# Experimental investigation of Dean-vortices oscillation downstream of a 90° Bend

Bilal Ben Haroual, Julie Albagnac, Pierre Brancher, Sébastien Cazin, Didier Boldo, Emmanuel Thibert, Romain Mathis

► To cite this version:

Bilal Ben Haroual, Julie Albagnac, Pierre Brancher, Sébastien Cazin, Didier Boldo, et al. Experimental investigation of Dean-vortices oscillation downstream of a 90° Bend. Experimental Thermal and Fluid Science, 2025, 163, pp.111402. 10.1016/j.expthermflusci.2024.111402. hal-04967563

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Submitted on 26 Feb 2025

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# Experimental investigation of Dean-vortices oscillation downstream of a 90° Bend

Bilal Ben Haroual<sup>a,b</sup>, Julie Albagnac<sup>a</sup>, Pierre Brancher<sup>a</sup>, Sébastien Cazin<sup>a</sup>, Didier Boldo<sup>b</sup>, Emmanuel Thibert<sup>b</sup>, Romain Mathis<sup>a,b,*</sup>

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

Keywords:

Pipe flow

90° bend

Dean vortices

Swirl-switching

## ABSTRACT

The present work experimentally investigates the dynamics of a pipe flow downstream of a 90° bend in turbulent regime. The study is carried out on two different test benches, enabling to cover a decade in Reynolds numbers ($Re \in [1.2 \times 10^4, 5.4 \times 10^5]$). Laser-based metrology techniques are employed to capture the three velocity components in several flow sections, namely, cross-sections orthogonal to the main flow and vertical diametral planes (parallel to the main flow direction). Time-resolved and long-time decorrelated measurements allow the characterisation of both the dynamics and the statistics of the flow. These measurements highlight the behaviour of the fully three-dimensional flow generated downstream of the bend. In particular, an oscillation of the dipolar structure generated by the bend, known as the Dean vortices, is measured and analysed using a Lamb-Chaplygin analytical model. The dependency of the flow behaviour as a function of both the Reynolds number and the distance downstream of the bend and the return to axisymmetric flow are evaluated.

## 1. Introduction

Due to their presence in many natural and industrial systems, hydraulic networks, whether closed or open, have been the subject of attention since the 19th century [1–3, to cite a few]. Particularly, hydraulic systems are often used in industry either to transport the fluid they contain (e.g., the petroleum industry), or as cooling systems (e.g., to control temperature in nuclear power plants). The systematic presence of bends, often due to spatial constraints, in hydraulic networks generates additional head losses and coherent vortical structures. These bend generated secondary motions, known as Dean vortices [4–6], are composed of two counter-rotating vortices. The presence of these vortices can reduce process efficiency, result in fluid-structure interactions causing structural fatigue, or may affect the measurements of certain devices such as flowmeters. In addition, whatever the application, flow control and knowledge of flow behaviour as a function of the system parameters are important for optimising the operation of the hydraulic network, particularly for high Reynolds numbers. This is an important motivation for deepening our understanding of these structures.

In the particular case of a 90° bend, Tunstall and Harvey [7] observed that the Dean cells exhibit a swirling motion when the flow regime is turbulent. They initially conceptualised it as a swirling vortex causing the Dean cells to rotate when both the bend's curvature and the

Reynolds number exceed critical thresholds. Brücker's [8] subsequent work, using Particle Image Velocimetry (PIV), reported a different global motion, featuring an oscillatory motion around the pipe's axis of symmetry. This dynamic behaviour, known as swirl-switching, was corroborated by other researchers like Rütten et al. [9] (LES) and Kalpakli et al. [10] (time-resolved stereoscopic PIV).

The present experimental study is intended to complement those mentioned above in a number of respects. This study focuses on the influence of two parameters, namely the Reynolds number and the distance downstream of a 90° bend, on the initial three-dimensionalisation of the flow, its dynamics, and its subsequent evolution towards an axisymmetric flow. A specific emphasis is given to the characterisation of the secondary flow and the swirl-switching dynamic. To our knowledge, previous investigations of the swirl-switching phenomenon have relied exclusively on cross-sectional flow data. In contrast, the present work uses measurements from a vertical diametral plane for the swirl-switching study.

This paper is organised as follows. Section 2 describes the experimental set-up, the relevant parameters and the measurement techniques. The experimental results are presented and analysed in Section 3. After consideration of the mean flow in the cross-section and in the vertical diametral plane, a specific focus is made on the Dean

<footnote><sup>a</sup> Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, Toulouse, 31000, France</footnote>

<footnote><sup>b</sup> EDF R&D, Chatou, 78400, France</footnote>

<footnote>* Corresponding author.
E-mail address: romain.mathis@imft.fr (R. Mathis).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Global schematic of the High Reynolds number test bench.</figcaption></figure>

vortices dynamics in the light of the Lamb Chaplygin's analytical model adapted to the problem under consideration. Then the evolution of the flow over a wide range of Reynolds numbers ($Re \in [1.2 \times 10^4, 5.4 \times 10^5]$) and as a function of the distance downstream of the bend (up to ten diameters downstream) is presented along with a quantification of the flow recovery. Conclusions are presented in Section 4.

## 2. Experimental facilities

### 2.1. The high Reynolds number test bench

The High Reynolds number test bench, referred to hereafter as HR test bench, is a closed-loop piping system made for liquid water flows with flow rate and temperature regulation. Owned by the R&D department of the French electricity company EDF, its primary purpose is to model real industrial systems, and to conduct metrology evaluations as well as equipment-calibration testings.

The facility, schematised in Fig. 1, consists of stainless steel pipes of various lengths with a common inner diameter $D_n = 110$ mm, a $1 \text{ m}^3$ storage tank containing the softened water used as the running fluid for the experiments, a variable speed pump providing flow rates between 13 and $150 \text{ m}^3/\text{h}$, and a heat exchanger allowing for fluid temperature control, here, $T = 21^\circ\text{C}$. The entire loop is monitored and controlled by a set of multiple valves with an automation system and two benchmark flowmeters using different technologies (Coriolis and ultrasonic).

This setup allows covering a wide range of Reynolds numbers, $Re_b = U_b D_n / \nu$ (where $U_b$ is the bulk velocity and $\nu$ the kinematic viscosity of the fluid), from $40 \times 10^3$ to $540 \times 10^3$. The flow regime is therefore always fully turbulent. The modular loop is equipped with a straight plexiglas sleeve with a circular inner cross section and a faceted outer section to allow optical measurements while reducing deformation (see Fig. 2). It should be noted that this test section can be placed at any desired distance downstream of the considered installation. For the present study, two specific locations are covered: $x/D_n = 3$ and $x/D_n = 10$ downstream of a 90° bend outlet.

### 2.2. The low Reynolds number test bench

The Low Reynolds number test bench, schematised in Fig. 3, is referred to hereafter as LR test bench and is also a closed-loop piping system for liquid flows.

The flow rates in the LR test rig range from 3.36 to $13.86 \text{ m}^3/\text{h}$, corresponding to Reynolds numbers $Re_b$ ranging from $11.8 \times 10^3$ to $49 \times 10^3$, ensuring fully turbulent flow regimes.

The main part of the test bench consists of a flow straightener chamber equipped with a honeycomb and a convergent, followed by

<figure><img src="image_4.png" /><figcaption>Fig. 2. Plexiglass test section of the HR test bench.</figcaption></figure>

(straight and/or bent) pipes of circular cross-sections with an inner diameter $D_n = 100$ mm. A flexible pipe segment upstream the straightener chamber ensures the modular characteristic of the loop, allowing for easy changes in pipe configurations including a long straight section, a 90° bend, etc. A volumetric screw pump is used on this bench, along with an electromagnetic flowmeter, pressure and temperature sensors, and a 180-liter storage tank to fill and drain the loop. Unlike the HR bench, the temperature is not controlled. However, according to the measurements, it is approximately constant and $T = 21 \pm 1^\circ\text{C}$.

Two fixed test sections have been designed for laser metrology measurements. They consist of a 3D-long cylindrical glass tube embedded in a rectangular glass view box ($30 \text{ cm} \times 30 \text{ cm} \times 28 \text{ cm}$) filled with the same fluid to minimise optical distortions (see Fig. 4). These test sections are centred around $x/D_n = 3$ and $x/D_n = 10$ downstream of the bend.

The use of two distinct benches enables multiple objectives to be met. The appropriate bench for high Reynolds numbers (HR bench) allows the study of operating points that can be assimilated to real industrial cases. The bench dedicated to moderate Reynolds numbers (LR bench) is conducive to the deployment of time- and space-resolved metrologies, and it allows for a more detailed study of the fundamental physical mechanisms involved, such as the phenomenon of swirl switching and the process of recovery towards uniformity downstream of a bend. Also, the two setups have a common Reynolds range that allows for the comparison of results. Finally, the Reynolds range targeted by the study spans more than two decades.

### 2.3. Geometrical characteristics and flow configurations

The geometrical characteristics of the 90° single bend are presented in Fig. 5. The radius of curvature is constant ($R_c = 1.5 \times D_n$) and the associated curvature ratio is $\gamma = R/R_c = 0.33$, where $R = D_n/2$ is the inner radius of the pipe. This curvature is very common in EDF's piping systems and is therefore of interest to the industrialist. The streamwise direction downstream of the bend is $e_x$, with $x = 0$ corresponding to the bend outlet (i.e., at the cross-section corresponding to $\alpha = 90^\circ$). The vertical and horizontal directions are respectively $e_y$, and $e_z$ such that $e_x = e_y \times e_z$ (see Fig. 5). The pipe centerline corresponds to $y = z = 0$. In the following, $(U_x, U_y, U_z)$ denote the velocity components along the directions $(e_x, e_y, e_z)$, respectively. The intrados refers to the inside of the bend, while the extrados corresponds to the outside of the bend (see Fig. 5). North and south refer to the upper and lower parts of the pipe respectively.

Measurements were conducted at $x/D_n = 3$ and $x/D_n = 10$ downstream of the 90° bend as well as in a straight pipe for comparison, in both LR and HR test benches. The range of Reynolds numbers covered in this study spans from $Re = 11.8 \times 10^3$ to $540 \times 10^3$.

### 2.4. Laser metrology techniques and measurement domains

Several laser based metrology techniques were deployed on the two experimental benches. Volumetric particle tracking velocimetry (3D-PTV) was carried out on a thickened illuminated cross-section of the
<figure><img src="image_5.png" /><figcaption>Fig. 3. Global schematic of the Low Reynolds number test bench.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Test section of the LR test bench.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Schematic of the 90° bend: top view.</figcaption></figure>

HR test bench, with the aim of performing a statistical study of the flow. For this purpose, LaVision's Shake The Box double-pulse module provided access to time-decorrelated velocity field measurements in the laser-illuminated cross-section orthogonal to the main direction of the flow at $x/D_n = 3$ and $x/D_n = 10$ as shown in Fig. 6. Four Imager SX CCD cameras with a resolution of 6 Mpixels (2752 px × 2200 px) and a double-image acquisition capability, were employed for this technique to image the volume illuminated by a dual-cavity Nd:YAG laser of the Bernoulli PIV 200-15 type (pulse energy of 200 mJ) at a frequency of 15 Hz. The illuminated volume is a cylinder whose diameter corresponds to that of the pipe and whose width, in the direction of flow, is 1.5 cm. The flow was seeded with fluorescent polymethyl methacrylate (PMMA) particles (20-50 µm, 1.19 g/cm³) allowing a very good traceability to the flow: the ratio of the bulk velocity to the sedimentation velocity is between 600 and $4.10^4$ for their range of tested Reynolds numbers. After data processing, the

<figure><img src="image_8.png" /><figcaption>Fig. 6. Laser-based measurement techniques and measurement domains for both HR (orange) and LR (green) test rigs: top view along with the sections of interest for each technique.</figcaption></figure>

spatial resolution achieved is 1.6 mm in the streamwise direction, with 36 divisions in the azimuthal direction and 30 meshes in the radial direction. The radial extension of the meshes along the radius varied such that the surface area of each mesh $dS = r\Delta r\Delta\theta$ remained constant in the cross-section. For the straight pipe, measurement uncertainties have been estimated by computing the difference between the flow rate determined by integrating the velocity field measured in the cross-section and the one supplied by the upstream flowmeter. For all the Reynolds numbers explored, the errors range between 4 and 12%.

The flow dynamics were addressed on the LR bench thanks to time-resolved stereoscopic particle image velocimetry (2D3C-PIV) realised in diametral vertical sections. The sections of interest were $1D_n$ high along the vertical direction $e_y$, $2D_n$ long in the streamwise direction $e_x$ and were centered around $x/D_n = 3$ and $x/D_n = 10$ downstream of the bend as sketched in Fig. 6 (green). For this technique, two high speed pco-dimax cameras (4 Mpixels and 1270 frames per second at full resolution), equipped with 105 mm lenses and oriented with an angle $\pm22.3^\circ$ with respect to the axis of the pipe, were imaging a plane illuminated by a high speed diode-pumped Nd:YLF laser (0.1 to 10 kHz, pulse energy of 30 mJ). The flow was seeded with glass hollow spheres (10 µm, 1.1 g/cm³) permitting excellent flow traceability, with bulk to sedimentation velocity ratio greater than $6.10^5$. The spatial resolution after processing the data was 0.57 mm in both the streamwise and the vertical directions. The errors range between 3 and 4.6% based on the difference between the measured flowrate and the one provided by the flowmeter.

The two measurement techniques deployed here address the issue of the flow downstream of a bend in a complementary manner. Indeed, the PTV conducted in the cross-section is necessary to characterise the three-dimensional flow and assess the influence of the Reynolds number on the shape and intensity of the coherent structures generated by the bend. On the other hand, this technique is local, and the TR-PIV performed in the vertical diametral plane $(x, y)$ allows, in complementarity, to evaluate, with a high temporal resolution, the streamwise evolution of the coherent structures.
<figure><img src="image_9.png" /><figcaption>**Fig. 7.** (a–b) Normalised mean streamwise velocity fields $\langle\bar{U}_x\rangle/U_b$, (c–d) and mean streamwise vorticity $\langle\bar{\omega}_x\rangle D/U_b$, along with in-plane velocity field; (left) straight pipe case, (right) at $x/D_n = 3$ downstream of the 90° bend. Results from HR test bench at $Re = 44 \times 10^3$.</figcaption></figure>

# 3. Results and discussion

## 3.1. Mean flow at $x/D_n = 3$ downstream of the bend

The statistical study using 3D-PTV measurements is based on physical quantities that result from a double averaging in time and space denoted $\langle \cdot \rangle$. A time-average of 1000 instantaneous fields is first performed (denoted by $\bar{\cdot}$), and the resulted time-averaged fields are then averaged in the streamwise direction over the measured volume thickness (indicated by $\langle \cdot \rangle$). Prior to performing the spatial average, it was verified that the velocity field is quasi-invariant in this thin layer. This allows for the obtaining of statistically significant values.

Fig. 7 presents the results obtained at $x/D_n = 3$ downstream of the bend (right side) and in the straight pipe (left side) for comparison. The top row shows the normalised mean streamwise velocity fields $\langle\bar{U}_x\rangle/U_b$, while the bottom one shows the normalised mean streamwise vorticity $\langle\bar{\omega}_x\rangle D/U_b$, along with the in-plane vector field. For the straight pipe configuration, the flow is mostly axisymmetrical, as expected, with non-significant near-wall variability probably due to minor defects in the experimental set-up and slight biases in near-wall measurements. The streamwise component of the vorticity has also been computed for the straight pipe configuration (Fig. 7 c) and reveals non-significant, uncorrelated signal. In contrast, the flow passing through the bend undergoes a major reorganisation, resulting in a strong persistent break in axisymmetry at $x/D_n = 3$. The centrifugal force that develops in the bend is responsible for an imbalance with the radial pressure gradient, which results in the generation of a secondary flow in the cross-section plane, from the intrados *I* to the extrados *E* (Fig. 7 d). This transverse flow is at the origin of Dean-like vortices [5,6], which are systematically observed in bend flows. It advects slow-moving fluid from the intrados to the center of the pipe, and fast-moving fluid from the center of the pipe to the extrados. This leads to high- and low-velocity zones, located respectively at the extrados and intrados, which is reflected in the crescent shape of the streamwise velocity field as shown in Fig. 7(b). This observation is in agreement with the measurements of Sudo et al. [11]. A complementary view of the velocity field is given in Fig. 8 which depicts the three time-averaged velocity components in the vertical diametral section from the LR test bench for approximately

<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Normalised time-averaged velocity components in the vertical diametral plane. (a, b, c) correspond respectively to $(\bar{U}_x, \bar{U}_y, \bar{U}_z)/U_b$. In each figure, the vertical profiles of the corresponding velocity component at different streamwise locations are superimposed (solid black line). Results obtained from the LR test bench at $Re = 33.6 \times 10^3$.</figcaption></figure>

the same Reynolds number. The crescent shape of the mean streamwise velocity component is materialised in this plane by two zones of high streamwise velocity at the south and north walls, and by a large zone of lower velocity in the center (Fig. 8 a). The vertical and horizontal components (Figs. 8 b and c, respectively) indicate the presence of Dean vortices. Indeed, the positive horizontal velocity $U_z$ in the neighbourhood of the centerline and the negative horizontal velocities at the south and north walls, along with the vertical velocities $U_y$ converging towards the centre are consistent with the vorticity and vectors fields measured in the cross-section, as shown in Fig. 7(d). This secondary motion is characterised by a quasi-symmetrical coherent vorticity field, with a clockwise circular motion in the northern hemisphere and a counterclockwise motion in the southern hemisphere (see Fig. 7d). It should be noted that the axis of symmetry of Dean vortices is not exactly aligned with the horizontal axis of the cross-section. This could be due to small alignment defects in the connection of the different upstream sections of the pipe.

## 3.2. Dean-vortices dynamics

Several studies of turbulent flows in bent pipes have revealed oscillations of the Dean vortices around the pipe axis [7–10,12,13]. In phase with the oscillation of the whole dipolar structure, each vortex alternately takes the lead over the other one in terms of vorticity intensity and radial extension. When the stagnation point at the extrados tilts northwards, the vortex in the southern hemisphere takes over from that in the northern hemisphere. Then, as the structure tilts back in the opposite direction, the intensity ratio reverses. This phenomenon, known as “swirl-switching”, induces oscillations around the central axis of up to several tens of degrees [8]. In the present study, the analysis of this dynamics was carried out in the vertical diametral plane, which does not give access to the streamwise vorticity. Therefore the swirl-switching phenomenon needs to be identified by another measurable quantity in this specific plane. This phenomenon can be identified by
<figure><img src="image_11.png" /><figcaption>Fig. 9. Instantaneous normalised spanwise velocity component ($U_z/U_b$) in the vertical diametral plane. The black thick line shows the vertical position of maximum velocity. $Re = 33.6 \times 10^3$ (LR bench).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Power spectral density of the spanwise velocity component. Blue curve: averaged spectrum over the central band; Grey curves: 10 single point spectra randomly chosen within the central zone. The green curve shows the equivalent to the blue one for the straight pipe configuration.</figcaption></figure>

considering the spanwise instantaneous velocity field $U_z$. Indeed, the region of high spanwise velocity (red region in Fig. 9) displays a large wavelength oscillation of the maximum of velocity around the $y/D_n = 0$ axis.

### 3.2.1. Spectral analysis

In order to identify the characteristic frequency associated with the swirl-switching phenomenon, a spectral analysis is performed on the time series of the spanwise velocity component from time-resolved Stereo-PIV measurements. Even with time-resolved PIV measurements, it can be difficult to achieve complete convergence of the spectrum, as time and/or sampling length may be insufficient. For the present data, the sampling frequency is $f = 500$ Hz, and the total acquisition duration is approximately 20 s. With these conditions, the smallest Strouhal number accessible is $St = f D_n / U_b = 0.0075$, which is sufficient to cover the Strouhal range associated with swirl-switching reported in the literature, namely $St \in [0.03, 0.18]$ [8,14], and others]. With the aim to increase the spectrum convergence, an ensemble of spectra calculated in the central zone of the field delimited by the dashed lines in Fig. 9 were averaged and result in the probability density function (PSD) given in Fig. 10. It should be noted, however, that spectra obtained by considering a single spatial point show no significant difference from the averaged spectrum, as highlighted in Fig. 10 through a set of ten single point spectra (grey colour) randomly chosen within the central zone. It should be noted that the same analysis performed for the straight pipe did not reveal any distinct peak (green curve in Fig. 10).

Four distinct Strouhal numbers emerge from the spectral analysis. This range of Strouhal number is in line with observations made by Carlson et al. 2015 [13] who distinguished high frequency Strouhal numbers (above $St \approx 0.5$), from low frequency ones (below $St \approx 0.01$)

and a peculiar peak corresponding to $St = 0.13$. According to their analysis, high frequencies are directly linked to bend curvature: the sharper the curvature, the higher the frequency. On the other hand, the very low frequencies would be a consequence of long streamwise streaks of streamwise momentum generated in the upstream wall-boundary layer turbulence and their amplitude would be directly linked to the length of the upstream straight pipe. The specific peak at $St = 0.13$ would correspond to shorter streaks responsible for the oscillation of the vortical structure, which in the present study seems to be identified by the peak at $St = 0.14$.

In the next section, an analytical model is formulated in order to evaluate the signature of an oscillating counter-rotating dipole projected onto the vertical diametral plane.

### 3.2.2. Asymmetric Lamb-Chaplygin analytical model

Time-resolved flow measurements are performed in the vertical diametral plane, in which the identification of the Dean vortices and their dynamics is not straightforward. In this context it may be helpful to estimate the velocity distribution associated with oscillating counter-rotating vortices in this specific plane. For this purpose, an analytical model is derived from the Lamb-Chaplygin dipole [15], which is a two-dimensional exact solution of the Euler equations describing a pair of counter-rotating vortices. Provided that the streamwise variations are small, the Navier-Stokes equations show that the flow in each cross-section is decoupled from the streamwise component. Under this assumption, the 2D Lamb-Chaplygin model is relevant for describing the dynamics of the flow in a cross-section at the main order. In all cases, this model provides a minima an analytical mathematical description of a velocity field mimicking the dipolar flow in a cylindrical pipe, from which geometric or structural information can be obtained to help understand the measurements in the various planes. A slight adaptation has been made to this analytical solution for the purposes of the present study, in order to mimic the dynamics of oscillating Dean vortices. The modifications to the original model consist of two contributions: (i) making the axis linking the front and rear saddle points of the vortex dipole oscillate about the center of the circle in which the structure is inscribed and (ii) making the dipole asymmetrical, with the most intense vortex alternating in phase with the alternating orientation of the overall structure. The modified stream function $\psi$ describing this solution is defined as follows:

$$ \psi(r, \theta, t) = \frac{\kappa}{J_0(\mu_1)} J_1 \left( \mu_1 \frac{r}{R} \right) \underbrace{\sin(\theta - \theta_0)}_{\text{inclination of angle } \theta_0} + \lambda \underbrace{\left( 1 - \frac{J_0 \left( \mu_1 \frac{r}{R} \right)}{J_0(\mu_1)} \right)}_{\text{asymmetry term}} \quad (1) $$

where $\kappa$ is a measure of the intensity of the dipole, $J_0$ and $J_1$ are the Bessel functions of order 0 and 1, and $\mu_1 = 3.8317$ is the first zero of $J_1$. $\lambda(t)$ and $\theta_0(t)$ define respectively the time evolution of the dipole asymmetry and of its angle of inclination with respect to the horizontal axis. From this streamfunction expression, the vorticity $\omega = -\nabla^2 \psi$ can be derived and is given by $\omega(r, \theta, t) = \mu_1^2 (\psi(r, \theta, t) - \lambda)$.

Fig. 11 illustrates the analytical model presented above, where the ensemble oscillates over a full period between -40° and 40°. The left-hand column shows the vorticity field and the right-hand column shows the spanwise velocity profile $U_z$ along the vertical diameter (shown as a green segment in the adjacent figure). Over a period, the overall shape of the velocity profile remains unchanged. However, the vertical position of the maximum velocity oscillates around the point $y = 0$ corresponding to the moment when the two vortices are symmetrical with respect to the horizontal axis, i.e. $\theta_0 = 0^\circ$. Then, for angles varying from $\theta_0 = 0^\circ$ to $\theta_0 = \pm \theta_{max}$ the spanwise velocity measured at $y = 0$ decreases monotonically. Following these observations, the idea is (i)
<figure><img src="image_13.png" /><figcaption>Fig. 11. Time evolution of an asymmetric oscillating Lamb-Chaplygin dipole over a complete period of oscillation between −40° and 40°. The left-hand column shows the axial vorticity field. The right-hand column gives the spanwise velocity profile along the vertical diameter (green line on the vorticity fields).</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12. Evolution of the SVDO law $f(\theta_0)$ for $\theta_{max} = 120°$ (Eq. (2)) relating the spanwise velocity at the center of the pipe to the angle of oscillation of the dipole.</figcaption></figure>

to consider the time evolution of the spanwise velocity at $y = 0$ and (ii) to find a relationship between the variation in spanwise velocity and the orientation angle of the dipole structure at a given position $x$, of the form:

$$f(\theta_0) = \frac{U_z(0, 0, x)\theta_0}{U_z(0, 0, x)\theta_0 = 0°} \quad (2)$$

This relationship will be referred to as SVDO law for “Spanwise Velocity Dipole Oscillation” law. The SVDO law, evaluated for the Lamb-Chaplygin analytical model oscillating between −120° and 120° is shown in Fig. 12.

All points lie on a same curve which is fitted with a second-order polynomial function. This suggests that the maximum angles reached by the oscillation of the vortex structure can be deduced from the maximum variations of the spanwise velocity at the center of the pipe. The idea is therefore to analyse the experimental data in this respect, by exploiting the evolution of the spanwise velocity values at $y = 0$, and to use the law resulting from the analytical study presented above.

### 3.2.3. Application of the SVDO law to the experimental data

In order to apply the SVDO law deduced above, the spanwise velocity field is phase averaged based on the period corresponding to the identified swirl-switching peak at $St = 0.14$, identified in Section 3.2.1 (Fig. 10). The phase-averaged spanwise velocity profiles are reported in Fig. 13 at $t = [0, T/3, 2T/3, T]$, where $T$ refers to the period of the complete phenomenon. It should be noted that a significant noise is present on the raw profile (black) due to the short duration of time-resolved Stereo-PIV. Indeed, high speed measurements are always a compromise between temporal resolution of the flow and obtaining a sufficiently long sample for the statistics to converge. Nevertheless, the overall behaviour of the oscillating Dean cells remains visible. In particular, the vertical position of the maximum spanwise velocity (red dashed line), along with that of the zero velocity (green dashed lines) follow a coherent oscillation. By following the time evolution of the spanwise velocity on the axis $U_z(y = 0)$ it is then possible to apply the SVDO law which indicates an oscillation of the secondary flow between $\theta \approx \pm 55°$.

## 3.3. Flow evolution downstream of the bend

### 3.3.1. Influence of the Reynolds number

Fig. 14 displays iso-contours of the normalised time-averaged streamwise velocity field $\langle U_x \rangle / U_b$ for the six Reynolds numbers investigated on the HR test bench, ranging from $Re = 9 \times 10^3$ to $Re = 540 \times 10^3$. The two lowest Reynolds numbers were obtained by using a bypass to achieve flows below the nominal flow range supplied by
<figure><img src="image_15.png" /><figcaption>**Fig. 13.** Phase averaged profiles of the spanwise velocity component $U_z$ based on $\Omega_x = 0.14$ at $x/D_n = 3$ for $Re = 33.6 \times 10^3$ (LR bench). The velocity profiles are represented from left to right at $t = [0, T/3, 2T/3, T]$, where $T$ refers to the period of the complete phenomenon.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 14.** Iso-contours of the streamwise time averaged velocity $\langle \bar{U}_x \rangle / U_b$ at $x/D_n = 3$ downstream of the bend's exit for Reynolds numbers $Re \in [9,540] \times 10^3$. Solid lines: $\langle \bar{U}_x \rangle / U_b = 1.05$, dotted lines: $\langle \bar{U}_x \rangle / U_b = 0.9$.</figcaption></figure>

the pump. For each Reynolds number, two iso-contours are plotted, namely, $\langle \bar{U}_x \rangle / U_b = 1.05$ in solid lines and $\langle \bar{U}_x \rangle / U_b = 0.9$ in dotted lines.

The lowest velocity contour is located near the centre of the pipe for low Reynolds numbers. For higher Reynolds numbers, this contour expands slightly and tends to move towards the extrados before converging towards the southern part of the pipe. In contrast, the highest velocity contour has a crescent shape lying on the extrados of the pipe (as previously observed in Section 3.1), whatever the Reynolds number. However, the extension of the high-velocity zone depends on the Reynolds number. Indeed, as the Reynolds number increases, the size of the zone encircled by the velocity iso-contour corresponding to a given proportion of the bulk velocity $\langle \bar{U}_x \rangle / U_b = 1.05$ decreases. In addition, the higher the Reynolds number, the more the barycenter of this zone shifts towards the upper part of the pipe. We postulated earlier (see Section 3.1) that the north-south asymmetry of the velocity iso-contour (or vorticity field) could be the consequence of small flatness defects in the joints of the different pipe sections. This new observation would suggest that the higher the Reynolds, the more sensitive the flow is to such misalignments.

In order to complete the above observation, Fig. 15 compiles the streamwise velocity profiles along the horizontal and vertical diameter

<figure><img src="image_17.png" /><figcaption>**Fig. 15.** Normalised streamwise time averaged velocity profiles $\bar{U}_x/U_b$ for all Reynolds numbers at $x/D_n = 3$. (a) horizontal axis, (b) vertical axis.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 16.** (a–b) Normalised mean streamwise velocity fields $\langle \bar{U}_x \rangle / U_{xs}$, (c–d) and mean streamwise vorticity $\langle \bar{\omega}_x \rangle D / U_b$, along with in-plane velocity field (black vectors). Left: straight pipe case, right: at $x/D_n = 10$ downstream of the 90° bend. The dashed black line and solid black line are the iso-contour $\langle \bar{U}_x \rangle / U_b = 1$ at $x/D_n = 3$ and $x/D_n = 10$, respectively. Results from HR test bench at $Re = 44 \times 10^3$.</figcaption></figure>

for all Reynolds numbers. It shows that the velocity profile becomes smoother with increasing Reynolds number. Indeed, peaks of velocity decrease and broaden with the Reynolds number. These results suggest a faster recovery towards a uniform streamwise velocity field with higher Reynolds numbers. This is in agreement with the idea presented by Hellström et al. [12] that the more turbulent the flow, and therefore the higher the Reynolds number, the faster the recovery of a fully developed flow due to the increased turbulent mixing.

### 3.3.2. Influence of the downstream distance

To investigate the flow's spatial evolution downstream of the bend, experiments were conducted for comparison at two downstream locations, $x/D_n = 3$ and $x/D_n = 10$. Fig. 16, which is similar to Fig. 7, presents the results obtained on the HR test rig at $z = 10D_n$, for $Re = 44 \times 10^3$. It can be observed that a return to an axial, fully developed,
<figure><img src="image_19.png" /><figcaption>Fig. 17. Evolution of the normalised velocity components along the pipe downstream of the bend: (a) streamwise component $\bar{U}_x$, (b) vertical component $\bar{U}_y$, and (c) spanwise component $\bar{U}_z$. Results from the LR test bench at $Re = 33.6 \times 10^3$.</figcaption></figure>

flow is underway: the high-velocity iso-contours closer to the exit of the bend are gradually transitioning from a crescent towards a circular shape further downstream. However, at $x/D_n = 10$ downstream from the exit plane of the bend, the axisymmetric profile has not been yet achieved and some persistent distortions in the streamwise velocity are still visible. Regarding the secondary flow, the Dean vortices measured at $x/D_n = 3$ persist at $x/D_n = 10$. Nevertheless, the vorticity magnitude has been divided by about a factor two.

The choice of the measurement plane for the LR experiments makes it possible to study the streamwise evolution of the velocity field more continuously downstream of the bend's exit plane. Fig. 17 illustrates the streamwise evolution of the normalised velocity profiles for all three components along the vertical diameter. The streamwise velocity profiles $\bar{U}_x/U_b$ show a gradual return to a top-hat shaped profile with a velocity decrease near the walls ($y/R = \pm 1$) and an increase in the middle section ($y/R \in [-0.5, 0.5]$), echoing the dissipation of the crescent-shaped high-velocity zone observed in the cross-section

<figure><img src="image_20.png" /><figcaption>Fig. 18. Uniformity index $\varphi$ for all studied configurations on HR test bench.</figcaption></figure>

$x/D_n = 10$ with the HR results (see Fig. 16). Regarding the secondary motion, a dissipation of the vertical $\bar{U}_y$ and spanwise $\bar{U}_z$ velocity components is measured in the streamwise evolution of the profiles (see Fig. 17 b and c). However, although a significant drop in velocity magnitude is observed, the general shape with positive and negative zones is still discernible, demonstrating the persistence of the Dean structure signature in the vertical diametral plane.

### 3.3.3. Flow recovery quantification

In order to quantify the return to an axisymmetric, fully developed turbulent, flow in the pipe, Bilde et al. [16] defined a uniformity index $\varphi$, that measures the difference between the time-averaged velocity and the bulk velocity in the cross-section:

$$ \varphi = 1 - \frac{1}{2AU_{\text{avg}}} \iint |\langle \bar{U}_x \rangle(y, z) - U_{\text{avg}}| \, dA, \quad (3) $$

where A corresponds to the effective measurement cross-section area, i.e. the field of view which captures 91% of the cross-section (boundary layer excepted), $\langle \bar{U}_x \rangle(y, z)$ is the double averaged mean streamwise velocity field as depicted in Figs. 7 and 16, and $U_{\text{avg}}$ is the spatial averaged of $\langle \bar{U}_x \rangle(y, z)$ in the effective measured cross-section A. The closer this coefficient is to 1, the more uniform is the velocity field.

Fig. 18 illustrates the assessment of this uniformity parameter as a function of the Reynolds number and the distance downstream from the exit plane of the bend. Regardless of the downstream distance, it can be observed that the flow tends to become more uniform as the Reynolds number increases. This can be explained by the fact that a higher Reynolds number corresponds to a larger turbulent intensity, which can be thought of as an equivalent increasing turbulent viscosity that tends to homogenise the flow. It is also observed that the uniformity index is larger for $x/D_n = 10$ than for $x/D_n = 3$. This observation is consistent with the homogenising role of turbulence along the straight part of the pipe after the bend. This trend becomes more pronounced as the Reynolds number increases.

Another parameter is the swirl intensity parameter:

$$ I_s = \frac{1}{AU_b^2} \iint (U_y^2 + U_z^2) \, dA \quad (4) $$

used, for example, by Kim et al. [17]. This parameter assesses the ratio between the magnitude or intensity of the in-plane velocity in the cross-section relative to the main flow in the streamwise direction, is also used to quantify the return to a steady and axisymmetric flow as a function of the downstream distance from the bend. This parameter can be calculated from the PTV performed on the HR experimental setup in the cross-section at $x/D_n = 3$ and $x/D_n = 10$ over the range of Reynolds number considered here. Complementarily, the stereo-PIV measurements, conducted on the LR experimental setup, allow for the evaluation of the evolution of this parameter as a function of
<figure><img src="image_21.png" /><figcaption>Fig. 19. Evolution of the swirl intensity parameter $I_{s_D}$ along the streamwise axis downstream of the bend. Results from the LR test bench at $Re = 33.6 \times 10^3$.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 20. Comparison of the swirl intensity parameters $I_s$ calculated on the whole cross-section and $I_{s_D}$ obtained along the diameter for several angles $\theta$. Results from the HR test bench at $Re = 43.9 \times 10^3$.</figcaption></figure>

the streamwise position downstream of the bend. However, given the vertical diametral plane of measurement, finding this value requires integration over the vertical diameter rather than the cross-section. This new understanding of the swirl intensity parameter, referred to as $I_{s_D}$, is defined as follows:

$$I_{s_D} = \frac{1}{D_h U_b^2} \int (U_y^2 + U_z^2) dy \qquad (5)$$

Fig. 19 depicts the swirl intensity parameter, $I_s$ and $I_{s_D}$ calculated from the cross section and from the vertical diametral plane measurements respectively, as a function of the distance downstream of the bend for several Reynolds numbers. The lower the swirl intensity, the lower the cross-sectional velocity compared to the dominant streamwise flow. Then, the decrease of $I_s$ and $I_{s_D}$ provides valuable insights into how the secondary motion decays downstream from the studied installation. In Fig. 19, the markers are calculated directly from the experiment, the solid lines are the best exponential fit passing through the data and the value of the swirl intensity at the bend exit $I_0$ is deduced from the exponential adjustment. A significant drop in the swirl intensity is observed moving away from the exit of the bend whatever the Reynolds number. Moreover, measurements performed on the LR test bench, which allows for the collection of swirl intensity over twice two diameters, remarkably follow a decreasing exponential law with distance from the bend, as suggested by Kim et al. [17]: $I_{s_D}/I_0 \approx e^{-\alpha(x/D_n)}$. Only small discrepancies between the experimental data and the fit are observed at the edges of the domain. The exponential law is not directly derived from a physical consideration, but rather stems

from an empirical observation. Nevertheless the trend is well captured, and an estimation of the number of downstream diameters needed to achieve a return to a symmetric flow seems reasonable. Note that the decay coefficient obtained in the present study, $\alpha$, varies from 0.1 to 0.3 compared to 0.21 in the study by Kim et al. [17]. Kim et al. suggested in their study that the swirl intensity was independent of the Reynolds number but depended on the bend curvature radius. The curves are indeed not arranged monotonically with the Reynolds number. On the other hand, the bend has the same radius of curvature on both experimental setups and cannot explain the variation of this coefficient.

Additionally, until now, the swirl intensity parameter was determined using integration over a whole cross-section. The modified swirl intensity used in the present study has the advantage of being applicable from a velocity profile measured along a diameter. This approach eliminates the need for cross-sectional measurements at numerous streamwise locations, reducing both the cost and time associated with data acquisition. The PTV measurements realised on the HR setup in the complete cross-section enable the calculation of the swirl intensity parameter $I_{s_D}$ along diameters with variable orientation $\theta$. In Fig. 20, the dashed black line is the classical swirl intensity $I_s$ computed over the cross-section, while each coloured line is a diametral swirl intensity $I_{s_D}$ derived from the cross-section and according to a particular orientation. The coefficient $\alpha$ of the exponential laws show variations with the diameter orientation $\theta$, namely, $\alpha \in [0.12, 0.19]$. The diametral swirl intensity closest to the global swirl intensity is obtained for $\theta = 90^\circ$. This corresponds to a measurement along the horizontal diameter which, on average, belongs to the symmetry plane of the flow, given the swirl switching. Finally, with all the precautions taken regarding the precise value of the coefficient $\alpha$, measuring the swirl intensity by measuring the velocity only along a diameter of the section is easier than measuring it in a plane and seems relevant for estimating the flow recovery.

## 4. Conclusion

In this work, the flow downstream of a single bend, a configuration widely encountered in industrial applications, was studied experimentally. By measuring the three velocity components in different sections of the flow, this experimental study examines the structure and dynamics of the flow as a function of Reynolds number and distance downstream of the bend. The global three-dimensionalisation of the flow, and in particular the generation of Dean vortices and their dependence on the control parameters, is discussed. Furthermore, the dynamics of these coherent vorticity structures and the phenomenon of “swirl-switching” are documented.

The observations are consistent with literature findings regarding the average flow downstream of a single 90° bend, namely a three-dimensionalisation of the flow with high- and low-velocity regions identified at the extrados and intrados of the bend, respectively. This dynamic results from the balance between centrifugal force and pressure gradient, generating a secondary motion in the form of Dean vortices, a well-known vortex structure in bent pipe flows. The signature of the oscillating aspect of the Dean structure is studied in a vertical diametral plane using an original variant of the oscillating Lamb-Chaplygin model. This enabled finding a mathematical relationship between the spanwise velocity and the orientation of the vortex pair. A spectral analysis is first performed in order to identify the oscillation. This has revealed a number of four dominant Strouhal numbers, ranging from 0.03 to 0.8. Based on the work of [13] the high- ($St > 0.5$) and low-frequency ($St < 0.05$) are respectively linked to the curvature of the bend and elongated streamwise wall-bounded streaks developing upstream the bend. The present work then focused on $St = 0.14$ which fits into the range of Strouhal numbers, in between 0.1 and 0.15, identified in the literature to be responsible for the swirl-switching behaviour [8,14, and others]. A phase-averaging operation based on this specific Strouhal number was carried out and,
thanks to the analytical law, enabled estimating the range of oscillation angles of the Dean structure, namely $\theta \in [-55^\circ, 55^\circ]$ for the studied configuration.

The recovery of the flow and dissipation of secondary motion down-
stream of the bend were investigated using the uniformity index $\varphi$
introduced by Bilde et al. [16] and the swirl intensity used for instance
by Kim et al. [17], respectively. It is found that the flow recovers
more rapidly as the Reynolds number increases, which seems to be
promoted by the increase of the turbulence intensity. Furthermore,
the swirl intensity parameter, which was originally estimated in a
cross-section, was demonstrated to be effective in determining how the
flow evolves downstream a bend when applied to a velocity profile
measured along a single diameter. It would be interesting to inves-
tigate, through dedicated precision measurements, the link between
the parameter estimated by a measurement in the cross-section and
the parameter evaluated using a single velocity profile, and then to
examine the dependence on orientation, i.e., on the azimuthal direction
of the chosen diameter. The simplicity of the method employed here,
compared to taking measurements in a cross-section, would certainly
make it a method of interest.

CRediT authorship contribution statement

**Bilal Ben Haroual:** Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis. **Julie Albagnac:** Writing – review & editing, Supervision, Project administration, Methodology, Formal analysis. **Pierre Brancher:** Writing – review & editing, Supervision, Project administration, Methodology, Formal analysis. **Sébastien Cazin:** Writing – review & editing, Visualization, Validation, Methodology, Investigation. **Didier Boldo:** Writing – review & editing, Supervision, Methodology. **Emmanuel Thibert:** Writing – review & editing, Supervision, Resources, Project administration. **Romain Mathis:** Writing – review & editing, Supervision, Methodology, Formal analysis.

Declaration of competing interest

The authors declare the following financial interests/personal rela-
tionships which may be considered as potential competing interests:
Bilal Ben Haroual reports financial support was provided by EDF. Bilal
Ben Haroual reports financial support was provided by ANRT. If there
are other authors, they declare that they have no known competing
financial interests or personal relationships that could have appeared
to influence the work reported in this paper.

Acknowledgements

The authors would like to acknowledge LaVision for providing the
circular 3D calibration plate and their assistance with the in-situ cali-
bration in the cross-section [18], transverse to the flow direction (HR

test bend). We would like to recognise the CNRS Research Federation FERMaT (FR 3089) for granting access to a high-speed pco-dimax camera used for the stereoscopic PIV experiments. Finally, we would also like to thank the technicians at EDF and IMFT for their invaluable assistance in setting up the experimental rigs. The financial support of EDF, France and ANRT, France is also acknowledged.

Data availability

The authors do not have permission to share data.

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