## Abstract

The wake flow around a circular cylinder at *Re*≈100 performing rotatory oscillations has been thoroughly discussed in the literature, mostly focusing on the modifications to the natural Bénard–von Kármán vortex street that result from the forced shedding modes locked to the rotatory oscillation frequency. The usual experimental and theoretical frameworks at these Reynolds numbers are quasi-two-dimensional, because the secondary instabilities bringing a three-dimensional structure to the cylinder wake flow occur only at higher Reynolds numbers. In this paper, we show that a three-dimensional structure can appear below the usual three-dimensionalization threshold, when forcing with frequencies lower than the natural vortex shedding frequency, at high amplitudes, as a result of a previously unreported mechanism: a pulsed centrifugal instability of the oscillating Stokes layer at the wall of the cylinder. The present numerical investigation lets us in this way propose a physical explanation for the turbulence-like features reported in the recent experimental study by the present authors.

## 1. Introduction

A circular cylinder performing rotational oscillations around its axis in an infinite viscous fluid produces an axisymmetric pulsed boundary layer, called a Stokes layer. This is a flow susceptible to generate centrifugal instabilities. The linear stability problem of this flow configuration has been studied by Hall [1], Seminara & Hall [2] using asymptotic methods. A threshold for the appearance of three-dimensional axisymmetric instability modes was determined. Riley & Laurence [3] did also stability calculations not directly on the Stokes layer problem but considering the modulated circular Couette flow under axisymmetric disturbances, in the narrow-gap limit. Later, Aouidef *et al.* [4], Ern [5], Ern & Wesfreid [6,7] considered this flow as a limit case for the stability problem of the classic geometry of two concentric cylinders with oscillation: the Taylor–Couette configuration (see e.g. Chandrasekhar [8] for a review). In both cases, the control parameter is the Taylor number, defined as
*R*_{i}=*ω*_{i}*r*_{i}*d*/*ν* is a Reynolds number based on the rotational angular velocity of the cylinder *ω*_{i}. We keep the notation of the Taylor–Couette configuration, where the subscript ‘i’ stands for *inner* cylinder, *r*_{i} thus being the radius of the cylinder. In addition, *ν* the kinematic viscosity. The characteristic length scale *d* in the Taylor–Couette case is the gap between the cylinders, which fixes the scale of the wavelength of the primary instability. For the case studied by Seminara & Hall [2], however, the instability occurs in the inner Stokes boundary layer of thickness *T* associated with the onset of a Taylor–Couette-type vortex flow. Vortices evenly spaced, with a critical length λ_{c} in the cylinder axial direction which is proportional to *δ*_{S}, are thus developed.

On the other hand, when a uniform flow comes across a cylinder, a prototypical two-dimensional wake flow takes place for moderate free-stream Reynolds numbers *Re*=*DU*_{0}/*ν*, where *D* is the diameter of the cylinder, *U*_{0} the free-stream velocity. The well-known Bénard–von Kármán (BvK) vortex street [9,10] results from the destabilization of the steady flow in the wake of the cylinder and produces the periodic shedding of opposite-signed vortices with a frequency *f*_{0}, that occurs above the threshold *Re*_{c}≈47 [11,12]. This flow is quasi-two dimensional up to *Re*=100. The rotational oscillation of the cylinder is prescribed by a forcing function of frequency *f* and amplitude *θ*_{0} that can be written as *A*=*u*_{θmax}/*U*_{0}, where *u*_{θmax}=*Dπfθ*_{0} is the maximal azimuthal velocity of the rotational oscillation, and the ratio *f*/*f*_{0}. We characterized the spatial development of the flow and its stability properties following previous studies by Thiria *et al.* [15], Thiria & Wesfreid [16]. A synthesis of the case study is presented in figure 1. From the analysis of power density spectra of the flow, we gave a detailed description of the forced wake, giving insight into the energy distribution, the different frequency components and in particular on a continuous spectrum observed for a high amplitude of the forcing oscillation. Furthermore, vortex structures revealed turbulence-like features such as splitting and mixing in a spatial cascade pattern. A question remained concerning the physical mechanism present in the bifurcation that triggers such behaviour of the wake.

We speculated on a three-dimensional centrifugal instability to be at the origin of this sequence of transitions. A natural first attempt to test this idea is shown in figure 1, where the critical Taylor number *T*_{c}=165 corresponding to the instability threshold of the pure rotatory oscillating cylinder case without crossflow studied by Seminara & Hall [2] is identified in the frequency–amplitude phase space ( *f*,*A*) of the forced wake of D'Adamo *et al.* [13]. This crude estimate for a threshold is compatible with the experimental points where the turbulent-like behaviour was observed (low frequencies and high amplitudes of the forcing oscillation). The purpose of this paper is to characterize in detail the existence of a three-dimensional instability and its centrifugal nature, using analytical estimations from the two-dimensional flow and from three-dimensional direct numerical simulations (DNS).

It is worth mentioning that centrifugal instabilities were also reported for forced flows with different configurations. For transverse oscillations of a cylinder in a fluid at rest, Honji [17], obtained visualizations that identified three-dimensional structures produced by centrifugal instabilities. Hall [18] performed a stability analysis of this configuration and gave a theoretical explanation. Tatsuno & Bearman [19] investigated in detail the patterns and the structure of the flows that result from these instabilities. Later, Elston *et al.* [20] addressed DNS calculations and Floquet stability analysis for this problem.

Three-dimensional instabilities in wake flows have been studied theoretically and numerically by Blackburn *et al.* [21] where it was determined that bifurcations to three-dimensionality can occur from a two-dimensional time-periodic base state with space–time reflection symmetry for the wake of symmetrical bluff bodies. More recently, for the case of the two-dimensional stationary flow past a rotating cylinder, Pralits *et al.* [22] suggest that the stationary unstable three-dimensional mode could be the result of a hyperbolic instability. Lo Jacono *et al.* [23] were interested on the role of rotationally oscillations can modify the three-dimensional transition in the wake of a cylinder. The frequency of oscillation was matched to the natural vortex-shedding frequency, *f*^{+}=1, for *Re*=300. They reported changes on the three-dimensional modes from Floquet stability analysis on two-dimensional periodic flow. They found that the rotational oscillation dramatically suppressed mode B, even for small amplitudes of oscillation. Mode A was also damped, but not as significantly as mode B. For what they considered high rotational oscillation amplitudes, in our notation *A*≃0.66 they identify a new three-dimensional transition mode, which they called D mode, that shares the same symmetries as mode A.

Three-dimensional characteristics of forced wakes have been recently studied by Kumar *et al.* [24] for the case of rotational oscillations at *Re*=185 near the transition, using flow visualization, hot-wire anemometry and PIV. Spatial distribution of lock-on regions and its relationship with the forcing frequencies and amplitudes was determined. They also found that for certain forcing parameters ( *f*^{+},*A*), the flow can be forced to become two-dimensional. Studied amplitudes were up to *A*=*π*, a value below the threshold found in [13] by means of spatio-temporal spectral analysis.

To summarize, this work sets up a new view about three-dimensional instabilities in wake flows, which have often been discussed in the case of the circular cylinder for *Re*>180 as secondary instabilities to the BvK vortex street. We organize the paper as follows: in §2, we describe the method used for the DNS; results are presented in §3 where we determine the three-dimensional stability threshold; in §4, we investigate the instability nature, using some concepts of centrifugal instabilities and propose, therefore, a reduction of the complex problem; lastly, we elaborate our conclusions in §5, showing analogies with the Taylor–Couette problem of eccentric cylinders.

## 2. Problem definition for direct numerical simulation

In order to study this problem, we performed two- and three-dimensional direct numerical simulations with Gerris free software, a parallelized tree-based adaptive solver for Navier–Stokes equations (Popinet [25]). The code combines an adaptive multi-grid finite volume method and the methods of immersed boundary and volume of fluid (VOF). The basic equations are the incompressible continuity equation and Navier–Stokes equations, which can be written in terms of the velocity **u**=(*u*,*v*,*w*) and pressure *p* fields as

The domain is spatially discretized using cubic finite volumes organized hierarchically as an octree. Along with the forcing problem parameters, a two-dimensional example of the spatial discretization is given in figure 2. The flow domain, shown in figure 2*c* is *L*_{x}×*L*_{y}=20*D*×10*D* for two-dimensional simulations and *L*_{z}=20*D* for the spanwise direction in three-dimensional simulations. As detailed in [25], the mesh can be refined near the solid boundary, and it can use vorticity gradients as an adaptive criterion. A cell is refined whenever
*Δx* is the size of the cell and *ξ* is a user-defined threshold which can be interpreted as the maximum angular deviation (caused by the local vorticity) of a particle travelling at speed *b*,*c*, where different box sizes are notable. In order to reveal BvK vortices as well as centrifugal structures, we choose a minimum grid size of *D*/51.2 for the solid boundary and *D*/12.8 to define vortex regions. The *ξ* threshold is set to 0.05 for three-dimensional simulations and to 0.01 for two-dimensional simulations. The flow parameters of the simulations are defined in order to match the experimental case of D'Adamo *et al.* [13]: cross flow velocity *ν*=10^{−3} and cylinder diameter *D*=0.1, giving a Reynolds number *Re*=100.

The boundary conditions are *u*=1 for *x*=−5*D*; *u*=1 for *y*=±5; the outflow condition is ∂*v*/∂*x*=0 and *p*=0 for *x*=15*D*; for three-dimensional simulations, a symmetry condition is used for the flow at *z*=20*D*; and at the cylinder surface, **u**=**u**_{solid,} where **u**_{solid} depends on the forcing. As depicted in figure 2*a,* rotatory oscillations are characterized by an angular coordinate *α*=2*πf*_{f}*t*, and tangential displacements *Δ*=*u*_{θ}/(2*πf*_{f}). Given *f*_{0} the natural frequency of vortex shedding, the forcing frequency *f*_{f} is written in dimensionless form as *f*^{+}=*f*_{f}/*f*_{0}. A non-dimensional number for the amplitude of oscillations is obtained by comparing the maximum tangential velocity *u*_{θmax} and the free flow velocity,

## 3. Results of the numerical simulation

We first performed three-dimensional DNS numerical simulations. Figure 3 shows a case with the forcing parameters ( *f*^{+}=0.75,*A*=4.00). The isosurface of vorticity modulus in figure 3*a* shows on one side the classic BvK wake structure synchronized with the forcing frequency. Additionally, a previously not reported effect is also clear: the modulation of the vorticity field along the direction of the cylinder axis. The two effects are depicted in figure 3*b*, revealing the three-dimensional vortex structure around the cylinder and a well-defined wavelength λ_{z}. Moreover, figure 4 shows the spatial distribution of *ω*_{x} along with *ω*_{z} for *f*^{+}=0.75, *A*=4.00, which allows us to consider the symmetry properties of the observed mode. The spatio-temporal symmetry, *H*, of the two-dimensional flow is defined as
*K*_{y} is a spatial reflection. For an *H*-symmetric flow, from (3.1), the *x*-vorticity changes sign with *x*,*z*). This is the case for mode A, whereas for mode B, the sign of *x*-vorticity does not change.

As studied by Blackburn *et al.* [21], there are exactly three codimension-one bifurcations from a two-dimensional time-periodic base state to three-dimensional flow that are observable with variations in a single parameter. In this regard, Lo Jacono *et al.* [23] showed that oscillatory forcing at *Re*=300 leads to the appearance of a different mode (mode D) which has the same symmetries of mode A. Considering the symmetries observed in the present case (figure 4), the identified structures are not *H*-invariant, and they share the same symmetry as mode B.

In what follows, we thoroughly scrutinize the onset of this three-dimensional pulsed instability. Figure 5*a* shows instantaneous contours of the spanwise velocity *w* for a plane at *y*=*D*/2, revealing Taylor–Couette-like vortices, with a wavelength λ_{z} that does not change with respect to the forcing amplitude within the range 3<*A*<4. We can describe the flow with a Taylor number based on equation (1.1) considering λ_{z} as a characteristic length scale and *w* and *T*_{f} the forcing period *T*_{f}=1/*f*^{+}. The result allows us to identify a maximum value that characterizes the intensity of the three-dimensional structure for the forcing case considered.

An additional characterization is possible by studying the amplitude of these fluctuations as a function of the forcing parameters ( *f*^{+},*A*). We use the three-dimensional DNS to study the flow modifications for two fixed forcing frequencies *f*^{+}=0.75 and *f*^{+}=1.00. A useful criterion to quantify the intensity of three-dimensional structures is to follow the evolution of *w*. Given that the three-dimensional structures are present for *A*=4.00 (case depicted on figure 3), we decrease the forcing amplitude from this value until they vanish. In figure 5*b*, the maxima of *w*_{rms}, _{z} is found to be 1.16*D*, and the corresponding forcing amplitudes. We can appreciate, looking at the square of the forcing amplitudes, that the three-dimensional structures become damped linearly as we approach a threshold at *T*=202 for *f*^{+}=0.75 and *T*=147 for *f*^{+}=1. The behaviour is common to supercritical bifurcations. Another scenario shows up when we follow the evolution of the intensity of three-dimensional structures for a fixed forcing amplitude. There is a range of frequencies for which the instability develops. This is shown in figure 5*d* where *w*^{2}_{rms} is observed for *A*=4.00 and the forcing frequency varying in a range 0.50<*f*^{+}<1.20. We can appreciate that for *f*^{+}≥0.55.

We perform simulations for different forcing frequencies at a fixed forcing amplitude *A*=4.00 in order to characterize the evolution of the wavelength λ_{z}. We observe in figure 6*a* that λ_{z} depends on *f*^{+} following a law ∝( *f*^{+})^{−1/2}. If we assume that the ‘gap’ size *d* is proportional to λ_{z}, from the Taylor number definition in equation (1.1), where *T* depends on *d*^{3/2}, then we expect that high forcing frequencies produce decreasing Taylor numbers. This could explain the damping of three-dimensional fluctuations for higher frequencies in figure 5*c*. In addition, we observe that the wavelength λ_{z} is practically invariant with respect to the amplitude for a given frequency.

In studies of pulsed centrifugal instabilities, Riley & Laurence [3], Carmi & Tustaniwskyj [26], Aoudief *et al.* [4] classified flow regimes based on a parameter *d*_{c} to the Stokes layer thickness. In our experiment, *γ* is limited to a range between 2 and 5, it does not depend on the forcing frequency and *d*_{c}∼λ behaves with respect to *f*^{+} as described in figure 6*a*, where λ decreases almost linearly as ( *f*^{+})^{−1/2}.

Even though the threshold for centrifugal instabilities determined in the Taylor–Couette pulsed flow is not directly applicable for a configuration with crossflow, the transformation of Taylor numbers based on the characteristic length *d*_{c} allows an approach for our results. This case presents similarity with the eccentric Taylor–Couette instability problem (see [27–29] and references therein). Indeed, in those problems, the axial wavelength of the critical perturbations is always of the same order of magnitude of the gap.

Figure 6*b* summarizes the stability curves (*γ*,*T*) for centrifugal pulsed flow determined by Aouidef *et al.* [4], Seminara & Hall [2] together with the values issued from our three-dimensional simulations. Two analytical curves show the solution corresponding to low values of *γ*, *T*_{c}=193.23*γ*^{−1} and high values of *γ*, *T*_{c}=15.28*γ*^{3/2}. The curves are supported with experimental data from [4]. On the other hand, within these reference threshold frames, we plotted from our results *T* against *γ* for a fixed forcing amplitude *A*=4.00, and for fixed forcing frequencies *f*^{+}=0.75 and *f*^{+}=1.00 (the same data used to construct figure 5). We observe that the points are contained in the unstable region defined by the analytical curves. For *A*=4.00, the instability develops for 0.55<*f*^{+}<1.16. When *f*^{+}=1.16, the critical point (*γ*=3.06, *T*=152) is in very good agreement with the experimental results from pure pulsed flows. For decreasing frequencies, *T* increases almost linearly regarding the estimated *γ* until for *f*^{+}=0.55, the flow stabilizes with respect to centrifugal disturbances (*γ*=4.4, *T*=433). For a fixed frequency *f*^{+}=0.75, *γ*=3.61, the flow destabilizes at *T*=202 and, with increasing forcing amplitudes, *T* eventually reaches the previous set of points at *A*=4.00. The same behaviour is found for the fixed forcing frequency *f*^{+}=1.00, where the flow is unstable from *T*=147.

We suggest that the centrifugal instability that develops in the forced wake can be thus considered in the context of pure rotatory pulsed oscillations. Nevertheless, the natural BvK dynamics plays an important role as the first bifurcation depends on the distance of the forcing state space parameters ( *f*,*A*, figure 1) to the resonance centred at ( *f*^{+}=1.00, *A*=0.00). This fact could explain that at *f*^{+}<0.55 the centrifugal instability is not strong enough even when *T* is high. Conversely, for ( *f*^{+}=1.16, *A*=4.00), the length *d*_{c} is significantly smaller and *T* decreases to the values predicted by the *pure* pulsed flow threshold.

We bring a quantitative picture of these ideas in the remainder of the paper, starting with a brief review of the criterion for centrifugal instability.

## 4. Centrifugal instability

The necessary condition for a three-dimensional centrifugal instability in flows with curved streamlines is given by Rayleigh [30] criterion for inviscid flow, see Drazin & Reid [31], which can be written for flows such as the Taylor–Couette flow in terms of the Rayleigh discriminant
*V* (*r*) is the two-dimensional velocity of an orthoradial base flow field. three-dimensional perturbations to this flow field are amplified if *ϕ*(*r*)<0, which translates the fact that the perturbed pressure field does not balance the centrifugal force, leading to flow instability. For a general profile *V* (*r*), the flow field can be subdivided into regions of different stability depending on the sign of *ϕ*(*r*): it will be unstable in the region where *ϕ*(*r*)<0 and stable when *ϕ*(*r*)>0. More generally, for other geometries described by a vorticity field *ω*_{z}, the Rayleigh discriminant can be written as *et al.* [34] made use of these expressions in order to identify potential instability regions in a backward-facing step flow and characterize the three-dimensional global instability. In the present case, we will see that the study of the local Rayleigh discriminant is a useful tool to predict the centrifugal stability of the forced cylinder wake problem, which lacks symmetry simplifications. In figure 7, the instantaneous flow streamlines along with the Rayleigh discriminant *ϕ*(*x*,*y*) are represented for a non-forced flow around a cylinder at *Re*=100, where the flow produces the Bénard–von Kármán vortex shedding. Two distinct regions of potential centrifugal instability exist: one near the stagnation point, where a concave streamline constitutes a Görtler-like geometry [35]; and another one in the near wake side, where the curvature of the streamlines around the vortex formation region corresponds to a Taylor–Couette geometry. At *Re*=100, nevertheless, viscosity prevents the development of three-dimensional instabilities, which never appear for the case shown in figure 7. When the rotational oscillatory forcing is applied, negative values of *ϕ* appear mostly in regions close to the cylinder. In what follows, we define the characteristic length scale *d*_{c} of regions potentially unstable giving a local Rayleigh criterion to analyse the stability properties of the forced wake.^{1}

Despite the flow complexity, it is possible to reduce the problem to investigate solely the centrifugal instability of the two-dimensional base flow and its relationship with the forcing parameters. We calculate the Rayleigh discriminant *Re*=100. In figure 8, we present three snapshots of *f*^{+}=0.75, *A*=4.00 and *Re*=100, where *α*∈[0,2*π*] is the forcing phase. For other forcing parameters, we obtain the same qualitative features than what we describe for figure 8. We observe that a ‘corona’-like region appears around the cylinder with negative values of *ϕ*. Figure 8*a* shows the phase when *ϕ* is the most negative, where we can expect the strongest possible centrifugal instability with the highest growth rate [36]. The location

Figure 8*b*,*c* describes the evolution of *x*-symmetric regarding the forcing phase, *ϕ*(*x*,*y*,*α*)=*ϕ*(*x*,−*y*,*α*+*π*). In figure 9, we show the variation of the lift coefficient *L* the resulting lift force, which is correlated with the phase reference *α*. The local radius

As we have already pointed out, we can extract a convenient length scale in order to adapt our problem to the pure centrifugal instability framework, allowing us to compare our results with previous works. If we consider the forcing phase that corresponds to *α*=4 in figure 9, the two-dimensional flow streamlines are depicted by figure 10*a*. (1.1), the Rayleigh discriminant is obtained and presented in figure 10*b*. It is worth mentioning that a *y*-symmetric field is retrieved for *α*≃1 that corresponds to the other maximum of |*ϕ*|. Image processing is used in order to extract a length scale from a contour plot of the Rayleigh discriminant obtained from equation (1.1) as shown in figure 10*c* (see appendix for details). The mean radius represented in figure 10*c* determines the length scale *d*_{c} related to the size of the unstable region for two-dimensional flow. It is shown in figure 11*a* for different forcing frequencies at a fixed forcing amplitude *A*=4.00 together with the size λ_{z}/2 of the centrifugal rolls that develop in the three-dimensional flow. Both *d*_{c} and λ_{z}/2 follow the same ( *f*^{+})^{−1/2} trend, supporting the idea of the pulsed Stokes layer. Their ratio, around a value of 3, is plotted in figure 11*b*. Given that the flow is under non-stationary forcing, the rolls are formed periodically symmetric with respect to the *x*-axis. Besides, *d*_{c} has been determined for a particular phase *α*≃4, where the instability is most intense, but the centrifugal instability region changes its size. These arguments may explain the difficulty for estimating *d*_{c} and the scale difference between λ_{z}/2 and *d*_{c}. Nevertheless, we can observe that the main behaviour is shared between λ_{z} and *d*_{c}, therefore, a centrifugal instability region observed in two-dimensional simulations is in agreement with the three-dimensional instability that develops in three-dimensional DNS.

## 5. Conclusion

This work gives a new view about three-dimensional instabilities in wake flows. In the context of forced wakes at moderate Reynolds numbers, we found a new transition that leads to the formation of three-dimensional structures. The instability shares aspects that were previously studied for centrifugal pulsed flows. Taylor–Couette-like vortices develop from a definite threshold of forcing parameters ( *f*,*A*) and these structures are modified by the incoming flow. For this complex instability, two-dimensional evaluation of the Rayleigh discriminant *ϕ* may give a fast criterion to determine whether a wake flow becomes three-dimensional or not. We found from streamline shapes and the spatial distribution of *ϕ* that the problem shares some analogy in relation to eccentric Taylor–Couette flows.

As two-dimensional forcing in wakes may indeed trigger three-dimensional structures, this behaviour must be taken into account in flow control schemes. Streamlines which become too ‘bent’ by forcing in wakes can make evident strong negative values of the Rayleigh discriminant *ϕ* and thus the possibility of a centrifugal instability.

On the other hand, this simple problem can offer an interesting benchmark to study instabilities and transition to turbulence from oscillatory rotation.

## Data accessibility

In order to reproduce all the calculations included in this paper, Gerris [25] is available free of charge under the Free Software GPL licence and our code files are available as electronic supplementary material. A video regarding three-dimensional vortices generation (figure 4*a*) is also accessible as electronic supplementary material.

## Funding statement

Concerning funding of travel and stays, all the authors had financial support from the LIA PMF-FMF (Franco-Argentinian International Associated Laboratory in the Physics and Mechanics of Fluids), Argentina (CONICET)- France (CNRS).

## Authors contributions

J.D. initially observed in preliminary results the three-dimensional modulation in the vicinity of the cylinder that defined the purpose of the present study. J.D. carried out the numerical simulations, participated in data analysis and discussion, figures elaboration and in writing the manuscript; R. G.-D. contributed to the analysis and interpretation of data, figures design and also writing the manuscript; J.E.W. gave important theoretical insights which led to the set-up of the numerical simulations. J.E.W. analysed and discussed the data and revised the manuscript critically. All authors gave final approval for publication.

## Conflict of interests

We have no competing interests.

## Acknowledgements

We thank B. Thiria for sharing the visualization images presented in the insets of figure 1. We acknowledge the organizers and participants of the 18th International Couette–Taylor Workshop at Twente University for fruitful discussions.

## Appendix

### Appendix A. Determination of the centrifugal instability region length

The choice of a characteristic length of the centrifugal instability region from the Rayleigh discriminant scalar fields is not straightforward as we observe figure 8. we choose to select the forcing phase that corresponds to the minimum value of *ϕ*, the most unstable state. Figure 10*b* presents such state, but the *ϕ* scalar field needs to be more clear in order to extract a length *d*_{c}. Simple image processing functions, erosion and dilation, are applied successively to the scalar field in order to obtain figure 10*c*, where a clear shape is noted. We found that such shape has an aspect that resembles an eccentric cylinder gap. Therefore, we choose as a characteristic length the mean radius of this gap *r* the shape radius varying with the angular coordinate *φ*.

#### (a) Convergence analysis for direct numerical simulation

In order to ensure that the results do not depend on the size of the domain we chose, we performed a convergence analysis for the two-dimensional case. Given that the domain size of the reference study is *L*_{x}×*L*_{y}=20*D*×10*D*, we label it as *L*1.00. As we selected larger domains which scale as [1.25;1.50;1.75;2.00] the reference study length, we label them *L*1.25, *L*1.50, *L*1.75 and *L*2.00. For these scaling lengths, we plotted mean flow profiles for the streamwise component of the velocity *u*_{m} at three different *x* positions *x*=0.5*D*, *D*, 2*D* in figure 12*a*. Figure 12*b* presents for the same direction, fluctuations intensity *u*_{rms} profiles for the same different *x* positions. We observe that changing the domain size does not modify the flow dynamics. Lift coefficient is also calculated for each case as it is presented in figure 13. We also observe good agreement between the different scaling domains.

## Footnotes

- Received January 7, 2015.
- Accepted April 1, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.