E Durán Venegas

S Le Dizès

Generalized helical vortex pairs

New solutions describing the interaction of helical pairs of counter-rotating vortices are obtained using a vortex filament approach. The vortices are assumed to have a small core size allowing the calculation of the self-induced velocities from Biot-Savart law using the cut-off theory. These new vortex structures do not possess any helical symmetry but they exhibit a spatial periodicity and are stationary in a rotating and translating frame. Their properties, such as radial deformation, frame velocity or induced flow, are provided as a function of the four geometric parameters characterizing each solution. Approximate solutions are also obtained when the mutual interaction is weak. This allows us to provide explicit expressions for the rotation and translation velocities of the structure in this limit. First order corrections describing helix deformation are also calculated and used for comparison with the numerical results. The variation of the vortex core size induced by the helix deformation is also analysed. We show that these variations have a weak effect on the shape and characteristics of the solutions, for the range of parameters that we have considered. The results are finally applied to rotor wakes. It is explained how these solutions could possibly describe the far wake of an helicopter rotor in vertical flight.

Introduction

Helical vortex structures are found in the wake of rotors in the context of helicopters [START_REF] Leishman | Principles of Helicopter Aerodynamics[END_REF], wind turbines [START_REF] Vermeer | Wind turbine wake aerodynamics[END_REF] and propellers [START_REF] Wald | The aerodynamics of propellers[END_REF]). In the present work, we propose generalized helical vortex solutions to describe these structures in the far field.

Having a good description of the wake is now recognized to be essential to optimize rotor efficiency. Since the early momentum theory by [START_REF] Rankine | On the mechanical principles of the action of propellers[END_REF] and [START_REF] Froude | On the elementary relation between pitch, slip and propulsive efficiency[END_REF], numerous improvements have been proposed by [START_REF] Betz | Windenergie und ihre Ausnützung durch Windmühlen[END_REF], [START_REF] Joukowski | Théorie tourbillonnaire de l'hélice propulsive[END_REF] and [START_REF] Goldstein | On the vortex theory of screw propellers[END_REF] (see [START_REF] Sørensen | General momentum theory for horizontal axis wind turbines[END_REF], for a recent review). Of particular interest for the present study is the model of [START_REF] Joukowski | Théorie tourbillonnaire de l'hélice propulsive[END_REF] (see [START_REF] Okulov | The rotor theories by Professor Joukowsky: Vortex Theories[END_REF] where the wake from each blade is described by a bound vortex on the blade, and two free vortices of opposite circulation detached from the hub and the tip of the blade. For a N blade rotor, these vortices form in the far-wake, a uniform helical braid composed of N vortices of circulation Γ plus a central vortex of circulation -N Γ .

Perfect helical vortex solutions have been the subject of numerous works since the early works by [START_REF] Kelvin | Vibrations of a columnar vortex[END_REF][START_REF] Da Rios | Vortici ad elica[END_REF], [START_REF] Levy | The steady motion and stability of a helical vortex[END_REF] and [START_REF] Joukowski | Théorie tourbillonnaire de l'hélice propulsive[END_REF]. For instance, [START_REF] Betchov | On the curvature and torsion of an isolated vortex filament[END_REF] and [START_REF] Kida | A vortex filament moving without change of form[END_REF] showed using the local induction approximation that these structures rotate and translate without changing form. When the vortex is (infinitely) thin, [START_REF] Hardin | The velocity field induced by a helical vortex filament[END_REF] † provided an exact expression for the induced velocity field inside and outside the cylinder containing the helix. [START_REF] Ricca | The effect of torsion on the motion of a helical vortex filament[END_REF], [START_REF] Kuibin | Self-induced motion and asymptotic expansion of the velocity field in the vicinity of a helical vortex filament[END_REF] and [START_REF] Boersma | On the self-induced motion of a helical vortex[END_REF] showed how the singularity of Hardin expression can be extracted to compute the self-induced motion of the helix. Velasco Fuentes (2018) recently applied these results to compute the motion of vortex elements on a helical vortex, emphasizing the role of tangential velocities. In these works, the vortex core size is implicitely assumed to be a small parameter. [START_REF] Lucas | A family of helically symmetric vortex equilibria[END_REF] and Selçuk et al. (2017b) have shown how helical vortices with a thick core can be obtained numerically by enforcing the helical symmetry in the governing equations.

Solutions with a more complex geometry are scarce in the literature. [START_REF] Walther | A numerical study of the stabilitiy of helical vortices using vortex methods[END_REF] had looked at equilibrium solutions composed of undeformed helical vortex pairs of same pitch but different radius. They demonstrated that undeformed helical vortex pairs of identical pitch and opposite circulation were possible for a particular radius ratio. This analysis was further pursued by [START_REF] Okulov | An acentric rotation of two helical vortices of the same circulations[END_REF] for helical vortices with same sign circulation. Reducing the framework to nearly parallel filaments [START_REF] Klein | Simplified equations for the interaction of nearly parallel vortex filaments[END_REF], [START_REF] Kwiecinski | Dynamics of nearly parallel interacting vortex filaments[END_REF] were recently able to provide more exotic solutions.

Our objective is to extend Joukowski model to configurations formed of N counterrotating helical vortex pairs when the root vortices are not on the axis but at a finite non-zero radius. Such a solution could be used to describe a rotor with a non constant circulation. As initially explained by [START_REF] Goldstein | On the vortex theory of screw propellers[END_REF], when the circulation is not uniform, vorticity is emitted all along the blade with a circulation per length equal to -∂ r Γ . The subsequent evolution is a complicated roll-up process of the vortex sheet into vortices of positive and negative circulation. In the simplest case where the circulation profile has a single maximum, each blade is expected to create two concentrated vortices of opposite circulation. In that case, after the roll-up phase, the flow is then composed of N pairs of vortices of opposite circulation for a N blade rotor. It is this flow that we want to analyse in the far field.

For this purpose, we use a free-vortex method. Such a method is now commonly used for direct numerical simulations [START_REF] Cottet | Vortex Methods: Theory and Applications[END_REF][START_REF] Winckelmans | Vortex methods and their application to trailing wake vortex simulations[END_REF]. Here, we use its simplest form by considering thin vortex filaments. Each vortex is discretized into straight-line segments and advected by the flow. This approach is explained in length in the textbook by [START_REF] Leishman | Principles of Helicopter Aerodynamics[END_REF].

The paper is organized as follows. In section 2, we introduce the vortex method framework ( §2.1 and §2.2) and review some preliminary results on helical vortices ( §2.3 and §2.4) in order to introduce the geometric parameters characterizing our solutions ( §2.5). In section 3, we first derive an approximation for the solutions by neglecting helix deformations ( §3.1). Then, we characterize the deformation of the helices ( §3.2) and the properties of the solutions in terms of structure velocities ( §3.3) and induced velocities ( §3.4). The relevance of the solutions for the description of helicopter wakes is addressed in section 5, before a brief conclusion provided in the last section.

Vortex filament framework

In this section, we describe the numerical method that we use. The method is based on a Lagrangian description of the vortices and a discretization of the vortices into segments. This discretization process allows us to use explicit expressions for the induced and selfinduced velocities. After having explained these two aspects, we apply the method to special cases: first to single helices to validate the method, then to non-interacting helical pairs to introduce the geometrical parameters that are used to define the solutions.

Lagrangian description

We consider small core size vortices which can be described as vortex filaments. In this framework, all the vorticity is concentrated along lines which move as material lines in

dt = U (ξ) = U ∞ + U ind (ξ), (2.1)
where ξ is the position vector of the vortex filament, U the velocity field, composed of an external field U ∞ and a field U ind (ξ) induced by the vortex filaments. When there are N vortices, this induced velocity is given by the Biot-Savart law

U ind (ξ) = N j=1 Γ j 4π (ξ j -ξ) × dτ j |ξ j -ξ| 2 , (2.2)
where the integrals cover each vortex filament defined by its circulation Γ j , its position vector ξ j and its tangent vector τ j . On the vortex line, the Biot-Savart integral is singular, and the self-induced velocity diverges. To avoid this singularity, one has to assume a small but finite core size a. The self-induced motion is then obtained by an integral of the same form but without considering the interval [-δa, δa] around the singular point. This so-called cut-off method is explained in length in textbooks (see [START_REF] Saffman | Vortex dynamics[END_REF]. The value of δ depends on the vortex core model. Here, we shall assume a Gaussian vorticity profile for which δ ≈ 0.8736.

Vortex discretization

We follow the vortex method approach described for instance in [START_REF] Leishman | Principles of Helicopter Aerodynamics[END_REF]. Each vortex filament is discretized in small segments in order to compute the velocity field and follow its displacement (see figure 1a).

The velocity field induced by a given segment [ξ n i , ξ n+1 i ] of the ith vortex at a point ξ m j can be calculated explicitly as

U seg i,n (ξ m j ) = Γ i 4π ((1 -|r j,m i,n | 2 )|r j,m i,n | + (1 -|r j,m i,n | 2 )|r j,m i,n+1 |)(r j,m i,n × r j,m i,n+1 )r j,m i,n • r j,m i,n+1 ((r j,m i,n • r j,m i,n+1 ) 2 -|r j,n i,n | 2 |r j,m i,n+1 | 2 )|r j,m i,n ||r j,m i,n+1 | , (2.
3) where r j,m i,n = ξ m j -ξ n i . This expression is defined everywhere except at the points ξ n formula. We obtain

U loc j,m = Γ j 4πρ m j ln ∆φ m j ρ m j δa b m j , (2.4)
where ρ m j and b m j are the curvature radius and binormal vector at ξ m j respectively, and ∆φ m j is the angle of the arc of circle (ξ m-1 j , ξ m+1 j ) as illustrated in figure 1b. The total induced velocity at ξ m j is then given by an expression of the form

U ind (ξ m j ) = U loc j,m + N n=1 pn i=1 U seg i,n (ξ m j ), (2.5)
where p n is the number of points discretizing the nth vortex, and assuming implicitly that U seg j,m (ξ m j ) = U seg j,m-1 (ξ m j ) = 0 . This formula is tested against direct calculation of the cut-off integral in figure 2 for a single helix (see also [START_REF] Gupta | Accuracy of the induced velocity from helical vortices using straight-line segmentation[END_REF]. We observe that a good approximation is obtained as soon as the helix is divided in 25 or more segments by turn when the local contribution is included. When the local contribution is not taken into account, a much larger number of segments by turn of order O(2πρ/a) is needed to obtain a good approximation. In practice, we use p n = 30 in most calculations.

Vortex ring and helical vortex

A vortex ring and a helical vortex are examples of vortex structures that move in space at a constant speed without deformation. For these particular vortices, there exist a unique moving frame where all the vortex elements are steady. For the ring, this frame is translating along the ring axis. For the helix, it is both translating and rotating along the helix axis.

The variation of the rotation rate and axial speed of a right-handed helix with respect to the pitch is given in figure 3 for a typical vortex core size. As already noticed by Velasco Fuentes (2018), it is interesting to see that the rotation changes sign as the pitch varies. For a left-handed helix, the rotation rate is the same but the axial speed is opposite. In figure 3, our numerical results are compared to theoretical approximations Figure 3: Non-dimensionalized rotation rate ΩR 2 /Γ (a) and axial speed W R/Γ (b) of a right-handed helix of circulation Γ , radius R, pitch h and core size a as a function of h/R for a/R = 0.03. The solid black curves correspond to the numerical results obtained in this work. The solid grey curves correspond to the theoretical expressions (A 1a,b), the dashed grey curves correspond to the results obtained by Velasco Fuentes (2018), that is the same expressions without the 1/4 terms. Both theoretical results are for an equivalent Rankine vortex of core size a/R = 0.0408. based on Hardin (1984) expressions. The dashed grey curves are obtained by taking the mean value of Hardin expression at Ra and R + a as it was done by Velasco Fuentes (2018). The solid dashed line is the same expression corrected by a 1/4 term associated with local curvature [START_REF] Ricca | The effect of torsion on the motion of a helical vortex filament[END_REF][START_REF] Kuibin | Self-induced motion and asymptotic expansion of the velocity field in the vicinity of a helical vortex filament[END_REF][START_REF] Boersma | On the self-induced motion of a helical vortex[END_REF]. This correction term permits to take into account the deformation of the vortex core induced by curvature. For completeness, we provide these expressions of Ω and W in appendix A. In these theoretical works, a Rankine vortex model (uniform vorticity in the core) is used, while we use a Gaussian vorticity profile. We have thus applied the correction factor a Rankine ≈ 1.36 a Gaussian to the core size to account for the different vortex models [START_REF] Widnall | The stability of a helical vortex filament[END_REF][START_REF] Saffman | Vortex dynamics[END_REF]. As it can be seen on this figure, the agreement between the numerical results and Velasco Fuentes (2018) is good and almost perfect for both Ω and W when the correction term is included. This comparison is a strong validation of our numerical approach.

For both rings and helices, there exist infinitely many other moving frames where the vortex structure is steady. The displacement associated with this frame just has to remain tangent to the structure. The condition of steadiness for the frame velocity V F can then be written as

(V F (ξ m j ) + U ind (ξ m j )) × τ m j = 0. (2.6)
In this frame, the vortex elements are moving along the vortex structure. For a ring, any rotation around the ring axis can for instance be added. For an helix of pitch h, any rotation and translation along the helix axis can also be added if the rotation rate Ω a and axial speed W a of this additional movement satisfy

W a /Ω a = ±h/2π, (2.7)
where the sign is + for right-handed helices, andfor left-handed helices.

An helical braid composed of N identical vortices of same axis, separated with each other by an azimuthal angle 2π/N also forms a steady solution in an adequate frame. When N = 1, a straight hub vortex placed on the central axis can be added without introducing any deformation on the helices. This is not possible when N = 1. The external helix indeed generates an horizontal velocity on the axis that induces a radial displacement of any straight structure place at this position. In that case, we expect the hub vortex to deform and to move out from the rotation axis. Our objective is to describe such a structure, as well as the structures composed of N vortex pairs with root vortices not on the rotation axis. We shall see that there still exist steady solutions in those cases.

Non-interacting helical pairs

In order to understand the parameters defining the solutions, it is useful to consider first a simple configuration composed of two co-axial helices of opposite circulation and same core size a. The internal and external helices are defined by their radii R int and R ext and pitches h int and h ext , as illustrated in figure 4. They also depend on their relative orientation characterized by a parameter κ, which is +1 if both helices have the same orientation, and -1 if they have opposite orientations.

If κ = -1 or if h ext = h int , this pair of vortices is no longer helically invariant. However, it exhibits a certain spatial periodicity. The periodicity is in the sense that there exists an axial distance L > 0 and an angle 0 φ < 2π such that the radial locations of each vortex are invariant by the double operation of translation by L and rotation by φ (see figure 4). The parameters L and φ are given by

1 L = 1 h int - κ h ext , (2.8a) φ = 2π L min(h int , h ext ) -E L min(h int , h ext ) , (2.8b)
where E(x) denotes the integer part of x. When κ = -1, L is smaller than h ext and h int : both helices have performed less than a complete rotation in one axial period L. When κ = 1, L is always larger than the smaller pitch; the helix with the smaller pitch has then performed more than a complete rotation in a period. The other helix has performed just one complete revolution less.

When h ext = h int , the undeformed helical pair corresponds to a steady solution only if the mutual induction of one helix on the other is negligible. This occurs when the core size becomes sufficiently small. In this limit, the locally induced velocity is the dominant contribution to the induced velocity which then becomes constant and oriented along the local binormal vector. Each helix then behaves as if the other helix was not present. A priori, they rotate and translate at different speeds. But, owing to the possibility to F and W (0) F are obtained. Solid and dashed lines represent the two helices at t = 0 and t = 1 respectively. The frame velocity is given by the vector connecting the crossing points of solid lines with dashed lines. The axial and azimuthal components of this vector provides

W (0) F and Ω (0) F respectively.
add any displacement along the helical line, it is possible to find a frame where both helical structures become steady. The frame selection is graphically explained in figure 5b. In that figure, we plot each helix in the (φ, z) plane at t = 0 (solid lines) and t = 1 (dashed lines) using two different colors. Each helix corresponds to a straight line with a slope equal to the helix pitch. The self-induced velocity of each helix, together with their decomposition on the axial and azimuthal direction is also indicated. Any vector that connects any two points from the lines at the two distinct times provides a possible frame velocity vector that keeps the considered helix steady. The vector that keeps both helices steady is the one that connects the crossing points associated with each instant. Such a vector exists as soon as the helix lines are not parallel in the (φ, z) plane, that is if

h ext = h int or κ = -1.
The angular velocity Ω (0) F and axial velocity W (0) F of the frame are given from the self-induced velocity of each helix by

Ω (0) F = 2π(W SI int -W SI ext ) + (Ω SI ext h ext -Ω SI int κh int ) h ext -κh int , (2.9a) W (0) F = W SI ext + h ext 2π (Ω (0) F -Ω SI ext ).
(2.9b)

The second equation shows that condition (2.7) is satisfied by the external vortex. It is immediate to obtain

W (0) F = W SI int + κh int 2π (Ω (0) F -Ω SI int ), (2.10)
that guarantees that (2.7) is also satisfied by the internal vortex.

Without restriction, we can assume the external helix to be right-handed with a positive circulation Γ . The internal helix has then a negative circulation -Γ . It is right-handed if κ = 1, left-handed if κ = -1.

Parameters defining the deformed helical structures

The two-helix structure obtained above is no longer a solution if the two helices interact. Indeed, the velocity field of one helix on the other contains a radial component that moves the structure radially. Each helix is therefore expected to be deformed by the field induced by the other helices. Being inspired by the non-interacting solutions, we shall search for steady solutions that still exhibit a spatial periodicity. We consider solutions composed of N pairs of counter-rotating vortices (the external vortex having a positive circulation Γ , the internal a negative circulation -Γ ), with a 2π/N azimuthal symmetry. By construction, we then assume that the solutions are invariant by the transform φ → φ + 2π/N . We also assume that there exist an axial distance L > 0 and an angle φ satisfying 0 φ < 2π/N , such that the solutions are invariant by the double operation z → z +L and φ → φ+ φ. We do not want the solution to repeat several times in a spatial period, so we further assume that there is a single location in an axial period L where internal and external vortices are at the same azimut. We shall choose this particular azimut to define from their radial positions the radius R int and R ext of the internal and external vortex (R int < R ext ). We also define the mean pitch h int and h ext for each vortex from the azimuthal angle covered in an axial period. For the external vortex, if this angle is φ ext , we have h ext = 2πL

φext . If we add the vortex core size a that we assume identical and constant for all the vortices, we obtain 5 spatial length scales from which we can form 4 independent non-dimensional parameters:

R * = R int R ext , h * = h ext R ext , α = h int h ext , ε = a R ext . (2.11)
To these 4 real positive parameters, we should add the number N of vortex pairs and the index κ = ±1 that defines the relative orientation of internal and external vortices.

In most cases, we shall keep κ = 1 and N = 1. The parameter R * will be varied between 0 and 0.75, h * between 0.1 and 2, α between 0.5 and 2. The parameter ε will always be considered small, and typically equal to 0.03. Even for this small value of ε, finite core size effects can become important if h and α are too small. This provides a limitation on the values of the parameters that we can consider. Here, only the extreme cases (h ≈ 0.1 and α ≈ 0.5) are expected to give rise to consequent finite core size effets.

In the paper, the vortex core size is also assumed to be constant. This approximation is discussed in section 4.

Note that both the normalized period L/R ext and the angle φ can be obtained from the above geometrical parameters:

L R ext = h * N |1/α -κ| , φ = 2π N 1 |1/α -κ| -E 1 |1/α -κ| . (2.12)
With each solution is associated a moving frame where the solution is steady. From the angular velocity Ω F and the axial velocity W F of the frame, we can construct two other dimensionless parameters using the external radius R ext and the circulation Γ of the vortices:

Ω = R 2 ext Ω F N Γ , W = R ext W F N Γ .
(2.13)

These two parameters characterizing the frame velocity are functions of the 6 geometrical parameters R * , h * , α, ε, N and κ.

Deformed helical vortex pairs

In this section, we describe the deformed helical structures as the geometrical parameters vary. After having introduced an approximated solution, we successively study the geometrical characteristics, the frame velocity and the induced velocity.

First order approximation

As already mentioned above, as soon as the mutual induction of one helix on the others is taken into account, the vortex structure does not remain helical. For a given vortex parametrized by its radial position r(z) and angular position φ(z) as a function of the z coordinate, the condition of steadiness (2.6) reduces to

dr dz = V ind r V ind z -W F , dφ dz = Ω ind -Ω F V ind z -W F . (3.1)
Each induced velocity component is composed of two contributions, a self-induced contribution V SI and a contribution induced by the other vortices V M I . The radial selfinduced velocity of an helix being null, we clearly see from the first equation that the radial deformation will be associated with the mutual induction, and more precisely with the radial component V M I r of the mutually induced velocity. A first order correction to the undeformed solution can be obtained by solving these two equations assuming that the velocity fields on the right-hand side are evaluated at the undeformed location:

dr (1) dz = V M I r (r 0 , φ 0 (z), z) V M I z (r 0 , φ 0 (z), z) + V SI z -W (1) F , (3.2a) dφ (1) dz = Ω M I (r 0 , φ 0 (z), z) + Ω SI -Ω (1) F V M I z (r 0 , φ 0 (z), z) + V SI z -W (1) F . (3.2b)
For the mutual induction, we use the formula given by [START_REF] Hardin | The velocity field induced by a helical vortex filament[END_REF] for a perfect helix.

The corrected frame velocities Ω

F and W

(1) F are obtained by using the definition of L for the internal and external vortex:

φ (1) int (L) -φ (1) int (0) = 2πL h int = L 0 Ω M I int (r (0) int , φ (0) int (z), z) + Ω SI int -Ω (1) F V M I z,int (r (0) int , φ (0) int (z), z) + V SI z,int -W (1) F dz , (3.3a) φ (1) ext (L) -φ (1) ext (0) = 2πL h ext = L 0 Ω M I ext (r (0) ext , φ (0)
ext (z), z) + Ω SI ext -Ω (1) F V M I z,ext (r (0) ext , φ (0)
ext (z), z) + V SI z,ext -W (1) F dz . (3.3b)
Note that equations (3.3a,b) give (2.9) if we neglect the mutual induction. The condition of periodicity of the radial deformation does not give an additional constraint because it is automatically satisfied for each vortex. This simple first order approximation for the helix deformation is compared to numerical results for two typical examples in figure 6. We clearly see that the agreement strongly depends on the pitch ratio α. This approximation tends to underestimate the deformation of the external vortex but to overestimate that of the internal vortex. The error is always larger for the internal vortex. The maximum error can then be quantified using

E (1) r,int = max(|r int -r (1) int |) R int (3.4)
which measures the maximum deviation between numerical and first order solutions. This quantity is plotted in figure 7 as a function of h * for N = κ = 1 and various α, R * and ε. This figure shows that except for very large R * (R * = 0.7), the error increases with h * . The error also increases with ε, R * and with the distance of α to 1. It becomes particularly large (superior to 30%) for large α and large h * . Note however that the error remains small for α ≈ 1, small R * and small ε.

Characterization of the helix deformation

The first order approximation clearly demonstrates that the radial deformation increases when the radial velocity induced by the other vortices grows. This occurs when internal and external vortices get closer to each other, that is when R * increases. This is illustrated on figure 8 where a perspective view of a single pair is shown for different values of R * , the other parameters being fixed.

In order to quantify the level of deformation, we introduce two quantities

∆r int max = max(|r int -R int |) R int , ∆r ext max = max(|r ext -R ext |) R ext (3.5)
that measure the maximum displacement of internal and external vortices. The growth of ∆r int max and ∆r ext max with respect to R * is quantified in figure 9c. The effect of the number of vortices is clearly visible. The vortex deformation strongly decreases with N . For R * = 0.7, the fluctuation of the internal vortex reaches 30% for a single vortex pair while it is less than 0.1% for three vortex pairs. Note that when R * goes to zero, the deformation of the external vortex vanishes whatever N . When N = 1, this is the same for the internal vortex. In that case, we recover the Joukowski's model with a straight vortex hub on the helix axis. For N = 1, the limit R * → 0 is by contrast singular.

The variations of ∆r int max and ∆r ext max with respect to h * and α are shown in figure 10. The increase of the vortex deformations with h * is associated with the decrease of the axial component of the induced velocity. Indeed, for large h * , the vortices get aligned with the respect to z axis. They therefore mainly induce a velocity field in the radial and azimuthal directions. The ratio V r /V z that defines the slope of the deformation thus gets large, implying large ∆r int max and ∆r ext max . In the opposite, when h * goes to zero, V r /V z goes to zero as well: the helical vortices are thus no longer deformed.

Concerning the effect of α, the increase of the deformation of the external helix with α can be understood by the same argument. It is associated with a decrease of h int , and therefore an increase of the ratio V r /V z . The variation of ∆r int max with respect to α is less simple. For h * < 1, ∆r int max is maximum for α close to 1. This value α = 1 is special for κ = 1 because the axial period L and the frame axial velocity W F get infinite [see equation (2.12)]. It therefore corresponds to a singular limit in our description. However, the radial positions r int and r ext of internal and external vortices can still be plotted as a function of z/L and we obtain a well-defined curve as α → 1. Similarly, W F /L converges to a non-zero constant as α → 1, which means that a finite time T = L/W F is needed to advect a perturbation on the period L at the velocity W F . The singular case α = 1 could therefore be described by an alternative way by considering the solution in the frame moving at the velocity W F . In this frame it should correspond to the temporal dynamics of perfect helices with the same R * , h * and ε. This is indeed what we have checked in figure 11. In figure 11(a), we show that the variation of r int as a function of z/L for α close to one, is well described by the variation of r int as a function of t/T in the temporal problem. We also check in figure 11(b) that L/W F converges to the temporal period obtained in the temporal problem as α → 1.

Structure velocities

As explained above ( §3.1), we can obtain different approximations of the frame velocity Ω F and W F by neglecting the helix deformations and/or the mutual induction. The leading order approximation (Ω F . In figure 12, we have compared both approximations to numerical results for a typical case. We clearly see that the leading order approximation does not capture the variations of Ω F and W F with h * , while the first order approximation follows both qualitatively and quantitatively these variations. By comparing other configurations, we have observed that the first order approximation provides a good approximation of Ω F and W F as soon as R * 0.5 and ε 0.1. In practice, these approximations have been used as guess values for the numerical calculation.

The variations of the frame velocities Ω and W with respect to h * and α are shown in figure 13. Both contour plots exhibit similar features: a same singularity line α = 1 and a single contour where Ω and W vanish. These zero level contours are different for Ω and W but both cross the singular line α = 1 at the same value (here h * ≈ 0.175). This special point on the singular line α = 1 corresponds to the particular solution obtained by [START_REF] Walther | A numerical study of the stabilitiy of helical vortices using vortex methods[END_REF]. For these parameters, the vortices are undeformed helices of same pitch. There therefore exist infinitely many frames where the helices are stationary as any values of axial speed W a and angular velocity Ω a can be added provided (2.7) is satisfied. This explains the degeneracy observed at this point in figure 13. These qualitative features do not depend on ε and R * . It is interesting to note that a contour of Ω may cross twice a contour of W (look at the saddle point region close to (α, h) ≈ (1.7, 0.4)). The coordinates of the two crossing points then correspond to couples of parameters (α, h) having the same frame velocities (Ω, W ).

By contruction, the vortex elements are advected along the stationary vortex structure. This tangential velocity is different for the internal and the external vortex and varies along the vortex structure, as illustrated in figure 14a. This variation is associated with the deformation of the helices. It is then important when the level of deformation is high. For each vortex, we define a mean tangential velocity Vtan and a measure ∆V tan of the fluctuation around this mean using

Vtan = 1 L z0+L z0 V tan (z)dz , ∆V tan = max |V tan -Vtan | | Vtan | . (3.6)
The measures ∆V ext tan and ∆V int tan for the external and internal vortex are plotted as a function of h * in figure 14b for a few cases. We do observe an increase of the tangential velocity fluctuation with h * and R * , in agreement with the increase of the vortex deformation (see figure 10).

The mean tangential velocity of each vortex is shown in the (α, h) plane in figure 15. A positive value corresponds to an advection in the positive axial direction, a negative value

= 0.5, α = 1.2, ε = 0.03, N = 1, κ = 1.
to an advection in the opposite direction. Not surprisingly, the mean tangential velocity blows up as α → 1 like Ω F and W F . It is also interesting to note that the contour curves are similar (in shape) for both vortices and close to those obtained for Ω F R 2 ext /Γ in figure 13(a).

In figure 16a, we have displayed on the same plot the parameters for which mean tangential velocities and Ω F vanish. We clearly see that mean tangential velocities and Ω F vanish for almost the same parameters. This means that there is a very small region of parameters around the line Ω F = 0 where internal and external vortices propagate in different directions. This region is delimited by the solid lines shown in figure 16a. Everywhere else, both vortices propagate in the same direction. We shall see in section 5 that this condition on the direction of propagation of the vortices is necessary for the solution to describe the flow generated by a rotor.

It is interesting to compare more precisely the mean tangential velocity with the velocity associated with the moving frame. Assuming that each structure is approximatively an helix, this tangential "frame" velocity is given by

V int F tan = - Ω F R ext R * + W F h * /(2παR * ) 1 + h * 2 /(2παR * ) 2 , V ext F tan = - Ω F R ext + W F h * /(2π) 1 + h * 2 /(2π) 2 (3.7)
for the internal and external vortex, respectively. The difference between V F tan and Vtan is associated with the vortex induction. In figure 16b, we can observe the similar values of Vtan and V F tan in the whole range of h * between 0.6 and 1.4 for a typical case. This means that the most important part of the tangential velocity is associated with the frame velocity and the vortex induction contribution remains in general small.

Induced flow

From the point of view of applications, it is useful to know the velocity field induced by the vortex structure. An illustration of the axial and angular components of such a field in a plane perpendicular to the structure axis is shown in figure 17. In these contour plots, the axial velocity W F and angular velocity Ω F have been subtracted such that the velocity field vanishes far from the center. The vortex cores where the velocity field is smoothed have also been indicated. We clearly see that the induced velocity field exhibits strong inhomogeneities.

In figure 18, we show the azimuthally averaged flow versus r at different axial locations. In the core region of each vortex (at a distance smaller that a for the vortex center), each velocity profile has been replaced by a linear fit. We do see small fluctuations of the profiles between different locations but the profiles remain close to the profiles generated by N pairs of perfect helices. These ideal profiles are given (for infinitely thin vortices) by [START_REF] Hardin | The velocity field induced by a helical vortex filament[END_REF] V

H z = N Γ h ext          1 - 1 α if r < R int , 1 if R int < r < R ext , 0 if r > R ext , (3.8)
ΩH = N Γ R 2 ext    - R 2 ext 2πr 2 if R int < r < R ext , 0
elsewhere .

(3.9)

The fluctuations are mainly associated with the radial displacement of the vortices. When the helices are less deformed, the fluctuations are smaller. It is interesting to note that the azimuthally averaged axial flow changes sign close to the axis when h int < h ext , that is α < 1.

For the applications, it is also useful to evaluate the mass flow rate induced by the structure. The mass axial flow rate is defined by

M = ρV ind z rdrdφ.
(3.10)

For N undeformed helical pairs, we get using (3.8)

M H = ρN Γ π R 2 ext h ext - R 2 int h int , (3.11) that is M H ρN Γ R ext = π h * 1 - (R * ) 2 α .
(3.12)

This expression provides a good approximation of the mass flow rate of the deformed structure, as observed in figure 19 for a typical example.

Note that M H changes sign when α < (R * ) 2 . In this regime, the induced axial flow is sufficiently negative close to the axis to inverse the positive mass flow rate occurring between the internal and external vortices. For the parameters of figure 19, this is expected for very small α (α < 0.25).

Analysis of the effect of a varying core size

We have seen that the tangential velocity varies along each vortex structure when it is deformed. This implies that each vortex experiences a stretching field that in principle induces a variation of its core size. In this section, our objective is to quantify this effect.

In a inviscid framework, the way the core size varies is simply given by the conservation of the mass flux in the vortex core which imposes

a 2 V tan = Cst. (4.1)
As soon as a changes, the self-induced velocity is modified and therefore a different equilibrium solution is obtained.

In figure 20, we have analysed the effect of a varying core size in an extreme case (R * = 0.8, h * = 1.4, ε = 0.03, α = 1.4, N = 1, κ = 1). As seen in figure 14(b), for this value of R * , the variations of V tan are the largest: they reach 32% for the internal vortex, and 23% for the external one. In figure 20(a), we have plotted the variation of the core size in a period for the converged solution in both vortices. The variations of the core size are weaker than of the tangential velocity as expected from (4.1). They are around 10% for the internal vortex, and 7% for the external vortex, with respect to the mean core size a = 0.03. In figure 20(b), the radial position of the vortices is shown. The solution with a varying core size is compared to the solution with a constant core size. We observe that the difference between both solutions is very small. The largest gaps between both radial positions are 0.78% and 0.66% for internal and external vortices respectively. In terms of moving frame velocities, the differences are also extremely small: we obtain the values W = 1.826 and Ω = 3.697 for a varying core size, while we had W = 1.809 and Ω = 3.610 for a constant core size.

This comparison garantees that the effects of a varying core size is negligible for all the cases that we have considered. It a posteriori justifies the use of the approximation of a constant core size in our work.

Discussion in the context of rotors

In this section, we discuss our solutions in the context of rotor wakes. Our objective is to identify as a function of the geometrical parameters which far wake situations the solutions could possibly describe. In other words, if a solution was able to represent the flow generated by rotor, which rotor configuration it would be.

In the previous sections, we have obtained the frame where each solution is stationary. If the flow was created by a rotor, this frame would necessarily be a frame attached to a blade. In other words, the angular velocity -Ω F would be the rotation rate of the rotor and -W F the external wind velocity. Moreover, in the far field, the vortices which have been created by the rotor, would have to move away from it. From the direction of propagation of the vortices, which is given by the sign of the mean tangential velocity, we can then deduce the side where the rotor should be. For example, if Vtan > 0, the vortices move in the positive direction, the rotor should then be located on the negative side.

By analysing the directions of propagation of the vortices and of the external wind, one can built the diagram shown in figure 21. The different domains are limited by the curves W F = 0 and Ω F = 0 and the line α = 1, which correspond to changes of signs of the direction of propagation of the vortices or of the external wind. In this figure, we have also indicated the typical azimuthally averaged axial flow corresponding to each regime. For example, in the white region on the right of the line α = 1, W F > 0, V ext tan < 0, V int tan < 0 and M > 0: both vortices moves in the same negative direction as the external wind, which is opposite to the direction of the mass flow rate. Assuming a positive axis downwards, this situation corresponds to the so-called windmill brake regime of an helicopter: the helicopter is going downwards, while the flow and the vortices are going upwards with respect to the helicopter rotor. This regime also corresponds to the wind turbine regime. The difference with the other windmill brake regime obtained for α < 1 is in the azimuthally averaged axial flow which is stronger than the external wind close to the axis in that case.

In the light gray regions, the vortex velocities are positive while W F < 0. This regime Light gray regions: ascending regime (both vortices are going downwards as the external wind). Dark gray regions: slow descending regime and vortex ring state (both vortices are going downwards while the external wind is going upwards). Close to the line Ω F = 0, there exist a small region where one vortex is going upwards while the other is going downwards: such a solution cannot describe the (far) wake of a rotor.

corresponds to a climbing regime: the vortices moves downwards as the external flow and the mass flow rate. In the dark gray regions, W F and the vortex velocities are positive as the mass flow rate. This situation corresponds to a slow descending regime: the vortices moves downwards while the external flow goes upwards. It does not tell us anything on the behavior of the vortices close to the rotor. It is not excluded that the vortices exhibit a complex pattern near the rotor as observed in the so-called vortex ring state [START_REF] Drees | The field of flow through a helicopter rotor obtained from wind tunnel smoke tests[END_REF][START_REF] Quaranta | Instabilities in a swirling rotor wake[END_REF][START_REF] Venegas | Structure and stability of rotor generated vortices[END_REF]. Note that the curve Ω F = 0 is not exactly the limit between the windmill brake regime and the slow descending regime. There is actually a small region close to this curve where the solution cannot correspond to any helicopter flight regime. This region has been displayed in figure 16a for a particular set of parameters. In this region, internal and external vortices move in opposite directions. For this reason, they cannot be created by a single rotor located far away.

Conclusion

We have obtained new numerical solutions that extend the uniform helices that are usually used to describe the far wake generated by a rotor in axial wind. These solutions are spatially periodic and steady in a rotating and translating frame. They describe a situation where each blade creates two counter-rotating vortices emitted at two different non-vanishing radii. We have shown that these solutions can be considered as deformed helices. They exhibit radial variations that increase as the internal vortex get closer to the external vortex. We have provided simpler numerical solutions that capture the main features of the solutions. We have also considered the variation of the vortex core size associated with the deformation and shown that it has a very weak effect on the main characteristics of the solutions. Finally, we have addressed the relevance of these solutions for the far wake description of helicopters in vertical displacement.

The stability of the solutions has not been considered. Long-wavelength instability is known to affect single helices [START_REF] Widnall | The stability of a helical vortex filament[END_REF][START_REF] Quaranta | Long-wave instability of a helical vortex[END_REF] as well as multiple helices [START_REF] Gupta | Theoretical analysis of the aerodynamic stability of multiple, interdigitated helical vortices[END_REF][START_REF] Okulov | On the stability of multiple helical vortices[END_REF]Selçuk et al. 2017a;[START_REF] Quaranta | Instabilities in a swirling rotor wake[END_REF]. Similar instabilities are expected to exist in the present solutions.

Note finally that we have not considered the inner structure of the vortices. [START_REF] Fukumoto | The velocity induced by a helical vortex tube[END_REF] and [START_REF] Blanco-Rodríguez | Internal structure of vortex rings and helical vortices[END_REF] among others have shown that the core of helical vortices deform due to the effects of curvature, torsion and strain. These deformations are responsible of short-wavelength instabilities that develop in vortex cores. Local curvature induces the curvature instability [START_REF] Hattori | Modal stability analysis of a helical vortex tube with axial flow[END_REF][START_REF] Blanco-Rodríguez | Curvature instability of a curved Batchelor vortex[END_REF] while strain causes the elliptic instability (Blanco-Rodríguez & Le Dizès 2016). Both instabilities are expected to be present in the inner core of our solutions.

Figure 1 :

 1 Figure 1: Discretization procedure of the vortex filaments. (a) Discretization in segments of two filaments of circulation Γ i and Γ j . (b) Arc of circle formed by three consecutive points of a discretized filament for the computation of the local contribution.

iFigure 2 :

 2 Figure2: Comparison of the cut-off formula (dashed red line) with the approximate formula (2.5) (solid black line). Binormal component of the induced velocity versus the number p n of segments by turn for an helix of circulation Γ = 1, pitch h = 1, radius R = 1, and core size a = 0.05. The two contributions U loc and U seg are also indicated in dash-dot and dotted lines, respectively.

Figure 4 :

 4 Figure 4: Configuration of two undeformed co-axial helical vortices. Here both helices have the same orientation: κ = 1.

Figure 5 :

 5 Figure 5: (a) Self-induced velocities of a pair of coaxial vortices when the mutual induction is negligible. (b) Schematic diagram showing how the moving frame velocities Ω (0)

Figure 6 :Figure 7 :

 67 Figure 6: Radial position of the internal and external vortices for a single counter-rotating helical pair for (a) R * = 0.5, κ = 1, h * = 1, α = 0.9 and ε = 0.03 (b) R * = 0.5, κ = 1, h * = 1, α = 1.4 and ε = 0.03. Solid line: numerical solution. Dashed line: first order approximation.

Figure 8 :Figure 9 :

 89 Figure 8: Representation of the deformed vortex structure for N = 1, κ = 1, h * = 2, α = 1.5, ε = 0.06 and different values of R *

Figure 10 :

 10 Figure 10: Maximum deformation contours in the (α, h) plane for R * = 0.5, ε = 0.03, κ = 1, N = 1. The thick solid line corresponds to the line ∆r int max = ∆r ext max .

Figure 11 :

 11 Figure 11: Characteristics close to α = 1 for h * = 1, ε = 0.01, N = 1, κ = 1. (a) Comparison of the radial positions of deformed helices versus z/L for α = 1.05 and α = 0.95 with the radial position of perfect helices versus t/T for α = 1. (b) Variation of the characteristic time scale L/W F versus α. The symbol represents the period T of the temporal evolution of perfect helices for α = 1.

Figure 12 :F

 12 Figure 12: Frame velocity versus h * for R * = 0.5, ε = 0.03, α = 1.4, κ = 1, N = 1. (a) Angular velocity Ω = R 2 ext Ω F /Γ , (b) axial velocity W = R ext W F /Γ . Solid line: numerical result. Dashed line: first order approximation (Ω (1) F , W (1) F ). Dash-dotted line: leading order approximation (Ω (0) F , W (0) F ).

Figure 13 :Figure 14 :

 1314 Figure 13: Contour values of the frame velocity in the (α, h) plane for R * = 0.5, ε = 0.03, N = 1 and κ = 1. (a) Angular velocity Ω = Ω F R 2 ext /Γ , (b) Axial velocity W = W F R ext /Γ . The dashed line (α = 1) indicates a line where Ω and W are not defined. The thick solid curve corresponds to the level zero.

Figure 15 :Figure 16

 1516 Figure 15: Contour values of the mean tangential velocities in the (α, h) plane for R * = 0.5, ε = 0.03, N = 1, κ = 1. (a) External vortex: V ext tan R ext /Γ . (b) Internal vortex: V int tan R ext /Γ . The dashed line (α = 1) indicates a line where V ext tan and V int tan are not defined. The thick solid curve corresponds to the level zero.

Figure 17 :

 17 Figure 17: Induced velocity contours in a cross section (z = 0 plane) for h * = 1, R * = 0.5, α = 1.5, ε = 0.03, N = 1 and κ = 1. In (a) colors are for |V ind z | with the same colormap as in (b).

Figure 18 :

 18 Figure 18: Azimuthally averaged induced velocity profile for the same parameters as in figure 17. Solid lines: numerical results at different axial locations. Dashed line: theoretical prediction for perfect helices using Hardin model. (a): Angular velocity Ωind R 2 ext /(N Γ ). (b) Axial velocity V ind z h ext /(N Γ ).

*Figure 19 :

 19 Figure 19: Contours of mass axial flow rate M (normalized by ρN Γ R ext ) in the (α, h) plane for R * = 0.5, ε = 0.03, N = 1 and κ = 1. Contours correspond from top to bottom to {1, 1.25, 1.5, 2, 2.5, 3, 4, 6, 10}. Solid lines: M ; red dashed lines: M H .

Figure 20 :

 20 Figure 20: (a) Variation of the core size a/R ext for the external (solid line) and internal (dashed line) vortex. The mean core size value in each vortex is a/R ext = 0.03. (b) Radial position of the internal and external vortices for a constant core size (dashed line) and for a variable core size obtained using (4.1) (solid line). The parameters in both figures are R * = 0.8, h * = 1.4, α = 1.4, N = 1, κ = 1 and a mean core size a/R ext = 0.03.

Figure 21 :

 21 Figure 21: Diagram of the rotor flow regimes. White regions: windmill brake regime or wind turbine regime (both vortices are going upwards as the external wind).Light gray regions: ascending regime (both vortices are going downwards as the external wind). Dark gray regions: slow descending regime and vortex ring state (both vortices are going downwards while the external wind is going upwards). Close to the line Ω F = 0, there exist a small region where one vortex is going upwards while the other is going downwards: such a solution cannot describe the (far) wake of a rotor.

Acknowledgments

This work received support from the French Agence Nationale de la Recherche under the A*MIDEX grant ANR-11-IDEX-0001-02, the LABEX MEC project ANR-11-LABX-0092 and the ANR TWIN-HELIX project ANR-17-CE06-0018-01. EDV also acknowledge DGA and Aix Marseille Université for his PhD grant.

Appendix A. Asymptotic expressions of Ω and W for a single helix

The angular rotation Ω and the axial velocity W of a fluid element on a helix of pitch h, radius R, core size a and circulation Γ are given by the expressions

where p = h/(2πR), = a/[R(1 + p 2 )] and W(p) is the function defined in [START_REF] Boersma | On the self-induced motion of a helical vortex[END_REF] by

These expressions are eqs. (2.13) and (2.15) of Velasco Fuentes (2018) corrected by a 1/4 term to account for the vortex deformation induced by curvature. This term is obtained by projecting on the azimuthal and axial direction a correction derived for the bi-normal velocity component by [START_REF] Boersma | On the self-induced motion of a helical vortex[END_REF]. The correction terms that would come from the tangential velocity component have never been computed, but we suspect that they are O( ).

In formulas (A 1a,b), it is implicitly assumed that the vortex core model is a Rankine vortex (uniform vortex). Using another vortex model amounts to change the definition of the vortex core size a as explained by [START_REF] Widnall | The stability of a helical vortex filament[END_REF] (see also [START_REF] Saffman | Vortex dynamics[END_REF].


Light Driven Control of an Aromatic Oligoamide Foldamer

Conformational Switch

Bappaditya Gole, [a] Brice Kauffmann, [b] Victor Maurizot, [a] Ivan Huc,* [c] and Yann Ferrand* [a] Abstract: We have investigated the conformation control of a foldamer switch by light. An aromatic oligoamide sequence composed of a light responsive diazaanthracene-based aromatic -sheet flanked by two variable diameter helical segments was prepared. Structural investigations revealed that such oligomers adopt two distinct conformations: a canonical symmetrical conformation with the two helices stacked above and below the sheet, and an unanticipated unsymmetrical conformation in which one helix has flipped to directly stack with the first helix. Photo-irradiation of the foldamer led to the quantitative, and thermally reversible, formation of a single photoproduct resulting from the [4+4] cycloaddition of two diazaanthracenes within the aromatic -sheet. NMR and crystallographic studies revealed a parallel (i.e. head-to-head) arrangement of the diazaanthracene photoproduct and a complete conversion to a symmetrical conformation requiring a rearrangement of all unsymmetrical conformers. These results highlight the potential of foldamers with structures more complex that isolated helices for the design of photo-switches showing nontrivial nanometer scale shape changes.

Molecular folding constitutes a formidable tool to implement switching, controlled motion and response to external stimuli at the nanometer scale. As best exemplified by the folded conformations of proteins, complex arrays of intramolecular interactions and a hierarchical structure composed of several sheet, turn, loop and helix sub-units, may give rise to sophisticated allosteric behavior; [1] signal transduction across e.g. bilayer membranes through subtle conformational changes mediated by guest binding or by a simple phosphorylation, as in G-protein coupled receptors; [2] or large scale controlled molecular motions coupled to the consumption of chemical energy as performed by kinesin [3] or myosin. [4] Likewise, synthetic foldamers [5] show promise in the design and fabrication of artificial systems endowed with comparable properties and constitute an underexplored approach to artificial molecular switches and motors. [START_REF]For reviews see[END_REF]7] Until now, most investigations have concerned isolated helically folded oligomers or polymers.

Helices may undergo handedness reversal upon interacting with chiral guests, thus leading to chiral amplification and, eventually, kinetic trapping of chiral helical states, [8] or to the transfer of chiral information from one end of the helix to the other. [9] They may also undergo equilibria between folded and unfolded states triggered by light, [10] and spring-like extensions and contractions mediated by metal ion binding, [11] or a change of redox state. [12] In addition, guest binding to, or release from, a helical host may be associated to single helix-double helix equilibrium, [13] to a light stimulus, [14] or to the ring contraction of a main chain aromatic unit. [15] Beyond isolated helices, one major challenge in foldamer science is the design of abiotic tertiary structures. Early attempts consisted in simply connecting several secondary folded modules with limited interactions between them. [16] Recent progress includes the consistent spatial arrangement of several abiotic helices, [17] as well as aromatic helix and sheet combinations. [18] The increasing size and sophistication of these structures provide opportunities for enhancing the amplitude and complexity of conformational responses to various stimuli. For instance, rotation about a single bond in a large folded object may give rise to a considerable and nontrivial change of molecular shape. Along this line, we now report the reversible, light-induced, switching between different conformations within a helix-sheet-helix aromatic amide foldamer. Specifically, we discovered a foldamer sequence that undergoes unexpected conformation dynamics leading to a non-canonical fold, and that these dynamics can be reversibly controlled by means of an intramolecular photoreaction between two anthracene subunits.

The dimerization of anthracenes under irradiation by light is one of the most studied photochemical reactions. The reversible [4+4] cycloaddition of two anthracene molecules results in a photodimer connected by covalent bonds. [19] For substituted anthracenes, syn/anti and parallel/antiparallel (i.e. head-to-head and head-to-tail) isomeric products may form. Products with opposite dipole orientations are generally favored unless molecular or supramolecular constraints guide the reaction. [20,21] Thus, related 1,8-diazaanthracenes (i.e. pyrido-[3,2-g]quinolines) undergo quantitative antiparallel photodimerization in solution (Figure 1a andS1). [22] Recently, we have shown that, when incorporated in aromatic oligoamide -sheet foldamers, 1,8diazaanthracene units formed parallel stacks favored by a better - overlap (Figure 1b). [18] This observation led us to consider the production of parallel photo-products guided by folding (Figure 1c). Model compound 1 was prepared to test this hypothesis (see the supporting information for details, Scheme S1). It comprises two 1,8-diazaanthracenes [23] A H separated by a dinitro-diaminobenzene turn 24 T. Upon irradiation under anaerobic conditions at 320<  < 390 nm using a 50W lamp and appropriate cutoff filters, 1 H NMR monitoring showed the quantitative conversion of 1 into a single new product having the same mass in 20 min (Figure S1). New signals of the former anthracene H10 and H9 protons were found as singlets at 5.35 and 5.70 ppm (Figure S1). These chemical values are consistent with a [4+4] photoreaction and the lack of H9*-H10* scalar coupling indicates that the parallel photoproduct 2 has formed (Figure 1c and S1-S2). [25] Such a quantitative parallel photoreaction guided by folding is remarkable. It reflects not only that the anti-parallel aromatic sheet is disfavored, but also that its conformation is not conducive of a photoreaction (Figure S16). As expected, the reaction was shown to be thermally reversible: in C2D2Cl4, 2 was stable at 328K but quantitatively converted back into 1 upon heating at 393K for 15h (Figure S3-S4). We then planned to test this photoreaction within a larger foldamer structure. Sequence 3 comprising two conical Q3PN2 helical hexameric segments attached on both sides of a central A H TA H sheet was designed and synthesized (Scheme S2-S3). We have previously observed that a related sequence having a longer, three-stranded ATATA sheet folds in a canonical manner with its two helical segments stacked on each side of the central aromatic -sheet. [18] With only two strands, as in 3, a -sheet is expected to mediate a helix handedness reversal leading to a plane symmetrical conformation noted (M,P)-s3 (see energy minimized models in Figure 2e and S17). In line with earlier observations, [18] the 1 H NMR spectrum of 3 in d6-acetone shows one set of sharp lines whose multiplicity is consistent with an overall symmetrical (M,P)-s3 structure (Figures 2a andS6). However, the 1 H NMR spectrum of 3 in CDCl3 revealed a more complex situation. The symmetrical species that prevails in d6acetone is also present but a second set of broad signals indicates coexistence with an other species in slow exchange on the NMR time scale (Figure 2b). Upon cooling in CDCl3 down to 233K, the proportion of this species increased to exceed 50% and signals remained somewhat broad (Figure 2c and S7-S8). In CD2Cl2, the signals sharpened at 243K and below (Figure S10) and revealed a number of amide resonances compatible with a dissymmetrical structure (Figure 2d and S10). Solid state investigations shed light on the conformational behavior of 3. X-ray quality single crystals were grown by slow diffusion of hexane into a chloroform solution. The solid-state structure revealed an unanticipated unsymmetrical conformation in which a helical domain has undergone a large flip to stack underneath the other helix (Figure 2e), and to which the dissymmetrical species observed in solution may tentatively be assigned. This conformer, which we named u3, is accessible from (M,P)-s3 through a 180° rotation about a single aryl-amide bond between two N units. Apart from that, both helical segments and the parallel aromatic -sheet fold as expected. The single bond rotation results in a repulsive electrostatic interaction between an amide oxygen atom and an adjacent naphthyridine endocyclic nitrogen atom (Figure 2e). This is apparently compensated by favorable contacts including through the antiparallel stacking of the second quinoline of one helix and a naphthyridine of the other helix and through the filling of the cavity of one helix by an isobutoxy side chain of the other helix (Figure S14). These contacts would not occur for -sheets with three or more aromatic strands, hence the absence of unsymmetrical conformer for longer parent sequences. [18] Conformation u3 in principle exists as a mixture of two degenerate (M,P) and (P,M) states. The mechanism of their interconversion is unclear, and in particular whether or not it must transit through s3 (equilibria 1 & 2 in Figure 3). Equilibria 1 and 2 do not require helix handedness inversion. Yet the ensemble of u3 and s3 species is at slow equilibrium with another mirror-imaged (thus degenerate) ensemble obtained through handedness reversal of both helical segments. The thermally reversible photoreaction of the two parallel A H units (equi. 3) locks the conformation in a plane symmetrical state. P and M helices are shown as blue and red tubes, respectively. The diazaanthracene sheet and its photo-product are color-coded in orange/grey and green/grey, respectively.

In both the crystal structure of u3 and the energy minimized model of s3, the two A H units have a parallel orientation but are significantly offset (Figure 2f,g) and not ideally preorganized for a [4+4] cycloaddition. Photoirradiation of 3 in CDCl3 was nonetheless performed. Thus, a 1 mM solution was irradiated under an argon atmosphere and a photoreaction took place, albeit at a slower rate than with 1. As followed by 1 H NMR, 80% of the starting oligomer 3 was converted into its photoproduct 4 after 1h of irradiation (Figure 4b). Complete conversion was reached within 4h (Figure 4c,e). The slower kinetics may be assigned to the not-ideal conformation mentioned above and to favorable interactions within u3 and s3 that must be disrupted for the reaction to take place. The parallel arrangement of the photoproduct was again ascertained by the emergence of two singlets at 4.8 and 5.5 ppm assigned to H10* and H9*, respectively. In addition, the 1 H NMR spectrum of 4 shows a single set of signals and indicates a symmetrical structure.

The symmetrical conformation with a P and an M helices, and the parallel arrangement of the photoproduct were confirmed by a solid state structure of 4. Crystals grew upon slowly diffusing methanol in a chloroform solution. The structure was solved in the P-1 space group and revealed that the helices remain above and below the butterfly-shaped photoreacted -sheet in a canonical structure even though proper stacking onto the A H units is no longer possible. Thus, the photoreacted turn is fully embedded inbetween the two helices, with H9* protons pointing towards the concave side of the sheet whereas H10* protons point towards to convex side of the sheet (Figure 4f-h). Starting from the equilibrating mixture of u3 and s3 conformers, light thus allows one to lock the molecule in a planesymmetrical conformation that has been expanded by the photoreaction, and thus to remove unsymmetrical conformers (Figure 3), as when longer sheets are present. The reaction between the two A H units is selective: cross-reaction with other heterocycles in the structure is avoided. [26] Furthermore, the photoproduct can be thermally reverted to the mixture of u3 and s3. Heating 4 in CDCl3 at 328K led to ~50% of conversion after 16 hours. Quantitative recovery of 3 required almost two days. One should note that, despite being slow, this reverse reaction is perfectly clean and anyway much faster than for photo-product 2 which is stable at this temperature. The lower stability of 4 possibly reflects a higher strain imposed by folding.

In summary, we have discovered an unanticipated unsymmetrical conformer in helix-sheet-helix foldamers, and found that the quantitative parallel photoreaction of diazaanthracenes within aromatic -sheets exclusively and reversibly produces symmetrical conformers. The cleanness of the photoreaction suggests that it could be implemented multiple times within a given aromatic foldamer sequence, thus leading to a global stiffening and length extension of the molecule, or that it could be reliably used to control guest binding and release. Efforts toward these objectives are currently in progress and will be reported in due course.

Figure 1 .

 1 Figure 1. Cartoon representations of: (a) an antiparallel diazaanthracene photodimer; (b) a parallel arrangement of strands in a diazaanthracene -sheet and (c) a parallel diazaanthracene photo-product in an aromatic -sheet. In (a,b,c) white arrows indicate local dipole orientation and blue spheres indicate nitrogen atoms. (d) Color coded formula and associated letters corresponding to amino acid, diamine, and diacid monomers. (e) Oligoamide sequences 14. Note that 2 and 4 are the photo-products of 1 and 3, respectively. The terminal A H and Q units of these sequences have a methyl ester group and an 8-nitro group (instead of an 8-amino function), respectively.

Figure 2 .

 2 Figure 2. Part of the 700 MHz 1 H NMR spectra of 3 (1 mM) at: (a) 298K in d6acetone; (b) 298K in CDCl3; (c) 233K in CDCl3 and (d) 213K in CD2Cl2. Signals assigned to symmetrical (M,P)-s3 and unsymmetrical (M,P)-u3 are marked with black () and empty () circles, respectively. The black and empty squares denote the amino protons of the turn unit for (M,P)-s3 and (M,P)-u3; respectively. (e) Side view of the energy-minimized molecular model using Merck Molecular Force Field static (MMFFs) of plane symmetrical (M,P)-s3. (f) Front view of the structure in the solid state of unsymmetrical (M,P)-u3. The zoom highlights the rotation of a helical segment about an amide-naphthyridine bond. A red double-headed arrow denotes a local electrostatic repulsion. (g,h) Zoom on the diazaanthracene -sheet turn unit in (M,P)-s3 (from e) and (M,P)-u3 (from f), respectively. The model and X-ray structures are shown in tube representation with color coded monomers as in Fig. 1. Blue balls indicate endocyclic nitrogens atoms. Transparent yellow isosurfaces represent the volume of the foldamer. Hydrogen atoms, side chains and solvent molecules are not shown for clarity.

Figure 3 .

 3 Figure 3. Schematic representation of the equilibria involved in the switching process. Equilibria 1 and 2 do not require helix handedness inversion. Yet the ensemble of u3 and s3 species is at slow equilibrium with another mirror-imaged (thus degenerate) ensemble obtained through handedness reversal of both helical segments. The thermally reversible photoreaction of the two parallel A H units (equi. 3) locks the conformation in a plane symmetrical state. P and M helices are shown as blue and red tubes, respectively. The diazaanthracene sheet and its photo-product are color-coded in orange/grey and green/grey, respectively.

Figure 4 .

 4 Figure 4. Part of the 700 MHz 1 H NMR spectra of at 298K of 3 (1 mM in CDCl3) under photoirradiation after (a) 0 min; (b) 60 min.; (c) 4 h. (d) Evolution after incubating sample (c) at 328K for 40 h. Signals of s3 and u3 are denoted with black () and white () circles, respectively. Black triangles indicate signals of photo-product 4. (e) Kinetic monitoring ( 1 H NMR) of: (left) the photoreaction of 3 yielding 4 and (right) the thermolysis of 4 back into 3. f) Crystal structure of 4. (g,h) Front and side views of the photoreacted -sheet highlighting the parallel arrangement. The structure is shown in tube representation and the monomers are color coded as in Fig. 1. Blue balls denote endocyclic nitrogen atoms. Side chains, solvent molecules and hydrogen atoms are not shown for clarity.

Acknowledgements

This work was supported by the post-doc program of the Excellence Initiative of Bordeaux University (B.G.). This work has benefited from the facilities and expertise of the Biophysical and Structural Chemistry platform (BPCS) at IECB, CNRS UMS3033, Inserm US001, and Bordeaux University. Keywords: foldamer • photochemistry • molecular switch • aromatic oligoamide • X-ray crystallography