Mohamed Abdi
email: abdi.mohamed1@live.fr

Meryem Ould-Rouiss

Abdelkader Noureddine

Hydrodynamic and Rheological Characteristics of a Pseudoplastic Fluid Through a Rotating Cylinder

Keywords: LES, Non-Newtonian, Pseudoplastic fluid, Turbulent flow, Rotating cylinder 1. Introduction:

The fully developed turbulent flow of pseudoplastic ( 0.75 n =

) and Newtonian fluids in an isothermal axially rotating cylinder has been carried out using a large eddy simulation (LES) with an extended Smagorinsky model. The simulation Reynolds number of the present predictions has been assumed to be Re 4000 s = at various rotation rates (0 3) N . This investigation seeks to assess the influence of the centrifugal force induced by the swirl on the mean flow quantities, turbulent statistics, and instantaneous turbulence structure to describe the rheological behavior and the turbulence features. The predicted results indicate that with increasing rotation rate, the pseudoplastic fluid tends to behave like a liquid when approaching the pipe center due to the lower apparent fluid viscosity in the logarithmic region as the pipe wall rotates. Moreover, the reduction in the pseudoplastic apparent viscosity in the core region induces a pronounced increase in the axial velocity profile further away from the pipe wall toward the core region. It is interesting to note that the growth of the centrifugal force induced by the swirl driven by the rotating pipe wall results in an apparent attenuation in turbulence intensities of the axial velocity fluctuation and, consequently, in the kinetic energy of turbulent fluctuations and the turbulent Reynolds shear stress of the axial-radial velocity fluctuations, as the pipe wall rotates. Moreover, the increased rotation rate leads also to a noticeable increase in the Root mean square (RMS) of the radial and tangential fluctuations. It can be said that the transport mechanism of turbulence intensities from the axial components to the other ones exhibits a marked increase with increasing pipe wall rotation.

Flow in swirling systems is not only of great importance to practical applications in mechanical and engineering fields (heat exchangers, combustion chambers, nuclear reactors, and turbomachines) but also of considerable fundamental interest. To better understand the laminarization phenomena and analyze the influence of the rotation pipe wall on the mean properties and turbulence statistics, many authors performed experimental [START_REF] Murakami | Turbulent Flow in Axially Rotating Pipes[END_REF], [START_REF] Kikuyama | Flow in an Axially Rotating Pipe : A calculation of flow in the saturated region[END_REF], [START_REF] Nishibori | Laminarization of turbulent flow in the inlet region of an axially rotating pipe[END_REF], [START_REF] Reich | Fluid flow and heat transfer in an axially rotating pipe-I. Effect of rotation on turbulent pipe flow[END_REF], and [START_REF] Imao | Turbulent characteristics of the flow in an axially rotating pipe[END_REF], theoretical and numerical [START_REF] Hirai | Predictions of the Laminarization Phenomena in an Axially Rotating Pipe Flow[END_REF], [START_REF] Tu | Fully developed periodic turbulent pipe flow. Part 1. Main experimental results and comparison with predictions[END_REF], [START_REF] Orlandi | Direct simulations of turbulent flow in a pipe rotating about its axis[END_REF], [START_REF] Yang | Large eddy simulation of fully developed turbulent flow in a rotating pipe[END_REF] and [START_REF] Feiz | Turbulence Statistics in a Fully Developed Rotating Pipe Flow[END_REF] studies. But most of them are based on the turbulent flow of Newtonian fluids. They found that the rotating pipe wall significantly influences the turbulence due to the centrifugal force, which has a stabilizing effect: the increased rotation velocity of the cylinder wall results in a pronounced increase in the mean axial velocity profile along the radial coordinate, where this profile progressively approaches to a laminar one; this is called the laminarization phenomena.

However, relatively few works focused on non-Newtonian fluids, whose viscosity depends on the shear rate. Some authors conducted theoretical and experimental investigations devoted to the turbulent flow of non-Newtonian fluids flow through a straight axial pipe [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], and [START_REF] Pinho | Flow of non-Newtonian fluids over a confined baffle[END_REF]. To provide detailed information on the turbulence statistics and flow characteristics, numerical investigations were performed [START_REF] Malin | Turbulent pipe flow of power-law fluids[END_REF], [START_REF] Rudman | Turbulent pipe flow of shearthinning 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 | Reynolds number effects in pipe flow turbulence of generalized Newtonian fluids[END_REF], [START_REF] Singh | The effect of yield stress on pipe flow turbulence for generalised newtonian fluids[END_REF], and [START_REF] Zheng | Direct numerical simulation of turbulent non-Newtonian flow using OpenFOAM[END_REF]. In their paper, Dodge and Metzner [START_REF] Dodge | Turbulent flow of non-newtonian systems[END_REF] developed for the first time a theoretical study for the Non-Newtonian fluids in turbulent flow through straight cylinders; they derived a semi-theoretical expression for the pressure loss and mean flow rate and predicted the non-Newtonian turbulent velocity profiles, as well as a correlation for friction factor. In 1990, Pinho and Whitelaw [START_REF] Pinho | Flow of non-Newtonian fluids over a confined baffle[END_REF] measured the streamwise velocity and the three normal stresses for the Non-Newtonian fluids in a fully developed flow for a Reynolds number ranging from 240 to 111000. Malin [START_REF] Malin | Turbulent pipe flow of power-law fluids[END_REF] investigated numerically the fully developed laminar and turbulent flows of shear-thinning, Newtonian, and shear-thickening fluids in a straight cylinder, at various values of Reynolds numbers, using a modified version of the Lam-Bremhorst K  -Model for the turbulent flow.

Rudman et al. [START_REF] Rudman | Turbulent pipe flow of shearthinning fluids[END_REF] reported one of the first numerical attempts to provide some important insights into the turbulence features and flow behavior of the shear-thinning and Herschel-Bulkley fluids through an isothermal axially pipe at different generalized Reynolds numbers. Their emerged predicted results suggest that the decreased flow behavior index results in a pronounced increase in the axial velocity profile further away from the pipe wall, and a reduction in the friction factor, due to the higher apparent viscosity of the shear-thinning fluid in the flow core region. Gnambode et al. [START_REF] Gnambode | Large-Eddy simulation of turbulent pipe flow of power-law fluids[END_REF] implemented a LES technique with an extended Smagorinsky model to perform numerically the turbulent flow of pseudoplastic and dilatant fluids with flow behavior index ranging from 0.5 to 1.4 at different Reynolds numbers (4000, 8000, and 12000). The findings suggest that the increased Reynolds number increases the velocity profiles along the pipe radius, especially in the logarithmic flow region. The friction factor decreases noticeably as the flow behavior index is decreased, while this friction also decreases with increasing Reynolds number for both shear-thinning and shear-thickening fluids. Moreover, a clear trend of decrease in the apparent fluid viscosity when approaching the core region when the flow behavior index is increased. 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] provided an extensive investigation of DNS of fully developed turbulent pipe flows of shear-thinning (0.4≤n≤1) fluids at two generalized Reynolds numbers 10000 and 20000. To critically assess the effects of the flow behavior index and the Reynolds number on the main turbulence features and the rheological behavior. The relevant predicted results are in accord with Rudman et al. [START_REF] Rudman | Turbulent pipe flow of shearthinning fluids[END_REF] study, indicating that the decreased flow behaviour index results in a pronounced increase of the apparent fluid viscosity further away from the pipe wall towards the pipe core region, where the shear-thinning tends to behave like a solid when approaching to the pipe centre. Which leads to increase the axial velocity in the core region, and attenuate significantly the friction factor [START_REF] Gavrilov | Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe[END_REF]. It is observed that with decreasing flow behavior index, the fluctuations of the radial and tangential velocity components exhibit marked attenuation. This deviation leads to an increase of the axial fluctuations, consequently integral fluctuations and kinetic energy of turbulent fluctuations. [START_REF] Singh | The effect of yield stress on pipe flow turbulence for generalised newtonian fluids[END_REF] [START_REF] Singh | Reynolds number effects in pipe flow turbulence of generalized Newtonian fluids[END_REF] reported a DNS investigation of turbulent power-law fluids across the axially stationary pipe at a friction Reynolds number of 323 over a flow behavior index range of (0.4≤n≤1.2). It is found that the decreased flow behavior index leads to a marked reduction in the Reynolds shear stress where the turbulence structures become finer with decreasing flow index. Moreover, the apparent fluid viscosity of the shear-thickening fluid is lower than those of the shear-thinning fluids out of the viscous sublayer. More recently, to ensure the accuracy and efficiency of the OpenFOAM library, Zheng et al. (2019) [START_REF] Zheng | Direct numerical simulation of turbulent non-Newtonian flow using OpenFOAM[END_REF] have performed numerical DNS of the fully developed turbulent flow of shear-thinning fluids through a pipe. The results show that the predicted velocity and viscosity profiles are well resolved, while the turbulence statistics have marked a noticeable difference compared with those obtained from the spectral element-Fourier DNS code. Interestingly, the turbulent intensities and Reynolds stresses profiles differ with a maximum of 16% and 10% at Reynolds numbers 5000 and 7500, respectively.

A literature survey reveals that very few studies deal with the turbulent swirling flow of non-Newtonian fluids. In contrast, the relevant numerical research aforementioned has been devoted to the turbulent flow of the non-Newtonian fluids inside a stationary pipe. Therefore, a noticeable paucity of studies describing the impact of the centrifugal force induced by the swirl driven by a rotating pipe wall on the rheological and turbulence behavior. It should be noted that the fully developed turbulent flow of the non-Newtonian fluid through the axially rotating pipe was treated theoretically by Vidyanidhi and Sithapathi [START_REF] Vidyanidhi | Non-Newtonian Flow in a Rotating Straight Pipe[END_REF] and theoretically and experimentally by Gunn et al. [START_REF] Gunn | On Newtonian and non-Newtonian flow in a rotating pipe[END_REF]. [START_REF] Gunn | On Newtonian and non-Newtonian flow in a rotating pipe[END_REF] [START_REF] Gunn | On Newtonian and non-Newtonian flow in a rotating pipe[END_REF] provide theoretically and experimentally an extensive analysis of laminar and turbulent Newtonian and non-Newtonian fluids through an axially rotating pipe to examine the effects of the rotation rate on the flow patterns. The experiment results indicate that in the laminar flow, the flow rate of the Newtonian fluid decreases significantly when the pipe is rotating, this trend is more pronounced as the Reynolds number is increased, and the flow rate becomes nearly independent of the rotation rate at the high Reynolds number in the laminar regime. In contrast, the flow rate of the non-Newtonian exhibits a marked increase when the pipe rotates, where the rotation rate induces a gradually increase in the flow rate in the laminar regime. In the turbulent flow, the rotation rate results a slightly increase in the pressure gradient of Newtonian fluid, while that this increase is more pronounced in the non-Newtonian fluid, it is interesting to note that this enhancement is more important as the flow rate increases.

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 performed for the first time a fully developed turbulent forced convection of thermally independent pseudoplastic fluid with a flow behavior index of 0.75 through an axially heated rotating pipe by mean 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 the increased rotation rate induces a marked attenuation in the temperature along the pipe radius. The centrifugal force induced by the rotating pipe wall leads to a noticeable increase in the mean axial velocity profile as the pipe wall rotates due to the reduction in the apparent fluid viscosity in the core region. It is also demonstrated that the turbulence intensities of temperature fluctuations and the axial turbulent heat flux exhibit an apparent attenuation when the pipe rotates. When the rotation rate increases, there is a clear reduction trend in the transfer mechanism of the temperature fluctuations between the conductive layer and the flow core region. Their findings indicate that the Nusselt number reduces when the rotation rate is less than 0.5, while it increases with an increasing rotation rate for N greater than 1. As for the higher-order statistics, it is demonstrated that the influence of the rotation rate on the skewness and flatness coefficients is mainly restricted in the nearwall region; these profiles seem nearly independent of the centrifugal force induced by the rotating pipe wall further away from the pipe wall.

It is clear that a comprehensive study on the turbulent pipe flow with a swirl of non-Newtonian fluids through an axially rotating pipe is still lacking for hydrodynamic and rheological investigations. To this end, under the same investigation hypothesis as that of 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], the present study deals numerically with the fully developed turbulent flow through an isothermal axially rotating pipe of a pseudoplastic (n=0.75) and Newtonian fluids by mean of LES with extended Smagorinsky model. The current study set out to reveal the effects of the centrifugal force induced by the rotating pipe wall on the hydrodynamic and rheological behavior as well as the turbulent statistics and instantaneous turbulence structure of the shear-thinning fluids, via analyzing and discussing critically the mean flow and rheological properties as well as the main turbulence characteristics.

The fluid modeled in this study is power-law (pseudoplastic) fluid, the shear stress is related to the shear rate by:

( ) n K  =
, where  is the shear rate K is the consistency index and n is the flow behavior in- dex. The apparent viscosity  is not constant for the power-law fluid; a function of the magnitude of the shear rate is written in the form:

1 n K  - =
. For 1 n  , the apparent viscosity decreases with increasing shear rate, and fluid is called shear-thinning (pseudoplastic).

1 n  , the apparent viscosity increases with the increased shear rate, and fluid is termed shear-thickening (dilatant), which the Newtonian flow behavior is expected for [START_REF] Metzner | Flow of non-newtonian fluids-correlation of the laminar, transition, and turbulent-flow regions[END_REF] [START_REF] Metzner | Flow of non-newtonian fluids-correlation of the laminar, transition, and turbulent-flow regions[END_REF] were the first to propose a definition of the generalized Reynolds number of the power-law fluids based on the effective viscosity ( ) ( )

1 n = .
1 3 1 4 8 nn eff K n n U D  - =+
; thus, the Metzner and Reed Reynolds number is defined as ( )

21 Re 8 3 1 4 n n n n MR U D K n n  -- =+ .
Another commonly used generalized Reynolds number Re g based on the apparent viscosity of the fluid at the wall ( )

1 n ww K  - =
, where this Reynolds number reflects the flow behavior in the vicinity of the wall. Moreover, the friction Reynolds is based on the wall friction velocity and the mean viscosity on the wall Re The present paper is organized as follows; the governing equations and numerical procedure are described in section 2. the effect of rotating pipe wall on the mean quantities and turbulent flow statistics (axial and tangential velocities, Root mean square of the fluctuating velocity, turbulent Reynolds stress and instantaneous turbulence structure), are presented in section 3. Section 4 summarizes the main findings of this study and concludes.

Governing equations and computational method:

Governing equations:

A large eddy simulation approach with an extended Smagorinsky model has been applied to study the turbulent flow of the pseudoplastic (n<1) fluid through a straight axially rotating cylinder of 20R as a length in the axial direction (see Fig. 1). The flow behavior index (n) was chosen to be 0.75 at the simulation Reynolds number of 4000, and the rotation rate in this paper set out to be (N=0, 0.5, 1, 2, and 3).

The filtered equations are written as:

0 i i u x  =  (1) 1 1 Re j i j j ij ij n i ijk k l i i s i j i i i u u u u T u p Nu t x x x x x x x   -          + + = - + + - +               (2)
Where 2

k k CL N R U =
is defined as the rotation rate and k  the rotational pipe wall velocity. Res is the Reynolds number of the simulations and is defined as

2 Re nn s CL U R K  - = .

Computational method:

Filtering the Navier-Stokes equations eliminates scales smaller than the filter . These eliminated scales, called subgrid-scale (SGS) motions (small eddies), are unresolved small-scale fluid motions in the filtered equations that govern the resolved large-scale scales. The SGS motions' effects on the resolved scales are modeled using an eddy-viscosity closure and the strain rate tensor. According to Ohta and Miyashita, the currently filtered equations were closed by an extended Smagorinsky model, where the additional terms were ignored [START_REF] Ohta | DNS and LES with an extended Smagorinsky model for wall turbulence in non-Newtonian viscous fluids[END_REF].

The present LES investigations were performed with a laboratory code; for more details on the computational procedure, see the investigation of Gnambode et al. [START_REF] Gnambode | Large-Eddy simulation of turbulent pipe flow of power-law fluids[END_REF]. The grid resolution 65 3 grid points in r, z, and directions, respectively, with a domain length of 20R in the streamwise direction, was chosen for all simulations presented in this work (this choice is based on the mesh independence study performed by Gnambode et al. [START_REF] Gnambode | Large-Eddy simulation of turbulent pipe flow of power-law fluids[END_REF] for n=0.75). We applied a non-uniform distribution grid defined by a hyperbolic tangent function in the radial direction, in addition to periodic boundary conditions and an equally spaced computational grid in the axial and the azimuthal directions.

It is worth noting that Montreuil [START_REF] Montreuil | Simulation numérique pour l'aérothermique avec des modèles sous-maille[END_REF] pointed out that the LES with high resolution can be carried out with z +  equal to 35 wall units and ( ) r  +  less than 10 wall unites, while Zang [START_REF] Zang | Numerical Simulation of the Dynamics of Turbulent Boundary Layers: Perspectives of a Transition Simulator[END_REF] reported that the adequate LES could be carried out using z +  and ( )

r  + 
less than 80 and 40 wall units, respectively. It can be seen from the data in Tab.1 that the first grid points away from the wall in the current work are placed at 1 Y +  , where the first mesh point is located at 0.0239 and 0.0241 wall units for the lowest (N=0) and the highest (N=3) rotation rate, respectively. In addition, 24 grid points at least are located within the viscous sublayer 5 Y +  for all simulations. One can also see from Tab.1 the maximum and the minimum values z +  are 66.01 and 70.55 wall units, respectively, while that r  +  ranging from 20.74 to 22.16 wall units. The stationary case (N=0) varies from 0.0506 at the pipe wall to 12.4 wall units at the pipe center. However, it varies from 0.0511 at the pipe wall to 12.5 wall units at the pipe center for the highest rotation rate case (N=3). The mesh discerption mentioned above shows that the current grid resolution used in the LES simulations can be considered reasonable.

The computational procedure was performed using a finite difference scheme, which was secondorder accurate in space and time. Convective terms were evaluated using Runge-Kutta explicit third-order schemes, while diffusive terms were evaluated using Crank-Nicolson implicit schemes. The time integral was calculated using a fractional step method. The time is made dimensionless using the pipe radius (R) and the maximum velocity of the laminar flow (UCL). A constant CFL (Courant, Friedrichs, and Lewy) condition was used in the calculations. The statistics are computed by averaging in the periodic directions and in time. The final data are obtained by ensemble averaging over the time interval from the dimensionless time t =250 until t =8000 for the lowest rotation rate. At the highest rotation rate, the statistics are sampled from the dimensionless time t = 250 until t =10000.

Results and discussion:

The current section seeks to analyze and discuss the influence of the rotation cylinder wall on the mean flow and turbulence statistics critically to describe the rheological and hydrodynamic behavior of the swirling flow of pseudoplastic fluid. The emerged LES predictions were validated by comparing some numerical findings results of the shear-thinning and Newtonian fluids with those available in the literature in the following cases: the experimental data of Rudman et al. [START_REF] Rudman | Turbulent pipe flow of shearthinning fluids[END_REF] in the stationery cylinder, the DNS of Rudman et al. [START_REF] Rudman | Turbulent pipe flow of shearthinning fluids[END_REF] for flow behavior index of 0.75 through a stationary cylinder, the experimental data by Reich and Beer [START_REF] Reich | Fluid flow and heat transfer in an axially rotating pipe-I. Effect of rotation on turbulent pipe flow[END_REF] for

1 n = at 0,1,3 N =
, the DNS data of Redjem-Saad et al. [START_REF] Redjem-Saad | Direct numerical simulation of turbulent heat transfer in pipe flows: Effect of Prandtl number[END_REF] for 1 n = at 0 N = , the experimental data by Eggels et al. [START_REF] Tu | Fully developed periodic turbulent pipe flow. Part 1. Main experimental results and comparison with predictions[END_REF] for Newtonian fluid at 0 N = , and the DNS data of Ould-Rouiss et al. [START_REF] Ould-Rouiss | Numerical predictions of turbulent heat transfer for air flow in rotating pipe[END_REF] for Newtonian fluid at 0,1,3 N = .

Apparent viscosity:

The Beyond approximately Y + =3, the shear rate profiles begin to drop progressively with the distance from the wall (Y + ) towards the core region, where this reduction is sharped in the buffer region (5≤Y + ≤30) for all cases. It is worth noting that the shear rate profiles seem slightly higher with an increasing rotation rate in the logarithmic region; this trend is more pronounced as the rotation rate increases.

It is apparent in Fig. 3 that no significant noteworthy difference is observed between the normalized apparent viscosity profile of the shear-thinning (n=0.75) fluid for all rotation rates in the near-wall region. In turn, the apparent viscosity profile of the Newtonian fluid is linear and equals the apparent viscosity at the wall along the pipe radius at every rotation rate. It can be said that the apparent viscosity of the pseudoplastic fluid seems independent of the rotation rate (N), over the viscous sublayer ( ) 05 Y +  . Beyond the buffer region (Y + >5), the apparent viscosity profiles begin to increase gradually far away from the pipe wall towards the core flow region with the distance from the wall () Y + , there is a clear trend of gradually increasing in the apparent fluid viscosity, where the pseudoplastic (n=0.75) fluid becomes more viscous in the buffer ( )

5 30 Y + 
and logarithmic ( )

30 200 Y + 
regions. In other words, the pseudoplastic fluid behaves like a solid when approaching the pipe center. On the other hand, it appears that the viscosity profiles of the shearthinning fluid are slightly affected by the rotating pipe wall; the profiles of the rotating pipe are very close to each other and sensibly higher than the stationary pipe in the buffer layer. As shown in Fig. 3, this deviation is more pronounced in the logarithmic region, where the increased rotation rate results in a marked attenuation in the apparent viscosity in this flow region. ) is not constant where is a function of the shear rate, where the apparent viscosity of the pseudoplastic fluid varies with the shear rate along the pipe radius (Fig. 4). It is evident that the shear rate attenuates gradually from the wall towards the core region (Fig. 2), where this decreased shear rate results in a marked increase in the apparent viscosity of the shear-thinning (Fig. 3). Moreover, the apparent viscosity of the pseudoplastic fluid is inversely proportional to the shear rate: the shear rate decreases as the apparent viscosity increases considerably for all rotation rates. Because of the reduced shear rate in the pipe core region, the fluid has a tendency to act more like a solid than a liquid as it moves away from the pipe wall. This causes the viscosity to rise.

On the other hand, with increasing rotation rate, the shear rate and the apparent pseudoplastic viscosity exhibit marked increase and attenuation, respectively, beyond the end of the buffer layer (logarithmic layer), as shown in Fig. 2 and Fig. 3. It can be said that the increased rotation rate induces in a noticeable increase in the shear rate in the logarithmic region (Fig. 2), leading to reduce the apparent viscosity (Fig. 3) in the core region. In other words, the pseudoplastic fluid behaves like a liquid approaching the pipe center as the pipe wall rotates (Fig. 4).

Mean velocity profile:

The streamwise velocity profile U U U  + = , normalized by the friction velocity ( )

12 w U   = , is
depicted in Fig. 5, for Re 4000 s =

. At a fixed pipe, the predicted velocity profile is in good agreement with those of Rudman et al. [START_REF] Rudman | Turbulent pipe flow of shearthinning fluids[END_REF] for pseudoplastic fluid and with the experimental results of Eggels et al. [START_REF] Tu | Fully developed periodic turbulent pipe flow. Part 1. Main experimental results and comparison with predictions[END_REF] and the DNS of Ould-Rouiss et al. [START_REF] Ould-Rouiss | Numerical predictions of turbulent heat transfer for air flow in rotating pipe[END_REF] and Redjem et al. [START_REF] Redjem-Saad | Direct numerical simulation of turbulent heat transfer in pipe flows: Effect of Prandtl number[END_REF] for a Newtonian fluid. The universal laws in the viscous sublayer and the log region are well predicted for 1 n = at 0 N = . The predicted Newtonian velocity profiles normalized by the mean velocity ( / ) zb UU, at various N , Fig. 6, agree very well with the experimental results of Reich and Beer [START_REF] Reich | Fluid flow and heat transfer in an axially rotating pipe-I. Effect of rotation on turbulent pipe flow[END_REF] and DNS predictions of Ould-Rouiss et al. [START_REF] Ould-Rouiss | Numerical predictions of turbulent heat transfer for air flow in rotating pipe[END_REF].

When the pipe rotates, the velocity profiles for the shear-thinning fluid, like the Newtonian fluid, deviate from the logarithmic law because of the flow laminarization, Fig. 7. Indeed, many investigations results [START_REF] Zhang | Simulation of pipe flow[END_REF], [START_REF] Orlandi | Turbulent budgets in rotating pipes by DNS[END_REF], [START_REF] Feiz | Turbulence Statistics in a Fully Developed Rotating Pipe Flow[END_REF], and [START_REF] Ould-Rouiss | Numerical predictions of turbulent heat transfer for air flow in rotating pipe[END_REF] reported such a deviation from the logarithmic law in the case of a Newtonian fluid. This trend is more pronounced as the rotation rate increases, meaning that the predicted axial velocity profiles have a logarithmic shape ln( ) A Y B + + , where the constants A and B depend on the rotation rate N . This tendency is due to the turbulence level attenuation in the pipe with increasing rotation rate [START_REF] Ould-Rouiss | Numerical predictions of turbulent heat transfer for air flow in rotating pipe[END_REF]. In the log regions, the predicted velocity for the Newtonian fluid at 3 N = is larger than that for the shear-thinning one because the apparent viscosity at 0.75 n = is greater than that at 1 n = . Indeed, the strain  decreases towards the pipe for the shear-thinning fluid, inducing an increase in the apparent viscosity : the shear-thinning fluid becomes more and more viscous when approaching the pipe center. The fluids become more viscous with decreasing shear rates, which induce a marked higher viscosity, where the pseudoplastic fluid does not flow more quickly towards the pipe center.

The streamwise velocity profile scaled by the centreline velocity versus le radial position, Fig. 8, clearly exhibits a turbulence reduction as the rotation rate N is increased. At very high N, the pseudoplastic mean axial profile gradually approaches the laminar non-Newtonian Poiseuille profile. This is called the laminarization phenomenon. This sort of laminarization has also been observed in the Newtonian fluid and has been pointed out by many researchers [START_REF] Nishibori | Laminarization of turbulent flow in the inlet region of an axially rotating pipe[END_REF], [START_REF] Hirai | Predictions of the Laminarization Phenomena in an Axially Rotating Pipe Flow[END_REF], [START_REF] Feiz | Turbulence Statistics in a Fully Developed Rotating Pipe Flow[END_REF], and [START_REF] Ould-Rouiss | Numerical predictions of turbulent heat transfer for air flow in rotating pipe[END_REF]. The LES predictions [START_REF] Ould-Rouiss | Numerical simulation of turbulent pipe flow[END_REF] for N up to 18 confirm these trends: the axial velocity profile 1 n = reaches the laminar parabolic distribution when 18 N  . Note that the axial velocity distribution moves towards the laminar profile more rapidly than 1 n = when approaching the pipe center because the shear-thinning fluid is more viscous in this region, and thus, the laminarization of the fluid flow is more quickly. Fig. 9 is depicted the tangential mean velocity profiles scaled with the wall rotational velocity. The profiles show a concave shape close to a parabolic curve, and the concavity seems to decrease with increasing rotation rates N for both fluids. Similar trends were also observed in the experimental and numerical studies 1 n = by other authors [START_REF] Imao | Turbulent characteristics of the flow in an axially rotating pipe[END_REF], [START_REF] Reich | Fluid flow and heat transfer in an axially rotating pipe-I. Effect of rotation on turbulent pipe flow[END_REF], and [START_REF] Feiz | Turbulence Statistics in a Fully Developed Rotating Pipe Flow[END_REF].

Turbulence intensity:

Fig. 10 and Fig. 11 present the evaluations of RMS of velocity fluctuations for the Newtonian and shear-thinning fluids, respectively at 0 N = . The predicted RMS are in satisfactory agreement with the results of Rudman et al. [START_REF] Rudman | Turbulent pipe flow of shearthinning fluids[END_REF] for pseudoplastic fluid and with DNS results [START_REF] Ould-Rouiss | Numerical predictions of turbulent heat transfer for air flow in rotating pipe[END_REF] and [START_REF] Redjem-Saad | Direct numerical simulation of turbulent heat transfer in pipe flows: Effect of Prandtl number[END_REF] for Newtonian fluid. The slight deviations may be raised because of the difference in the Reynolds value or/and the different numerical methods.

The influence of rotation rate on the RMS of the fluctuating axial velocity z U  normalized by the friction velocity against the wall distance Y + is shown in Fig. 12 at various values N . In the non-rotating case, the maximum in the axial velocity fluctuations occurs at a distance 17

Y +  for 1 n = and 19 Y +  for 0.75 n = .
The peak reaches the value of 2.54 for 1 n = and 2.91 for 0.75 n =

. The peak value of 2.54 for the Newtonian fluid and its position 17 Y +  is in accordance with the results of literature [START_REF] Redjem-Saad | Direct numerical simulation of turbulent heat transfer in pipe flows: Effect of Prandtl number[END_REF] and [START_REF] Ould-Rouiss | Numerical predictions of turbulent heat transfer for air flow in rotating pipe[END_REF]. In the vicinity of the wall, the RMS of the streamwise velocity fluctuations is reduced with increasing rotation rate N in the buffer , see Fig. 7, it decreases when N raised beyond these regions. Thus, when N increases, the turbulent fluctuations are reduced near the wall on a widened zone for 0.75 n = .

It can be seen that the fluctuations of the axial velocity are larger than the azimuthal and radial ones, (which correspond to the peaks of the axial velocity fluctuations). However, beyond these positions, both RMS is increased with increasing N , when N ranges from 0 to 2, and reduced for 2 N  .

This trend also appears clearly on the turbulent kinetic energy profile, Fig. 15. This observation is similar to that reported for the Newtonian case by some authors [START_REF] Ould-Rouiss | Numerical predictions of turbulent heat transfer for air flow in rotating pipe[END_REF], [START_REF] Feiz | Turbulence Statistics in a Fully Developed Rotating Pipe Flow[END_REF], and [START_REF] Orlandi | Turbulent budgets in rotating pipes by DNS[END_REF], who showed, in their numerical investigations, that the turbulence is suppressed for high rotation rates. Note that the wall peaks of the axial velocity RMS, Fig. 12, and the turbulent kinetic energy, Fig. 15, are reduced until a flat region 2 N  is achieved. A similar outcome was found by Ould-Rouiss et al. [START_REF] Ould-Rouiss | Numerical predictions of turbulent heat transfer for air flow in rotating pipe[END_REF] and by Orlandi and Fatica [START_REF] Orlandi | Direct simulations of turbulent flow in a pipe rotating about its axis[END_REF] in their numerical simulations for the Newtonian fluid.

The axial-radial distribution of the Reynolds shear stress rz UU at 0 N = of the shear-thinning fluid agrees reasonably well with the DNS results [START_REF] Rudman | Turbulent pipe flow of shearthinning fluids[END_REF], Fig. 16. This stress is the only non-null stress in the stationary pipe. With increasing rotation rate, rz UU  it is reduced far from the wall, Fig. 17 , and it gets stronger and stronger with increase N . Further away from the wall, the normal-wall oscillations of z UU  becoming noticeable at high rotation rates, and the increased apparent fluid viscosity towards the core region generates larger streamwise vortices at high rotation rates.

Friction factor:

The friction factor of the pseudoplastic (n=0.75) fluid for the stationary pipe (N=0), is compared to different correlations of the literature for validation (see Tab.2), where DM f FSG f and HR f correspond, respectively, to Dodge and Metzner [START_REF] Dodge | Turbulent flow of non-newtonian systems[END_REF], Gomes [START_REF] Gomes | Hydraulic, Power Law models, calculation method using rational polynomial models[END_REF] and Hanks and Ricks [START_REF] Hanks | Transitional and Turbulent Pipeflow of Pseudoplastic Fluids[END_REF] The present LES friction factor is in satisfactory agreement with the literature results: it is within 3.8% and 6.5% , respectively, and 15.3% accuracy compared with DM f . The present friction factor is slightly overestimated in comparison to DM f . Rudman et al. [START_REF] Rudman | Turbulent pipe flow of shearthinning fluids[END_REF] reported a similar observation: their predicted friction factor is higher than Dodge and Metzner's correlation [START_REF] Dodge | Turbulent flow of non-newtonian systems[END_REF] by about 10-15% (Tab.2).

For the pseudoplastic fluid (n=0.75), table 1 shows a slight diminution of the friction factor when N it varies from 0 to 0.5 and augmentation when. 1 N  , with increasing N . For the pseudoplastic fluid, the augmentation of the rotation rate N induces a rise of the shear stress at the cylinder wall leading to an increase in the friction factor because the fluid becomes more viscous near the wall. As it approaches the cylinder center with increasing rotation pipe wall rate, the attenuation in the apparent fluid viscosity generates less large and less weak axial vortices, leading to less dissipation and increasing the wall drag.

Visualizations:

The effects of the rotation rate on the resolved axial and radial velocity fluctuations are depicted in Fig. 20 and Fig. 21, respectively, for pseudoplastic and at the lowest (N=0) and highest (N=3) rotation rates.

Contours of the resolved axial velocity in r  -a plane at the stationary pipe, Fig. 20a, show that the turbulence activity is attenuated in the core region of the pipe where the shear rate is lower (and thus the viscosity higher) than that near the pipe wall. On the contrary, due to the higher shear rate near the wall, the turbulent structures are essentially located close to the pipe wall. In the rotating case, the shear rate is slightly increased (beyond 5 Y +  ) towards the pipe center (see Fig. 2), inducing an augmentation of the turbulent structures in the core region, Fig. 20b, giving a clear picture that the pipe's rotation is responsible for the transport of the fluctuations from the wall towards the core region. A similar picture is given by the contour lines of the radial velocity fluctuations when the rotation rate increases, Fig. 21a, Fig. 21b. These figures also show that for 3 N = (Fig. 21b), the number of contour levels r U  is more than that for 0 N = (Fig. 21a), which explains the increase of the peak of the RMS of r U  in Fig. 14 when the rotation rate varies from 0 to 3. Indeed, in the core region, the viscosity of the shear-thinning fluid is slightly decreased (see Fig. 3) with an increased rotation rate (while the shear rate is slightly increased), inducing a rise of the turbulent structures in the core region and hence an augmentation in radial momentum transfer: the turbulent structures are shorter and stronger and bring more high-speed fluid from the wall region to the core region, reducing the production of turbulent energy near the wall and an increase in the core region (see Fig. 15).

Conclusions:

In the present work, a large eddy simulation is performed to investigate the turbulent flow of pseudoplastic fluid in an axially rotating cylinder by using an extended Smagorinsky model. Focus is made on the influence of the rotation rate on the mean flow and turbulence statistics. Results indicated a significant dependence of the dynamic field on this parameter. Some of the results are pointed out below:

The centrifugal force induced by the swirl flow induces a marked increase in the shear rate of the pseudoplastic fluid far away from the vicinity of the pipe wall, resulting in a pronounced reduction in the apparent viscosity of the shear-thinning fluid in the logarithmic region, where this trend is more noticeable as the rotation rate increases. It should be noted that the variation in the apparent viscosity leads to a redistribution of the axial velocity profile along the pipe radius. The decreased apparent viscosity of the pseudoplastic fluid in the logarithmic layer with an increasing rotating rate results in a gradual increase in the axial velocity profile far away from the pipe wall towards the core region as the pipe wall rotates.

On the other hand, the swirl driven by the rotating pipe wall results in an apparent attenuation in the generation and the transport mechanism of turbulence intensities of the axial velocity fluctuation from the wall vicinity towards the core region for the pseudoplastic fluid. This reduction in the RMS of the axial velocity fluctuations leads to a decrease in the kinetic energy of turbulent fluctuations and, consequently, in the turbulent Reynolds shear stress of the axial-radial velocity fluctuations as the pipe wall rotates. In turn, the turbulence intensities of the radial and tangential velocity fluctuations exhibit a pronounced increase when the pipe wall is rotating, where this trend is more evident as the rotation rate increases. In other words, the centrifugal force induced by the swirl flow results in a noticeable amelioration in the transport mechanism of the energy fluctuations from the axial one to the other as the pipe wall rotates. Furthermore, the friction factor for the pseudoplastic fluid exhibits a reduction when the rotation rate varies from 0 to 0.5 and increase when 1 N  with an increasing rotation rate.

Further studies are needed to study the influence of other rotation rates, flow behavior index ranges, and other effective parameters such as the Reynolds and Prandtl numbers. Moreover, further research could also be conducted to determine the effect of the rotation rate on another model of Non-Newtonian fluids.

  in wall units Y + is also based on the viscosity of the fluid at the wall, it thus follows that

  present subsection seeks to reveal the effect of the rotating pipe wall on the different fluid rheological characteristics to describe the rheological behavior of the pseudoplastic fluid. The main emerged rheological properties, such as the apparent fluid viscosity and a shear rate of the shear-thinning, have been critically analysand and discussed in the following paragraphs.The normalized shear rate  and apparent fluid viscosity  distributions of the shear-thinning (n=0.75) and Newtonian (n=1) fluids along the pipe radius (R) are depicted in Fig.2and Fig.3, respectively, against the distance from the wall-in-wall units Y + at a simulation Reynolds number of 4000 and over a rota-tion rate range of ( )03 N .It can be seen from Fig.2that the shear rate of the shear-thinning and Newtoni- an fluids is nearly linear and remains constant along the viscous sublayer ( ) 05 Y +  for all rotation rates. The effect of the flow behavior index on the shear rate distributions is significant in the vicinity of the wall, where the shear-thinning profiles lie above those of the Newtonian ones along the near-wall region for all rotation rates. It can be said that the decreased flow behavior index results in a marked increase in the shear rate distributions over the pipe radius. On the other hand, the rotating pipe wall significantly affects the shear rate distributions of both fluids in the viscous sublayer; the increased rotation rate (N) induces a pronounced attenuation in the shear rate for the Newtonian fluid.

Fig. 4

 4 Fig.4 presents the distribution of apparent fluid viscosity normalized by viscosity at the wall ( ) w

Fig. 12 .

 12 Beyond the log region, the RMS of the axial velocity is increased with increasing rotation rate. This is because the axial velocity z U + diminishes with increasing rotation rate N for the Newtonian fluid in the buffer region(5 35) Y + and the shear-thinning fluid in the buffer and log regions

Fig. 13 ,

 13 Fig.13, Fig.14. It is worth noting that both azimuthal and radial turbulence intensities increase with increasing rotation rate in the vicinity of the wall, the azimuthal up to 10 Y +  for 1 n = and 13 Y +  for 0.75 n = (which

  , and its distribution is moved to the pipe wall for both fluids, while the stress r UU  is increased, Fig.18, and the stress z UU   shows an oscillatory behavior, Fig.19. Near the wall, the peak of z UU  is related to the tilting of the near- wall vortical structures that increase the amount of correlation between z U  andU   by [8], Fig.19. Near the rotating wall, higher values of z UU   the shear-thinning fluid than those for the Newtonian one, especially at 3 N = (in comparison to 1 N = ), suggest that the amount of correlation between z U  and U   is stronger for 0.75 n =

Fig. 1 :Fig. 2 :Fig. 3 :Fig. 4 :Fig. 5 :Fig. 6 :Fig. 7 :Fig. 8 :

 12345678 Fig. 1: Geometry of problem

Fig. 9 :Fig. 10 :Fig. 11 :Fig. 12 :Fig. 13 :Fig. 14 :Fig. 15 :

 9101112131415 Fig. 9: Tangential velocity distribution

Fig. 16 :Fig. 17 :

 1617 Fig. 16: Validation of Reynolds shear stress

Fig. 18 :Fig. 19 :

 1819 Fig. 18: Reynolds shear stress of tangential and normal-wall velocity fluctuations

Fig. 20 :Fig. 21 :Fig

 2021 Fig. 20: Contour lines of the axial velocity fluctuations

  correlations (Tab.2). , being respectively the LES and DNS predictions by Rudman et al.[START_REF] Rudman | Turbulent pipe flow of shearthinning fluids[END_REF] and Gnambode et al.[START_REF] Gnambode | Large-Eddy simulation of turbulent pipe flow of power-law fluids[END_REF].

	f	DM	=	/ a Re MR	b	where	0.0665 an =+ 0.01175	b	=	0.365	-	0.177	n	+	0.062	n	2
									f	FSG	=	0.110	0.616 n Re -0.287 MR
							f	HR	=	0.0682	n	0.5 -	/	( 1/ 1.87 2.39 MR Re +	n	)
	PG f and R f

Table 1 :

 1 parameters of present LES simulations and mean flow quantities for

	n =	0.75	and Re 4000 s =

Table 2 :

 2 Friction factor at Re 4000 s =

	f	.10	3	f	DM	.10	3	f	FSG	.10	3	f	HR	.10	3	f	PG	.10	3	f	R	.10	3
	9.20	7. 79			8.05			7.75			8.85			8.6

Conflict of interest:

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Hydronautics, vol. 9, no. 1, pp. 39-44, Jan. 1975.

NOMENCLATURE


domain: phys spi

text:
Riveiro Moreno
email: carmen.riveiromoreno@onera.fr

M Couliou
email: marie.couliou@onera.fr

N Fabbiane
email: nicolo.fabbiane@onera.fr

O Marquet

R Bur

AERO2023-86-RIVEIRO MORENO Interaction of shock-waves with a compliant wall

The Shock/Boundary Layer Interaction (SBLI) over a rigid plate with an upstream 1.35 Mach number is characterized experimentally by unsteady pressure measurements and schlieren visualizations. Spectral Proper Orthogonal Decomposition (SPOD) is used on the schlieren videos to characterize the flow. The shock displacement spectrum is then used to chose a compliant wall. The interaction of the SBLI the compliant wall is analysed with schlieren imagery. The interaction consists on a lock-in of the flow at the structure's vibration frequency.

INTRODUCTION

Shock-waves are inherent to transonic, supersonic and hypersonic regimes, and thus are present in numerous locations in high speed vehicles: aircraft wings, air engine inlets, and control surfaces to name a few. In the presence of sufficiently strong shock-waves, the adverse pressure gradient causes boundary layer separation, prompting an unsteady flow, characterised by low-frequency broadband shock oscillation, two orders of magnitude lower than that of the small-scale structures present in the incoming boundary layer. The resulting unsteady flow can give rise to transonic buffet on wings, buzz on supersonic engine air intakes, and unsteady pressure and thermal loads on the structure among others [START_REF] Clemens | Lowfrequency unsteadiness of shock wave/turbulent boundary layer interactions[END_REF].

Another consequence of Shock/Boundary Layer Interaction (SBLI) dominated flows can be structural fatigue failure. Due to lightweight constrains, high speed vehicles are characterized by the use of thin flexible panels that become compliant under the presence of the flow, that is, they passively deform due to the fluid structure interaction (FSI). In the presence of Shock/Boundary In-teraction (SBLI), panels dynamically respond to low frequency pressure fluctuations which can lead to failure due to high-cycle fatigue when the flow couples with the panel's resonance modes. Even if the mechanism behind the SBLI's low frequency fluctuations is not yet fully understood, a significant amount of research has focused on the interaction of flexible thin plates with shock-waves in the last decade.

Experimental campaigns have characterised the SBLI in the presence of an impinging shock [START_REF] Brouwer | Experiments on a Thin Panel Excited by Turbulent Flow and Shock/Boundary-Layer Interactions[END_REF][START_REF] Tripathi | Response of a compliant panel to shock boundary layer interaction at mach 2[END_REF][START_REF] Varigonda | Investigations of fsi generated by an impinging sbli on a thin panel using multivariate imaging of flow/structural properties[END_REF] as well as compression ramps [START_REF] Eitner | Effect of Shock-Wave Boundary Layer Interaction on Vibratory Response of Compliant Panel[END_REF][START_REF] Ahn | Experimental Investigation of Flow-Structure Interaction of a Compliant Panel under Mach 2 Compression-Ramp Interaction[END_REF]. In these studies, the dynamics of the flow and the compliant wall were measured using non intrusive techniques such as Digital Image Correlation (DIC), schlieren visualizations, Particle Image Velocimetry (PIV) or Pressure Sensitive Paint (PSP). They investigated the influence on the interaction of shock impingement position and the cavity pressure under the plate [START_REF] Tripathi | Response of a compliant panel to shock boundary layer interaction at mach 2[END_REF], thickness of the panel [START_REF] Eitner | Effect of Shock-Wave Boundary Layer Interaction on Vibratory Response of Compliant Panel[END_REF], thermal loads and shock strength [START_REF] Brouwer | Experiments on a Thin Panel Excited by Turbulent Flow and Shock/Boundary-Layer Interactions[END_REF] among others. Works from [START_REF] Eitner | Effect of Shock-Wave Boundary Layer Interaction on Vibratory Response of Compliant Panel[END_REF][START_REF] Ahn | Experimental Investigation of Flow-Structure Interaction of a Compliant Panel under Mach 2 Compression-Ramp Interaction[END_REF][START_REF] Musta | Investigation of flow-structure coupling for a compliant panel under a shock/boundary-layer interaction using fastresponse PSP[END_REF][START_REF] Varigonda | Investigations of fsi generated by an impinging sbli on a thin panel using multivariate imaging of flow/structural properties[END_REF] highlighted the presence of fluid-structure coupling by showing the presence of natural frequency peaks of the compliant wall on the flow spectrum. In their experiment. Brouwer et al. [START_REF] Brouwer | Experiments on a Thin Panel Excited by Turbulent Flow and Shock/Boundary-Layer Interactions[END_REF] found limit cycle oscillations of the thin plate under the influence of a weak shock.

In transonic flows, investigations on the interaction of a normal shock with a thin metallic plate [START_REF] Gramola | Experimental FSI study of adaptive shock control bumps[END_REF][START_REF] Jinks | Wind tunnel experiments with flexible plates in transonic flow[END_REF][START_REF] Gramola | Offdesign performance of 2D adaptive shock control bumps[END_REF] explored the use of panel deformation to recreate the effect of Shock Control Bumps (SCB) [START_REF] Bruce | Unsteady shock wave dynamics[END_REF]. Even in a bistable behaviour of the shock was observed at large plate deformations [START_REF] Gramola | Experimental FSI study of adaptive shock control bumps[END_REF], the dynamic coupling mechanism is not explained.

The aim of this study is to shed new light in the fluid structure interaction with the presence of a normal shockwave (transonic regime). For this purpose, the present work focuses on experimentally characterizing the interaction between a compliant wall and a normal unsteady shock with a 1.35 upstream Mach number. Firstly, the interaction of the shock-wave with the incoming turbulent boundary layer on the rigid wall is analysed by unsteady pressure measurements and high speed schlieren imagery. Knowing the dynamics of the flow, a compliant wall is designed so that its natural frequency is of the same order of magnitude as the shock oscillations. Lastly, the fluid structure interaction of the oscillating shock and the compliant wall is studied by high speed schlieren imagery.

WIND TUNNEL FACILITY

Experiments are conducted in S8Ch ONERA's transonic open circuit wind tunnel in Meudon. This wind tunnel, has been largely used for experimental studies on passive and active (fluidic) control of SBLI [START_REF] Bur | Study of passive control in a transonic shock wave/boundary-layer interaction[END_REF]. The stagnation temperature in the settling chamber is 300 ± 10 K and the stagnation pressure 96.0 ± 0.3 kPa.

For this study, S8Ch is equipped with a half nozzle that generates an unsteady normal shock-wave with a nominal upstream Mach number of 1.4. The flow is chocked by means of a second throat located at the outlet which isolates the test section from downstream perturbations. By regulating the cross section of the second throat, it is possible to adjust the position of the shock in the channel. The test section's span is 120 mm, its height 100 mm and the end of the nozzle is located at 369 mm from the end of the converging part of the channel. All throughout the experimental campaign, the shock was located at 370 mm from the end of the converging part of the channel, which will be considered as location x = 0 as shown in Fig. 1. At this location, the Mach number upstream the shock is equal to 1.35.

SHOCK-WAVES OSCILLATION

Pressure wall measurements

Pressure is measured on the centerline of the bottom wall using 30 static pressure sensors and 3 unsteady pressure sensors as illustrated in Fig. 1. The three unsteady pressure sensors are Kulite transducers type XCQ-093-15A which sampled pressure fluctuations at 100kHz for 60 s with a high pass filter at 10kHz. Fig. 2 shows the pressure ratio P/P i0 along the lower wall of the test section, where the red dots indicated the time-averaged measurements made with the unsteady pressure sensors and the blue dots correspond to the static pressure measurements. This pressure distribution reveals the high adverse pressure gradient the normal shock imposes on the flow. At the location of the shock (x = 0 mm), no pressure plateau is visible meaning that the shock does not induce a massive separation or recirculating flow.

To confirm that purpose, oil visualizations were conducted, and it was found, that there exists only a small separation bubble around the centerline of the test section's lower wall. The existence of this incipient separation is consistent with the notion that the onset of shockinduced separation that is expected for Mach numbers ranging from 1.3 to 1.35 [START_REF] Babinsky | Shock Wave-Boundary-Layer Interactions[END_REF].

From Fig. 1, it can be seen that solely the center unsteady pressure sensor named K2 is located at the rear leg of the shock foot. The other two pressure sensors K1, K3 are located further upstream and downstream (respectively 50 mm upstream and downstream K2 location.).

In order to capture the pressure dynamics along the shock foot and add locations of dynamic pressure measurements, the shock is displaced 21.4 mm downstream and then 11.4 mm upstream, which is equivalent to add two pressure measurement locations named respectably K2' and K2". As the Mach number upstream of the shock is only sightly modified, the physics of the flow remains unchanged, especially the boundary layer separation region. As a result, one has access to the unsteady pressure measurement at 5 positions as visible in Fig. 3. The first unsteady sensor, K1 is located in the incoming boundary layer, K2' is under the shock foot, K2 within the separation region, K2" is at reattachment and K3 is further downstream of the interaction region. Fig. 3 displays the premultiplied spectra for the unsteady pressure measurements at these five locations. The spectra were obtained through the Welch method using 24576 samples per block, and a Hanning windowing with a 50% of overlap. Low frequency oscillations, below 20 Hz, were found to be the result of fluctuations of the stagnation pressure in the settling chamber. Consequently, the spectra below 20 Hz is cut out in Fig. 3.

The premultiplied spectrum of the pressure at the location K1 (Fig. 3a) shows a high frequency bump with its maximum around 8kHz. We associate it to the high frequency pressure fluctuations caused by the small structures contained in the incoming boundary layer. Note that the amplitude of the oscillations is two orders of magnitude lower than the other spectra that will be presented.

Under the first leg of the shock foot, the premultiplied spectrum (K2', Fig. 3b) shows a low frequency bump centered around 200 Hz. This spectrum is representative of the shock oscillations. The lack of high frequencies on the spectrum as the one present on the incoming boundary layer (Fig. 3a), suggests that the shock behaves as a low pass filter with a cut off frequency around 200 Hz. Additionally, it can be seen that amplitude of the shock oscillations are the highest of all spectra presented in Fig. 3.

Within the separation region, the premultiplied spectrum of K2 (Fig. 3c), shows a broad low frequency bump accompanied by a high frequency bump located around 6kHz. As the flow undergoes the extreme pressure gradient because of the front leg of the shock, the flow detaches generating a recirculating zone with a mixing layer developing on top of it. The appearance of this high frequency bump in Fig. 3c can be linked to the footprint of the small structures of the mixing layer [START_REF] Clemens | Lowfrequency unsteadiness of shock wave/turbulent boundary layer interactions[END_REF]. On the other hand, the low frequency content can be attributed to the motion of the shock. Note that the energy of the low frequency oscillations is lower than the one obtained under the shock foot (K2', Fig. 3b). These results, suggest that in the recirculating region, where K2 is placed, both the influence of the shock oscillations as well as that of the mixing layer coexist [START_REF] Sartor | Unsteadiness in transonic shock-wave/boundary-layer interactions: experimental investigation and global stability analysis[END_REF].

At the reattachment region, the premultiplied spectrum, represented by K2" in Fig. 3d, consists in a high frequency bump centered at 6kHz. The high frequency bump is the same as the one found in the spectrum of K2 (Fig. 3c), but with higher amplitude. As the bound-ary layer reattaches, the structures in the mixing layer are brought closer to the wall, which increases the pressure fluctuations felt at the wall. This could account for the increase in amplitude of the high frequency bump observed in Fig. 3d. At the location K2", the amplitude of the low frequency oscillations has decreased with respect to the K2, indicating that there is little influence of the shock oscillations at the reattachment point.

Further downstream of the interaction, the premultiplied spectrum obtained from K3 (Fig. 3e), displays a high frequency bump located at 6kHz. This is the same high frequency bump as in K2 and K2" (Fig. 3c-d). Therefore, the bump is still related to the structures in the mixing layer, which can be felt downstream the interaction. It can be noted that its amplitude has increased with respect to K2". The reason behind comes from the spatial development of the structures shed in the mixing layer as they move downstream. As they become larger, so will the pressure fluctuation associated with it. Hence, the reason for the increase in amplitude of the high frequency bump in Fig. 3e. Same trends have been found in other experimental studies in transonic flow [START_REF] Sartor | Unsteadiness in transonic shock-wave/boundary-layer interactions: experimental investigation and global stability analysis[END_REF], and even in different configurations such as shocks generated by a compression ramp [START_REF] Erengil | Unsteady wave structure near separation in a Mach 5 compression rampinteraction[END_REF] or shock generators [START_REF] Dupont | Space and time organisation of a shock wave/turbulent boundary layer interaction[END_REF].

High speed Schlieren visualizations

Schlieren visualizations are conducted with a Z-type Schlieren setup with a high speed Phantom camera and a blue LED source. A vertical knife orientation was used, which yields images whose light intensity is proportional to horizontal density gradients.

The schlieren image in Fig. 4 presents an instantaneous image of the flow in a zoom region around the shock. In it, it is possible to observe a normal shock with its λ shaped shock foot, which hints the presence of shock induced separation. Note that the apparent thickness of the shock is due to a spanwise integration of light. The horizontal density gradients reveal the shock as well as the shocklets downstream the shock foot, which appear in white, indicating a positive density gradient. The presence of shocklets indicates the existence of secondary supersonic regions downstream of the lambda region [START_REF] Babinsky | Shock Wave-Boundary-Layer Interactions[END_REF]. The alternating areas of positive and negative gradients downstream of the shock foot correspond to structures developing in the mixing layer. These structures seem to generate noise that can be seen in the shape of vertical lines right above them, downstream the shocklets.

Spectram Proper Orthogonal Decomposition (SPOD) [START_REF] Towne | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis[END_REF] is computed on the schlieren images with the purpose of finding oscillating coherent structures in the flow, and link them to the findings obtained with the unsteady pressure sensors.

Due to camera memory limitations, it was not possible to obtain a long video of the whole flow field sampled at a high frequency, so that both the low and high frequency content of the flow, already displayed in Fig. 3, could be captured. Consequently, two different set of images were acquired. The first set was framed at the shock (Fig. 4, blue frame), which has associated low frequencies as seen from the spectrum under the shock foot (Fig. 3b). These videos were acquired at a sampling frequency of 7kHz for a duration of 16.6 s. Another set of images, framed at the shock foot (Fig. 4, red frame), where higher frequency content is expected as seen in the spectra at the recirculation region and downstream of it (Fig. 3), were sampled at 20kHz for 5.8 s. Both set of images have a magnification ratio of 445 pixel to 1 mm, and the image size are of 160x448 pixels in the shock framed images and 480x152 in the shock foot framed images.

In the SPOD calculation, 112 blocks of 2048 samples with a 50% of overlap were used for the shock framed videos, and 55 blocks of 4096 samples with a 50% of overlap for the shock foot framed one. In this manner, the frequency resolution for both calculations is similar, being 3.41 Hz and 4.88 Hz respectively, while ensure a sufficient amount of blocks for convergence of the SPOD computation.

The black line in Fig. 5a presents the premultiplied spectrum of the first eigenvalue for the image data set framed at the shock (Fig. 4, blue frame). Being the first eigenvalue the most energetic, it is therefore the most representative of the flow dynamics. The spectra below 20 Hz is cut out not to display the low frequency oscillations, which are due to fluctuations of the stagnation pressure in the settling chamber. The spectrum for this set of images consists in a low frequency bump.

For a given frequency, the complex eigenvector of the first eigenvalue (from here onwards referred as the first mode) indicates which areas in the flow move coherently with the most energy. So as to analyse which areas of the flow oscillate at low frequency, the first mode at frequencies within the low frequency bump were extracted and When compared to the schlieren image (Fig. 4), it can be inferred that the real and imaginary parts of the mode are located on the shock and to a lesser extend on the shocklets downstream of the shock foot. This implies that the shock presents a pure oscillation. It moves as a rigid structure around an average position. Consequently, it confirms the notion that the pressure spectrum at the shock foot (position K2', Fig. 3b) is representative of the shock motion.

The two areas at the shock on the real and imaginary parts correspond to the most upstream and downstream position of the shock during its oscillation (Fig. 5b). Same patterns and conclusion can be made for the shocklets. Their motion consist on an upstream-downstream translation, synchronized with that of the shock.

Same results were obtained at different frequencies within the bump. Hence, these results confirms the notion that the shock oscillations are broadband and low frequency, as presented in the results of the pressure spectrum (Fig. 5a, blue line). Nonetheless, the low frequency oscillations on the pressure spectrum are broader with respect to the ones obtained with SPOD. The differences with respect to the unsteady pressure sensor are due to the different nature of both measurement techniques.

The premultiplied spectrum of the first eigenvalue for the data set focused at the shock foot (Fig. 4, red frame), is presented in Fig. 6 (black line) along side the pressure spectrum at the reattachment (position K2", Fig. 3d) plotted in red for comparison purposes. For this type of image, spectrum displays a low frequency bump as well as high frequency peaks

At frequencies within the low frequency bump, the real and imaginary parts of the mode show a pure oscillation of the shock foot and at the shocklets, similarly to the real and imaginary parts displayed in Fig. 5b. Thus, it is not presented for brevity reasons. On the higher frequency part of the spectrum, it is worth noticing the existence of three distinctive peaks at frequencies 1235 Hz (Fig. 6, vertical dashed blue line), 1665 Hz and 1977 Hz on the spectrum. The real and imaginary parts of the first mode at frequency 1235 Hz (Fig. 7a) are located mainly downstream the shock, above the boundary layer. Within this region, the real and imaginary parts are not coincident. This suggests the presence of upstream traveling waves, which could be linked to upstream acoustic propagation. Traveling waves above the mixing layer are also found on the modes at the other two peaks (1665 Hz and 1977 Hz) as well as the two consecutive bumps below 4kHz. These peaks and bumps are not present on the pressure spectra obtained within the reattachment area (position K2", Fig. 3d) nor in locations further downstream (position K3, Fig. 3e). Being the sensors under the mixing layer and the upstream traveling waves located above it, the pressure sensors do not detect the acoustic waves, or at least, their influence at the boundary layer is not strong enough to overcome the pressure fluctuations coming from the mixing layer.

On the real and imaginary parts at frequency 1235 Hz (Fig. 7a) it is also possible to observe part of the shock foot. As the flow downstream the shock is subsonic, it is possible for downstream pressure perturbations to propagate upstream and interact with the shock, affecting its position and strength. Numerous studies have been conducted on the influence of downstream perturbation generated by a rotating shaft, that generates quasi-sinusoidal pressure signals, on the dynamics of the shock in transonic flow [START_REF] Edwards | An experimental study of the interaction of an unsteady shock with a turbulent boundary layer at Mach numbers of 1.3 and 1.5[END_REF][START_REF] Bur | Experimental and numerical study of forced shock-wave oscillations in a transonic channel[END_REF][START_REF] Bruce | Unsteady shock wave dynamics[END_REF]. All coincide that, as the frequency of the shaft was increased, keeping pressure oscillation constant, the amplitude of the shock displacement decreased. Therefore, even if on the first mode's real and imaginary parts there is an area concerning the shock, its oscillation amplitude is smaller than for lower frequencies. This explains why high frequency peaks related to acoustic perturbations are not present in the unsteady pressure spectrum obtained at the shock foot (Fig. 5a, blue). Additionally, as the location of the upstream traveling waves are cut out of the image frame, these peaks are not present on the SPOD results with the image data set focused on the shock (Fig. 5a, black). At higher frequencies, above 4KHz, as illustrated by the first mode extracted at frequency 7749 Hz (Fig. 6, vertical dashed red line), the real and imaginary parts are lo-cated downstream, in the region coinciding with the mixing layer and on the flow immediately above (Fig. 7b). As the real and imaginary parts are not coincident, it suggests that there is a traveling motion of structures contained in the mixing layer. It seems therefore, that there is a propagation of vortical structures that spatially develop as they move downstream. Judging by their vortical shape, at this level of frequency, these structures could be the result of a Kelvin-Helmholz instability.

These results are consistent with the pressure spectra obtained within the separation region and further downstream i.e from K2 to K3 (Fig. 3c to Fig. 3e). The high frequency bump present in these spectra, related to the mixing layer, is of the same order of magnitude as the frequency range where these vortical structures in the mixing layer are present on the first mode. Furthermore, the increase in size of the vortical structures as the move downstream is coherent with the increment on the high frequency amplitude observed from K2' to K3 (Fig. 3, red, green).

The coherent vertical lines above the vortices presented in the real and imaginary parts of the mode (Fig. 7b) are probably related to noise radiation from the mixing layer. The interaction of the shock with the incoming boundary layer developing on a rigid wall is characterized by several flow phenomena at different time scales, as sum-marized in Tab. 1. The frequencies characterizing the shock oscillation are lower than those existing in the incoming boundary layer or in the shear layer downstream the shock.

INTERACTION OF A SHOCK WAVE WITH A COMPLIANT WALL

We are now interested in analysing the interaction of the shock with a compliant wall. With that aim, the latter will be designed so that its vibration modes of lowest frequency lie in the range of frequencies where the shock oscillates, i.e. between 70 Hz and 300 Hz as shown in Tab. 1.

Description of the compliant wall

The compliant wall is made of a polyurethane block that is inserted in an aluminium container. The elastic properties of the polyurethane were characterized with a rehometer giving a Young modulus equal to 3 MPa. This material is clamped to the four lateral surfaces of the container but is free to move at the bottom and top surfaces, the latter being the interface with the flow. The compliant wall is 160 mm long, 85 mm wide, and 20 mm thick.

As shown later, this thickness allows the lowest natural vibration modes of the compliant wall to be between 100 -200 Hz, i.e. in the range of frequency characterizing the shock oscillation. As shown in the picture of Fig. 8, the container is placed on the lower wall of the test section. Its mid point coincides with the location of the shock, denoted as x = 0 in Fig. 1. In the experiment described below, the shock will be therefore located at the center of the compliant wall, as shown in the sketch of Fig. 8. Finally, it should be noted that, under the compliant wall, the pressure is not equal to the atmospheric pressure, but rather corresponds to an average value of the pressure upstream and downstream the shock, due to leaks (mechanical joints, gaps, ... ) in the test section of the wind tunnel.

Experimental analysis of the shockwave/compliant-wall interaction

The schlieren images are first used to study the interaction of the compliant wall with the shock wave. They are sampled at 7 kHz for a duration of 3.87 s. Their width is here increased to view the entire length of the compliant wall. Moreover, a vertical knife orientation is used so that only the horizontal density gradients are retrieved. Fig. 9 displays a series of instantaneous schlieren visualizations of the interaction of the shock with the compliant wall (t n , n = [START_REF] Ahn | Experimental Investigation of Flow-Structure Interaction of a Compliant Panel under Mach 2 Compression-Ramp Interaction[END_REF][START_REF] Bur | Experimental and numerical study of forced shock-wave oscillations in a transonic channel[END_REF]), as well as a snapshot of the rigid wall for comparison purposes. On the interaction with the compliant wall, we clearly observe an oscillation of the shock position as well as a large displacement of the wall. An upwards displacement of the later is represented by a dark zone in the schlieren image due to the shadow projected by the wall onto the camera plane. On the other hand, the downward motion of the wall is not visible due to the nature of the setup. The first schlieren image (Fig. 9, t 1 ) displays an upward deformation of the wall between x = -80 mm and x = 0 mm with a maximum located around x = -40 mm. At this instant, the shock is located at x = 0 mm and it is non symmetric with a large λ shape, indicating the existence of a large separation region. Note that the shock foot shape on the rigid wall case is smaller. The instant t 2 shows that the upward deformation of the wall has decreased and the shock has undergone an upstream motion. It can also be noted that the shock foot has reduced in size, which suggests a smaller recirculating bubble underneath the shock foot. The schlieren image at t 3 presents no visible upward deformation at the wall and a flow similar to the snapshot at t 2 . Later at t 4 , while upstream the shock there is no visible displacement of the compliant wall, an upward deformation can be observed from 0 to 80 mm, with a maximum around 40 mm. The flow displayed in the schlieren image at t 4 presents a more symmetric shock with a smaller λ region and small structures being shed downstream. The topology of the shock at this instant resembles that found in the rigid case. At t 5 , no deformation is observed at the wall indicating that the deformation from x = 0 to x = 80 mm has decreased. The shock is still around the same position as on the snapshots from t 2 to t 4 and the flow resembles that at instant t 4 . On the last schlieren image, at t 6 , an upward deformation of the compliant wall is visible between x = -80 mm and x = 0 mm. This upward motion is accompanied by an downstream movement of the shock, which at t 6 is located at x = 0 mm. For the time being, at t 6 , the shock still displays a similar λ region as shown at t 5 . The results here presented suggest that there is a strong link between the motion of the compliant wall and the dynamics of the flow.

In order to better characterize that interaction, the frequency spectra of the flow and wall displacement are now analyzed and compared. The frequency content of the flow is first retrieved by performing a SPOD analysis on a set of schlieren images that exclude the wall. For this computation, 25 blocks of 2048 samples per block with 50% overlap are used. The blue line in Fig. 10a corresponds to the first eigenvalue of the SPOD analysis, premultiplied by the frequency. A dominant peak is clearly visible at f = 153 Hz and the other peaks of smaller magnitude corresponds to superharmonics, i.e. for frequencies 2 f , 3 f , . . . . The real and imaginary parts of the mode at f = 153 Hz, shown in Fig. 10b, both exhibit a change of sign upstream and downstream of the shock as well as at the shock foot. The opposite signs between the real and imaginary parts indicate that this mode is a standing wave creating an oscillation of the shock position after superposition with the time-averaged schlieren image. Note that the weak oblique shock, visible in the imaginary part of the mode as well as in the instantaneous images of Fig. 9, is due to a slight discontinuity of the rigid and compliant wall. It was also observed in other cases (not reported here) where the compliant wall does not display such large amplitude displacement so it has no major impact on the flow dynamics. From the schlieren SPOD analysis, we can finally conclude that the compliant wall strongly modifies the shock-wave dynamics since it oscillates periodically while it is characterized by a broadband frequency spectrum in the rigid wall case (see Fig. 5a).

The oscillation of the compliant wall is now analyzed by extracting from the schlieren images the y position of the red point in Fig. 9 that is located at x = -40, where the maximum upward deformation of the wall occurs. The power spectrum density of the extracted temporal signal is computed with a Welch method using 2048 samples per block with 50%overlap. The resulting premultiplied spectrum is shown with red lines in Fig. 10a. The largest amplitude peak is also found at 153 Hz, the same frequency as for the shock oscillation. This suggests that the shock-motion lock-in at the vibration frequency of the compliant wall.

To confirm the lock-in phenomenom, we determine the natural frequencies of the compliant wall by performing a finite-element modal analysis. The compliant structure, clamped at the four lateral boundaries and free to move at the bottom and top surfaces, is modeled with the threedimensional linear elasticity equations. The vibration modes of lowest frequency, obtained by solving an eigenvalue problem, are shown in Fig. 11. The first, second and third modes oscillate at the natural frequencies 143 Hz, 182 Hz and 245 Hz, respectively. They display no spatial oscillation in the transverse direction (z) but have zero-node, one-nodes and two-nodes of oscillations in the streamwise direction (x). The frequency of the shock and compliant wall oscillations observed in the experiments is 153 Hz, so in between the first and second natural frequencies of the structure. Nonetheless, the displacement of the compliant wall observed in the experiments is closer to that of the second vibration mode. This is shown in Fig. 12, which displays the spatio-temporal evolution of the wall displacement at the centerline (z = 0) using a xt/T diagram. Fig. 12a corresponds to the second vibration mode (oscillating at 182 Hz) while the experimental results are shown in Fig. 12b.

The displacement of the second mode (Fig. 12a) exhibits an anti-symmetric pattern with respect to x = 0 that corresponds to the most upstream positions in the experiments as shown in Fig. 9 at t 1 . For a fixed time, an upward displacement (red) of the compliant wall for x < 0 is asso- ciated to an equal downward displacement for x > 0, and vice versa. For a fixed spatial position, let us say x = -40 where the largest amplitude is obtained, the wall displacement is alternatively positive and negative with identical extreme values. The wall displacements observed in the experiment (Fig. 12b) share some similarities with the pattern of the second mode. Upstream the shock position (x < 0), upwards displacement (red) are measured every half-period of oscillation. The downward displacement (blue) are not visible in the figure since they can not been measured from our schlieren visualization, but we may legitimately assume that white patterns correspond to negative displacements. Interestingly, downstream the shock x > 0, the upward displacement of the wall is of lower amplitude and for a shorter duration. This clearly indicates that, in the experiments, the compliant wall does not symmetrically oscillate around the y = 0 positions, but rather around a mean position that is below the y = 0 axis downstream the shock, and above that axis upstream the shock.

For a better picture of the fluid-structure interaction, the displacement signal at two points of the compliant wall, (Fig. 9, red and green dots) and at a point on the shock (Fig. 9, blue dot) are reported in Fig. 13. The points at the compliant wall correspond to the locations where the maximum displacement upstream and downstream are found, i.e. x = -43 mm and x = 34 mm. As all signals have been retrieved on the same set of images, they are synchronized and can be compared in time. As shown in Fig. 9, Fig. 13 illustrates an upward movement of the material upstream the shock coincides with a downstream translation of the shock and vice-versa. When the downstream part of the material presents an upward motion, the shock presents a small movement upstream, almost a plateau, as it is for the case between t 2 till t 5 . This is consistent with the observation made in Fig. 9, where the shock seemed to remained in the same positions from t 2 till t 5 . When looking at the signals, it is possible to see that the oscillation of the material presents a lag with respect to that of the shock. This implies that the most downstream position of the shock happens before the material has reached its maximum deformation. In order to asses the lag, the cross spectral density of both signals was computed. At the dominant frequency of the interaction, 153 Hz, the phase between both wall displacement signals and the shock signals is of 28 • .

CONCLUSION

The interaction of an unsteady normal shock with the incoming turbulent boundary layer has been experimentally characterised by unsteady pressure measurements and high speed schlieren visualizations. The pressure spectra showed that the shock oscillations ranged from 70 to 300 Hz, whereas oscillations related to the mixing layer ranged from 4 to 8 kHz. SPOD calculations were conducted on the schlieren videos. Low frequencies were related to modes associated to the displacement of the shock. High frequency associated modes displayed vortical structures shed downstream in the mixing layer. With SPOD it was also possible to find high frequency modes related to upstream traveling waves, which are thought to be of acoustic nature.

A compliant wall was chosen so that its natural frequencies are within the range of the shock oscillations. The fluid-structure interaction consists in a lock-in of the flow at the vibration frequency of the compliant wall. The dominant does not correspond to the natural frequencies of the unforced structure response. Nonetheless, the response of the compliant wall resembles the shape of the second mode. A further analysis of the response of the compliant wall to the forcing of the flow is needed. During the interaction, an upward deformation of the material coincides with an upstream displacement of the shock. A cross spectral analysis showed that the movement of the shock and the compliant wall are out of phase.

Figure 1 :

 1 Figure 1: Shock location in test section of S8Ch. In blue are the location of static pressure sensors. In red are the location of the three unsteady pressure sensors (K1, K2, K3)

Figure 2 :

 2 Figure 2: Pressure ratio profile along lower wall of the test section. Red dots correspond to the location of the unsteady pressure sensors. Blue dots correspond to the static pressure sensors.

Figure 3 :Figure 4 :

 34 Figure 3: Premultiplied spectra from unsteady pressure sensor at the position a) K1 = -50 mm (purple), b) K2' = -21.4 mm (blue), c) K2 = 0 mm (orange), d) K2" = 11.4 mm (red) and e) K3 = 50 mm (green). f) Sketch of the pressure sensors localisation superimposed with a Schlieren visualisation

Figure 5 :

 5 Figure 5: (a) Premultiplied spectra of first eigenvalue obtained with SPOD for frame centered at the shock (black) (Fig. 4, blue frame) and from the unsteady pressure sensor located at the position K2' under the shock foot (blue). Blue dashed line corresponds to frequency 112 Hz. (b) Real (left) and imaginary (right) parts of the mode for the first eigenvalue at a frequency of 112 Hz.

Figure 6 :

 6 Figure 6: Premultiplied spectrum of first eigenvalue obtained with SPOD for frame centered at the shock foot (black) (Fig. 4, red frame) and of the pressure sensor located at the position K2" in the reattachment region (red). Blue dashed line corresponds to frequency 1235 Hz. Red dashed line corresponds to frequency 7749 Hz

Figure 7 :

 7 Figure 7: SPOD's real (top) and imaginary (bottom) parts of the mode for the first eigenvalue at a frequency of a) 1235 Hz and b) 7749 Hz.

Figure 8 :

 8 Figure 8: View of the compliant wall on the lower wall of the test section (top). Sketch of a lateral view along the center line (white dashed line on top image) of the compliant wall with the location of the shock (bottom)

Figure 9 :

 9 Figure 9: Instantaneous schlieren visualizations during a oscillation cycle of the compliant wall. Red, green and blue dots illustrate the displacement of the compliant wall and the shock. Last schlieren image corresponds to the rigid wall case.

Figure 10 :

 10 Figure 10: Spectral analysis of the compliant wall configuration. a) Premultiplied spectrum of (blue) the first SPOD eigenvalue of the schlieren vizualisation and (red) the compliant wall displacement at streamise location x = -40. b) Real (left) and imaginary (right) parts of the first mode at frequency 153 Hz

Figure 11 :Figure 12 :

 1112 Figure 11: Mode shapes of the first natural frequencies obtained with a finite-element modal analysis of the compliant wall structure.

Figure 13 :

 13 Figure 13: Displacement signals of the compliant wall at x = -43 mm (red) and x = 34 mm (green) and of the shock (blue).

Table 1 :

 1 Summary of frequency content in the shock boundary layer interaction

		Incoming BL	Shock	Mix. layer
	Freq. (kHz)	[5, 10]	[0.07, 0.3]	[4, 8]
	(f*PSD) max	5 × 10 2	3.3 × 10 5	1.7 × 10 5

ACKNOWLEDGEMENTS

This work is co-funded by the Direction Générale de l'Armement (AID Convention N°2020 65 0022) and ONERA.