Lingli Ni

Abraham Chemtob
email: abraham.cherntob@uha.fr

Céline Croutxé-Barghorn
email: celine.croutxebarghorn@uha.fr

Nadia Moreau

Thierry Bouder

Sébastien Chanfreau

Nadine Pébère

Direct-to-metal UV-cured hybrid coating for the corrosion protection of aircraft aluminium alloy

Keywords: A, Aluminium A, Polymer B, EIS

Finding eco efficient and environmentally viable alternat ives to chromate coatings represents a funda mental milestone in the aerospace industry. Here , we show a chromate free approach to protective hybrid coatings on aluminium alloy (M2024 T3) departing from photoinduced sol gel and cationic polymerizations. Beginning with a film of n alkyltrimethoxysilane and diepoxy monomer, we rely on photogenerated superacids to induce the single step formation of two inorganic and organic barrier networks. Such system combines the unique aspects of photopolymerization including fast reactions , temporal control , solvent free composition and temperature independence. Used without chemical conversion coating or anodizing, some films have passed 2000 h of sait spray testing.

B. IR spectroscopy C. Passive film s C. Polymer coatings

Introduction

T3 aluminium alloy (AAT3

) is widely employed in the aeronautical industry because of its unique set of properties, such as a well balanced weight to strength ratio, high fracture toughness and low cost [1). In this alloy, the presenoe of copper and magnesium intermetallic particles allows a mechanical rein forcement of the aluminium matrix, but at the expense of a higher sensitivity to localised corrosion [2,[START_REF] Ghali | Corrosion Resistance of Aluminum and Magnesium Alloys: Understanding, Performance, and Testing[END_REF]. To protect the alloy sub strate against corrosion, the usual strategy relies on a stack of three different layers. Exoept for the top organic layer (top coat1 hexava lent chromate compounds are generally present as salts in the first surface treatment layer (anodizing or conversion film), and as pig ments in the second layer (primer) [START_REF] Ghali | Corrosion Prevention and Protection: Practical Solutions[END_REF]. Cr(VI) is highly effective in promoting passive and active corrosion protection. However, it is now widely recognized as a taxie, carcinogenic, mutagenic and environmentally hazardous substance [START_REF] Twite | [END_REF]. Due to health issues and stricter legislations, the replacement of chromates has become a critical issue for the aircraft industry. Severa! approaches have been developed over the last 20 years in attempts to replace chromate based surface treatments and primers, but very few offer equivalent technical performances in regard to corrosion protection, adhesion and solvent resistance [6 9). Among them, sol gel inorganic coatings have attracted con siderable interest mainly as pre treatment [10 12). The pioneering studies of Sugama [13,14) and Blohowiak [15,16) in the mid 90s have revealed the numerous advantages pertaining to alkoxysi lane based sol gel chemistry: a low temperature prooess prevent ing the thermal modification of metal [START_REF] Hatch | Aluminum: Properties and Physical Metallurgy[END_REF], a dense silica network acting as a physical passive barrier [START_REF] Van Ooij | [END_REF] and a strong interfacial adhesion to the substrate [19). Subsequently, the addition of organic moieties has enabled to increase film thickness, hydropho biàty and resistance to cracking [20 22). The building black for the formation of these hybrid organic inorganic structures are organ oalkoxysilane precursors of the form R [SiX 3 ) n , wherein Ris a cova lently bound organic fragment and n is the number of SiX 3 hydrolysable group (X= OCH 3 , OCH 2 CH 3 ). Their sol gel polymeri zation proceeds by hydrolysis and sequential Si O Si condensation reactions to afford a R [SiO1.s) n polysilsesquioxane structure con taining a polysiloxane network covalently connected to the organic units. In the field of corrosion protection, the most common organosilanes used are mono silylated precursors (n • 1) bearing alkyl [23,24), epoxy [25 27) or vinyl [28 30) groups, and to a minor extent bis silylated (n•2) precursors [31 34). More recently, corrosion inhibitors were added to hybrid sol gel films either directly [35] or as micro or nano capsules [36,37], these lat ter serving as host reservoirs in order to provide additional active corrosion protection when the coating is locally damaged.

Despite the abundance of literature in the subject, very few hybrid sol gel coatings have successfully met the stringent aircraft specifications. For instance, organosilicate films such as Socogel Ò or Boegel Ò are exploited industrially as adhesion promoters to enhance the protection performance of chromated primers [38]. At the lab scale, the salt spray resistance provided by sol gel films alone vary generally from 100 to 500 h on AA2024 T3 depending on the thickness (1 10 lm) and film structure [8,39]. With regard to processing, sol gel chemistry raises problems of low reactivity and high volatile organic compounds concentrations since the sol is prepared in diluted alcoholic solutions [START_REF] Brinker | Sol-Gel Science[END_REF]. Other challenges are related to the mixing of several components (catalyst, solvent, and precursors), the sensitivity to temperature or humidity, and the short pot life of the resultant preparation. In order to be more efficient and environmentally compliant, we have turned our attention to a solvent free UV driven technology yielding hybrid barrier films in a fast single step, and without the requirement of surface treatment. As depicted in Scheme 1, our approach for the synthesis of this ''superprimer'' is based on a twin sol gel and organic photopolymerization catalyzed by highly efficient photo generated superacids. To be as close as possible to industrial requirements, irradiation is performed under a UV conveyor used conventionally for the series production of UV curable industrial coatings on metal (can, coil and automotive coatings) but also wood and plastic [START_REF] Schwalm | UV Coatings Basics Recent Developments and New Applications[END_REF]. Applied directly on metal (without surface treatment layer), the liquid film is initially comprised of diepoxy monomer, n alkyltrimethoxysilane precursor (C n H 2n+1 Si(OCH 3 ) 3 , n = 4 16) and diaryl iodonium hexafluorophosphate salt acting as a photoacid generator (PAG), which are all stable, easy to use and commercially available products [START_REF] Ni | [END_REF]. Upon UV exposure, in situ photogenerated H + PF 6 À catalyst triggers both sol gel process of trimethoxysilyl functions and cationic ring opening polymeriza tion (ROP) of epoxy functions [43,44]. This results in the one step formation of a 13 lm thick poly(n alkysilsesquioxane)/polyether nanocomposite film providing, under certain conditions, promising barrier protection even in the absence of corrosion inhibitors [45,46]. Using a range of spectroscopic and microscopic tech niques, we show that corrosion resistance is determined by the structural characteristics of the two non covalently bound poly mers. Clearly, hydrophobicity of the alkyl chains (i) and cross link ing of the two organic inorganic networks (ii) are essential features to prevent ingress and diffusion of aggressive species (water, ions, and oxygen). In addition, thermodynamic and kinetic considerations on network compatibility (iii) and formation rate (iv) respectively have been addressed to give a comprehensive pic ture of the protection mechanism. Understanding the effect of these four key features has paved the way to a novel hybrid photopolymer coating delivering corrosion resistance and adhe sion for aluminium substrate.

Experimental section

Chemicals

All n alkyltrimethoxysilane precursors were purchased from ABCR: n butyltrimethoxysilane (C 4 TMS, 97 wt.%), n octyltrimeth oxysilane (C 8 TMS, 97 wt.%), n dodecyltrimethoxysilane (C 12 TMS, 95 wt.%), n hexadecyltrimethoxysilane (C 16 TMS, 90 wt.%). Their chemical structures are shown in Table 1. The epoxy resin based on hydrogenated diglycidyl ether of bisphenol A (Epalloy 5000) was provided by CVC Chemicals. The photoacid generator 4 methylphenyl[4 (2 methylpropyl) phenyl] iodonium hexafluoro phosphate was supplied by BASF. Byk 333 (BYK Chemie) is a surface wetting agent based on a polyether modified polydimethylsiloxane. All chemicals were used as received, without any further purification.

2024 T3. aluminium alloy panel preparation

The 2024 T3 aluminium panels (AA2024 T3, 125 Â 80 Â 1.6 mm) were provided by Airbus Group Innovations and prepared with adapted home made and confidential Airbus Group Innovations surface preparation. Although the surface prep aration is critical for anticorrosive properties, the procedure used herein is general and has not been modified specifically for the present hybrid sol gel coating.

Preparation of poly(n alkylsilsesquioxane)/polyether hybrid films

The hybrid precursor was mixed with Epalloy 5000 organic resin at a concentration between 0 and 60 wt.%. Subsequently, 0.5 wt.% of Byk 333 and 2 wt.% of PAG were added to this homoge neous solution labelled E 100Àx C n TMS x with x the weight fraction in organosilane. Despite the use of hydrophobic n alkylsilane and epoxy resins, there is no insolubility issues to worry about because solvent and hydrolysed products are absent from the initial formu lation. The resultant photolatent formulation was applied onto prepared AA2024 T3 aluminium alloy substrates using an Elcome ter 4340 automatic film applicator equipped with a wire wound bar. A reproducible liquid film thickness of ca. 13 ± 3 lm was obtained whatever the formulation composition. The films were irradiated at room temperature through 5 successive passes under a UV conveyor (Fusion UV) including a microwave H lamp (Fusion) to yield cross linked hybrid films. A belt speed of 10 m min À1 was selected to ensure an overall light intensity of 1.46 J cm À2 per pass [UV C (250 260 nm) = 0.09 J cm À2 , UV B (280 320 nm) = 0.42 J cm À2 , UV A (320 390 nm) = 0.45 J cm À2 , UV V (395 445) = 0.50 J cm À2 ]. During the irradiation process, the relative humidity, checked with a hygrometer, was found to be between 30% and 35%.

Characterization techniques 2.4.1. Salt spray test (SST)

Corrosion protection performance of the different coatings was evaluated by neutral SST according to ASTM B117 Salt Spray Test standard (and ISO 9227) in the same fog chamber. For each film composition, three specimens were exposed under similar condi tions to a salty fog atmosphere generated by a 5 wt.% aqueous NaCl solution (pH range from 6.5 to 7.2 and temperature: 35 °C ± 2 °C). ASTM B117 specifies the apparatus, the reagents and the procedure to be used in conducting the neutral salt spray. In our case, the per iod of exposure was varied (400 2800 h) without interruption and the test specimens had typically a size of 120 Â 80 Â 1.2 cm. Prior to exposure, the coated substrate was scratched (cross scratch). Whilst the use of scratched specimens in SST is not common for the characterization of anodizing or conversion film alone, it is widespread for primers. Since the objective of this study is the syn thesis of a ''superprimer'' that can substitute both for the first sur face treatment layer surface and the second layer of primer, a scratch was systematically carried out for all samples. After a given exposure time, the three samples were washed in clean running water to remove salt, and then a picture was taken immediately without drying. The reproducibility was first checked, and a detailed examination of the corrosion points (number, size and position), blisters as well as change of colour was made. The partic ular requirements are the absence of blistering, and extended corrosion on either side of the scratch (no more than 1.25 cm).

Electrochemical impedance spectroscopy (EIS)

A three electrode electrochemical cell was used in EIS measure ments. A coated specimen was used as working electrode. A cylin drical plexiglas tube was assembled on top of the coated sample (exposed surface area: 24 cm 2 ) and filled with the aggressive solu tion prepared from distilled water by adding 0.5 M NaCl (reagent grade). A large platinum sheet and a saturated calomel electrode were used as counter and reference electrodes, respectively. The electrochemical cell was kept at room temperature and open to air. Electrochemical impedance measurements were carried out using a Biologic VSP apparatus. The impedance diagrams were obtained under potentiostatic conditions at the corrosion potential over a frequency range of 200 KHz to 3 MHz with 10 points per decade using a 20 mV peak to peak sinusoidal voltage. The elec trochemical behaviour of the coatings was characterized for differ ent exposure times to the aggressive solution ranging from 2 h to 336 h (14 days).

Real time Fourier transform infrared spectroscopy (RT FTIR)

Epoxy photopolymerization and methoxy hydrolysis conver sion time curves were obtained by RT FTIR performed in transmis sion. In a typical experiment, the liquid C n H 2n+1 Si(OCH 3 ) 3 / monomer/PAG film was exposed at room temperature to UV light and also to an IR analytical beam. A particular UV irradiation source was implemented to allow the acquisition of infrared spec tra throughout the irradiation process. The C n TMS x E 100Àx film was deposited on BaF 2 pellets and irradiated by the polychromatic light of a mercury xenon lamp (Hamamatsu, L8252, 200 W) coupled with a flexible light guide at an incident light intensity of 200 mW cm À2 . The end of the optical guide was placed at a dis tance of 3 cm from the film and directed at an incident angle of 90°onto the sample. The IR spectra were recorded with a Bruker Vertex 70 equipped with a liquid nitrogen cooled mercury cad mium telluride detector. The resolution of the infrared spectra was 4 cm À1 . During UV irradiation, the decreased absorbance of the C H epoxy stretching vibration band at 3050 cm À1 was assessed to monitor epoxy conversion of the Epalloy 5000 monomer. The decay of the symmetrical C H vibration (t sym (Si O CH 3 )) IR absorption at 2840 cm À1 was used to determine the hydrolysis degree throughout the irradiation time. The relative rate of methoxysilyl hydrolysis and epoxy conversion furnished by RT FTIR was taken as an indicator of the concomi tancy of both polymerizations.

Solid state nuclear magnetic resonance (NMR)

The 29 Si ( 13 C) cross polarization magic angle spinning (CP MAS) solid state NMR spectra was performed on a Bruker Avance II 400 spectrometer with a Bruker double channel 7 (4) mm probe. Zirco nium rotors were employed at 79.48 (100.6) MHz using a recycling delay of 5 (5) s, a spinning frequency of 4 (12) KHz and a contact time of 4 (1) ms. 29 Si chemical shift is relative to tetramethylsilane.

Water contact angle

The contact angle measurements were performed with a KRUSS DSA 100 equipped with a video camera. Each analysis was made at room temperature by means of the sessile drop technique. Ten measurements were performed on the same sample to obtain an averaged value.

Scanning electron microscopy ene,gy dispersive X ray analysis (SEM EDX)

Morphology of the samples was characterized by SEM (FEI Quanta 400 microscope working at 30 kV). Before the analysis, the samples were coated with a 15 nm thick layer of gold to reduce the charging effect on the surface. Cllemical composition of film interfaces was determined by EDX acquired as point and/or line scans. Ali analyses were repeated four times and the results were averaged out.

UV Visible spectrophotometry

UV Visible spectra were recorded from 300 nm to 800 nm on a Shimadzu UV 1601 spectrometer with UV cured films prepared on borosilicate glass substrate.

Results and discussion

Hybridfilm characterization

Nanometre scale structure is primarily responsible for the cor rosion performance of hybrid sol gel films. ln our case, the relative simplicity of the three component system (hybrid precursor/ organic monomer/PAG) is conducive to an in depth study of the structure properties relationship. Four structural data having a presumed protective effect are successively investigated including network cross linking, silsesquioxane hydrophobicity, photopoly merization kinetics and the thermodynamic compatibility between the two organic and inorganic networks.

Double cross linked networks

After UV ex posure of a C n H 2 ,,. 1 Si(OCH 3 h/monomer/PAG film, the extent of cross linking of the polysilsesquioxane and polyether networks was examined using 29 Si and ne CP MAS NMR respectively. The 29 Si NMR spectra in Fig. 1A were obtained with four different alkyl chain lengths (n = 4, 8, 12 and 16) at a constant weight concentration in alkyltrimethoxysilane (50 wt.%). ln most cases, the spectra are dominated by T 3 (-Si(OSih) and T 2 (-Si(OHXOSi) i ) species, which are the most condensed siloxane structures. Clearly, these results demonstrate the use of diaryl iodonium hexafluorophosphate as an efficient in situ generated cat alyst for the hydrolysis and condensation of these organosilanes. Furthermore, a strong dependence of the substituent hindrance on the degree of condensation is observed. Thus, the ratio of the Jess substituted T 1 siloxane speàes (-Si(OH) i (OSi)) is significantly stronger upon increasing the chain length. In Fig. 18, the ne spec trum of the n C 8 TMS 50 E 50 film shows a complete epoxy ring open ing, retlected by the absence of epoxy carbons arising only in the monomer at ô ::,, 49 ppm. ln contrast with the siloxane network, only qualitative evidence of epoxy cross linking is provided by two broad peaks at 71.0 ppm and 79.6 ppm assigned to the two polyether carbons ( CH 2 CHR O i In addition, the breadth of this latter signais and the emergence of an additional peak at ca. 58 ppm ascribed to methoxy carbons are indicative of competitive transfer reactions with methanol molecules released by hydrolysis reac tions (sol gel process). Propagation reactions leading to chain growth, and transfer reactions interrupting this growth are described in the inset of Fig. 1 B. The presence of deleterious transfer reactions involving hydroxyl containing compounds is well described in cationic polymerization [ 47,48 ). However, it is difficult to quantify precisely their impact on the cross linking of the polyether network. ln addition, reducing their level remains tricky in our tandem photopolymerization process because they are initiated by adventitious alcohol species continuously released by the sol gel process.

Film hydrophobidty

Contact angle measurements were performed to assess the change of hydrophobiàty resulting from the addition of n alkyltri methoxysilane. As shown in Table 2, the incorporation of only 5 wt.% n C 8 TMS is sufficient to increase the water contact angle from 76 ° to 101 °. To enhance the concentration even further hardly change the wetting properties (::,,100 °), showing that a surface saturation is achieved at rather low organosilane content. ln several studies, minimization of interfacial energy by migration of low energy chemical groups to the air/film interface has proven to be an efficient method to limit water condensation, which is the first step before its ingress through the film [49). Another way to evidence the preferential location of organosilane molecules at the air/film interface is to examine the difference of chemical corn position between the two sides of the film after delamination on the aluminium substrate. As shown in Fig. 2, the smooth upperside and the rough underside of the n C 8 TMS 50 E 50 film have been exam ined by SEM compositionally probed by EDX. A higher Si/C weight ratio (0.3) is found at the air/film interface compared with the substrate/film interface (0.1 ), supporting a preferential concen tration of hybrid precursor at the externat surface. Mechanistically, this result is consistent with the formation of transient mobile amphiphilic species (C, ,H 2n+1 Si(OH )x(OSih-xl arising from hydroly sis and moving to the surface like surfactant before they are fixed by siloxane cross linking [50). Additionally, the weight concentra tion in iodine at the metal/film interface (0.6 wt.%) is double than that detected at the air/film interface (0.3 wt.%), showing a higher local concentration in PAG (Ar 2 i + Pfs) at the metal/film interface ( vide infra, part 3.2 ).

Organic inorganic polymerization kinetics

ln situ RT FTIR spectroscopy was used to monitor the kinetics of hydrolysis and ROP for three distinct CnTMSsoEso films (n = 4, 8 and 16). ln ait instances, the conversion time curves in Fig. 3 show a fast and complete consumption of the epoxy and trimethoxysilyl functions. Whilst the ring opening is relatively insensitive to alkyl substituent hindrance, hydrolysis becomes slower with longer car bon chain lengths. But the most salient feature is that these two photoinduced processes take place on a similar time scale. Whilst the concomitancy between both reactions is particularly marked with the n butyl precursor, difference in conversion rates tends to widen with longer alkyl chains. In a dual process involving two (a priori) incompatible organic and inorganic networks, reaction kinetics are assumed to have strong impact on the final structure. In particular, the minimization or elimination of macrophase sep aration can be favoured kinetically through two cross linking reac tions occurring at similar time scales and rapidly enough. In the present case, concomitant and fast organic inorganic polymeriza tions can drive the formation of kinetically trapped nanocompos ites, Whilst thermodynamically driven process would have promoted more composite like or segregated mixtures.

Network compatibility

Although all n alkylsilane precursors are fully miscible with the diepoxy resin before reaction, the subsequent cross linking reac tions and the absence of covalent bonds between the two polymer networks are clear challenges for the preservation of film homoge neity. SEM examination of the C 8 TMS based film (C 8 TMS 50 E 50 ) reveals a uniform and homogeneous surface morphology (Fig. 4A). By contrast, its replacement by C 12 TMS and C 16 TMS at similar con centration results in significant signs of phase separation increasing with the length of the alkyl chain (Fig. 4B andC). Accordingly, the optical properties of the film change from high transparency in the UV Vis range to much higher absorbance values when increas ing the alkyl chain length (Fig. 5). The formation of a phase segre gated structure in the presence of the longest alkyl chains (C 12 TMS and C 16 TMS) results in light scattering and significant light attenuation. Despite the relative simultaneity of both reactions for these three precursors, the incorporation of long alkyl chain (n P 12) drives a phase separation consistent with a decreased affinity between the two networks. This suggests that not only kinetics but also thermodynamics determine the final structure.

Corrosion protection properties

A first diagnostic of the protecting ability of the films is given by the number and size of defects (etch pits) appearing on the coated surface after 400 h of exposure in the salt spray chamber (Fig. 6). For organic polyether films (sample A), more than 150 corrosion points are observed on coated AA2024 T3 substrate and the corro sion protection is considered ineffective. These features are circular with diameters between 1 and 3 mm. Their density is drastically reduced by the addition of n alkylsilane (samples B E). Examina tion of the effect of the alkyl chain length (C n TMS, n = 4, 8, 10 and 16) reveals that the alkyl chain length of n = 8 (sample C) yields the best protection, with no corrosion on the surface. In addition, the corrosion protection is strongly dependent on organosilane concentration and an optimal content in C 8 TMS is found at ca. 50 wt.% (see Fig. S1 of electronic supporting information). spray of at least 3000 h for the full 3 layer protective system (surface treatment layer+ primer+ top coat). Note that such per formance can be achieved without topcoat by a protective chromated primer applied on a conventional surface pretreatment like Alodine ( chromated) or tartaric acid anodisation layer (non chromated). ln our case, the protective performance of the "super primer" alone was assessed. The first corrosion points appear after 670 h. in particular close to the scratch. For longer exposure times, the defects increase only slightly in number and in size. However, there is no sign of blistering, Joss of adhesion, change of colour, and no corrosion extending further than 2 mm on either side of the scratch. We think that 2800 h of SST gives already a good insight into the potential of this nove) photopolymer hybrid coating. Given the simpliàty of our system, these results are promising, whilst offering a great scope for improvement with a more complex mul ticomponent formulation. Furthermore, the corrosion resistance of a complete system including both a "superprimer' and a topcoat will need to be tested, particularly because the hydrophobicity of the hybrid coating may raise some issues about top coat adhesion. The behaviour of the C 8 TMS 50 E 50 hybrid film has been also characterized by electrochemical impedance spectroscopy (EIS). structure remains difficult to prove, but could be supported by the higher concentration in iodine found in the vicinity of the substrate (Fig. 2). This suggests a locally higher photoacid concentration at the metal/film interface, driving further condensation than in the bulk and the formation of a highly cross linked inner layer.

Conclusions

A UV driven methodology having many process advantages has allowed the synthesis of a direct to metal protective hybrid coating providing efficient barrier properties on aluminium alloy substrate. The use of photoacid species catalyzing simultaneously diepoxy ROP and organosilane condensation enables fast polymerization rates and highly cross linked siloxane network without the need of solvent or even water. This opens the door to a novel ''superpri mer'' capable of replacing both surface treatment and primer. A combination of SST and EIS analysis has revealed the best performances of the poly(n octylsilsesquioxane)/polyether coating. Comparison with a pure polyether film shows that n alkyltrimeth oxysilane is responsible for the enhanced corrosion protection. Its first mode of action is to increase film hydrophobicity. Hydrocarbon chains are able to provide low surface energy, and effectively ''shield'' hydrophilic groups such as alcohols (polyether network) or silanols (polysiloxane network). A long enough alkyl chain (n P 8) and sufficient organosilane loadings (50 wt.%) are necessary to increase the barrier properties. Yet, enrichment in alkyl groups on the air side interface is noticeable at low concentration, this means that further alkyl chains embedded inside the film are also essential to deliver an efficient corrosion resistance. A second effect of n alkyltrimethoxysilane is to generate a silsesquioxane network that complements the barrier effect already provided by the poly ether network. The high degree of condensation attainable under photoacid catalysis is also instrumental in ensuring a minimal number of residual hydrophilic silanols. In addition to these two structural features, further experiments have shown the impor tance of having a relative compatibility between the two noncova lently bound networks. To favour the formation of a homogenous interpenetrated system, too long alkyl chains (n P 12) raising phase segregation and leading to inefficient protection must be avoided. Thus, the best performances are achieved with the use of the intermediate C 8 H 17 Si(OCH 3 ) 3 precursor. Another way to mitigate the incompatibility between the two networks is to pro mote a rapid single step formation. In situ kinetic measurements by RT FTIR has supported the occurrence of kinetically driven path way in which the two poorly miscible polyether and silsesquioxane structures could be ''fixed'' at the nanometre scale through fast and concomitant cross linking reactions.

+(Scheme 1 .

 1 Scheme 1. Photoinduced formation of two non-covalently bound networks via photoacid-catalyzed sol-gel and epoxy ROP.

  to the sum of monomer and precursor)

Fig. 1 .

 1 Fig. 1. (A) 2 9Si CP-MAS NMR spectra of UV cured C n TMS 50 E,; 0 films (n • 4. 8, 12 and 16); (B) 13C CP-MAS NMR spectrum of the UV-0ned CaTMS50Es0 film (inset: propagation and transfer reactions to methanol in a cationic polymerization process, R • � � 0 . . _,}

Fig. 2 .Fig. 3 .

 23 Fig. 2. SEM images showing the two sides of a UV-cured C 8 TM5,; 0 Es., film after delamination on an aluminium alloy substrate: metal/film (A) and air/film (8) interfaces.

Fig. 4 .

 4 Fig. 4. SEM im<lges of the hybrid films containing three dilferent n-alkylsilane (50 wt.%): (A) Ca TMSsoEso, (8) C12TMSsoEso and (C) C,6TMSsoEse>

Fig. 7 Fig. 5 .Fig. 6 .Fig. 7 .

 7567 Fig. 5. UV-Visible spectra of the hybrid coatings: (-) C 8 TMS 50 E 50 , (---) C 12 TMS 50 E 50 , and (ÁÁ Á) C 16 TMS 50 E 50 .

Fig. 8 .

 8 Fig. 8 reports impedance diagrams obtained after different immer sion times in a 0.5 M NaCI solution. The results, presented in Bode coordinates, show three time constants in the high (HF), medium (MF) and low frequency range (LF). The LF time constant is attrib uted to the presence of a thin oxide film covering the AA2024 T3.More interestingly, the two other time constants are considered to be representative of the coating and suggestive of a double layered morphology. lmpedance modulus decreases with increasing immersion time, particularly at the beginning of immersion, which indicates a water uptake and a decrease in the barrier properties. A decrease of the impedanoe modulus in the LF range is also obseived which may be attributed to the increase of the surface area at the metal/coating interface in contact with the electrolyte. However, it can be seen that the diagrams are poorly modified between 168 h and 336 h of immersion, showing that the system evolves slowly. High impedance modulus values (10 7 10 8 n cm 2 ) are measured, indicating that high level of corrosion protection can be achieved. The MF time constant at 1 10 Hz can be represen tative of a first inner layer at the oxide/coating interface, Whilst the HF time constant at 100 1000 Hz is rather ascribed to a second outer layer in contact with the electrolyte. The two layered

Table 1

 1 Chemical structures of the n-alkyltrimethoxysilane precursors, diepoxy monomer and photoacid generator used for the direct-to-metal hybrid coating.

	Compound	Chemical structure	Weight ratio (wt.%)
	n-Butyl trimethoxysilane		OCH 3	0-60
		H 3 CO	Si
			OCH 3
	n-Octyl trimethoxysilane		OCH 3
			Si

Table 2 Erfect of the concentration of Ca TMS on the surface properties of CnTMS,E10O_, films.

 2

	Ca TMS concentration	Contact
	( wt:X )	angles ( 0 )
	0	76 2
	5	100 2
	10	102 2
	20	99 2
	40	101 2
	50	101 2
	60	102 2

Acknowledgements

This work was carried out in the framework of the MHYRCEA project, with the financial support of Agence Nationale de la Recherche (ANR) under contact ANR 08 MAPR 0003. The authors gratefully acknowledge the partners of the project: Airbus France (Dr. Mélanie Bedu), Airbus Group Innovations (Dr. Sophie Senani), Institute Charles Sadron in Strasbourg (Prof. Christian Gauthier, Dr. Vincent Le Houérou) and ABB (Christophe Jouve).

Appendix A. Supplementary material

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.corsci.2014.09.006.


Benoît Pouffary

Regiane Fortes

Jean-Luc Reboud

Pierre-Alain Lambert

Numerical Simulation of 3D Cavitating Flows: Analysis of Cavitation Head Drop in Turbomachinery The numerical simulation of cavitating flows in turbomachinery is studied at the Turbomachinery and Cavitation team of Laboratoire

des Ecoulements Géophysiques et Industriels (LEGI), Grenoble, France in collaboration with the French space agency (Centre National d'Etudes Spatiales, CNES), the rocket engine division of Snecma and Numeca International. A barotropic state law is proposed to model the cavitation phenomenon and this model has been integrated in the CFD code FINE/TURBO™. An analysis methodology allowing the numerical simulation of the head drop induced by the development of cavitation in cold water was proposed and applied in the case of two four-bladed inducers and one centrifugal pump. Global results were compared to available experimental results. Internal flows in turbomachinery were investigated in depth. Numerical simulations enabled the characterization of the mechanisms leading to the head drop and the visualization of the effects of the development of cavitation on internal flows.

The considered numerical approach uses a single fluid model to describe the liquid-vapor mixture, with a mixture density varying in the flow field between the vapor density and the liquid

Introduction

Cavitation is an important phenomenon for the design of rocket engine turbopumps and remains a major issue for the space industry. That is why the CNES and the Rocket Engine Division of Snecma have supported research in order to make progress in the understanding and prediction of the mechanisms associated with cavitation for many years. In this frame, one main purpose of the research performed at the Turbomachinery and Cavitation team of Laboratoire des Ecoulements Géophysiques et Industriels ͑LEGI͒ consists in developing and validating a 3D Reynolds-averaged Navier-Stokes ͑RANS͒ code, enabling the simulation of the cavitating operation of turbopumps. Through collaboration between LEGI and Numeca International, a physical model of cavitation was implemented in the commercial code FINE/TURBO™ ͓1-3͔.

The applied numerical model is very similar to the one proposed by Kunz et al. ͓4,5͔ to calculate cavitating flows in centrifugal pumps. The main difference between the present work and those studies is the physical model applied to describe the cavitation phenomenon. In this study, the barotropic cavitation model initially proposed by Delannoy and Kueny ͓6͔ is considered and presented in the first section of this paper.

The 3D RANS code, described in Sec. 3, allows the steady and unsteady cavitating flow calculations in the 2D or 3D geometries, as presented in Refs. ͓1-3,7͔, and could be used as a tool to analyze inducer instabilities ͓8͔.

In complement to previous work ͓1,7,9͔, the main target of the present study is to propose a qualitative and quantitative analysis of the cavitation head drop in turbomachinery. As a matter of fact, some recent numerical works have been developed to predict cavitation inception, cavity dimensions, and/or the threshold corresponding to pump head drops. For example, Ait-Bouziad et al. ͓10,11͔ applied off-the-shelf software to investigate cavitating flows in a three-bladed industrial inducer. The authors present interesting global and local analyses of the cavitating flow field, but they noticed that the absolute value of head drop was not well predicted by the numerical simulations. Mejri et al. ͓12͔ applied the same off-the-shelf software to analyze the overall performances, as well as the location and sizes of blade cavities for the three three-bladed inducers identified by different blade angles at the leading edge. A qualitative agreement between the experimental and predicted results was found for two inducers. The authors concluded that the implemented cavitation model overpredicted the cavitating flow, mainly in partial load conditions.

We cite also the important numerical work developed by Hosangadi and Ahuja ͓13͔ concerning cavitating flows over a hydrofoil and a cylindrical headform. In Ref. ͓14͔, Hosangadi et al. presented a good comparison between the simulated and experimental data concerning breaking down of a helical flat-plate inducer configuration in cold water.

In spite of those relevant works, more studies are needed to improve and validate simulated and physical approaches in order to obtain reliable quantitative predictions of overall performances, especially in the case of inducer geometries. Moreover, detailed cavitating flow analyses are also required for the better understanding of the head drop hydrodynamic mechanism and of energy transfers in centrifugal and axial pumps.

In this context, the developed numerical model has been applied to calculate 3D steady cavitating flows in a centrifugal pump and in two rocket engine turbopump inducers, whose geometries are described in Sec. 4. The influence of steady cavitation behavior on the pump characteristics and on the final head drops has been simulated and compared to experimental results for different operating points. The observations of the pump global performances and associated local flow fields led us to carry out a detailed analysis of the influence of cavitation in the three different geometries ͑Sec. 5͒. From these results, the origin of the cavitation head drop is discussed and an analysis of its mechanism is proposed. The study is improved by the investigation of the repartition of the total pressure rise, the blade loads, as well as by the evaluation of cavitation influence on secondary flows.

Barotropic State Law

density. The equivalent void ratio ␣ of the liquid-vapor mixture relates to the varying specific mass by = ␣ v + ͑1-␣͒ l .I nt h e mixture, the velocities of liquid and vapor phases are the same and we obtain only one set of equations for the mixture mass, momentum, or turbulence ͑k-͒, written in their conservative form.

The void ratio ␣ of the mixture depends on the local static pressure. As initially proposed by Delannoy and Kueny ͓6͔,w e use a barotropic state law ͑P͒ to manage the relation between pressure and mixture density ͑Fig. 1͒. A smooth arbitrary law was chosen, rapidly varying between liquid density l and vapor density v when the local static pressure P is around the vapor pressure P v . The law is characterized by its maximum slope at P = P v , which is related to the minimum speed of sound c min in the two-phase homogeneous medium ͓8͔.

In the present work, based on experimental previous work concerning Venturi type sections ͑Stutz and Reboud ͓15,16͔͒, the parameter c min is taken equal to 2.35 m / s. The density ratio v / l is imposed to be 0.1. The compressibility of pure liquid and vapor phases is not taken into account. These choices were made to improve the robustness of the numerical code. Extensive numerical tests were carried out in other works to evaluate the influence of the speed of sound c min and of the ratio v / l . Most important results obtained from these influence analyses are described in Refs. ͓17-19͔.

Numerical Model

The barotropic cavitation model was implemented in the commercial code FINE/TURBO™ developed by Numeca International. FINE/TURBO™ is a three-dimensional structured mesh code that solves the time dependent RANS equations, with artificial compressibility method. A detailed description of the initial code is given in Hakimi ͓20͔.

Time-accurate resolutions use the dual time stepping approach. Pseudotime derivative terms are added to the equations. They enable equation convergence at each physical time step. The initial set of mass and momentum equation can then be written under the following form: The space discretization is based on a finite volume approach. A second order central scheme is associated with two artificial dissipation terms, respectively, of second and fourth orders. The first one is activated in the strong pressure and density gradient areas. The other one is used in the whole domain, and it results in a second order space accuracy.

In the case of low-compressible or incompressible flows, the pseudotime derivatives are multiplied by a preconditioning matrix, based on the studies of Turkel ͓21͔, Choi and Merkle ͓22͔. The modifications of the preconditioning technique developed for cavitating flows are detailed in Refs. ͓2,23͔.

The pseudotime integration is made by a four-step Runge-Kutta procedure. In noncavitating conditions, the code resorts to a multigrid strategy to accelerate the convergence. The different computations presented hereafter have been performed in the steady case, without taking into account the real time derivative term.

A more detailed description of the code is given in Refs. ͓1-3͔. A meaningful numerical work has been performed by Pouffary ͓23͔ to improve the preconditioner and the stability of numerical code for calculations of cavitating flows. It is worth noting that these numerical modifications led to major improvements in accuracy and numerical stability in comparison with our previous computations of cavitating flows in turbomachinery ͓7,9,24,25͔. An illustration and quantification of these numerical improvements will be presented in Sec. 5.

Studied Geometries

Centrifugal Pump. The first considered geometry is the SHF centrifugal pump tested by Combes and Archer ͓26͔. The SHF pump has a shrouded seven-bladed centrifugal impeller. The scale and the calculation configuration are chosen as close as possible to the experimental conditions to simplify comparisons in cavitating conditions. The outlet radius is 330 mm, the inlet pipe length is 200 mm, the rotation speed equals to 3000 rpm, and the nominal flow rate is 0.157 m 3 / s. Grid resolution tests have been done in cavitation-free conditions. After comparing pump head and pressure distribution on the blade obtained with different multiblock structured meshes ͑I and H meshes with 170,000, 250,000 and 500,000 cells͒, an I mesh of 250,000 cells for one blade-to-blade channel was chosen ͑Fig. 2͒. Yang-Shih's k-turbulent model with extended wall functions is used ͓27͔.

Turbopump Inducers. To achieve high rotational speed and low inlet pressure, rocket engine pumps are equipped with an axial stage. Two different four-bladed inducers have been tested. Inducer 1 was designed by Snecma ͑Fig. 3͑a͒͒. The influence of the mesh and of the turbulence model ͑i.e., Baldwin-Lomax and three different k-models͒ was firstly studied in noncavitating conditions, mainly by comparing the static pressure profile at the shroud with experimental measurements performed at different flow rates at the CREMHyG Laboratory ͑Grenoble, France͒.A n example at nominal flow rate is given on Fig. 4. From these comparisons, a mesh of about 500,000 cells for one blade-to-blade channel was chosen, associated with a k-model with wall functions ͓27͔. Tip leakage was taken into account in noncavitating conditions, but to increase the numerical convergence speed, it was suppressed for the cavitating flow computations, by locally modifying the mesh. Analysis of the influence of the tip leakage on cavitating flows is in progress.

The geometry of Inducer 2 shows a pure radial leading edge shape ͑Fig. 3͑b͒͒ and has been experimentally studied at the University of Osaka ͓28͔. The mesh has 250,000 cells for one bladeto-blade channel, and the tip leakage is not taken account of.

Simulation results presented in this paper for the three pump geometries are 3D and steady. Used meshes were chosen after a detailed study of mesh influence based on cavitation-free global performances of the inducer ͑head coefficient, torque, and effi-ciency͒.

Analysis of Cavitating Flows

Numerical Conditions. Steady calculations of cavitating and noncavitating flows were performed in these three geometries for different cavitation numbers.

In cavitation-free conditions, two sets of upstream and downstream boundary conditions have been compared in the case of Inducer 1 geometry: either the flow rate is imposed upstream and the static pressure downstream ͑with or without the radial equilibrium condition͒ or total pressure is imposed upstream and flow rate downstream. Influences of the rotation of hub and shroud boundaries upstream and downstream from the impeller have also been analyzed ͓23͔. With a good convergence level of the computations, no major influences of the boundary conditions tested were observed on the evaluation of global performances.

In cavitation conditions, the flow rate is imposed upstream and the static pressure downstream. The different analyzed cavitation numbers ͑or corresponding NPSH͒ are obtained by varying the pseudovapor pressure P v involved in the definition of cavitation number , while the downstream pressure is kept constant.

It is worth noting that each calculated steady state solution was carefully converged. Convergence criteria of cavitating flow calculations were based on the comparisons between inlet and outlet mass-flow rates ͑allowed maximum error Ͻ0.4%͒, on the analyses of the torque, the static and total heads obtained through numerical iterations ͑pseudotime integrations͒, and on the verification of the mass-flow rate through different sections inside the pumps, located in cavitating flow areas.

In relation to previous work ͓2͔, numerical modifications were performed by Pouffary ͓23͔, on the preconditioning technique and associated stability of the numerical algorithms, to deal with very large variations of compressibility of the liquid-vapor mixture in cavitating conditions. "Local velocity scaling" strategy was used for SHF pump calculations. For inducer computations, preconditioning coefficients were calculated as a function of the reference velocity U ref and of the barotropic state law. These improvements allowed a very good convergence level, even in the cases of very large cavitation structures and associated strong modifications of the pump head. Indeed, operating points in cavitating regime leading to a 10% head drop or more could be accurately simulated in the three pump geometries considered in this paper.

Centrifugal Pump. For the SHF centrifugal pump, the cavitating behavior obtained at nominal flow rate Qn shows thin attached sheet cavities on the suction side of the impeller blades, their length increasing with the decreasing cavitation number ͑Fig. 5͒. Head begins to approximately drop when the cavitation sheet reaches the throat between two adjacent blades and rapidly decreases by more than 10% of its noncavitating value ͑Fig. 6͒. The simulated prediction of head breakdown is found for the critical NPSH close to 6.3 m, which corresponds to the NPSH 3% experimentally found at Electricité de France ͓26͔. Flow rates superior to the nominal flow rate lead to larger critical NPSH, head breakdown being observed at 1.22Qn and 1.3Qn for NPSH values, respectively, of about 18 m and 26 m.

Simulation results obtained at nominal flow rate were analyzed in detail. It can be seen on Fig. 7 that the head drop can be mainly associated with a decrease of the torque ͑at the NPSH with 7% total head decrease, the calculated torque decreases by about 5% and the efficiency by 2%͒.

The torque decrease can be illustrated by Fig. 8, where the static pressure distributions around a blade section at midspan are drawn for different NPSH values. When the net pressure P ref -P v becomes smaller, a long sheet cavity develops along the blade suction side, reducing the suction effect near the blade leading edge, without significantly modifying the pressure distribution along the wetted surfaces, and thus clearly reducing the blade load. At head breakdown, it can be observed that cavitation also appears on the upstream part of the blade pressure side ͑NPSH ϳ 6.3͒. Such observations can also be made along other sections of the pump blades.

To study the energy transfers within the pump, an analysis of (b) (a) (c) If one observes the meridian repartition of the total pressure rise corresponding to Calculations 3-6, one can note also, downstream of the cavitation sheet, an augmentation of the total pressure rise. This phenomenon of compensation is mainly observed between Cuts IV and VI.

For a better understanding, a local analysis of the influence of cavitation on the flow structure in the different sections has been also performed. In the present paper, the analysis of the secondary flows is based on the dimensionless relative helicity, drawn for the four upstream sections ͑I-IV͒: Fig. 11͑a͒ illustrates the results concerning cavitation-free condition and Fig. 11͑b͒ corresponds to a regime close to the cavitation breakdown. The blue colors represent the anticlockwise vorticity and the yellow to red ones indicate the clockwise vorticity. In noncavitating conditions, one can observe that the secondary flows in the rotating frame mainly develop along the blade suction side ͑blue color͒, progressively evolving to a hub vortex ͑Section IV͒. Clockwise vorticity ͑red color͒ can also be observed along the shroud.

In cavitating conditions ͑Fig. 11͑b͒͒, secondary flows are strongly modified along the cavitation sheet surface: in Section I, at the suction side, one observes a well organized structure of clockwise vorticity corresponding to the region of the cavitation sheet. In Section II, this structure disappears. The cavity wake generates a large anticlockwise structure, which seems to be related to the centrifugation of the cavity wake. In Sections III and IV, these vorticity structures due to the presence of the cavity sheet progressively dissipate. Inducer 1. An analogous analysis has been performed for the Inducer I geometry. Development of cavitation structures starts from the blade tip and rapidly extends along the leading edge to the hub side. Then the cavitation sheet length progressively increases until it reaches the throat ͑Fig. 12͒. The head drop abruptly occurs when the cavity enters into the blade-to-blade channel.

Comparison of the simulated cavitation performance chart with experiments performed at the CREMHyG Laboratory is drawn on Fig. 13. For a high cavitation number, the obtained simulated value is found about 5% higher than the experimental inducer head, which can be mainly attributed to the fact that the tip leakage flow is not taken into account by numerical calculations. Successive cavitating flows were computed and a good convergence could be obtained even with more than 10% head loss. The critical cavitation number is very well predicted ͑Fig. 13͒ and the improvement of this prediction is significant with respect to the preceding code version ͓29͔.

According to the analyses done, detailed in Ref. ͓23͔, the inducer head drop seems to be mainly associated with a drop of the torque: For the critical cavitation number, the calculated torque decreases by about 10% of the noncavitating value, while hydraulic efficiency decreases by about 3%.

Static pressure distributions around the blade are drawn on Fig. 14 side distribution near the leading edge. The load reduction is observed when the cavitation sheet developed on the upstream blade suction side reaches the entry of the blade-to-blade channel. Perturbations of the trailing edge pressure distribution can also be observed. These perturbations can be associated with noticeable modifications of secondary flows, as will be discussed hereafter.

The mass-flow averaged total pressure rise between successive sections in Inducer 1 is drawn on Fig. 15. Analyzed sections are perpendicular to axial direction, as schematized in Fig. 21 for the Inducer 2 geometry. One can observe that the head decrease is mainly caused by the decrease in the upstream region.

The secondary flows in cut Sections I and II ͑presented in Fig. 12͒ are analyzed in Fig. 16. In noncavitating condition, an intense anticlockwise vorticity structure near the shroud is observed, which seems to be mainly due to the kinetic energy of the tip leakage flow. Another anticlockwise vorticity structure is observed near the suction side and a clockwise one is present close to the pressure side; both structures are related to the centrifugation of the boundary layer along the blades. There is no relevant vorticity effect in the middle of the analyzed sections.

In cavitating conditions ͑Fig. 16͑b͒͒, near the head breakdown, the secondary flow in Section I is slightly modified. On the other hand, the outlet flow is highly influenced by the important cavity wake structure. This can be seen in Section II: the clockwise vorticity structure observed near the pressure side is amplified; the vorticity structure close to the suction side is convected through the middle of the section; a small secondary structure appears near the hub. Figure 16 shows that the downstream flow is highly influenced by the cavitation development: Increasing vorticity observed near the pressure side ͑left part of Plane II in Fig. 16͑b͒͒ interacts with the suction side in the trailing edge region of the inducer ͑right part of Plane II͒, and seems to lead to pressure distribution perturbations observed in Fig. 14 in this zone.

It is worth reminding that calculations performed under cavitating conditions do not take into account tip leakage. It is the reason why intense anticlockwise vorticity structure observed near the shroud in noncavitation condition ͑Fig. 16͑a͒͒ does not appear on Fig. 16͑b͒. Improvements of methodology for tip leakage meshing, in association with multigrid strategy to accelerate the convergence of cavitating flow calculations, are needed to progress on secondary flow analyses and on the evaluation of the tip clearance effect for cavitating cases. Such improvements are still in progress.

Inducer 2. The second analyzed inducer geometry shows a quite different behavior in cavitating conditions. According to our numerical study, as illustrated by Fig. 17, the development of the cavitation structures appears much more pronounced at the inducer periphery. We can observe also a large backflow structure in front of the blades. On the blade suction side, the cavitation sheets show a triangular shape, increasing from hub to periphery, and merging with the cavitating backflow structure.

On the performance chart, illustrated by Fig. 18, the head drop much more gradually appears than for Inducer 1. The last point calculated shows a 20% head drop, mainly due to the drop of the torque. The efficiency decreases by about 3%.

The fact that the cavitation head drop is less abrupt can be explained by analyzing the blade load evolution drawn on Fig. 19, and comparing it to Fig. 14. In noncavitating condition, the blade loads of the two inducers appear very different: Inducer 2 blades are mainly loaded near the leading edge, while the load remains approximately constant along the Inducer 1 blades. In cavitating regime, the development of the sheet cavity from the leading edge of the Inducer 2 blade suction side gradually modifies the blade load, while in Inducer 1, the blade load decreases only when the cavity closure reaches the blade-to-blade passage.

It can be clearly seen on Fig. 20 that the main part of the total pressure rise in Inducer 2 is provided by the upstream part of the inducer. As a consequence, cavitation head drop mainly occurs due to a progressive decrease of the pressure rise between Sections I and II. The blade load at the downstream part of the inducer is not very significant and a compensation phenomenon cannot appear with this inducer design.

Conclusion

A numerical model of 3D cavitating flows has been developed to predict the cavitation behavior in turbomachinery. Compared to the previous studies ͓7,9,24,25͔, thanks to recent improvements of numerical stability and algorithm reliability, very good conver-gence and accuracy of the simulation of cavitating flows in turbomachinery can now be obtained, even for operating conditions very close to the machine head drop ͑Fig. 13͒.

Simulation results were presented and analyzed for a centrifugal pump and two different four-bladed inducers. Noncavitating and cavitating conditions were investigated and compared. The vapor structure extensions and associated pump head breakdowns seem to be correctly simulated by the numerical model and favorably compare with available experimental results.

Local analyses of the results provide interesting information on the breakdown mechanisms.

• In the three geometries, head breakdown is clearly associated with a decrease of the torque provided by the impeller, through a strong variation of the blade load. Otherwise, only a small decrease of the machine efficiency can be associated with the development of the cavitation structures. • An important difference between the cavitation behaviors of the two inducers can be observed, which can be explained by an analysis of their respective blade load and total pressure rise repartition. • Local visualizations of the flow fields and their evolution with cavitation can also be obtained. For example, the analysis of dimensionless relative helicity maps gives useful information about secondary flows in the machines and enables the identification of the location and influence of the cavity wakes in the blade-to-blade channels.

The methodology of analysis proposed in the present paper shows that numerical simulation can provide useful information for the design of turbomachinery, and more particularly of inducers. Applications performed on unsteady flows are in progress and the first results presented in Refs. ͓8,17͔ are very promising for the analysis and prediction of unstable cavitation behavior in turbomachinery.

Improvements of the cavitation model are also in development, mainly to take into account the thermodynamic behavior of fluids, such as hydrocarbons or cryogenic propellants.

  of the conservative variables, F = F c -F v notes the convective and diffusive fluxes across the surface dS = n dS,STis the source term, and the pseudotime.

Fig. 1

 1 Fig. 1 Graph of the barotropic state law = "P…. Illustration of the speed of sound influence for water at 20°C. For more information, see Ref. †8 ‡.

Fig. 2

 2 Fig. 2 Mesh of the SHF centrifugal pump

Fig. 3

 3 Fig. 3 Inducer geometries: "a… Inducer 1; "b… Inducer 2

Fig. 5

 5 Fig. 5 SHF centrifugal pump, nominal flow rate, for decreasing NPSH= "P ref -P v … / "g… É 9.5 m, 7 m, and 6.3 m. The yellow color corresponds to a 5% equivalent void ratio. Comparison experiment/calculation of the suction side cavity extension. Experimental visualizations by Electricité de France †26 ‡.

Fig. 7 Fig. 8

 78 Fig. 7 Nominal flow rate. Evolution of mechanical and hydraulic power calculated; mechanical power= T hydraulic power = ⌬P tot . Q.

  Fig. 11 "a… Secondary flows: Dimensionless relative helicity, varying from -1 "blue, maximum anticlockwise vorticity… to 1 "red, maximum clockwise vorticity…. Noncavitating flow. Planes I to IV. "b… Secondary flows "dimensionless relative helicity…. Cavitating flow corresponding to 7% total head decrease. Planes I to IV.

Fig. 12

 12 Fig. 12 Inducer 1, nominal flow rate, for decreasing cavitation number. The blue color corresponds to a 5% equivalent void ratio. Upstream cavitation numbers: "a… 0.0196; "b… 0.016; "c… 0.014; "d… 0.012.

Fig. 16

 16 Fig. 16 Secondary flows "dimensionless relative helicity…, and noncavitating "a… and cavitating "b… flows. Planes I and II.

Nomenclaturec

  min ϭ minimum speed of sound in the medium ͑m/s͒ Hr norm = W . rot͑W͒ ʈWʈ . ʈrot͑W͒ʈ ϭ dimensionless relative helicity L ref ϭ geometry reference length ͑m͒ P ref ϭ reference pressure= total upstream pressure ͑Pa͒ P up ͑P dw ͒ ϭ static upstream ͑downstream͒ pressure ͑Pa͒ P v ϭ vapor pressure ͑Pa͒ Q ϭ flow rate ͑m 3 / s͒ t ϭ physical time ͑s͒ T ϭ torque ͑Nm͒ u , v , w ϭ velocity components ͑m/s͒ U ref ϭ reference velocity ͑generally U bladeគtip ͒͑ m/s͒ W ϭ relative velocity vector ͑rotating frame͒͑ m/s͒ ␣ ϭ void ratio

Fig. 19

 19 Fig. 19 Nominal flow rate. Inducer 2. Blade load at mid span, for decreasing cavitation numbers.

Fig. 21

 21 Fig. 21 Illustration of Planes "a"-"f" in the meridian view of Inducer 2

Acknowledgment

This research was supported by the French space agency Centre National d'Etudes Spatiales ͑CNES͒ and Snecma. The authors wish also to express their gratitude to Electricité de France and CREMHyG for providing experimental results, to Professor Tsujimoto for providing the Inducer 2 geometry, and to Numeca International for its cooperation to the development of the numerical code.