly, the degree of methacrylation could be enhanced by using DMF as co-solvent and the percentage of derivatization of HA for the four feed ratios was between 16 and 24 % of the added AMA after 16 h of reaction (Table 1). Actually, such conditions allow the solubilization of methacrylic anhydride in the reaction medium resulting in better control over the DM. This method compared favorably with that performed with GMA in PBS/DMF 22 which required much longer reaction times (5-10 days) and higher GMA/HA molar ratios (with respect to the AMA/HA ratios) to obtain DM in the range of 0.32-0.60.

Hydrogel formation and rheological properties.

The gels were formed by photopolymerization at 365 nm of HA-MA macromers in the presence of photoinitiator Irgacure 2959 which was selected for its known biocompatibility with cells. 43 The gelation process was monitored in situ by photorheometry. In order to optimize the conditions for the synthesis of HA networks, we investigated the effect of various inserm-00712208, version 1 -26 Jun 2012 experimental parameters that control photogelation, i.e. photoinitiator concentration and power of light used for irradiation. Figure 1 shows the effect of (A) photoinitiator concentration and, (B) power of light used for irradiation on the time sweep profiles of the storage modulus (G') and loss modulus (G'') obtained from a solution of HA-MA 2b with a DM of 0.20.

The samples were equilibrated for 1 min 30 s before being illuminated for a time of 20 or 40 min (see Materials and Methods). Initially, G'' is larger than G', reflecting the viscous behavior of the sample.

After an induction period, which depends on the photoinitiator concentration and power of light, the storage modulus increases sharply due to the formation of elastic effective intermolecular cross-links whereas the loss modulus increased negligibly during this time. Consequently, there is a cross-over point where G' becomes higher than G''. The time required for this crossover to occur is sometimes referred to as the gelation time although this point depends on the applied frequency. [44][45] In the postgelation period, the G' curve levels off indicating the end of the gelation process. In this context, the steady-state value of G′ was used as a metric for hydrogel elasticity. The loss modulus increased negligibly during this time for all hydrogels.

From Figure 1A, it can be seen that the time required for the beginning of gelation, corresponding to the time where G' becomes equal to G'' as mentioned above, is dependent on the photoinitiator concentration. In these experiments, the power of light used for irradiation was fixed at 50 mW/cm 2 . For concentrations in the range of 0.05-0.2 % (w/v, i.e. [photoinitiator]/[MA] ratios ranging from 0.30 to 1.23), the gelation times are around 2 min 40 s. However, when the initiator concentration was decreased to 0.025% or lower, one can notice significant increase in the gelation time going from 3 min 20 s for the concentration of 0.025 % (i.e. a [photoinitiator]/[MA] ratio of 0.15) to 6 min 30 s for a concentration of 0.01 % ([photoinitiator]/[MA] ratio = 0.06). Moreover, a significant decrease in the G' modulus can be observed for the latter two concentrations. For the lowest initiator concentration, the G' modulus thus tends to a value which is ~ 10 times lower than those measured for the three higher concentrations. This suggests that the initial photoiniator concentration is too low to activate all inserm-00712208, version 1 -26 Jun 2012 methacrylate groups for radical polymerization. The activation of the methacrylate groups is particularly delicate as it strongly depends on the diffusion of the photoiniator in the HA chemical network. On the other hand, the slight decrease in the G' values with the time observed for the higher concentrations may be related to the slight degradation of HA network due to the high concentration of radicals produced by the photoinitiator. Indeed, the same experiment performed with a film of silicone oil lead to a similar curve indicating almost no solvent evaporation during the experiment performed under these conditions (Supporting Information, Figure SI-4). All together, these data suggest that below a critical [photoinitiator]/[MA] ratio situated between 0.06 and 0.15, the completion of network formation requires much longer times of irradiation.

The decrease in the light power from 144 to 15 mW/cm 2 also delayed the beginning of gelation whereas it did not have striking effect on the steady-state value of G' (Figure 1B). As can be seen on Figure 1B, the time required for the G' modulus to reach a plateau was significantly increased when the light power was decreased to 15 mW/cm 2 . Indeed, the G' modulus was found to level off after 40 min of irradiation, whereas the steady-state value of G' was reached after ~ 20 min of irradiation for light powers > 50 mW/cm 2 . Therefore, from these results, all subsequent oscillatory time sweep experiments were performed using a photoinitiator concentration of 0.05 % and a light power of 50 mW/cm 2 . One can notice a progressive decrease of the G' modulus after reaching its maximum for the sample illuminated with UV light at a power of 144 mW/cm 2 . Such conditions likely result in sample heating, causing partial evaporation of water. Indeed, when the same experiment was carried out with a film of silicone oil, the decrease was much less pronounced (Supporting Information, Figure SI-5). This result supports our hypothesis that almost no solvent evaporation occurs when polymerizations are conducted with UV light at a power of 50 mW/cm 2 .

We next focused our attention on some factors that may affect mechanical strength, namely macromer concentration and degree of methacrylation. values become constant ~ 20 min after the beginning of irradiation (i.e. 21 min 30 s after the beginning of the oscillatory shear experiment) for the hydrogels made from the macromer 2a whatever its concentration, it appears that for the hydrogel prepared from 2b at a macromer concentration of 50 g/L, the G' values level off only ~ 35 min after the beginning of experiment. Moreover, a similar behavior to 2b was observed for HA-MA 2c. These results indicate that from DS values  0.2 and macromer concentrations  50 g/L, the time of irradiation performed using a photoinitiator concentration of 0.05 % must be increased from ~ 20 min to ~ 40 min to reach a plateau for G'. It should be noted that in the case of 2b, this time could be decreased to ~ 30 min by increasing two fold the photoinitiator concentration. However, the steady-state value of G' was found to be 20 % lower than that obtained with a photoinitiator concentration of 0.05 %, which was attributed to degradation of HA at high radical concentrations.

To verify if the time for G' to level off corresponds to the total conversion of the methacrylate groups, we analyzed by 1 H NMR samples of HA-MA 2b (C p = 50 g/L) which were photopolymerized for 20 and 40 min and subsequently degraded by hyaluronidase. After 20 h incubation at 37 °C in 500 U hyaluronidase/mL, the swollen networks were converted into solutions due to the cleavage of internal β-N-acetyl-D-glucosaminidic linkages in HA chains. After 20 and 40 min of irradiation, traces of methacrylate double bonds could be observed from the NMR spectra (Supporting Information, Figure SI-6). Digital integration of the proton signals of the double bonds and the anomeric proton signals of HA indicated that increasing time of irradiation from 20 to 40 min lead to an increase in the crosslinking conversion from 80 to 90 %. This result suggests that the time required for G' to become nearly constant can be considered as an indication of the time for the photopolymerization to reach near 100 % conversion. This method can have some advantages compared to 1 H NMR analysis considering its simplicity, rapidity and sensitivity. It should be noted that Fourier Transform Infrared spectroscopy failed to reveal traces of methacrylate double bonds in the sample. The fact that 100 % conversion is not inserm-00712208, version 1 -26 Jun 2012 strictly obtained can be attributed to severe mobility restrictions after the gel point is reached, which hamper reactions between all methacrylate groups especially when their concentration is very high as discussed below.

Comparing HA-MA hydrogels prepared from solutions of macromers at the same concentration (15 g/L) but with different DM, an increase in the storage modulus values measured 21 min after the beginning of irradiation could be observed as the DM of macromers was increased from 0.12 to 0.28 (Figure 3). This is consistent with theory which predicts that for crosslinked rubberlike networks, the elasticity is related to the concentration of crosslinks by the relation

G = nRT/V ( 1
)
Where n is the number of moles of elastic chains per volume V, and R is the gas constant. 46 It should be noted however, that in this work, n cannot be directly related to the DM of HA macromers owing to the complexity of the network structure consisting of HA chains cross-linked by oligo(methacrylate) chains.

A lower storage modulus value was found when the DM reached 0.49. As such DM means that on average one over two repeating units is substituted by a methacrylate group, it can be reasonably assumed that the 100 % conversation of methacrylate groups is hindered by accessibility problems due to the high DM and severe mobility restrictions after the gel point is reached. This assumption was supported by other experiments performed at higher macromer concentrations which afforded hydrogels with significantly lower storage modulus values compared to those of hydrogels prepared from the derivatives 2a-c as well as 1 H NMR analysis of hydrogels after degradation by hyaluronidase. Indeed, signals corresponding to the protons of methacrylate double bonds could be observed from the NMR spectra of samples prepared from 2c at concentrations of 15 and 30 g/L (Supporting Information, Figure SI-7). Digital integration of these signals and the anomeric proton signals of HA indicated cross-linking conversions of 82 and 61 % for gels made at macromer concentrations of 15 and 30 g/L, respectively.

In order to determine conditions to prepare gels in the physiologically relevant stiffness, we then compared the evolution of G' by varying the concentration of macromers 2a-c. Matrix elasticity can inserm-00712208, version 1 -26 Jun 2012

vary from 100 to a few thousands Pa depending on the natural tissue. 5,47 Figure 4 plots the storage modulus as a function of methacrylate concentration (C MA , calculated from the average molar mass of the repeating unit of HA-MA and the DM) for hydrogels made from solutions of HA-MA 2a-c as well as a HA-MA derivative (1b) prepared according to a literature procedure 28 at concentrations in the range 15-50 g/L. As can be seen on this Figure, a large range of physiologically relevant stiffness can be produced from solutions of HA-MA derivatives with DM between 0.12 and 0.28. Interestingly, a unique curve is obtained for C MA ranging from 4.4  10 -3 mol/L to ~ 20  10 -3 mol/L, thus indicating that HA macromers 1b, 2a-c can independently be used for preparing hydrogels having well-defined storagemoduli included between 500 and 21000 Pa. All together, these results demonstrate that thanks to optimized procedures for the synthesis of HA-MA derivatives and their photopolymerization, hydrogels with tunable stiffness could be prepared with good reproducibility by varying the DM or macromer concentration.

Cellular responses in vitro. Next, we wanted to check the ability of mammalian cells to interact with our methacrylated HA hydrogels. In order to provide stability during manipulation and experimentation with cells, the hydrogels were immobilized to glass substrates during their synthesis by photopolymerization under the optimal conditions discussed above. Therefore, from the unique curve depicted in Figure 4, several macromer solutions were prepared, deposited on the non-sanded quartz plate of the rheometer and were gelled in contact with glass coverslips modified with methacrylate units.

Three different cell lines were subsequently seeded on the hydrogels as described under Materials and Methods and cultivated overnight in DMEM under standard conditions. Figure 5 shows that under these experimental conditions, the different cell types were able to adhere firmly on the hydrogels and were no longer floating in the medium. Surprisingly, all cell types were able to spread to various degrees onto these substrates in a manner that was dependent on the storage modulus in a very similar way that was observed in the classical integrin mediated adhesion to ECM proteins, i.e., favored by the increase in the storage modulus. 48

inserm-00712208, version 1 -26 Jun 2012

Cell spreading is the read-out of cytoskeleton reorganization and establishment of internal tensions. It was markedly dependent on the cell type. HeLa cells were poorly spread even at the highest storage modulus although some small membrane extensions were occasionally detected at 17000 Pa suggesting some initiation of the spreading process. However the quantification of the distribution of the cell projected areas (n=60) showed that those epithelial cells did respond to the storage modulus although to a lesser extent. Conversely, NIH 3T3L cells were fully spread at this storage modulus and exhibited a polarized morphology suggesting a high motility activity. Finally pre-osteoblast exhibited an intermediate situation with some spreading that was increased from 600 to 17000 Pa. The differences observed according to the cell types suggested that the attachment and spreading observed was not due to some nonspecific binding of adsorbed ECM proteins since all cell types attached and spread similarly on ECM protein coated surfaces (not shown). Cell type dependence also ruled out a possible cell surface interaction with unreacted methacrylate moieties. In addition, trypan blue staining showed a good viability (94%) of the cell spread on HA methacrylates gels (not shown). It should be noted that no difference was observed by comparing the biological response of HA-MA gels having the same stiffness (2500 Pa) but prepared from different HA-MA samples, i.e. 2a (40 g/L) and 2d (15 g/L), for which the methacrylate conversion after photopolymerization was different (~ 100 % for 2a and 82 % for 2d)

(Figure SI-8).
To confirm that cell behavior on methacrylated HA hydrogels was due to a specific interaction of the cells with HA, we pre-incubated for one hour at 4°C pre-osteoblasts with blocking antibodies directed against the two major HA receptors at the concentration of 30 µg/mL Subsequently overnight culture in DMEM containing the antibodies was carried out on HA methacrylate gels.

After 12 hours of culture, the cells were fluorescently labeled with a lipophilic tracer Vybrant™ DIL as described under Materials and Methods, and the fluorescent cell images were taken using a Provis 70 Olympus inverted microscope.

Figure 6 shows that even at high storage modulus, cell attachment and spreading to HA-MA gels could be almost fully inhibited by adding either anti CD44 or anti RHAMM. It is notworthy that incubation of inserm-00712208, version 1 -26 Jun 2012

the cells with the antibodies did not affect cell spreading of cell on the culture plastic surrounding the HA gels in the 6 well plates, ruling out a possible toxicity of these antibodies. These results indicated that both receptors were functional on these cells and strongly suggested that cell spreading was due to the interaction of the gel HA moiety with specific receptors and not to interferences with either methacrylate moieties or adsorbed proteins.

CONCLUSIONS

By improving the synthesis of methacrylated HA based on its reaction with methacrylic anhydride as well as the photopolymerization conditions of the resulting macromers for gel formation by in situ photorheometry measurements, we showed in this work the ability to prepare transparent hydrogels with good reproducibility and a large range of elasticity. Interestingly, by plotting the storage modulus of the hydrogels as a function of the initial concentration of methacrylate functions linked to HA macromers, we obtained a single curve indicating that HA macromers with DM in the range of 0.1-0.30 can independently be used for the synthesis of HA matrices with storage-moduli included between 500 and 21000 Pa. This result was then exploited to synthesize series of HA hydrogel substrates for 2dimensional cell culture experiments using various cell types. Very interestingly, we showed that the cells not only interacted with the HA hydrogels but were able to spread on these surfaces in a manner that was favored by the increase in the gel rigidity. This interaction was specific and involved on preosteoblast the HA receptors CD44 and RHAMM. Rigidity dependence of integrin mediated cell adhesion through ECM protein has been extensively described and the mechanism of integrin mechanotransduction partly unraveled. However, our data indicated for the first time cell types are sensitive to the substrate rigidity when cells are only using HA receptors indicating that these transmembrane proteins also play a role of mechanotransducers and induced the rearrangement of actin cytoskeleton.

Cell interaction with a nonspecific matrix such as poly(L-lysine) results in an isotropic spreading of the cells and indeed, the initial stage of cell spreading seems to be independent on integrin activation. 49 inserm-00712208, version 1 -26 Jun 2012

However in the case of HA hydrogels the final stages of cell spreading was strongly anisotropic with highly polarized cells indicating that HA receptors are likely to engage active processes within the cells that support migration. Other reports have mentioned the lack of cell spreading of human dermal fibroblasts and NIH3T3 cells, respectively. 33,36 Our data show that cell spreading on HA gels is cell type and storage modulus dependent. In addition for most of the cells, this spreading however is quite smaller than integrin mediated spreading (for instance when hydrogels are functionalized with RGD peptides). Such limited spreading has been neglected in former works. Earlier studies with NIH3T3 cells were carried out in the absence of serum. 36 These experimental conditions switched off the regulatory GTPase RhoaA and consequently myosin II activity, resulting in the loss of cellular contractility that is required for cell spreading. In summary, HA hydrogels with a large range of mechanical properties seem to be very powerful tools for Cell Biology and provide a microenvironment suitable for a fine tuning of Cell Physiology.

ACKNOWLEDGMENT

This work was financially supported by the "Agence Nationale pour la Recherche" (grant ANR P3N 2009 to RAV). HVDB gratefully acknowledges the CNRS for a post-doctoral fellowship. RAV is indebted to the Institut Universitaire de France for financial support.

Supporting Information Available:

Figure SI- 2 )

 2 Figure SI-2). It should be noted that the treatment of the quartz plate had no effect on the values of both

  10% fetal calf serum. Blocking HA receptors CD44 and RHAMM were from Beckton Dinkinson (Le Pont de Claix, France) and Santa Cruz Technologies (Tebu Bio, Le Perray en Yvelines France), respectively, and the toxic preservative NaN 3 was removed by spin desalting prior to use. Cell labeling and quantification of cell spreading. For quantifying the projected areas of spread cells, the cells were fixed 10 min with a solution of 4% (w/v) of paraformaldehyde in 0.2 M phosphate buffer (pH= 7.2) for 10 min at room temperature. After 3 washes in PBS, the cells were incubated in DMEM supplemented with 0.5% of Vybrant™ DIL cell labeling solution for 15 min at 37°C according to the manufacturer (Life Technologies, St Aubin, France). The cells were washed 3 times in PBS and mounted under a coverslip with 10 µL of mounting solution for epifluorescence observations. Observation were carried out with a Provis 70 microscope (Olympus Europe Hamburg, Germany) equipped with a Plan NeoFluar 20X (N.A. 0.5) objective. Areas were determined with Metamorph software (Molecular Devices, Sunyvale, CA USA) at a magnification of 200X. Statistical analyses were carried out with R software (The R Project for Statistical Computing, http://www.r-project.org/)

Figure 2

 2 Figure2shows the influence of time of irradiation and macromer concentration on the storage modulus

Figure 1 .

 1 Figure 1. In situ photorheometric measurement of hydrogel formation from methacrylate modified-HA:

Figure 2 .

 2 Figure 2. Effect of time of irradiation and macromer concentration on the storage modulus of hydrogels

Figure 3 .

 3 Figure 3. Effect of the degree of methacrylation on the mechanical properties of HA hydrogels prepared

Figure 4 .

 4 Figure 4. Storage modulus of hydrogels made of methacrylated HA 2a () 2b (), 2c () and 1b ()

Figure 5 .Figure 6 .

 56 Figure 5. Cell attachment and spreading on methacrylated HA hydrogels as a function of the G' values.

Figure 6

 6 Figure 1

Table 1 .

 1 Reaction conditions for the modification of HA with methacrylate groups through the hydroxyl functions.

	Reaction a Sample	AMA b	Solvent	DM
				(Vol %)
				Water	DMF
	1	1a	2	100	0	0.10
	2	1b	20	100	0	0.20

H NMR spectrum of of HA-MA derivatives

2a-d; in situ photorheometric measurement of hydrogel formation from methacrylate modified-HA 2c performed on a non-sanded and sanded quartz plate; in situ photorheometric measurement of hydrogel formation from methacrylate modified-HA 2b performed on a non-sanded and sanded quartz plate; in situ photorheometric measurement of hydrogel formation from methacrylate modified-HA 2b performed in the absence and in the presence of a film of inserm-00712208, version 1 -26 Jun 2012

silicone oil; in situ photorheometric measurement of hydrogel formation from methacrylate modified-HA 2b performed in the absence and in the presence of a film of silicone oil; 1 H NMR spectra of initial HA and HA-MA hydrogels after digestion by hyaluronidase; 1 H NMR spectra of hydrogels prepared from HA-MA 2d at 15 g/L and 30 g/L in PBS after digestion by hyaluronidase; pre osteoblast spreading on HA-MA gels with a storage modulus of 2500 Pa prepared with HA-MA 2a (40 g/L) and HA-MA 2d (15 g/L). This material is available free of charge via the Internet at http://pubs.acs.org.


L R Pastur

Y Fraigneau

F Lusseyran

J Basley

From linear stability analysis to three-dimensional organisation in an incompressible open cavity flow

Keywords: 47.10.ad -Navier-Stokes equations 47.11.Df -Finite volume methods 47.15.Rq -Laminar flows in cavities, channels, ducts, and conduits 47.20.Qr -Centrifugal instabilities

Three-dimensional direct numerical simulations of an incompressible open square cavity flow are conducted. Features of the permanent (non-linear) regime together with the linear stability analysis of a two-dimensional steady base flow are discussed. Spanwise boundary conditions are periodic and control parameters set such that the shear layer is stable with respect to Kelvin-Helmholtz modes. Three branches of destabilising modes are found. The most destabilising branch is associated with steady modes, over a finite range of spanwise wavenumbers. The two other branches provide unsteady modes. Features of each branches are recovered in the permanent regime: wavelength of the most powerful spanwise Fourier mode, swaying phenomenon, angular frequencies, indicating that modes of each branches are selected and interact in the permanent flow.

Introduction. -Open cavity flows belong to the family of impinging flows, which are known to develop self-sustained oscillations. For decades, an extensive literature has been devoted to the study of such flows [?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?]. The main features of open cavity flows, in the incompressible limit, may be roughly summarized as follows. The shear layer that forms between the outer and inner flows is known to be unstable with respect to streamwise Kelvin-Helmholtz modes, beyond some critical value of the Reynolds number. Kelvin-Helmholtz modes develop, downstream in the shear layer, from perturbations initiated at the upward corner. The unstable shear layer rolls up on itself to produce vortices that cyclically collide with the downward edge. Changes in the pressure field at impingement are instantaneously fed back to the upward edge, due to incompressibility or, equivalently, when the cavity length is small in regard to the acoustic wavelength and the material velocity small before sound celerity. Feedback then initiates the growth of a new perturbation at the upward edge. Therefore, both upward and downward edges are locked in phase, giving rise to strong self-sustained oscillations, whose frequency both depends on the incoming flow velocity, the cavity length, and the boundary layer thickness at the upward edge.

What has been far less studied is the cavity inner-flow, which exhibits non-trivial three-dimensional organisation [?, ?, ?, ?]. Centrifugal instabilities were shown to give rise to Taylor-Görtler-like rolls beyond a critical Reynolds number [?,?]. The main inner-flow recirculation bends the pathline of material particles, which are subject to inertial forces. When viscous terms cannot overcome centrifugal effects, instabilities may develop, eventually giving rise to the formation of stable raw of pairs of counter-rotating vortices, in the spanwise direction, as observed in experiments or direct numerical simulations of the flow [?, ?, ?]. Recently, numerical studies have been carried out in unsteady cavity flows driven by Kelvin-Helmholtz instabilities, in compressible [?] and incompressible [?] regimes.

Our present contribution relies on a detailed analysis of intrinsic features of the inner flow, in an open cavity flow at a Reynolds number below the threshold of Kelvin-Helmholtz instability. Three-dimensional direct numerical simulations of the cavity flow are conducted with peri- odic spanwise boundary conditions, in order to get rid of border effects. A linear stability analysis is conducted, on a two-dimensional steady state with respect to threedimensional perturbations. As it will be shown, three branches of (spanwise) growing modes are found, whose features are recovered in the permanent regime with little distortion, though coupled through non-linear terms.

Cavity flow features. -Instabilities in the shear layer primarily depend on the dimensionless cavity length Γ θ0 = L/θ 0 , ratio of cavity length, L, to boundary layer momentum thickness, θ 0 , at the upward edge, the Reynolds number Re=U 0 L/ν, based on L and incoming velocity, U 0 , at the upstream cavity wall, and, for a lower part, the cross-stream aspect ratio Γ L = L/H, defined as the ratio of cavity length, L, to cavity height, H.

The curvature induced by the main flow recirculation, inside the cavity, shown in fig ??a, is responsible for the development of centrifugal instabilities, over some range of the control parameters. Relevant parameters, there, are the Reynolds number based on cavity height, Re H = U 0 H/ν, Γ L and H/θ 0 [?]. In the non-linear saturated regime, centrifugal instabilities give rise to spanwise alley of pairs of counter-rotating vortices, as shown in the spanwise cut (top view) of fig ??b (top). Each pair wraps the main inner-flow recirculation, forming torus-like patterns with major axis in the streamwise plane [?, ?]. As shown in [?], spanwise walls, at z/S = ±1/2 where S is the span, generate a Bödewadt pumping, responsible, at Γ L = 1, for a drift of the Taylor-Görtler rolls toward q /H (edge values ±5.8). Only one half of the spanwise direction is represented for the sake of clarity. Spanwise boundary conditions are periodic. Top pictures are side-views of the structure, slightly seen from above and downstream (flow direction is indicated by the arrow). Bottom picture is a front view from the cavity downstream wall. Black corresponds to negative values of helicity, white to positive values.

one wall or the other, as can be seen in the experimental space-time diagram of fig ??b (bottom). In this figure, oblique lines correspond to drifting structures, and it is clear, from the different slopes, that inner spanwise dynamics is associated with several drift velocities, depending on the spanwise position, z. Henceforth, drift motions introduce additional unsteadiness in the flow, enriching the low-frequency range of power spectra.

Direct numerical simulations. -In order to get the intrinsic features of the inner flow, a three-dimensional direct numerical simulation is performed with periodic spanwise boundary conditions.

Numerical methods. The study addresses an incompressible and isothermal flow whose governing equation can be described by the non-dimensional Navier-Stokes equations:

∂U ∂t + (U • ∇) U = -∇P + 1 Re ∆U ∇ • U = 0 (1)
where U is the velocity field and P the pressure field. Numerical simulations are performed with the OLORIN code based on an incremental prediction -projection method [?]. Momentum equations are discretised with a finite volume approach on a staggered structured grid. The spatial discretisation of fluxes is carried out with a second-order centred scheme in a conservative form and time derivation is approximated by a second-order backward differentiation formula. Viscous terms are implicitly evaluated whereas convective fluxes are explicitly estimated at time t n+1 by means of a linear Adams-Bashford extrapolation.

The discretised form of the Navier-Stokes equations yields a Helmholtz-type problem of the form:

   I - 2∆t 3Re ∇ 2 U n+1 = -∇P n+1 + S n,n-1 ∇ • U n+1 = 0 (2)
where superscript n tags time t n , ∆t is the time step and S n,n-1 is the source term gathering all explicit quantities, evaluated at times t n and t n-1 . The integration is performed with an ADI (Alternating Direction Implicit) method [?]. The resulting velocity field U * is not yet divergence-free. Incompressibility is imposed by using an incremental projection method [?]. The projection step requires to resolve a Poisson-type equation, using a relaxed Gauss-Seidel method coupled to a multigrid method, in order to accelerate convergence, where the source term relies on non-zero divergence of the predicted velocity field:

∇ 2 φ = ∇ • U * (3)
where ∇φ is the correction term. The Poisson equation is commonly solved with Neumann-type boundary conditions, where the normal derivative on the domain limits is zero. By doing so, the boundary condition, on the corresponding normal velocity component, is not affected by the correction term.

Numerical simulation setup.

The geometric setup consists in an open cavity capped with a parallelepipedic duct in which is generated the channel flow driving the inner cavity flow. A cartesian coordinate system (x, y, z), for streamwise, crosswise and spanwise directions, respectively, is set midspan at the top of the upstream cavity wall. The cavity dimensions are L = 5 cm, L/H = 1, S/H = 6 (span over height). Upstream and downstream lengths of the duct are respectively L u /L = 1 and L d /L = 3 and its height is H v /H = 3. The Reynolds number is set to Re = 3 850 (bulk velocity U q = 1.2 m/s), and the boundary layer momentum thickness is θ 0 /H = 26×10 -3 .

The total domain is meshed on 160 × 128 × 192 nodes, among which 64 × 64 × 192 are devoted to the cavity. The mesh, regular in the spanwise direction, is particularly refined close to the walls and at the cavity top plane in order to enhance the spatial resolution of boundary layers and shear-layer.

The inlet flow is determined by Dirichlet boundary conditions. The inlet velocity profile is preliminary calculated by means of a 2D simulation of a laminar channel flow in spatial development, representative of the experimental upstream vein. The profile is then extracted out of the appropriate cross-section of the channel-flow and extruded in the periodic spanwise direction. Usual non-sliding conditions are applied at the walls. The numerical simulation is carried out over a time-duration of 60 s (1440 time-units H/U q ), after overtaking the numerical transitory state of the flow.

Intrinsic features of Taylor-Görtler-like vortices. Raw of Taylor-Görtler-like vortical structures take place inside the cavity flow and wind around the main recirculation vortex, as shown in fig ??. However, contrary to the wallbounded configuration, no spanwise drift of the structure towards the spanwise walls is observed. Although the flow is not fully steady, it is insightful to consider streamlines released backward and forward to a seeding line at the upstream bottom quadrant of the cavity (materialized line in fig ??), and consider streamlines as approximate material trajectories. Trajectories gather into beams and draw funnel-like shapes, materializing the coherent structures highlighted by iso-surfaces of helicity (see fig ??). In addition, although particles may start aligned in the spanwise direction z, their trajectories coil around each other, which reveals an helical flow in Taylor-Götler vortices. There is no mass flux between two adjacent counter-rotating vortices. The only spanwise transfer occurs inside both corner vortices, at the bottom of the cavity. Particles seeded inside the core of the main recirculation describe concentric circles, characteristic of a solid rotation (not shown in the

figure).

A spectral Fourier analysis on velocity components, in the spanwise direction, yields another insight on the spatial arrangement of the flow (see fig ??a). The most energetic Fourier modes, in space, are found for spanwise wavelengths, λ/H ≃ 0.42 (λ/S ≃ 0.07) and 0.84 (λ/S ≃ 0.14). Wavelength λ/H = 0.42 is therefore associated with the rake of vortex pairs, while its second harmonic, λ/H = 0.84, is connected to single vortices inside pairs. The streamwise (x, y) structure of the most energetic mode is shown in fig ??b.

Spectral Fourier analysis on temporal samplings of velocity, at probes set in the cavity, shows that the flow is not quite stationary and contains a slight oscillating component, see fig ??. This oscillation, found all over the cavity, is characterized by a angular frequency, ωH/U q = 0.16. A flow inspection reveals that this oscillating mode is associated with a swaying motion of Taylor-Görtler-rolls, around the main recirculation. In fig ??, the slightly swaying motion is mainly observed on the inner blobs of helicity, close to the upstream cavity wall (better seen on the top right picture).

It is worthwhile noticing that the mode with largest growth-rate found in [?], for Γ L = 1 and Mach numbers M > 0.3 has a wavenumber λ/H ≃ 0.5, similar to our λ/H = 0.42. This mode, in [?], is expected to be a steady mode (ω = 0). In our configuration, the non-linearly saturated regime is found slightly unsteady, with ωH/U q = 0.16. This discrepancy motivates a threedimensional linear stability analysis of the flow in the incompressible limit (M = 0). Linear stability analysis. -Stability properties of a two-dimensional steady base state, with respect to spanwise perturbations, are now considered. Principle.

The method is described in [?]. This is aimed at characterizing the time-evolution of infinitesimal perturbations, u ′ , with respect to an unstable steady base flow, U 0 , by means of linearised Navier-Stokes equations. In compact form, the system rewrites:

∂u ′ ∂t = N U u ′ + Du ′ ≡ Au ′ , (4)
where N U is the linearised operator of evolution associated with the eulerian part of the Navier-Stokes equations and D is the linear operator associated with viscous terms.

The main instability features are depicted by the leading eigenpairs of the linear evolution operator, A, namely eigenpairs with the greatest real part. Leading eigenpairs are determined by the Arnoldi method.

Choice of the base flow and initial perturbation form. A preliminary two-dimensional numerical simulation, in the streamwise plane (x, y), for the same Reynolds number, Re = 3 850, exhibits a stable steady-state flow, U 2D 0 . As a consequence, in the three-dimensional configuration under study, the instability of the base-flow must occur with respect to spanwise modes, as it is observed in threedimensional numerical simulations of the Navier-Stokes equations (??). Therefore, the two-dimensional base-flow, U 2D 0 , is perturbated with initial conditions of the form:

u ′ (x, y, z) = u 0 (x, y) exp i 2π λ j z exp (µ j t) , (5)
that is, a mode of wavelength λ j = L/m, m ∈ N ⋆ , in the spanwise direction, and (complex) growth-rate µ j = σ j + iω j , with σ j the temporal growth-rate and ω j the angular frequency. Only stationary wave-like modes are considered since no spanwise drift is expected in the permanent flow.

Linearly growing modes.

The base flow loses stability with respect to modes with positive growth-rate, σ > 0. Three branches of linearly growing modes are found, whose temporal growth-rate σ and angular frequency ω, as functions of λ, are illustrated in fig ??. Modes with positive growth-rates have wavelength in the range [0.24H, 1.32H], confirming the base-flow stability with respect to streawise perturbations (vanishing wavenumbers).

One branch (black squares) is associated with a stationary bifurcation (ω = 0), on the wavelength range λ/H ∈ [0.24, 0.618]. The largest growth-rate on this branch is found at λ/H = 0.402, close to the wavelength found for the raw of Taylor-Görtler-like vortices (λ T G /H ≃ 0.42). We will refer to this mode as mode (i). A spatial representation of mode (i) is shown in fig ??.

In fig ?? is also found, over the range of wavelength λ/H ∈ [0.24, 0.642], another branch of solutions (green triangles) with positive growth-rates, and angular frequencies roughly constant over the range, at ω ii H/U q = 0.16, very close to the one of Taylor-Görtler-like vortices. For this branch, the highest growth-rate is found, again, for λ/H ≃ 0.4, and we will refer to this mode as mode (ii). A comparison of both real and imaginary parts of mode (ii), in fig ??, reveals a swaying motion of the structure around the main recirculation.

When the growth-rate of the stationary branch crosses zero, at λ/H = 0.618, a new branch rises, on the range λ/H ∈ [0.618, 1.32], characterized by a non-zero angular frequency (red circles). Following this oscillatory branch, the angular frequency increases from 0 to about ωH/U q = 0.092, with a linear variation beyond λ/H = 0.78. In this family, the mode with the largest growth-rate, referred to as mode (iii), is found for λ iii /H ≃ 0.82 with ω iii H/U q ≃ 0.075.

From instability to permanent flow.

The inner flow most energetic structure, in the permanent flow, has both a spanwise wavelength, λ T G /H, and a streamwise structure, in close accordance with those of mode (i), the most linearly unstable mode in our linear stability analysis (compare figs ??b and ??). However, mode (i) is steady, whereas the permanent regime exhibits a slightly unsteady swaying motion. Note that transient dynamics, although not fully excluded, are very unlikely, since simulations in the permanent flow cover a time-range of 1 440 time-units H/U q . In fact, when considering the power spectral density of fig ??, in the permanent regime, a main peak is found at ω T G H/U q ≃ 0.16, which precisely is the angular frequency of modes of the second family. Moreover, the streamwise structure of mode (ii), in that family, exhibits a swaying-like motion, as shown in fig ??, when comparing both real and imaginary parts. Such features suggest that this mode, as mode (i), is also selected by the flow. Note that linear modes are recovered in the permanent regime without significant distortion, despite of non-linear effects.

It is worth to note that the angular frequency of the third family mode, with the highest growth-rate, is ω iii H/U q ≃ 0.07. Would such a mode be present in the flow, it could couple to mode (ii) and generate components at (ω ii ± ω iii )H/U q , i.e. 0.09 and 0.23. In the power spectral density of fig ??, a lobe is actually found around ωH/U q ≃ 0.25, suggesting a non-linear (quadratic) coupling of mode (iii) with mode (ii). In addition, the spanwise Fourier mode associated with m = 30, in fig ?? for the permanent regime, is not found in the linear stability analysis, indicating that this mode is rather generated by quadratic non-linear self-interaction of mode m = 15. The fastest (linearly) growing spanwise mode, in a linear stability analysis of the steady base state, has a wavelength λ/H = 0.4, fairly close to the wavelength associated with the spanwise alley of vortices. However, from stability analysis, this mode is expected stationary. Yet, the swaying motion is caught by a mode of the second family, whose growth-rate is maximal for wavelengths of the order of λ T G . Surprisingly, streamwise structures, wavelength and angular frequencies of linear modes are not much distorted in the non-linear regime. Non-linear effects are revealed by coupling modes of two families, namely the modes with largest growth-rates from families (ii) and (iii).

Conclusion

Note that beyond the drift motion initiated by the pumping effect at the spanwise walls, a secondary bifurcation towards an unsteady oscillating phenomenon of the alley of vortices has been reported in experiments, see for instance [?]. Although the streakline features of this secondary instability could possibly be understood as the result of a swaying motion, it is not possible, yet, to ascertain whether both phenomenons, in simulations and experiments, be related or not. * * *

Fig. 1 :

 1 Fig. 1: Top: spanwise topview of the flow. Bottom: space-time diagram of the Taylor-Görtler rolls dynamics. The space-time diagram is made of the vertical concatenation of the spatial line shown in the top picture (the line intercepts the mushroomlike structure, close to the cavity upstream wall). From an experimental cavity flow, at ΓL = 1, ReH ≃ 5 800.

Fig. 2 :

 2 Fig. 2: Three views of Taylor-Görtler structure, as isosurfaces of dimensionless helicity hU 2q /H (edge values ±5.8). Only one half of the spanwise direction is represented for the sake of clarity. Spanwise boundary conditions are periodic. Top pictures are side-views of the structure, slightly seen from above and downstream (flow direction is indicated by the arrow). Bottom picture is a front view from the cavity downstream wall. Black corresponds to negative values of helicity, white to positive values.

Fig. 3 :

 3 Fig. 3: Streamlines stem from a spanwise line, at the bottom of the cavity, close to the upstream wall (the seeding line is materialised in pictures). (a) Downstream half topview, (b) topview. Inlet flow is oriented along x. Trajectories forward in time form inner loops (gray in (a)), trajectories backward in time form larger loops (black in (b)). Isosurfaces of fig ?? are shown in transparency. One hundred particles seed the flow.

Fig. 4 :

 4 Fig. 4: (a) Each (x, y) slice represents the power structure, in the (x, y) plane, of spanwise Fourier modes e ikmz from a discrete Fourier decomposition of the permanent flow, where km = 2πm/S is the m th mode wave-vector. Slices of fifty modes are shown and connected by isosurfaces with values 1.2ux/Uq = 10 -4 (blue), 10 -3 (yellow), 10 -2 (red). Most energetic modes are found for m = 15 and m = 30 (red colors). (b) Streamwise structure of the most energetic spanwise Fourier mode, m = 15. Colorscale encodes power spectral density of ux, in units of Uq.

Fig. 5 :

 5 Fig. 5: Power spectral density computed from velocity timerecording at a probe inside the cavity.

Fig. 6 :

 6 Fig. 6: Two-dimensional steady base-flow, solution of equations (??) for Re = 3 850 and ΓL = 1.

Fig. 7 :

 7 Fig.7: Dimensionless growth-rate (a) and angular frequency (b) with respect to dimensionless spanwise wavenumber λ/H. Three families are found: i) steady growing modes (black squares), ii) oscillatory growing modes, with a constant angular frequency, ωH/Uq ≃ 0.16, and small wavelength, λ/H < 0.62 (green triangles), iii) oscillatory growing modes of larger wavelength, with an angular frequency increasing with wavelength (red circles). Mode with the highest growth-rate belongs to family i) with a dimensionless wavelength λ/H ≃ 0.4.

  . -Three dimensional direct numerical simulations of an open square cavity flow are performed in the incompressible limit and at Reynolds number Re = 3 850, L/H = 1, below the threshold of shear layer instability. They reproduce most of the inner flow characteristic features observed in experiments. As a result of centrifugal instabilities, vortical structures develop and form spanwise alley of pairs of counter-rotating vortices [?, ?]. When spanwise boundary conditions are periodic, the alley of vortices is fixed. However, in the non-linearly saturated regime, a closer inspection of the flow reveals a slightly swaying motion of the structure of vortices around the main recirculation flow.

Fig. 8 :

 8 Fig. 8: Streamwise structure (modulus) of the spanwise mode with largest growth rate of the first family (ω = 0), referred to as mode (i).

This work has been supported by the ANR project CORMORED. J.B. gratefully acknowledges fruitful discussions with J. de Vicente Buendia. The authors wish to thank Th.M. Faure for fruitfull discussions.