Robert Fagiewicz

Corinne Crucifix

Torben Klos

Célia Deville

Bruno Kieffer

Yves Nominé

Johan Busselez

Paola Rossolillo

Helgo Schmidt
email: schmidth@igbmc.fr

In vitro characterization of the full-length human dynein-1 cargo adaptor BicD2

INTRODUCTION

Cytoplasmic dynein drives many retrograde microtubule transport events in eukaryotic cells. The main isoform -cytoplasmic dynein-1 (hereafter dynein-1) is involved in cell division, the transport of organelles and vesicles, brain and muscle development, and can also be hijacked by pathogenic viruses to reach cellular locations (Reck-Peterson et al., 2018; , Paschal and Vallee, 1987; , Dodding and Way, 2011; , Wilson and Holzbaur, 2012; , Fu and Holzbaur, 2014). It is also a particularly interesting motor in neurons, as it carries essential signals and organelles from distal axon sites to the cell body [START_REF] Schiavo | Cytoplasmic dynein heavy chain: The servant of many masters[END_REF].

Dynein-1 motility and cargo specificity depends on cargo adaptor proteins. Currently there are eleven dynein-1 cargo adaptors known and this number is continuously growing [START_REF] Olenick | Dynein activators and adaptors at a glance[END_REF]. Cargo adaptors are characterized by α-helices forming extended coiled-coil domains and they can interact with multiple proteins. Generally, the N-terminal region of the cargo adaptor associates with the dynein motor while the Cterminal domain recruits cargos. Selectivity towards their cargos is achieved by adaptorspecific cargo binding mechanisms (Schroeder et al., 2014; , Schroeder and Vale, 2016; , Gama et al., 2017).

The best studied family of cargo adaptors are the BicD proteins. Mammals have two BicD orthologues, BicD1 and BicD2. BicD's form homodimers and based on secondary structure predictions, they fold into the three coiled-coil domains CC1, CC2, and CC3 (Figure 1 A). The N-terminal CC1 coiled-coil domain was shown to bind dynactin and dynein via the ARP1 subunit of dynactin and the N-terminus of the dynein heavy chain [START_REF] Chowdhury | Structural organization of the dynein-dynactin complex bound to microtubules[END_REF][START_REF] Urnavicius | The structure of the dynactin complex and its interaction with dynein[END_REF] as well as LIC1 of the dynein motor complex [START_REF] Schroeder | A Ras-like domain in the light intermediate chain bridges the dynein motor to a cargo-binding region[END_REF][START_REF] Lee | A conserved interaction of the dynein light intermediate chain with dynein-dynactin effectors necessary for processivity[END_REF]. These multiple interactions increase the affinity of dynein for dynactin and are required for stable dynein/dynactin/BicD2 complex (DDB) formation [START_REF] Schlager | In vitro reconstitution of a highly processive recombinant human dynein complex[END_REF][START_REF] Splinter | BICD2, dynactin, and LIS1 cooperate in regulating dynein recruitment to cellular structures[END_REF][START_REF] Mckenney | Activation of cytoplasmic dynein motility by dynactin-cargo adapter complexes[END_REF]. It was also shown that the N-terminal region of BicD2 alone is able to induce DDB complex formation with higher efficiency than the full-length protein [START_REF] Hoogenraad | Bicaudal D induces selective dynein-mediated microtubule minus end-directed transport[END_REF]. The CC1 domain of BicD's contains the dynein and dynactin interacting motifsthe CC1 box [START_REF] Gama | Molecular mechanism of dynein recruitment to kinetochores by the Rod-Zw10-Zwilch complex and Spindly[END_REF][START_REF] Lee | A conserved interaction of the dynein light intermediate chain with dynein-dynactin effectors necessary for processivity[END_REF][START_REF] Celestino | A transient helix in the disordered region of dynein light intermediate chain links the motor to structurally diverse adaptors for cargo transport[END_REF] and the spindly motif [START_REF] Gama | Molecular mechanism of dynein recruitment to kinetochores by the Rod-Zw10-Zwilch complex and Spindly[END_REF][START_REF] Zhang | Cryo-EM Reveals How Human Cytoplasmic Dynein Is Auto-inhibited and Activated[END_REF][START_REF] Torisawa | Autoinhibition and cooperative activation mechanisms of cytoplasmic dynein[END_REF]. The C-terminal domains of BicD's, CC2 and CC3, are usually implicated in cargo recruitment. The BicD1 C-terminal CC3 domain can interact with GTP-bound RAB6B [START_REF] Wanschers | A role for the Rab6B Bicaudal-D1 interaction in retrograde transport in neuronal cells[END_REF][START_REF] Hoogenraad | Mammalian golgi-associated Bicaudal-D2 functions in the dynein-dynactin pathway by interacting with these complexes[END_REF][START_REF] Short | The Rab6 GTPase Regulates Recruitment of the Dynactin Complex to Golgi Membranes[END_REF][START_REF] Schlager | Pericentrosomal targeting of Rab6 secretory vesicles by Bicaudal-D-related protein 1 (BICDR-1) regulates neuritogenesis[END_REF][START_REF] Matanis | Bicaudal-D regulates COPI-independent Golgi-ER transport by recruiting the dynein-dynactin motor complex[END_REF] and cytomegalovirus/HHV-5 protein UL32 [START_REF] Indran | Bicaudal D1-dependent trafficking of human cytomegalovirus tegument protein pp150 in virus-infected cells[END_REF]. BicD2 is able to recruit RAB6A [START_REF] Peeters | Molecular defects in the motor adaptor BICD2 cause proximal spinal muscular atrophy with autosomal-dominant inheritance[END_REF], NEK8 [START_REF] Holland | Purification, cloning, and characterization of Nek8, a novel NIMA-related kinase, and its candidate substrate Bicd2[END_REF], DNAI1 [START_REF] Oates | Mutations in BICD2 cause dominant congenital spinal muscular atrophy and hereditary spastic paraplegia[END_REF][START_REF] Peeters | Molecular defects in the motor adaptor BICD2 cause proximal spinal muscular atrophy with autosomal-dominant inheritance[END_REF], and kinesin KIF5A [START_REF] Splinter | Bicaudal D2, dynein, and kinesin-1 associate with nuclear pore complexes and regulate centrosome and nuclear positioning during mitotic entry[END_REF]. BicD2 C-terminus is also able to interact with RANBP2 [START_REF] Splinter | Bicaudal D2, dynein, and kinesin-1 associate with nuclear pore complexes and regulate centrosome and nuclear positioning during mitotic entry[END_REF] (or NUP358), a component of the nuclear pore complex. The subsequent BicD2 mediated recruitment of dynein-dynactin (Figure 1B ) allows the tethering of the centrosome to the nucleus prior to mitosis [START_REF] Splinter | Bicaudal D2, dynein, and kinesin-1 associate with nuclear pore complexes and regulate centrosome and nuclear positioning during mitotic entry[END_REF] and drives interkinetic nuclear migration during the development of the human neocortex [START_REF] Hu | Dynein recruitment to nuclear pores activates apical nuclear migration and mitotic entry in brain progenitor cells[END_REF][START_REF] Baffet | Cdk1 Activates Pre-mitotic Nuclear Envelope Dynein Recruitment and Apical Nuclear Migration in Neural Stem Cells[END_REF]. It has been shown in vivo and in vitro that BicD2 is recruited to RanBP2 in a CDK1-dependent manner [START_REF] Baffet | Cdk1 Activates Pre-mitotic Nuclear Envelope Dynein Recruitment and Apical Nuclear Migration in Neural Stem Cells[END_REF] (Figure 1B).

Cargo adaptors of the BicD family also interact with themselves. It has been observed by several groups that the C-terminal part is able to make contact with the N-terminal part leading to an autoinhibited state in BicD1 [START_REF] Terawaki | Structural basis for cargo binding and autoinhibition of Bicaudal-D1 by a parallel coiled-coil with homotypic registry[END_REF] and BicD [START_REF] Liu | Bicaudal-D uses a parallel, homodimeric coiled coil with heterotypic registry to coordinate recruitment of cargos to dynein[END_REF][START_REF] Wharton | Structure of the Drosophila BicaudalD protein and its role in localizing the posterior determinant nanos[END_REF]) (Figure 1C). There is little known about human BicD2 autoinhibition and many features are assumed to be common with the better-studied Drosophila BicD ortholog [START_REF] Sladewski | Recruitment of two dyneins to an mRNA-dependent bicaudal D transport complex[END_REF][START_REF] Stuurman | Interactions between coiled-coil proteins: Drosophila lamin Dm0 binds to the Bicaudal-D protein[END_REF]. It has also been suggested that the autoinhibited state can be relieved by cargo binding [START_REF] Liu | Bicaudal-D uses a parallel, homodimeric coiled coil with heterotypic registry to coordinate recruitment of cargos to dynein[END_REF][START_REF] Huynh | Disease-associated mutations in human BICD2 hyperactivate motility of dynein-dynactin[END_REF][START_REF] Mcclintock | RNA-directed activation of cytoplasmic dynein-1 in reconstituted transport RNPs[END_REF][START_REF] Sladewski | Recruitment of two dyneins to an mRNA-dependent bicaudal D transport complex[END_REF]. There is evidence of cargo-induced autoinhibition release in Drosophila BicD with Egl and mRNA as a cargo [START_REF] Sladewski | Recruitment of two dyneins to an mRNA-dependent bicaudal D transport complex[END_REF]. So far no such studies have been done for human full-length BicD1 or BicD2.

Understanding the structure, dynamics and reactivity of cargo adaptor proteins is required to advance our understanding of cargo selection, recruitment, and release. Most of the in vitro reconstitution experiments of the human BicD2 with cargo and/or dynein-dynactin were performed on a truncated protein or the Drosophila ortholog. As it is an extensively studied dynein adaptor, there is a need for a protocol for robust production of the fulllength human BicD2. The long-standing challenges in characterising cargo adaptors like BicD2 are limited expression levels and solubility which makes them poor targets for structural and functional analyses. Here we describe the recombinant production of human full-length BicD2 and provide novel insights to its in vitro behaviour. We validate the dimeric nature and correct folding of our recombinant BicD2. Our results also reveal unexpected pH-dependent conformational changes of BicD2 in the pH range from 6 to 8.

We also address the biochemical activity of BicD2 by interaction studies with RanBP2.

Our findings lay the ground for further studies on BicD2 reactivity, regulation, and role in motor protein complexes formation.

RESULTS

Biochemical characterization of full length BicD2

We succeeded in producing full-length human BicD2 isoform 2 under mild solubilizing conditions (see STAR Methods) and subsequently characterized its biochemical and functional properties. The addition of mild zwitterionic surfactants like CHAPS to the lysis buffer was crucial for obtaining sufficient protein amounts for large scale purifications. Our size-exclusion multi-angle Light scattering (SEC-MALS) analysis indicates a molecular weight of 192 (±0.7%) kDa (Figure S1A), which is slightly larger than the expected value of 186 kDa for a full-length BicD2 dimer The higher molecular weight is likely caused by the presence of residual CHAPS detergent. BicD2 also elutes at a volume of 1.35 mL on a Superose® 6 Increase 3.2/300 size exclusion column, which is earlier than expected and most likely caused by the increased hydrodynamic radius due to the extended coiledcoil fold of BicD2.

We evaluated the effect of buffers on the stability of BicD2 by differential scanning fluorimetry (nanoDSF) to establish an optimal buffer composition. The pH screen revealed that the BicD2 stability varies considerably in the range from pH 6.0 to pH 8.0 (Figure 2A).

The melting curves at pH 7.0 and pH 8.0 show clear but distinct inflection points, while the curve at pH 6 indicates partly unfolded protein. Full-length BicD2 carries a net negative charge at all these pH values (Figure S1B). We also performed three SEC runs on full-length BicD2 at pH 6.0, 7.0, and 8.0. The first run after affinity purification at pH 7.0 and pH 8.0 was characterized by an elution peak for dimeric BicD2 between 1.3 -1.4 mL and a broad peak of degraded BicD2 around 1.8 mL (Figure S1C). The second SEC run yielded clean BicD2 fractions (Figure 2B). The slightly different SEC elution volumes and the distinct inflection points in the DSC melting curves suggest the possibility of different BicD2 conformations at pH 7.0 and pH 8.0. Moreover, the SEC run at pH 7.0 had a significantly higher ratio of full length BicD2 to degraded BicD2 compared to the SEC run at pH 8.0, where degradation proceeds more rapidly (Figure S1C). At pH 6.0

BicD2 eluted almost at the end of the column suggesting partial denaturation (Figure 2B).

Re-running the BicD2 peak fractions revealed elution peaks at the expected volumes without additional degradation peaks. These results were further confirmed by circular dichroism (CD) experiments (Figure S1D). The spectra for pH 7.0 and pH 8.0 suggest almost pure α-helical content, consistent with the predicted BicD2 fold. The CD spectrum for pH 6.0 does not resemble the α-helical model, but also does not fit into random-coil models. Heating the protein up to 60 °C did not reveal the random coil characteristic peaks (Figure S1E). This suggests that the protein at pH 6.0 adopts a partially unfolded structure.

Influence of pH value on BicD2's conformation and stability

In order to obtain insights into the conformation of BicD2 at pH 7.0 and pH 8.0, we directly visualized BicD2 at these pH values using a cryo-electron microscope. We collected two datasets and applied identical data processing workflows to avoid bias in the interpretation. The representative 2D classes showed that most molecules at pH 8.0 fall into an extended conformation (Figure 2C), whereas the vast majority of BicD2 particles adopt more compact conformations (Figure 2D) at pH 7.0. These results are consistent with the SEC runs where BicD2 elutes at 1.4 mL at pH 7.0 but 1.35 mL at pH 8.0 suggesting a shorter hydrodynamic radius at pH 7.0.

Next, we wanted to identify amino-acid residues that potentially could be responsible for the pH dependent conformational changes of BicD2. We analysed multiple sequence alignments and focused on conserved histidines as they often have crucial functions in pH-sensitive proteins. An alignment of BicD, BicD1, and BicD2 sequences from all animal kingdoms showed only a poor sequence identity of less than 10%. However, in the Cterminal part of the CC2 domain there is a conserved 'YH' sequence motif and an additionally conserved histidine (H640 in human isoforms) that is present in all vertebrates (Figure 3A). We generated a structure prediction with AlphaFold2 [START_REF] Jumper | Highly accurate protein structure prediction with AlphaFold[END_REF] (AF2) for the CC2 and CC3 domains of human BicD2 (Figure 3B) to evaluate where these residues are potentially located within the predicted structure. Our analysis suggests that they are not involved in dimer interactions, but that they are located within the tri-helical interface of H3, H4, and H5. We compared the YH-containing motif of hsBicD2 (Y538-H539) with its two distant isoforms -D. melanogaster BicD and C. elegans BicD1. The architecture is very similar for all isoforms. In the case of hsBicD2, the additionally conserved histidine residue H640 on helix H5 could interact with residue Y538 on helix H3 (Figure 3C). In order to validate the structure predicted by AF2, we have also compared it to the negative stain class averages reported for D. melanogaster BicD [START_REF] Sladewski | Recruitment of two dyneins to an mRNA-dependent bicaudal D transport complex[END_REF] (Figure S2A, B andC).

To test if the YH motif and the interhelical tyrosine-histidine pair (Y538-H640) interfere with BicD2's pH dependent behaviour in-vitro, we generated the H539A/Y538A and H539A/Y538A/H640A mutants and characterized them by SEC at pH 7.0 and pH 8.0. The fact that the differences in the BicD2 elution peaks do still exist suggests that also the underlying conformational differences are still present. Furthermore, the mutant BicD2 proteins are more prone to degradation because the equilibrium between the peaks for full-length BicD2 and its degradation products has shifted towards the degradation products. This effect is more pronounced for the H539A/Y538A/H640A triple mutant (Figure 3D). At pH 7.0 the BicD2 fraction is the most abundant, while at pH8.0 the BicD2 fraction significantly decreases, and the protein is prone to degradation. Even though H539/Y538/H640 does not seem to be involved in the pH-induced conformational changes of full-length BicD2, these residues nevertheless are clearly important for the stability of the protein.

BicD2's interaction with RanBP2

RanBP2 (or Nup358) is known to recruit BicD2 through a small disordered region between two Ran binding domains as demonstrated in yeast two-hybrid assays [START_REF] Splinter | Bicaudal D2, dynein, and kinesin-1 associate with nuclear pore complexes and regulate centrosome and nuclear positioning during mitotic entry[END_REF]. The BicD2 binding domain of RanBP2 (hereafter RanBP2BBD) interacts with the C-terminus of BicD2 (715-804) [START_REF] Gibson | Coil-to-Helix Transition at the Nup358-BicD2 Interface for Dynein Recruitment and Activation[END_REF] and contains several consensus CDK1 phosphorylation sites, which have been shown to be important for BicD2 binding [START_REF] Baffet | Cdk1 Activates Pre-mitotic Nuclear Envelope Dynein Recruitment and Apical Nuclear Migration in Neural Stem Cells[END_REF]. The individual contribution of these phosphorylation sites to BicD2 binding is unknown. A recent in-vitro experiment demonstrated that the RanBP2BBD-BicD2 complex can also be formed in the absence of phosphorylation [START_REF] Gibson | Coil-to-Helix Transition at the Nup358-BicD2 Interface for Dynein Recruitment and Activation[END_REF]. Since all these experiments used the truncated BicD2 C-terminus, the binding of the full-length BicD2 to RanBP2BBD remains to be characterized. In order to clarify the role of phosphorylation in BicD2-RanBP2 BBD complex formation and to confirm the biochemical activity of our recombinant full-length BicD2, we evaluated the interaction with its binding partner RanBP2.

We performed double pull-down assays with full length BicD2 and RanBP2BBD as well as with a phosphomimetic variant of RanBP2 BBD where the known CDK1 phosphorylation sites had been mutated to aspartates (T2153D, S2246D, S2251D, S2276D, and S2280D). We were able to obtain a BicD2-RanBP2 BBD complex in both cases (Figure S3A). However, subsequent SEC runs of the purified complexes indicated that they are not stable and that they dissociate. In order to investigate if RanBP2BBD phosphorylation leads to more stable complex formation, we produced recombinant CDK1 (Veld et al., 2021) and performed an in vitro kinase assay on RanBP2 BBD . The mass spectrometry analysis revealed eight S/T consensus sites in the BicD2 binding region of RanBP2 of which five overlapped with earlier findings [START_REF] Baffet | Cdk1 Activates Pre-mitotic Nuclear Envelope Dynein Recruitment and Apical Nuclear Migration in Neural Stem Cells[END_REF] (Figure S3B). We mixed in vitro phosphorylated or non-phosphorylated RanBP2 BBD with full length BicD2 and analysed the samples by SEC. Successful complex formation was only observed for in vitro phosphorylated RanBP2 (Figure 4A). Next, we investigated if BicD2 can be phosphorylated by CDK1 and if such a modification could potentially strengthen complex formation with RanBP2BBD. The mass spectrometry analysis after BicD2 in vitro phosphorylation indicated five phosphorylation sites in regions predicted to be structurally disordered demonstrating that BicD2 can be phosphorylated by CDK1 (Figure S3B). We mixed the in vitro phosphorylated BicD2 with non-modified RanBP2 BBD and subsequently analysed the sample by SEC. There was no indication of complex formation (Figure 4A).

The equivalent experiment with in vitro phosphorylated RanBP2BBD did not show any evidence of increased BicD2-RanBP2BBD complex formation compared to non-modified BicD2 -in vitro phosphorylated RanBP2 SEC run (Figure 4A). These results suggest that BicD2 phosphorylation does not contribute to the BicD2-RanBP2 BBD complex formation.

We also analysed the influence of pH on BicD2-RanBP2BBD complex formation. To this end, we mixed full-length BicD2, RanBP2BBD and CDK1 at pH 7.0 and pH 8.0 to induce complex formation and analysed the samples by SEC. We also carried out the corresponding control experiments in the absence of CDK1 (Figure 4B). From the fractions analysed by SDS-PAGE, it is evident that complex formation is enhanced with CDK1-treated samples. Interestingly, the two runs at pH 7.0 (with and without CDK1) showed a broad distribution of RanBP2BBD throughout the elution and only minor increase in the RanBP2 intensity at BicD2 peak fractions in the presence of CDK1. Taken together, these observations suggest that the BicD2-RanBP2 BBD complex forms best under slightly alkaline pH which narrows the pool of RanBP2 BBD species.

DISCUSSION

The ability of Dynein-1 to bind a wide range of cargoes is mediated by its compatibility with many cargo adaptor proteins, such as BicDs, Hooks, NINs, Spindly, TRAKs, RILP or JIP3 [START_REF] Olenick | Dynein activators and adaptors at a glance[END_REF]. Investigating in vitro reconstituted complexes between cargo adaptors and dynein/dynactin has led to valuable insights into dynein motor recruitment and activation [START_REF] Terawaki | Structural basis for cargo binding and autoinhibition of Bicaudal-D1 by a parallel coiled-coil with homotypic registry[END_REF][START_REF] Liu | Bicaudal-D uses a parallel, homodimeric coiled coil with heterotypic registry to coordinate recruitment of cargos to dynein[END_REF][START_REF] Noell | Role of Coiled-Coil Registry Shifts in the Activation of Human Bicaudal D2 for Dynein Recruitment upon Cargo Binding[END_REF].

However most of these studies were done with truncated BicD2 constructs that lack either the dynein or the cargo -binding domains. Many biophysical and structural studies have been hampered by the lack of sufficient amounts of full-length cargo adaptors. Here we report a robust method for the recombinant production of a full-length BicD2 and demonstrate its correct fold and biochemical activity towards its RanBP2 cargo. We believe that other full-length cargo adaptors might benefit from the expression and purification protocols described here and that our results will facilitate in vitro reconstitution approaches of motor-cargo complexes.

One of the unexpected findings was the strong influence of the pH value on the conformation and stability of full-length BicD2. At pH 6.0, 7.0 and 8.0 BicD2 switches from partially unfolded to compact and to an open conformation. Our cryoEM 2D classification of full-length BicD2 at pH 7.0 indicates that over 50% of the particles fall into compact classes while the majority of particles at pH 8.0 adopt an open conformation. Recent work has provided evidence that the autoinhibited state is characterized by an interaction between the CC2 and CC3 domains [START_REF] Sladewski | Recruitment of two dyneins to an mRNA-dependent bicaudal D transport complex[END_REF]. Although we were not able to assign individual coiled-coil domains in our cryoEM 2D class averages due to the intrinsic flexibility of BicD2, there is the possibility that the more compact 2D class averages at pH 7.0 might represent the autoinhibited state and the extended conformation at pH 8.0 the open, activated state. A similar autoinhibited conformation has also been recently described for cargo adaptor Spindly (d'Amico et al., 2022) which suggest that autoinhibition might be a common feature of cargo adaptors.

It is not clear if these pH driven conformational changes are relevant only in vitro or if they also have a physiological relevance. In principle, the cellular pH value could also trigger pH dependent conformational changes of the BicD2 cargo adaptor. At the cytosolic pH value of ~7.2 most BicD2 molecules would adapt the compact, potentially autoinhibited form and require additional, activating signals to recruit motor proteins and cargo.

Conditions that lead to an increase in the intracellular pH, like mitosis and growth factor signalling [START_REF] Gagliardi | Is intracellular pH a clock for mitosis?[END_REF][START_REF] Pouysségur | Cytoplasmic pH, a key determinant of growth factor-induced DNA synthesis in quiescent fibroblasts[END_REF][START_REF] Chambard | Intracellular pH controls growth factor-induced ribosomal protein S6 phosphorylation and protein synthesis in the G0→G1 transition of fibroblasts[END_REF], could bypass the need for such signals and activate the BicD2 cargo adaptor on a globular level. Adaptor autoinhibition is not a BicD2-unique phenomenon. A recent study describes a similar behavior for the dynein-1 adaptor Spindly (d'Amico et al., 2022). Cargo adaptors alone are not extensively studied but our work and the one presented by D [START_REF] Amico | Conformational transitions of the mitotic adaptor Spindly underlie its interaction with Dynein and Dynactin[END_REF] suggests that this behavior might be common for some cargo adaptors and play a role in recruitment of partner proteins.

In the case of BicD2 one of the activating signals is believed to be the interaction with its cargos like RanBP2 [START_REF] Liu | Bicaudal-D uses a parallel, homodimeric coiled coil with heterotypic registry to coordinate recruitment of cargos to dynein[END_REF][START_REF] Sladewski | Recruitment of two dyneins to an mRNA-dependent bicaudal D transport complex[END_REF][START_REF] Huynh | Disease-associated mutations in human BICD2 hyperactivate motility of dynein-dynactin[END_REF][START_REF] Mcclintock | RNA-directed activation of cytoplasmic dynein-1 in reconstituted transport RNPs[END_REF]. Furthermore, post-translational BicD2 modifications might also influence its reactivity. BicD2 contains several consensus phosphorylation sites [START_REF] Holland | Purification, cloning, and characterization of Nek8, a novel NIMA-related kinase, and its candidate substrate Bicd2[END_REF] and an acetylation site in its N-terminal part [START_REF] Van Damme | N-terminal acetylome analyses and functional insights of the N-terminal acetyltransferase NatB[END_REF].

We also have identified three conserved residues, H539/Y538/H640 that contribute to the stability of BicD2. Mutating these amino-acid residues to alanines clearly increased the tendency of BicD2 for degradation, which could be caused by auto-cleavage or residual protease amounts in the sample. Our template-free AF2 prediction suggest the possibility of Y538/H640 stabilizing the H3-H4-H5 α-helical bundle via an intramolecular hydrogen bond (Figure S2D). Disrupting it could trigger partial unfolding and subsequent protein degradation. A similar architecture (Figure S2D) was also observed in a pH-sensitive endotoxins and mutating the equivalent histidine-tyrosine pair mutation lead to protein destabilization and degradation [START_REF] Seale | The role of a conserved histidine-tyrosine interhelical interaction in the ion channel domain of δ-endotoxins from Bacillus thuringiensis[END_REF].

We characterized the BicD2 reactivity with its known cargo protein RanBP2. There are conflicting results in the field with respect to the question whether RanBP2 phosphorylation is a requirement for the interaction with BicD2 [START_REF] Baffet | Cdk1 Activates Pre-mitotic Nuclear Envelope Dynein Recruitment and Apical Nuclear Migration in Neural Stem Cells[END_REF][START_REF] Gibson | Coil-to-Helix Transition at the Nup358-BicD2 Interface for Dynein Recruitment and Activation[END_REF]. Our results now reconcile these reports, because they show that BicD2 can bind both non-phosphorylated as well as phosphorylated RanBP2DDB. In double pulldowns we observed that BicD2 readily interacts with RanBP2 without any protein modification at pH 7.0 and 8.0. These complexes are nevertheless unstable over SEC. This suggests that there is a weak association of these proteins in vitro in a wide range of conditions. The CDK1 phosphorylation of RanBP2 BBD (but not BicD2) significantly increased the stability of the complex and increase in the pH value improved the complex homogeneity.

ACKNOWLEDGEMENT

AUTHOR CONTRIBUTIONS

RF: conceptualization, data curation and analysis, validation, investigation, methodology, funding acquisition and manuscript writing. CC: methodology, investigation and data analysis. TK: methodology, investigation and data analysis. CD & BF: data analysis and discussion. YN, JB and PR: investigation and data analysis. HS: conceptualization, data analysis, supervision, funding acquisition, methodology, manuscript writing and project administration.

DECLARATION OF INTEREST

The authors declare no competing interests.

STAR METHODS

RESOURCE AVAILABILITY

Lead contact

Further information and requests for resources and reagents should be directed to and will be fulfilled by the lead contact, Helgo Schmidt (schmidth@igbmc.fr)

Materials availability

Plasmids generated in this study are available upon request.

Data and code availability

No cryoEM or x-ray crystal structures were generated in this study. This paper does not report original code. Any additional information required to re-analyze the data reported in this paper is available from the lead contact upon request.

EXPERIMENTAL MODEL AND SUBJECT DETAILS

E. coli BL21(DE3) cell were obtained from New England Biolabs, product number C2527H. The Sf9 insect cells used in this study were obtained from Sigma-Aldrich, product number 71104-M.

METHOD DETAILS

Cloning and mutagenesis

Human BicD2 isoform 2 and RanBPBBD genes were ordered from Epoch LifeScience and codon-optimized for Escherichia coli expression. The genes were subsequently cloned into the pNHD expression vector, a vector used for site-specific pSer incorporation [START_REF] Rogerson | Efficient genetic encoding of phosphoserine and its nonhydrolyzable analog[END_REF], but both proteins express equally well in a commercial pMMS vector). The BicD2 plasmid was modified to introduce an N-terminal STREP-tag followed by a short 'GSGSG' linker. The RanBPBBD plasmid was fused with a C-terminal double TEV cleavage site, a GSGSG linker, and a 6xHis tag. The H539A/Y538A and H539A/Y538A/H640A mutants were generated by rolling circle PCR. The co-expression plasmid encoding CDK1-cyclinB for insect cell expression was a kind gift from the Andrea Musacchio lab.

Protein expression and purification

Plasmids for bacterial expression were transformed into E. coli BL21 (DE3) chemically competent cells and grown overnight on tetracycline plates. The plate was used for inoculation of the 50 mL LB medium pre-culture which was grown overnight at 37°C for further scale-up. Protein expression was induced at 25°C with 1mM isopropyl-1-thio-β-Dgalactopyranoside (IPTG) and incubated for 3h at 37°C. Cells were harvested, washed with 1xPBS, and frozen. BicD2 WT and mutants were re-suspended in lysis buffer containing 1xPBS pH8.0, 10% glycerol, 7 mM CHAPS, 2 mM DTT, 2mM PMSF and cOmplete EDTA-free protease inhibitor cocktail tablets (Roche), and incubated on a roller for 1-2h at 4°C. Cells were lysed by sonication (5x30s with 30s breaks on ice) and centrifuged at 30000 rpm at 4°C for 1 hour. The supernatant was mixed with preequilibrated STREP-tactin 4XL beads and incubated on a roller for 30 minutes at 4°C before loading on a gravity flow column. The resin was washed with 10 CVs of the lysis buffer, 10 CVs of the wash buffer (1xPBS pH8.0, 10% glycerol, 7 mM CHAPS), and eluted with elution buffer (1xPBS pH8.0, 10% glycerol, 7 mM CHAPS, 50 mM biotin). Pooled fractions were concentrated to 2 mg/mL and snap-frozen in 100 µL aliquots for further experiments and gel filtration. RanBP2 BBD was purified in the same manner as BicD2 with a difference in the buffer used (20 mM Tris-HCl pH8.0 instead of PBS). Both constructs (WT) yielded around 1 mg of protein per litre of culture after the affinity purification step.

The CDK1 CyclinB plasmid was transformed into E. coli DH10MB-MCherry chemically competent cells with a heat shock at 42°C for 30s followed by 6h recovery in LB medium at 37°C with shaking. The cells were then plated on agar plates containing kanamycin (50 μg ml -1 ), gentamicin (7 μg ml -1 ), tetracycline (10 μg ml -1 ), Xgal (600 μg ml -1 ) and IPTG (40 μg ml -1 ) and positive clones were identified by blue/white selection after 24h incubation at 37°C followed by 24h at RT. The positive clones were grown in LB media containing kanamycin (50 μg ml -1 ), gentamicin (7 μg ml -1 ), and tetracycline (10 μg ml -1 ) overnight. Bacmids were purified using the isopropanol precipitation method. 2ml of Sf9 cells at 0.5×10 6 cells per ml were transfected with 2 µg of fresh bacmid DNA and FuGene HD transfection reagent (Promega) at a ratio of 3:1 transfection reagent to DNA. After 72h the 6-well plate was analysed for mCherry fluorescence signal, and the positive conditions (V 0 ) were directly used for further virus amplifications by adding 2mL of V 0 to 50 mL of Sf9 cells at 1x10 6 cells per mL (V 1 ) and grown for 72h. The V 1 was then used for infecting another 50 mL of Sf9 at 1x10 6 cells/mL (V2). The V2 virus was further used to infect 500 mL of the Sf9 culture 1x10 6 cells/mL. After 72h the cells were collected at 1000 rpm for 10 min at 4°C. The pellet was flash-frozen in liquid nitrogen and stored at -80°C until purification. The CDK1 CyclinB pellet was re-suspended in the lysis buffer containing 50mM Tris-HCl pH8.0, 10mM MgCl 2 , 10% glycerol, 0.1% NP40, 0.1 mM EDTA, 2mM DTT, 2 mM PMSF, protease inhibitors, and incubated on a roller for 1h at 4°C. The cells were lysed by sonication (2x30s with 30s breaks on ice) and centrifuged at 40000 rpm at 4°C for 1 hour. The supernatant was mixed with pre-equilibrated GST-sepharose beads for 30 minutes at RT and loaded onto a gravity flow column. The resin was washed with 10 CV of the lysis buffer, 10 CV of the wash buffer (Tris-HCl pH8.0, 150mM NaCl, 10% glycerol, 0.1mM EDTA), and eluted with elution buffer (Tris-HCl pH8.0, 150mM NaCl, 10% glycerol, 0.1mM EDTA, 25mM reduced GST). Pooled fractions were concentrated to 0.5 mg/mL and snap-frozen in 50µL aliquots for further experiments.

SEC-MALS

Prior to SEC-MALS analysis, the BicD2 dimer was purified on Superose® 6 Increase

Circular dichroism

CD experiments were recorded on a Jasco J-815 spectropolarimeter (Easton, MD) equipped with an automatic 6-position Peltier thermostated cell holder. The instrument was calibrated with 10-camphorsulphonic acid. Samples (65 μL) were prepared in PBS supplemented with CHAPS 7 mM and buffered at various pH (6.0, 7.0 or 8.0). Far-UV CD data were collected in the 182-270 nm range using a 0.1 mm pathlength cell (Quartz-Suprasil, Hellma UK Ltd) at 20.0 °C ± 0.1 °C. Spectra were acquired using a continuous scan rate of 50 nm/min and are presented as an average of 10 successive scans. The response time and the bandwidth were 1.0 s and 1 nm, respectively. The spectra were corrected by subtracting the solvent spectrum obtained under identical conditions.

CryoEM sample preparation

A 100µL BicD2 sample was thawed on ice and equilibrated with 5xPBS at either pH 8.0 or pH 7.0. The Superose ® 6 Increase column was equilibrated for 16h at a specific pH (in PBS buffer only) and the dimer fraction from the SEC taken for grid preparation. Cu/Rh 1.2/1.3 300mesh grids were plasma cleaned for 90s at 30% plasma power (80:20 argon:oxygen). The protein concentration was adjusted to 1µM and applied on the grid and plunge-frozen into liquid ethane in a Vitrobot Mark IV robot (FEI), maintained at 100% humidity and 10 °C.

CryoEM data collection and image analysis

Datasets for BicD2 at pH 8.0 and pH 7.0 were collected using a Glacios™ Cryo-TEM operating at 200keV with a K2 Summit direct electron detector (Gatan). Videos were collected in counting mode, with a final calibrated pixel size of 1.078 Å/pixel, 8s exposure, and total dose of ~ 54 e -/Å 2 . SerialEM [START_REF] Mastronarde | Automated electron microscope tomography using robust prediction of specimen movements[END_REF] was used for automated data collection and the videos were processed using Relion3.0 (Zivanov et al., 2018). The micrographs were manually filtered giving each dataset ~700 micrographs. Particles were picked using the general model in TOPAZ [START_REF] Bepler | Positiveunlabeled convolutional neural networks for particle picking in cryo-electron micrographs[END_REF] and 2D classified twice. Each dataset yielded around 150k particles used for 2D classification. All data processing steps were done reference-free.

In vitro kinase assays and phosphorylation mapping

BicD2 or RanBP2 BBD were mixed with CDK1 at 1:100 molar ratio with 0.1mM MgATP and incubated for 30min at RT. Each sample had a negative control that did not contain MgATP. Each reaction mixture was loaded on SDS-PAGE and stained with pro-Q diamond stain to check for phosphorylated proteins. Phosphorylation sites were analysed with trypsin digest MS using an Orbitrap Elite and Acclaim Pepmap 100 column. The PSM values corresponded to the number of MS2 spectra which made it possible to identify peptides. In the case of BicD2 75% of the sequence were covered (PSM's: 2087) and for RanBP2 85% of the sequence were covered (PSM's: 1195).

AlphFold2 structure prediction

We used the whole CC2 and CC3 domain of a human BicD2 (residues 265-855) to model a homooligomer. We generated 5 models using the AlphaFold2 [START_REF] Jumper | Highly accurate protein structure prediction with AlphaFold[END_REF] advanced notebook from Google Colab. We used MMseqs2 (Steinegger and Söding, 2017) (UniRef+Environmental) for MSA generation.

QUANTIFICATION AND STATISTICAL ANALYSIS

No quantification or statistical analysis was carried out in this study.

  This research was supported by a LabEx start up grant (ANR-10-LABEX-30-HS) to HS and Boehringer Ingelheim Fonds as well as FRM PhD fellowships to RF. This study was further supported by the grant ANR-10-LABX-0030-INRT, a French State fund managed by the Agence Nationale de la Recherche under the frame program Investissements d'Avenir ANR-10-IDEX-0002-02. The authors acknowledge the support and the use of resources of the French Infrastructure for Integrated Structural Biology (FRISBI) ANR-10-INBS-05 and of Instruct-ERIC. We also thank the IGBMC molecular biology and virus service (Nicole Jung and Thierry Lerouge). This work of the Interdisciplinary Thematic Institute IMCBio, as part of the ITI 2021-2028 program of the University of Strasbourg, CNRS and Inserm, was supported by IdEx Unistra (ANR-10-IDEX-0002), and by SFRI-STRAT'US project (ANR 20-SFRI-0012) and EUR IMCBio (ANR-17-EURE-0023) under the framework of the French Investments for the Future Program.

  3.2/300 column in SEC buffer (50 mM Tris pH 7.5, 150 mM Potassium acetate, 10 mM Magnesium acetate, 7 mM CHAPS). 90µL of BicD2 sample (1mg/mL) was injected into FPLC Ettan Micro LC (formerly GE Healthcare, now Cytiva) coupled with MiniDAWN TREOS MALS detector (Wyatt Technology), Optilab T-rEX (Wyatt Technology) RI detector, and Superdex® S200 10/300 GL (formerly GE Healthcare, now Cytiva) column and run at 25°C in SEC buffer (50 mM Tris pH 7.5, 150 mM Potassium acetate, 10 mM Magnesium acetate). The data was processed using the ASTRA software (Wyatt Technology).

  S.H.W.(2018). New tools for automated high-resolution cryo-EM structure determination in RELION-3. Elife 7. https://doi.org/10.7554/ELIFE.42166.

Figure 1 .

 1 Figure 1. Schematic representation of BicD2 domain organization and function. (A) Domain organization of the BicD2 cargo adaptor dimer. The N-terminal part is responsible for binding dynein and dynactin while the C-terminal part binds the plusend directed motor kinesin-1 and cargos like RanBP2. (B) BicD2 associates with its RanBP2 nuclear pore cargo and dynein-dynactin. The CDK1 CyclinB mediated phosphorylation of RanBP2 triggers the binding of BicD2 to recruit dynein-dynactin to the nucleus during interkinetic nuclear migration in the developing neocortex. (C) Hypothetical BicD2 autoinhibited conformations from the previously reported data.

Figure 2 .

 2 Figure 2. Biochemical and structural characterization of human full-length BicD2 at different pH values. (A) First derivative of the DSF melting curves at three different pH values. The curves at pH 7.0 and pH 8.0 display main inflection point in the range of 47-50°C. The curve at pH 6.0 does not show a distinct inflection point. Each measurement was repeated 3 times. (B) SEC profiles of BicD2 run at three different pH values. Each sample was run twice on a Superose ® 6 Increase column to ensure clean peak fractions. (C) and (D) Representative cryoEM 2D classes from ~700 micrographs (per pH condition). The yellow box represents classes from pH 8.0 while the grey box from pH 7.0. In both cases the data was processed with Relion3.1. See also Figures S1 and S4.

Figure 3 .

 3 Figure 3. The conserved amino-acid residues Y538,H539 and H640 contribute to BicD2 stability. (A) Sequence alignment of the tri-helical domain (H3, H4, and H5 helix) of D. melanogaster, C. elegans, and mammalian BicD1 and BicD2 isoforms. The conserved amino-acid residues Y538, H539 and H640 are marked by black dots. (B) Structure of the BicD2 CC2 and CC3 dimer predicted by AlphaFold2. The C-terminal loop of CC3 has been deleted for clarity. (C) Enlarged view of the conserved histidine and tyrosine residues that are crucial for protein stability. (D) Chromatograms of affinity purified BicD2 variants. Yellow -WT, orange -Y538A/H539A double mutant, dark

Figure 4 .

 4 Figure 4. SEC profiles of the BicD2 reaction with RanBP2BBD in the absence and presence of CDK1 CyclinB . (A) Purified proteins were mixed at a 1:1 molar ratio and incubated for 30 minutes at RT. In light grey a chromatogram of

  .


Mohamed Abdi
email: abdi.mohamed1@live.fr

Khaled Chaib

Slimane Benferhat

Large Eddy Simulation of Turbulent Flow of Pseudoplastic and Dilatant Fluids: Rheological and Hydrodynamic Behaviour

Keywords: LES, Shear-thinning, Shear-thickening, Fully developed, Turbulent flow

A fully developed turbulent pipe flow of non-Newtonian fluids was investigated numerically using the large eddy simulation (LES) method. The present investigation mainly focuses on the rheological and hydrodynamic behaviour of the pseudoplastic (shear-thinning) and dilatant (shear-thickening) fluids over a wide range of flow behaviour indexes 0.75, 0.8, 1, 1.2, 1.4, and 1.6 at a simulation Reynolds number of 12000. The computations are based on a finite difference scheme, second-order accurate in space and time. The numeric resolution is 65 3 grid points in r, θ and z directions, respectively. According to the results, the shear rate profiles of the pseudoplastic decreased while the dilatant fluid improved with a higher flow index along the pipe radius. Additionally, the elevated flow index caused a rise in the fluid viscosity close to the wall, which raised the friction factor.

INTRODUCTION

Numerous industrial applications have emerged that enhance technology in the flow of non-Newtonian fluids. These fluids are used in various technical applications, including cement and paint production, oil field operations, the petroleum, pharmaceutical, and polymer processing industries, transportation, and food and polymer solutions. Over the past few decades, there has been an increasing interest in the turbulent flow of non-Newtonian fluids through axial pipes. Various experimental, theoretical, and numerical studies have been conducted to understand how the fluids' rheological properties affect the hydrodynamic and thermal characteristics. The flow of power-law fluids in an isothermal axial pipe has been studied in a significant quantity of literature, either experimentally [START_REF] Metzner | Flow of non-Newtonian fluids-correlation of the laminar, transition, and turbulent-flow regions[END_REF][START_REF] Metzner | Non-Newtonian Fluid Flow. Relationships between Recent Pressure-Drop Correlations[END_REF][START_REF] Dodge | Turbulent flow of non-newtonian systems[END_REF][START_REF] Tomita | A Study on Non-Newtonian Flow in Pipe Lines[END_REF][START_REF] Vidyanidhi | Non-Newtonian Flow in a Rotating Straight Pipe[END_REF][START_REF] Pinho | Flow of non-newtonian fluids in a pipe[END_REF] or numerically [START_REF] Malin | Turbulent pipe flow of power-law fluids[END_REF][START_REF] Rudman | Turbulent pipe flow of shear-thinning fluids[END_REF][START_REF] Ohta | DNS and LES with an extended Smagorinsky model for wall turbulence in non-Newtonian viscous fluids[END_REF][START_REF] Gnambode | Large-Eddy simulation of turbulent pipe flow of power-law fluids[END_REF][START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF][START_REF] Gavrilov | Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes[END_REF][START_REF] Singh | The effect of yield stress on pipe flow turbulence for generalised newtonian fluids[END_REF].

One of the critical studies to fill in the gap and advance understanding was that of Gavrilova and Rudyak [START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF][START_REF] Gavrilov | Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes[END_REF]. They conducted a DNS (Direct Numerical Simulations) at two generalised Reynolds values, 10000 and 20000, over the power-law index 0.4-1. The authors presented the distributions of Reynolds stress tensor components, averaged viscosity, viscosity fluctuations, and measures of turbulent anisotropy in 2016 [START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF], concentrating on turbulent mean values. The identical research was described a year later [START_REF] Gavrilov | Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes[END_REF]. But this time, they presented the distributions of the turbulent stress tensor components and the shear stress and turbulent kinetic energy balances, focusing on the energy balance and the shear stresses. To understand how shear thinning or thickening affects first and second-order flow statistics, including turbulent kinetic energy production, transport, and dissipation in such flows, Singh et al. [START_REF] Singh | The effect of yield stress on pipe flow turbulence for generalised newtonian fluids[END_REF] recently investigated the effect of the flow index parameter of power-law fluids in turbulent pipe flow using direct numerical simulation at a friction Reynolds number Re=323.

More recently, Abdi et al. [START_REF] Abdi | Numerical simulation of turbulent forced convection of a power law fluid flow in an axially rotating pipe[END_REF] have conducted a fully developed turbulent forced convection of thermally independent pseudoplastic fluid with a flow behaviour index of 0.75 through an axially heated rotating pipe, by means of LES with an extended Smagorinsky model. With a rotation rate ranging from 0 to 3, the simulation Reynolds and Prandtl numbers of the working fluid were assumed to be 4000 and 1, respectively. It is observed that as the pipe wall rotates, it is seen that the temperature along its radius noticeably decreases as the rotation rate increases. This is because the apparent fluid viscosity in the pipe's core region decreases, which causes a centrifugal force that causes the mean axial velocity profile to increase noticeably.

The present investigation focuses numerically on the fully developed turbulent flow of power-law fluids in an axially stationary pipe using a large eddy simulation (LES) with a standard dynamic model. The flow behaviour index was set to 0.75, 0.8, 1, 1.2, 1.4, and 1.6 at a simulation Reynolds number of 12000. The specific objective of the present investigation is to discern the effects of the flow behaviour index of the shear-thinning and shear-thickening fluids on different fluid rheological characteristics and hydrodynamic properties to describe the turbulent flow patterns and rheological behaviour of this kind of non-Newtonian fluid.

GOVERNING EQUATIONS AND NUMERICAL PROCEDURE

Governing Equations

The present study deals numerically with a fully developed turbulent flow of power-law fluids in a pipe at different flow behaviour indexes (0.75≤ n ≤1.6) at a simulation Reynolds number of 12000, employing the LES approach with a standard dynamic model, with a computational domain length of 20R (Fig. 1). The filtered equations were expressed as (Equations 1 and 2): Where Res=ρUCL 2-n R n /K is the simulation Reynolds number, where (UCL) is the centreline axial velocity of the analytical fully developed laminar profile and is defined as UCL=(3n+1).Ub/(n+1). The constitutive equation for the power-law model is given by:

0 i i u x  =  (1) 1 1 Re j i j j ij n i i j s i i j i u u u u u dP t x x x x x x   -         + = - + + +              (2)
n K  = (3)
The variables K and n are the fluid consistency index and the flow behaviour index, respectively. The shear-thinning behaviour occurs for 1 n  , shear-thickening for n>1, and n=1 the fluid shows Newtonian behaviour.

Numerical Procedure

In the non-Newtonian Smagorinsky model, the subgrid stress tensor ij  is linked to the strain rate tensor by 𝜏̅ 𝑖𝑗 =2t𝑠̅ 𝑖𝑗 . The turbulent viscosity is computed by t=Csfs (fn)  𝑠̅ 𝑖𝑗 , where  is the computational filter, Cs the model constant, fs the van Driest wall damping function, and fn is the correction function for the change in viscosity.

A no-slip boundary condition was applied at the pipe wall to perform a realistic numerical simulation, and a periodic boundary condition was applied in the axial and circumferential directions. As for the grid resolution, a grid of 65 3 grid points in axial, radial, and circumferential directions, respectively, was found to provide an accurate prediction of the rheological and hydrodynamic characteristics, in agreement with the available data of the literature and to give a good compromise between the required CPU-time and accuracy. Moreover, a uniform computational grid was utilised for the axial and circumferential directions, while a non-uniform mesh specified by a hyperbolic tangent function was applied in the radial direction.

RESULTS AND DISCUSSION

The present section largely analyses and discusses the mean quantities and the rheological properties, such as the evolution of the apparent fluid viscosity, the shear stress, the shear rate, the mean axial velocity, and the friction factor. This section seeks to explore the effect of the flow behaviour index of the power-law fluid on the rheological and hydrodynamic properties to describe the turbulent flow pattern and rheological behaviour of this kind of non-Newtonian fluid in the present computational domain.

Validation

The findings were compared with DNS data available in the literature to ascertain the accuracy and reliability of the predicted results of the LES laboratory code. Fig. 2 compares the turbulent axial velocity of a shear-thinning fluid with a flow behaviour index of 0.75 at a Metzner Reed Reynolds number of 4255 with a direct numerical simulation flow behaviour index of 0.75 at Metzner Reed Reynolds number of 3935 [START_REF] Rudman | Turbulent pipe flow of shear-thinning fluids[END_REF]. It should be said that no significant noteworthy differences were observed, where the predicted profile almost coincides with that of DNS data [START_REF] Rudman | Turbulent pipe flow of shear-thinning fluids[END_REF] along flow regions. There is a minor discrepancy in the logarithmic region because of the difference in the Reynolds number value and the numerical solution procedure. The increase in shear-thinning (i.e., decreasing n) and the rise in shear-thickening (i.e., increasing n) resulted in a marked attenuation and enhancement in the shear rate, respectively, along the pipe radius, especially in the near-wall region. It should be noted that for the pseudoplastic fluids, the increased flow behaviour index leads to a noticeable reduction in the shear rate along the pipe radius: the flow behaviour index increased as the shear rate was attenuated. In contrast, for dilatant fluids, the increased flow behaviour index induced a significant enhancement of the shear rate along the pipe radius: the flow behaviour index increased as the shear rate was enhanced. As shown in Fig. 4, the apparent viscosity of the Newtonian fluid was linear along the pipe radius and equalled the apparent viscosity at the wall ( w  ). In turn, the apparent viscosity of the shear-thinning and shear-thickening fluids were identical and equal to the viscosity at the wall along the viscous sublayer (0≤ Y + ≤5) for all flow behaviour indices. The flow behaviour index significantly affected the apparent viscosity of the shear-thinning and shear-thickening fluids in the near-wall region, where the increased flow index yielded to enhancement in the fluid viscosity in the viscous sublayer.

The apparent viscosity of the power-law profiles deviated significantly from each other, only starting from the buffer region; this deviation became more distinct with distance away from the wall toward the core region for all flow behaviour indices. Beyond Y + =5, the apparent viscosity of the pseudoplastic fluids enhanced gradually with the distance from the wall toward the core region; it was evident that the as the flow behaviour index (n) decreased significantly as the pseudoplastic apparent viscosity increased in the logarithmic region (30≤ Y + ≤200). In contrast, the apparent viscosity of the dilatant fluids was attenuated gradually with distance from the wall (Y + ) towards the core region. The increased flow behaviour index (n) yielded a marked reduction in the apparent viscosity of the viscous sublayer, especially in the logarithmic region. It should be noted that with decreasing flow behaviour index, the pseudoplastic and dilatant fluids tended to behave like solids when approaching the pipe centre. These fluids behaved like liquids rather than solids, with an increasing flow behaviour index approaching the pipe core region. Fig. 5 depicts the apparent viscosity profiles scaled by the viscosity at the wall ( w  ) of the shear- thinning and shear-thickening fluids against the shear rate scaled by the shear rate at the wall ( w  ). The apparent viscosity of the Newtonian fluid was constant along the pipe radius. It equalled the wall viscosity, which means the Newtonian apparent viscosity was independent of the flow shear rate. As observed in Fig. 5, The apparent viscosity of the power-law profiles deviated significantly from each other, only starting from the buffer region; this deviation became more distinct with distance away from the wall toward the core region for all flow behaviour indices. Beyond Y + =5, the apparent viscosity of the pseudoplastic fluids increases gradually with the distance. As observed in Fig. 5, the apparent viscosity of the non-Newtonian fluids was not constant where it was a function of the shear rate, where the apparent viscosity of the pseudoplastic and dilatant fluids varied with the shear rate along the pipe radius. It was evident that the shear rate attenuated gradually from the wall towards the core region for all flow behaviour indices (Fig. 3), where this decreased shear rate resulted in a marked enhancement and reduction in the apparent viscosity of the shear-thinning and shear-thickening (Fig. 4). This trend was more pronounced as the flow behaviour index (n) increased. It should also be noted that this finding confirmed the association between the apparent fluid viscosity and the power-law fluids' shear rate

1 n K  - = .

Axial Velocity Profiles

Figure 6 illustrates the turbulent axial velocity profile scaled by the friction velocity versus the distance from the wall in wall units Y + of the shear-thinning and shear-thickening fluids. The flow behaviour index was chosen to be 0.75, 0.8 (shear-thinning) and 1.2, 1.4, 1.6 (shear-thickening) in addition to the Newtonian fluid (n=1) at a same simulation Reynolds number (Res=12000). Where the dash lines represented the universal velocity distributions in the viscous sublayer (0≤ Y + ≤5) and in the logarithmic layer (30≤Y + ≤200).

As shown in Fig. 6, the turbulent axial velocity profile of the Newtonian fluid agreed entirely with the universal linear (U + = Y + ) and logarithmic (U + =2.5lnY + +5.5) laws in the viscous sublayer and logarithmic layer, respectively. As for the power-law fluids, the turbulent axial velocity profiles of the pseudoplastic (n<1) and dilatant (n>1) fluids were identical for all flow behaviour indices in the viscous sublayer, where they were also in good agreement with the universal linear (U + = Y + ) in the vicinity of the wall up to approximately (Y + = 10). It can be said that the mean axial velocity was almost independent of the flow behaviour index in this flow region.

Further away from the pipe wall, the effect of the flow behaviour index began to become apparent with the distance from the wall (Y + ), and the turbulent axial velocity profiles of the pseudoplastic and dilatant fluids deviated significantly from each other beyond (Y + =20) with the distance from the wall. Away from the wall towards the pipe centre, this deviation was more pronounced in the logarithmic layer (30≤Y + ≤200), where this discrepancy was attributed to the influence of the apparent viscosity and the shear rate of the power-law fluids in this region.

It can be seen from Fig. 6 that the turbulent axial velocity profiles of the shear-thinning fluids were somewhat larger than the Newtonian one, where the pseudoplastic profiles lay above the corresponding Newtonian fluid in the logarithmic layer. In contrast, the turbulent axial velocity profiles of the shearthickening fluids lay down the Newtonian one in this region. It should be noted that the decrease in the flow behaviour index resulted in an enhancement in the turbulent axial velocity profile with distance from the wall (Y + ) away from the wall towards the logarithmic region, where this trend was more pronounced as the flow behaviour index (n) decreased.

Friction factor

As seen in Table1, the friction factor (f) enhanced monotonically with the flow behaviour index (n); it can be said that the increased flow behaviour index results in a marked enhancement in the friction factor for the pseudoplastic and dilatant fluids. This discrepancy was attributed to the influence of the apparent fluid viscosity: the high apparent viscosity at the wall increased the friction factor. As the flow behaviour index rose as the apparent fluid viscosity was enhanced (Fig. 5), the friction factor was also improved (Table1).

CONCLUSIONS

The numerical analysis of the fully developed turbulent flow of pseudoplastic and dilatant fluids in an isothermal stationary pipe using a large eddy simulation and a conventional dynamic model was presented in this work over a wide range of flow behaviour indexes 0.75, 0.8, 1, 1.2, 1.4, and 1.6 at a simulation Reynolds number of 12000.

The results showed that when the flow behaviour index was raised, the shear rate attenuated, which resulted in a notable drop in the shear rate profile throughout the pipe radius for pseudoplastic fluids. On the other hand, for dilatant fluids, an increase in the flow behaviour index resulted in a significant amplification in the shear rate throughout the pipe radius; that is, as the flow behaviour index rose, so did the shear rate. The apparent viscosity of the pseudoplastic fluids increased away from the pipe wall as the flow behaviour index decreased, and the fluid tended to behave like a solid as it got closer to the pipe core area. When approaching the pipe core area, the fluid behaved more like a liquid, whereas the apparent viscosity of the dilatant fluids progressively decreased with distance from the wall. This tendency was more noticeable in the flow behaviour index. Likewise, when reaching the pipe core region, the pseudoplastic and dilatant fluid acted like a solid and a liquid, respectively. The shear rate increased as it moved out of the viscous sublayer through the remaining flow regions. Particularly in the logarithmic region, the turbulent axial velocity of the shear-thinning fluid was greater than that of the shear-thickening fluid; this tendency became more pronounced as the flow behaviour index was reduced. This discrepancy was related to the power-law fluids' apparent viscosity and shear rate in this region. Finally, due to a higher flow behaviour index, the apparent fluid viscosity close to the wall was significantly enhanced, which raised the friction factor.

Apparent viscosity profile of the pseudoplastic, Newtonian, and dilatant fluids versus Y + .

Apparent viscosity of the pseudoplastic, Newtonian, and dilatant fluids versus shear rate.

Figures list.

Tables Captions

Table 1 Friction factor

Tables list.

Figure 1 :

 1 Figure 1: computational domain

Figure 2 :

 2 Figure 2: Validation of the present results.

Figure 3 :

 3 Figure 3: Shear rate profile of the pseudoplastic, Newtonian, and dilatant fluids against the Y + .

Figure 4 :

 4 Figure 4: Apparent viscosity profile of the pseudoplastic, Newtonian, and dilatant fluids versus Y + .

Figure 5 :

 5 Figure 5: Apparent viscosity of the pseudoplastic, Newtonian, and dilatant fluids versus shear rate.

Figure 6 :

 6 Figure 6: Turbulent axial velocity profile of the pseudoplastic, Newtonian, and dilatant fluids.

2

 2 Figures CaptionsFigure 1computational domain Figure 2 Validation of the present results.

Figure 3 Shear

 3 Figure 3

Figure 6

 6 Figure 6Turbulent axial velocity profile of the pseudoplastic, Newtonian, and dilatant fluids.

Table 1 .

 1 Friction factor

	n	0.75	0.8	1	1.2	1.4	1.6
	f E-01	0.0819	0.0845	0.108	0.125	0.141	0.147


Mohamed Abdi
email: abdi.mohamed1@live.fr

Meryem Ould-Rouiss

Fatima Zohra

Nedjda Bouhenni

Abdelfettah Menouer

Large Eddy Simulation of Turbulent Flow of Pseudoplastic and Dilatant Fluids: Turbulence Characteristics

Keywords: Large eddy simulation (LES), Shear-thinning, Shear-thickening, Fully developed, Turbulent flow

The current inquiry analysed the turbulent flow pattern of power-law fluids to shed light on the critical turbulence characteristics of non-Newtonian fluids. The flow behaviour index of the power-law fluid in the current investigation was set at 0.75, 0.8, 1, 1.2, 1.4, and 1.6 at a simulation's Reynolds number of 12000. This investigation was focused on large eddy simulation (LES) of turbulent pipe flow of pseudoplastic and dilatant fluids passing through an axially stationary pipe. The results point to a significant increase in the generation of axial turbulence and a noticeable decrease in the transfer of tangential and radial turbulence intensities from the axial intensities with a drop in the flow behaviour index. A substantial reduction of the turbulent kinetic energy and the Reynolds shear stress of velocity variations in the radial direction was another effect of the lower flow behaviour index.

INTRODUCTION

Many fluids with potential for industrial applications exhibit non-Newtonian fluid behaviours, including polymer, petroleum, plastics, cement, food products, paper, and paint. Fully developed turbulent flow through a smooth pipe is a canonical problem in fluid mechanics and is significant in engineering. According to a literature review, several studies have been done on the turbulent flow of non-Newtonian fluid in recent years, including theoretical and experimental work [START_REF] Metzner | Flow of non-Newtonian fluids-correlation of the laminar, transition, and turbulent-flow regions[END_REF][START_REF] Metzner | Non-Newtonian Fluid Flow. Relationships between Recent Pressure-Drop Correlations[END_REF][START_REF] Dodge | Turbulent flow of non-newtonian systems[END_REF][START_REF] Tomita | A Study on Non-Newtonian Flow in Pipe Lines[END_REF][START_REF] Vidyanidhi | Non-Newtonian Flow in a Rotating Straight Pipe[END_REF][START_REF] Pinho | Flow of non-newtonian fluids in a pipe[END_REF] and numerical work [START_REF] Malin | Turbulent pipe flow of power-law fluids[END_REF][START_REF] Malin | The Turbulent Flow of Bingham Plastic Fluids in Smooth Circular Tubes[END_REF][START_REF] Malin | Turbulent pipe flow of Herschel-Bulkley fluids[END_REF][START_REF] Rudman | Turbulent pipe flow of shear-thinning fluids[END_REF][START_REF] Ohta | DNS and LES with an extended Smagorinsky model for wall turbulence in non-Newtonian viscous fluids[END_REF][START_REF] Gnambode | Large-Eddy simulation of turbulent pipe flow of power-law fluids[END_REF][START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF][START_REF] Gavrilov | Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes[END_REF][START_REF] Singh | The influence of shear-dependent rheology on turbulent pipe flow[END_REF][START_REF] Singh | The effect of yield stress on pipe flow turbulence for generalised newtonian fluids[END_REF].

Rudman et al. [START_REF] Rudman | Turbulent pipe flow of shear-thinning fluids[END_REF] applied direct numerical simulation (DNS) to simulate a fully developed turbulent flow of shear-thinning at various generalised Reynolds numbers for distinct flow behaviour indices (0.5, 0.69 and 0.75). DNS research was conducted by Gavrilova and Rudyak [START_REF] Gavrilov | Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes[END_REF] at two generalised Reynolds values, 10000 and 20000, throughout a 0.4-1 power-law index range. Gavrilova and Rudyak [START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF] presented distributions of Reynolds stress tensor components, averaged viscosity, viscosity fluctuations, and measures of turbulent anisotropy with an emphasis on turbulent mean variables. By providing the distributions of the turbulent stress tensor components, the shear stress and turbulent kinetic energy balances, Gavrilova and Rudyak [START_REF] Gavrilov | Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes[END_REF] published the same findings one year later with a focus on the energy balance and the shear stresses.

More recently, Abdi et al. [17] have conducted a fully developed turbulent forced convection of thermally independent pseudoplastic fluid with a flow behaviour index of 0.75 through an axially heated rotating pipe, by means of LES with an extended Smagorinsky model. With a rotation rate ranging from 0 to 3, where the simulation Reynolds and Prandtl numbers of the working fluid were assumed to be 4000 and 1, respectively. It is observed that as the pipe wall rotates, it is seen that the temperature along its radius noticeably decreases as the rotation rate increases. This is because the apparent fluid viscosity in the pipe's core region decreases, which causes a centrifugal force that causes the mean axial velocity profile to noticeably increase.

The present paper aimed to examine the effect of the power-law fluid's flow behaviour index (n) on turbulence statistics and instantaneous turbulence structure. Toward this end, a large eddy simulation (LES) with a standard dynamic model was devoted to a fully developed turbulent flow of shear-thinning (n<1) and shear-thickening (n>1) fluids across an isothermal stationary pipe. This study was performed at a simulation's Reynolds number of 12000, and the flow behaviour indexes were chosen to be 0.75, 0.8, 1, 1.2, 1.4 and 1.6.

GOVERNING EQUATIONS AND NUMERICAL PROCEDURE

Governing Equations

The study dealt numerically with a fully developed turbulent flow of power-law fluids in a pipe at different flow behaviour indexes (0.75≤ n ≤1.6) at a simulation Reynolds number of 12000, using the LES approach with a standard dynamic model with computational domain length of 20R (Fig. 1). The filtered equations were expressed as follows (Equations 1 and 2):

0 i i u x  =  (1) 1 1 Re j i j j ij n i i j s i i j i u u u u u dP t x x x x x x   -         + = - + + +              (2)

Numerical Procedure

The mathematical model was implemented in a finite difference laboratory code. The governing equations were discretised on a staggered mesh in cylindrical coordinates with a computational length in the axial direction 20R. The numerical integration was performed using a finite difference scheme, second-order accurate in space and time. The time advancement employed a fractional-step method. A third-order Runge-Kutta explicit scheme and a Crank-Nicholson implicit scheme were used to evaluate the convective and diffusive terms, respectively. In the non-Newtonian Smagorinsky model, the subgrid stress tensor ij  is linked to the strain rate tensor by

2 ij t ij S   =- .

RESULTS AND DISCUSSION

The goal was to reveal the influence of the flow behaviour index of the pseudoplastic (shear-thinning) and dilatant (shear-thickening) fluids on the turbulent flow statistics and instantaneous turbulence structure along the radial direction, especially in the vicinity of the wall. Fig. 3 compares the turbulence intensities of the Newtonian fluid with that of Redjem et al. [18] at the Reynolds number of 5500; the results present the root mean square (RMS) distribution of the axial, radial and tangential velocity fluctuations of the Newtonian fluid (n=1). As shown in Fig. 3, the RMS of the axial components were in excellent agreement along the pipe radius with that of the literature; the tangential and radial components were also in accord with the results of Redjem et al. [18]. There was little discrepancy due to the difference in the Reynolds number value and the numerical solution procedure.

Root mean squares of velocity fluctuations

The root mean square (RMS) distribution of the axial, tangential and radial velocity fluctuations of the pseudoplastic and dilatant fluids along the pipe radius (R) are presented in Figures 345, respectively. These profiles were scaled by the friction velocity versus the distance from the wall in wall units Y + . The flow behaviour indexes were set at 0.75, 0.8 (shear-thinning), 1.2, 1.4, 1.6 (shear-thickening) and 1 (Newtonian) at a simulation Reynolds number of 12000.

Based on the DNS data of Redjem et al. [18], the accuracy and reliability of the predictions and numerical method have been confirmed decisively. As shown in Fig. 2, the current RMS is in excellent agreement with the results of Redjem et al. [18], where no significant differences were found between them.

As shown in Fig. 3, the power-law and Newtonian fluids exhibited a similar trend along the pipe radius, where there was a clear trend of increase in these profiles over the radial direction. The RMS of the axial components was almost identical for all fluid behaviour indices in the vicinity of the wall up to approximately (Y + = 10), where it could be said that the RMS of the axial component was almost independent of the flow behaviour index in this region. These profiles began to deviate from each other further away from the wall towards the core region; this deviation became more distinct with distance from the wall (Y + ), especially in the logarithmic region (30≤ Y + ≤200). As seen in Fig. 3, the RMS of the axial component enhanced or increased gradually away from the wall with the wall distance up to approximately (Y + = 10) because of axial fluctuation generation in the near-wall region, as pointed out by Gavrilov and Rudyak [START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF]. At a large wall distance, the axial profiles began to drop rapidly to lower values in the core region for all fluid behaviour indices; this meant that the axial fluctuations vanished gradually in this flow region.

It was evident that the decreased flow behaviour index resulted in a pronounced enhancement in the RMS of the axial velocity fluctuations beyond the buffer region (5≤Y + ≤30). Moreover, this reduction in the flow index led also to a shift in the peak location slightly away from the wall toward the core region. The axial velocity fluctuations were generated in the wall's vicinity and transported from the near-wall region toward the core region, as pointed out by previous researchers [START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF]. It can be said that with a decreasing flow behaviour index, the generation of the axial velocity fluctuations enhanced considerably, where this decrease in the flow index also led to ameliorating or improving the transport of the axial velocity fluctuations from the wall vicinity towards the core region. As shown in Fig. 4, the same trend of RMS of tangential fluctuations was observed along the pipe radius for all flow behaviour indices; the tangential fluctuations were almost negligible in the near-wall region, where this was ascribed to the molecular shear stress being the dominant force compared to the turbulent one in this region. These profiles began to increase gradually and deviated from each other away from the nearwall region with distance from the wall (Y + ). They dropped rapidly and fell to lower values beyond approximately (Y + = 30) for all flow behaviour indices. As seen in Fig. 4, the RMS of tangential fluctuations of the dilatant fluids lay above the pseudoplastic along the pipe radius, as the flow behaviour index was decreased as the RMS of tangential fluctuations was attenuated or diminished along the pipe radius. The peak locations were shifted towards the core region. It was evident that the decreased flow index induced a noticeable reduction in the transfer of the tangential fluctuations from the axial ones, consequently in the tangential fluctuations. This decreased flow behaviour index also ameliorated the transport of the tangential fluctuations from the wall vicinity towards the core region. It was worth noting that these predicted results were in line with those of Gavrilov and Rudyak [START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF]. As seen in Fig. 5, the RMS of radial fluctuations was identical for all flow behaviour indices and equal to zero in the viscous sublayer (5≤ Y + ≤30) up to approximately (Y + = 3). Beyond Y + =3, these profiles exhibited a significant enhancement further away from the wall towards the core region. Beyond (Y + = 50), these profiles fell off rapidly to lower values after reaching their maximum values. It should be noted that with a reducing flow behaviour index (n), the radial fluctuations reduced significantly along the pipe radius, and the predicted peak locations shifted away towards the core region, as Gavrilov and Rudyak [START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF]. It can be said that the decreasing flow behaviour index yielded an attenuation in the radial fluctuation, which meant reducing the fluctuations transfer from the axial to the radial one and ameliorating the transport of the radial fluctuations from the near-wall towards the core region.

Turbulent Kinetic energy

The turbulent kinetic energy of the pseudoplastic and dilatant fluids along the pipe radius (R) versus the distance from the wall in wall units Y + are depicted in Fig. 6, at a flow behaviour index of (0.75≤n≤1.6) and at a simulation Reynolds number of 12000. Overall, the kinetic energy of turbulent fluctuations exhibited almost the same trend as the axial turbulence intensities, which were almost independent of the flow behaviour index and equal to zero in the near-wall region: molecular shear stress was the dominant force in this region. It was important to note that the observed increased and decreased turbulent kinetic energy with the wall distance were related to the enhancement and reduction, respectively, of the axial turbulence intensities over the pipe radius, the increased turbulent kinetic energy beyond the buffer region was raised because the generation of the axial turbulence intensities and transfer of these fluctuations to the tangential and radial ones, as pointed out by other studies [START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF]. It was worth noting that the turbulent kinetic energy attenuated pronouncedly over the radial direction with a decreasing flow behaviour index.

Turbulent Reynolds shear stress

The turbulent Reynolds shear stress of the axial and radial velocity fluctuations of the shear-thinning, shear-thickening and Newtonian fluids were also assessed. In the viscous sublayer, the Reynolds shear stress profiles were identical and equal to zero. This was due to the absence of velocity fluctuations near the wall region. Out of the viscous sublayer, the Reynolds stress profiles deviated from each other and increased sharply to reach their peak value in the buffer region, as shown in Fig. 7. This enhancement was related to the generation of the axial fluctuations further away from the wall (Fig. 3). Moreover, these profiles fell off rapidly to a zero value in the core region. This was attributed to a vanishing of the axial and radial velocity fluctuation in this region (Fig. 3 and Fig. 4). It was worth noting that when the flow behaviour index decreased, the Reynolds shear stress of the axial, radial velocity fluctuations reduced considerably along the pipe radius (Fig. 7).

CONCLUSIONS

Using a large eddy simulation (LES) and a universal dynamic model, a fully developed turbulent flow of pseudoplastic and dilatant fluids through an isothermal stationary pipe was examined numerically. This investigation was conducted using a simulation with a Reynolds number of 12000 and a flow behaviour index of 0.75<n<1.6. The results indicated that the flow behaviour index dropped, which caused an increase in the generation of axial turbulence and a noticeable decrease in the transfer of tangential and radial turbulence intensities from the axial intensities. Additionally, this drop in the flow behaviour index improved the conveyance of the axial, radial and tangential turbulence intensities from the wall proximity to the core region. The turbulent kinetic energy and Reynolds shear stress of velocity variations decreased significantly in the radial direction with a decreasing flow behaviour index. On the other hand, the pseudoplastic and dilatant fluids' skewness and flatness seem nearly independent of the flow behaviour index.

Figure 1 :

 1 Figure 1: computational domain

Figure 2 :

 2 Figure 2: Validation

Figure 3 :

 3 Figure 3: RMS of axial component

Figure 4 :

 4 Figure 4: RMS of the tangential component

Figure 5 :

 5 Figure 5: RMS of the radial component

Figure 6 :

 6 Figure 6: Turbulent kinetic energy

Figure 7 :

 7 Figure 7: Turbulent Reynolds stress 3.4. Higher-order statistics Fig.8 and Fig.9 display respectively the skewness and flatness of the axial velocity fluctuations distribution against the distance from the wall in wall units Y + at Res=12000 for 0.4≤ n ≤1.6. The skewness

Figure 8 :

 8 Figure 8: Skewness of axial fluctuations Figure 9: Flatness of axial fluctuations

  Figure 1 computational domain Figure 2 Validation.

Figure 3 Figure 3 :

 33 Figure 3Figure3: RMS of axial component.

Figure 4 RMS

 4 Figure 4RMS of the tangential component.

Figure 5 RMS

 5 Figure5RMS of the radial component.

Figure 6

 6 Figure 6Turbulent kinetic energy.

  Figure 7 Turbulent Reynolds stress Figure 8 Skewness of axial fluctuations Figure 9 Flatness of axial fluctuations