Delphine Bresch-Pietri

Miroslav Krstic

Adaptive compensation of diffusion-advection actuator dynamics using boundary measurements

For (potentially unstable) Ordinary Differential Equation (ODE) systems with actuator delay, delay compensation can be obtained with a prediction-based control law. In this paper, we consider another class of PDE-ODE cascade, in which the Partial Differential Equation (PDE) accounts for diffusive effects. We investigate compensation of both convection and diffusion and extend a previously proposed control design to handle both uncertainty in the ODE parameters and boundary measurements. Robustness to small perturbations in the diffusion and convection coefficients is also proved.

I. INTRODUCTION

For linear systems subject to input-delay, prediction-based control strategies, more commonly known as Smith Predictor (see [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF], [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF], [START_REF] Smith | A controller to overcome dead time[END_REF]) are state-of-the-art for systems with constant input time-delays (see for instance [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF], [START_REF] Bresch-Pietri | Adaptive control scheme for uncertain time-delay systems[END_REF], [START_REF] Gu | Survey on recent results in the stability and control of time-delay systems[END_REF], [START_REF] Jankovic | Recursive predictor design for linear systems with time delay[END_REF], [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF], or [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF] and the references therein). Grounding on the use of a prediction of the system state on a time horizon equal to the delay, this technique aims at compensating it, which notably improves transient performances.

This strategy has been analyzed in [START_REF] Krstic | Boundary Control of PDEs: a Course on Backstepping Designs[END_REF] in the light of distributed parameters systems techniques. In details, representing the delay as a first-order hyperbolic PDE, predictionbased control has been interpreted as the result of a backstepping transformation of the distributed input. This maps the original system into an ODE-PDE cascade in which the output of a finite-time stable transport PDE feeds a stable ODE.

In this work, we propose to study an extension of this technique for diffusion PDE-ODE cascade. Applying the same ideas as in [START_REF] Krstic | Boundary Control of PDEs: a Course on Backstepping Designs[END_REF] to construct an infinite-dimensional transformation and a corresponding control law, it has been shown in [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF] that the closed-loop system can be equivalently represented by a diffusion process, with zero boundary condition, and with the boundary opposite to the controlled one feeding a stable ODE. In this sense, the proposed control law compensates the actuator diffusion. However, while delay dynamics can be finitely stabilized, diffusion compensation should then be understood differently as it introduces an infinite relative degree. Correspondingly, while delay compensation requires to predict future values of the ODE state, in the diffusion case, the control law mainly accounts for an inversion of the diffusion dynamics.

In this work, we are interested in a convection/diffusion PDE-ODE cascade with parameters uncertainties. We pro-D. Bresch-Pietri (corresponding author) is with CNRS at the Department of Automatic Control, Gipsa-lab, 11 rue des Mathématiques, 38000 Grenoble, France Email: delphine.bresch-pietri@gipsa-lab.fr M. Krstic is with the Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla CA 92093, USA pose to extend the control law from [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF] to the case of boundary measurement and to handle ODE parametric uncertainties in an adaptive manner. We consider that only the ODE state and one boundary condition of the PDE, namely, the control input, are measured. Taking into account the stability of the actuator dynamics, we propose to use a simple openloop observer for the PDE state. Applying classical adaptive techniques [START_REF] Ioannou | Adaptive Control Tutorial[END_REF], [START_REF] Ioannou | Robust Adaptive Control[END_REF], [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF], we chose the parameter update law grounding on a Lyapunov design and prove closed-loop stability. Finally, we show that the proposed technique is robust to PDE parametric uncertainties. These are the main contributions.

The paper is organized as follows. In Section II, we start by presenting the problem under consideration along with the nominal control design proposed in [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF]. Then, in Section III, we detail the proposed boundary adaptive control law and our main result before providing its proof. Robustness to PDE parametric uncertainties is shown in Section IV and illustrated in a numerical example in Section V. We conclude by sketching directions of future work.

Notations. For

u : [0, 1] × R → R, we write u(t) = 1 0 u(x,t) 2 dx (1)

II. PROBLEM STATEMENT

In this work, we consider the following PDE-ODE cascade

       Ẋ(t) = A(θ )X + B(θ )u(0,t) u t = u xx + au x u x (0,t) = 0 u(1,t) = U(t) (2)
in which X ∈ R n , U is scalar, a > 0 and the dynamics matrices A and B are linearly parametrized with respect to an unknown parameter θ ∈ R p , that is,

A(θ ) =A 0 + p ∑ i=1 θ i A i and B(θ ) = B 0 + p ∑ i=1 θ i B i (3)
As always in indirect adaptive control, certain a priori assumptions on the parameter values are needed in order to ensure stabilizability under parameters estimation. This is why we first characterize further the dynamics under consideration.

Assumption 1: There exists a bounded and known convex set Π = {θ ∈ R p |P(θ ) ≤ 0}, in which the convex function P is smooth, such that θ ∈ Π.

Assumption 2: The pair (A(θ ), B(θ )) is controllable for each θ ∈ Π. Furthermore, we assume that there exists a triple of functions θ → (K(θ ), P(θ ), Q(θ )) such that K ∈ C 1 (Π), P ∈ C 1 (Π), Q ∈ C 0 (Π), the matrices P and Q are symmetric positive definite and the following equation holds

P(θ )(A + BK)(θ ) + (A + BK)(θ ) T P(θ ) = -Q(θ ) , θ ∈ Π (4)
To make stabilization possible in the presence of ODE parametric uncertainties [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF], we assume that the ODE state is known.

Assumption 3: The state X is measured.

This assumption is standard but quite restrictive. Future works will focus on alleviating it by studying normal ODE forms [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF].

In the nominal case, that is, when the system parameters are known and the distributed PDE state u is measured, diffusion can actually be compensated by the controller proposed in [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF] in which a purely diffusive process is studied. Furthermore, this control strategy can be extended to dynamics including advection, that is, the case under consideration here, to compensate both convection end diffusion effects, as follows

U(t) = 1 0 k(θ , 1, x)u(x,t)dx + γ(θ , 1)X(t) (5)
γ(θ , x) = (K(θ ) 0) exp 0 A(θ ) I -aI x I 0 (6) k(θ , x, y) = x-y 0 e -a(x-y-s) γ(θ , s)B(θ )ds (7)
In other words, it was shown in [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF] (for purely diffusive dynamics, i.e., a = 0) that this control law (5) maps (2) into the target system

       Ẋ(t) = (A + BK)(θ )X(t) + B(θ )w(0,t) w t = w xx + aw x w x (0,t) = 0 w(1,t) = 0 (8)
in which the diffusion and advection effects have been compensated for in the ODE state.

In this paper, we aim at extending this controller to handle (i) ODE parameters uncertainties; and (ii) boundary measurement and control, which means that we assume that the boundary u(1,t) = U(t) is the only element of the PDE state which is measured.

III. ADAPTIVE BOUNDARY CONTROL DESIGN

We consider the following open-loop observer

   ût = ûxx + a ûx ûx (0,t) = 0 û(1,t) = U(t) (9)
along with the adaptive control law

U(t) = 1 0 k( θ , 1, x) û(x,t)dx + γ( θ , 1)X(t) (10)
which is based on the certainty equivalence principle and involves an estimate θ of the uncertain parameters. Its update law is chosen based on a Lyapunov design (presented in the following section) as

θ (t) =γProj Π (τ θ (t)) (11)
in which the adaptation gain γ is a positive constant which has to be chosen small enough (in a sense defined in the sequel) and

τ θ ,i (t) = 2 X(t) T P( θ ) + b 1 ŵx (1,t)γ( θ , 1) -b 1 1 0 [γ( θ , x) T ŵ(x,t) + γ x ( θ , x) T ŵx (x,t)]dx × A i X(t) + B i û(0,t) 1 + N(t) , i = 1, . . . , p (12)
N(t) = X(t) T P( θ )X(t) + b 1 1 0 ( ŵ(x,t) 2 + ŵx (x,t) 2 )dx (13)
where the matrix P is defined in Assumption 2, the function γ is given in ( 6), the constant b 1 is chosen such that

b 1 > 4 max θ ∈Π |P( θ )B( θ )| 2 a min θ ∈Π min λ (Q(θ )) (14)
the standard projector operator is given by

Proj Π {τ θ } = (15)
τ θ      1, θ ∈ Π or ∇ θ P T τ ≤ 0 I - ∇ θ P∇ θ P T ∇ θ P T ∇ θ P , θ ∈ ∂ Π and ∇ θ P T τ > 0 .
and the distributed variable ŵ is defined as

ŵ(x,t) = û(x,t) - x 0 k( θ , x, y) û(y,t)dy -γ( θ , x)X(t) (16)
Theorem 1: Let Assumptions 1-3 hold and consider the closed-loop system consisting of (2) with a > 0, the control law defined through ( 6)-( 7) and [START_REF] Krstic | Boundary Control of PDEs: a Course on Backstepping Designs[END_REF], and the update law defined by ( 11)- [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF]. There exists γ * > 0 such that, for any γ ∈ (0, γ * ), there exist positive constants R 1 and R 2 such that, for all initial conditions satisfying (X 0 , u 0 , û0

, θ 0 ) ∈ R n × H 2 (0, 1) × H 2 (0, 1) × Π, the following holds Γ 1 (t)e Γ 2 (t) ≤R 1 Γ 1 (0)e R 2 Γ 2 (0) (17)
where

Γ 1 (t) =|X(t)| 2 + 1 0 û(x,t) 2 dx + 1 0 ûx (x,t) 2 dx (18) Γ 2 (t) = 1 0 (u(x,t) -û(x,t)) 2 dx + 1 0 (u x (x,t) -ûx (x,t)) 2 dx + |θ -θ (t)| 2 (19)
Furthermore, both ODE and PDE states asymptotically converge, i.e.,

X(t) → t→∞ 0 , max x∈[0,1] u(x,t) → t→∞ 0 and u(1,t) = U(t) → t→∞ 0 (20)

A. Comments

Before providing the proof of this theorem, we point out a few elements to ease the understanding of this control design.

It is worth underlining the fact that the entire control strategy can be computed with only the measurement of the ODE state and the control law.

As mentioned above, the choice of the update law is based on the Lyapunov design provided below. This explains the appearance of the infinite-dimensional backstepping transformation [START_REF] Smith | A controller to overcome dead time[END_REF] which is an analysis tool used in the sequel. As common in adaptive control, a projection operator is used in [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF] to obtain a physically meaningful value for the parameters estimation. In addition, normalization [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF] aims at limiting the rate of change of the parameter estimate, which could otherwise act as a disturbance. This technique is also standard in adaptive control [START_REF] Ioannou | Adaptive Control Tutorial[END_REF], [START_REF] Ioannou | Robust Adaptive Control[END_REF], [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF] to obtain global stabilization.

As a final remark, notice that the previous design only holds for a > 0 to allow for the scalar parameter b 1 to be finite. However, one could straightforwardly modify the stability analysis provided in the sequel when the convection can be neglected compared to the diffusion effect (i.e. a < 1). This should lead to an update law of the form (12) but with a kernel cos(x + α) under the integrals and a factor cos(α) multiplying the backstepping variable ŵx , in which the kernel parameter α should be chosen into ]0, arccotan(a) -1[. One could infer from the following analysis that the conclusion of Theorem 1 would still hold under similar assumptions but with a lower bound on b 1 which does not tend to infinity while a tends to zero contrary to [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF].

B. Proof of Theorem 1

We start our analysis by expressing the closed-loop dynamics into the new set of variables (X, ũ, ŵ) in which ũ = uû is the distributed estimation error. First, subtracting (9) to (2) and using the definition of the backstepping transformation ( 16) along with (6) for x = 0, one obtains

           Ẋ(t) = (A + BK)( θ )X(t) + B( θ ) ŵ(0,t) +A( θ )X(t) + B( θ ) û(0,t) + B(θ ) ũ(0,t) ũt = ũxx + a ũx ũx (0,t) = 0 ũ(1,t) = 0 (21)
in which we introduced the parameter estimation error θ (t) = θ -θ (t) and the matrices

A( θ ) = p ∑ i=1 A i θi and B( θ ) = p ∑ i=1 B i θi (22)
Second, taking time-and space-derivatives of ( 16), one gets

       ŵt = ŵxx + a ŵx -θ (t) T g(x,t) -γ( θ , x)A( θ )X(t) -γ( θ , x)(B ũ(0,t) + B( θ ) û(0,t)) ŵx (0,t) = 0 ŵ(1,t) = 0 (23) with g(x,t) = x 0 k θ ( θ , x, y) û(y,t)dy + γ θ ( θ , x)X(t) (24)
in which k θ and γ θ are continuous functions from the definitions ( 6)-( 7) and Assumption 2. In the sequel, we also need the governing equation of the spatial-derivative of these distributed variables which can be obtained straightforwardly as

   ũxt = ũxxx + a ũxx ũxx (0,t) = ũt (0,t) ũx (1,t) = -1 a ũxx (1,t) (25)
and

             ŵxt = ŵxxx + a ŵxx -θ (t) T g x (x,t) -γ x ( θ , x)A( θ )X(t) -γ x ( θ , x)(B ũ(0,t) + B( θ ) û(0,t)) ŵxx (0,t) = ŵt (0,t) ŵx (1,t) = 1 a (-ŵxx (1,t) + θ T g(1,t) + γ( θ , 1)A( θ )X(t) +γ( θ , 1)(B( θ ) û(0,t) + B ũ(0,t)) (26)
We are now ready to carry out the Lyapunov analysis. Consider the following Lyapunov functional candidate

V (t) =log (1 + N(t)) + b 2 1 0 ũ(x,t) 2 dx + b 2 1 0 ũx (x,t) 2 dx + 1 2γ | θ (t)| 2 (27)
in which N has been introduced in ( 13), b 2 is a positive constant and γ and b 1 are the tuning parameters involved in the update law [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF]. Taking a time-derivative and using ( 21)-( 26) and integrations by parts, one gets

V (t) = 1 1 + N(t) -X(t) T Q( θ )X(t) + 2X(t)P( θ )(B( θ ) ŵ(0,t) + A( θ )X(t) + B( θ ) û(0,t) + B(θ ) ũ(0,t)) -2b 1 ŵx (t) 2 -ab 1 ŵ(0,t) 2 -2b 1 ŵxx (t) 2 -ab 1 ŵx (1,t) 2 + 2b 1 ŵx (1,t) × ( θ (t) T g(1,t) + γ( θ , 1)(A( θ )X(t) + B( θ ) û(0,t) + B ũ(0,t)) -2b 1 θ (t) T 1 0 (g(x,t) ŵ(x,t) + g x (x,t) ŵx (x,t))dx -2b 1 1 0 γ( θ , x)(A( θ )X(t) + B ũ(0,t) + B( θ ) û(0,t)) ŵ(x,t)dx -2b 1 1 0 γ x ( θ , x)(A( θ )X(t) + B ũ(0,t) + B( θ ) û(0,t)) ŵx (x,t)dx - p ∑ i=1 θi (t) T X(t) T dP d θi ( θ )X(t) -ab 2 ũ(0,t) 2 -2b 2 ũx (t) 2 -ab 2 ũx (1,t) 2 -2b 2 ũxx (t) 2 - 1 γ θ T θ (t) (28)
Using the projection operator properties, one obtains

V (t) ≤ 1 1 + N(t) -λ |X(t)| 2 + 2X(t)P( θ )(B( θ ) ŵ(0,t) +B(θ ) ũ(0,t)) -2b 1 ŵx (t) 2 -ab 1 ŵ(0,t) 2 -2b 1 ŵxx (t) 2 -ab 1 ŵx (1,t) 2 + 2b 1 ŵx (1,t)( θ (t) T g(1,t) + γ( θ , 1)B ũ(0,t)) -2b 1 1 0 θ (t) T g(x,t) + γ( θ , x)B ũ(0,t) ŵ(x,t)dx -2b 1 1 0 θ (t) T g x (x,t) + γ x ( θ , x)B ũ(0,t) ŵx (x,t)dx + max 1≤i≤p, θ ∈Π dP d θi ( θ ) p ∑ i=1 θi (t) T |X(t)| 2 -ab 2 ũ(0,t) 2 -2b 2 ũx (t) 2 -ab 2 ũx (1,t) 2 -2b 2 ũxx (t) 2 (29)
in which λ = min θ ∈Π min λ (Q(θ )) > 0. Taking into account the fact that the inverse of the backstepping transformation ( 16) is (see [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF], [START_REF] Susto | Control of PDE-ODE cascades with neumann interconnections[END_REF])

û(x,t) = ŵ(x,t) + x 0 ( θ , x, y) ŵ(y,t)dy + β ( θ , x)X(t) (30) with β ( θ , x) = K( θ ) 0 exp 0 (A + BK)( θ ) I -aI x I 0 (31) ( θ , x, y) = x-y 0 e -a(x-y-s) β ( θ , s)B( θ )ds (32)
one can observe, with the help of Young and Cauchy-Schwartz inequalities, that there exists M > 0 such that

2 ŵx (1,t) θ (t) T g(1,t) ≤γM ŵx (1,t) 2 (33) 2 ŵx (1,t)γ( θ , 1)B ũ(0,t) ≤ a 2 ŵx (1,t) 2 + M ũ(0,t) 2 (34) | θi (t)| ≤γM , i = 1, . . . , p (35)
2| θ (t)| 1 0 g(x,t) ŵ(x,t)dx + 1 0 g x (x,t) ŵx (x,t)dx ≤ 2Mγ(|X(t)| 2 + ŵ(t) 2 + ŵx (t) 2 ) (36) 2 1 0 γ( θ , x)B ũ(0,t) ŵ(x,t)dx ≤ ŵ(t) 2 8 + M ũ(0,t) 2 (37) 2 1 0 γ x ( θ , x)B ũ(0,t) ŵx (x,t)dx ≤ ŵx (t) 2 2 + M ũ(0,t) 2 (38)
Using those inequalities and the fact that N(t) > 0, one gets

V (t) ≤ 1 1 + N(t) -ab 1 - 4 λ |P( θ )B( θ )| 2 ŵ(0,t) 2 - λ 2 -γM p max 1≤i≤p, θ ∈Π dP d θi ( θ ) + 2 |X(t)| 2 - b 1 8 -2γM ( ŵ(t) 2 + ŵx (t) 2 ) -b 1 ŵxx (t) 2 - ab 1 2 -γM ŵx (1,t) 2 - b 2 4 ( ũ(t) 2 + ũx (t) 2 ) -ab 2 ũx (1,t) 2 -2b 2 ũxx (t) 2 -ab 2 -3b 1 M - 4 λ |P( θ )B(θ )| 2 ũ(0,t) 2 (39)
in which we have also used the Poincaré inequality (taking advantage of the fact that ŵ(1,t) = 0)

ŵ(t) 2 ≤4 ŵx (t) 2 (40)
and a similar one for the distributed estimation error ũ. Consequently, by choosing b 1 such that ( 14) holds and

b 2 > 1 a 3b 1 M - 4 λ max θ ∈Π |P( θ )B( θ )| 2 (41) γ <γ * (42) = min    λ 2M p max 1≤i≤p, θ ∈Π dP d θi ( θ ) + 2 , a 0 b 1 16M    in which a 0 = min {a, 1}, one finally obtains V (t) ≤V (0) , t ≥ 0. ( 43
)
Using ( 16) and its inverse [START_REF] Smith | A controller to overcome dead time[END_REF], one can show that there exist strictly positive constants c 1 , c 2 , c 3 and c 4 such that û(t

) 2 ≤c 1 (|X(t)| 2 + ŵ(t) 2 ) (44) ŵ(t) 2 ≤c 2 (|X(t)| 2 + û(t) 2 ) (45) ûx (t) 2 ≤c 3 (|X(t)| 2 + ŵ(t) 2 + ŵx (t) 2 ) (46) ŵx (t) 2 ≤c 4 (|X(t)| 2 + û(t) 2 + ûx (t) 2 ) (47)
and ( 17) follows from there. This last inequality guarantees that |X|, ŵ , ŵx , ũ , ũx and | θ | are uniformly bounded. Now, integrating (39) between zero and infinity, one obtains that X, ŵ and ŵx are square integrable. Hence, from the following Agmon inequality, max

x∈[0,1] ŵ(x,t) 2 ≤ 2 ŵ(t) ŵx (t) (48)
which uses the fact that ŵ(1,t) = 0, one infers that ŵ(x, •) is also square integrable and uniformly bounded for all x ∈ [0, 1]. Thus, using (44),( 46) and the backstepping definition ( 16), the same properties hold for û , ûx and û(x, •) for all x ∈ [0, 1]. Similarly, the same properties hold for ũ , ũx and ũ(x, •) for all x ∈ [0, 1] and thus for u , u x and u(x, •) for all x ∈ [0, 1] by definition of ũ. Further,

d dt |X(t)| 2 =2X(t) T (AX(t) + Bu(0,t)) (49) d dt ŵ(t) 2 = -2 ŵx (t) 2 -ŵ(0,t) 2 -2b 1 1 0 γ( θ , x) × (A( θ )X(t) + B ũ(0,t) + B( θ ) û(0,t)) ŵ(x,t)dx -2b 1 θ (t) T 1 0 g(x,t) ŵ(x,t)dx (50) d dt ũ(t) 2 = -2 ũx (t) 2 -ũ(0,t) 2 (51)
in which the right-hand terms are bounded applying Young and Cauchy-Schwartz inequality and the previous considerations. Applying Barbalat lemma, it follows that |X|, ũ and ŵ converge to zero as t tends to infinity. Using Agmon inequality, one obtains that max x∈[0,1] ũ(x, •) and max x∈[0,1] ŵ(x, •) also converge to zero, as ũx and ŵ are bounded. From there, using the inverse backstepping transformation (30), one concludes that max x∈[0,1] u(x, •) converge to zero which concludes the proof.

IV. ROBUSTNESS TO DIFFUSION/ADVECTION PARAMETERS UNCERTAINTY

In this section, we study robustness of the following closed loop system, in which U(t) is given in [START_REF] Krstic | Boundary Control of PDEs: a Course on Backstepping Designs[END_REF],

       Ẋ(t) = A(θ )X(t) + B(θ )u(0,t) u t = (1 + ε 1 )u xx + a(1 + ε 2 )u x u x (0,t) = 0 u(1,t) = U(t) (52)
with respect to the parameters ε 1 and ε 2 which are assumed to be either positive of negative but small a priori. We have the following result.

Theorem 2: Let Assumptions 1-3 hold and consider the closed-loop system consisting of the dynamics (52), the controller defined through ( 6)-( 7) and [START_REF] Krstic | Boundary Control of PDEs: a Course on Backstepping Designs[END_REF], and the update law defined by ( 11)- [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF]. Consider also the functionals Γ 1 and Γ 2 introduced in (18). There exist decreasing functions

ε * 1 , ε * 2 : R 2 → R * such that, if |ε 1 | < ε * 1 (Γ 1 (0), Γ 2 (0)) and |ε 2 | < ε * 2 (Γ 1 (0), Γ 2 (0))
, then the conclusions of Theorem 1 still hold.

Proof: Using the inverse backstepping transformation (30), the error system now writes

   ũt = (1 + ε 1 ) ũxx + a(1 + ε 2 ) ũx + ε 1 g 1 (x,t) + aε 2 g 2 (x,t) ũx (0,t) = 0 ũ(1,t) = 0 (53) in which g 1 (x,t) = ŵxx (x,t) + KB( θ ) ŵ(x,t) + x 0 xx ( θ , x, y) ŵ(y,t)dy + β ( θ , x)(A + BK)( θ )X(t) -aβ x ( θ , x)X(t) (54) g 2 (x,t) = ŵx (x,t) + x 0 x ( θ , x, y) ŵ(y,t) + β x ( θ , x)X(t) (55)
and its space-derivative satisfies

               ũxt = (1 + ε 1 ) ũxxx +a(1 + ε 2 ) ũxx + ε 1 g 1,x (x,t) + aε 2 g 2,x (x,t) ũx (1,t) = -1 a(1+ε 2 ) (1 + ε 1 ) ũxx (1,t) +ε 1 g 1 (1,t) + aε 2 g 2 (1,t)
Thus, using the following integration by parts, and using the same techniques as in the previous section, one obtains the existence of a constant

M 0 > 0 such that d dt ( ũ(t) 2 + ũx (t) 2 ) ≤ -2(1 + ε 1 )( ũx (t) 2 + ũxx (t) 2 ) -a(1 + ε 2 ) ũ(0,t) 2 -a (1 + ε 2 ) ũx (1,t) 2 + M 0 (|ε 1 | + |ε 2 |) × (|X(t)| 2 + ŵ(t) 2 + ŵ(0,t) 2 + ŵx (t) 2 + ŵx (1,t) 2 + ŵxx (t) 2 + ũ(t) 2 + ũ(0,t) 2 + ũx (t) + ũx (1,t) 2 + ũxx (t) 2 ) (57)
Hence, equations ( 29) and (39) given in the proof of Theorem 1 now rewrite as

V (t) ≤ 1 1 + N(t) -ab 1 - 4 λ |P( θ )B( θ )| 2 ŵ(0,t) 2 - λ 2 -γM p max 1≤i≤p, θ ∈Π dP d θi ( θ ) + 2 |X(t)| 2 - b 1 8 -2γM ( ŵ(t) 2 + ŵx (t) 2 ) -b 1 ŵxx (t) - ab 1 2 -γM ŵx (1,t) 2 - b 2 4 ( ũ(t) 2 + ũx (t) 2 ) -ab 2 ũx (1,t) 2 -2b 2 ũxx (t) 2 -ab 2 -3b 1 M - 4 λ |P( θ )B(θ )| 2 ũ(0,t) 2 + b 2 (M 0 + 2 + a)(|ε 1 | + |ε 2 |)(|X(t)| 2 + ŵ(t) 2 + ŵ(0,t) 2 + ŵx (t) 2 + ŵx (1,t) 2 + ŵxx (t) 2 + ũ(t) 2 + ũ(0,t) 2 + ũx (t) + ũx (1,t) 2 + ũxx (t) 2 (58) Define V 1 (t) =|X(t)| 2 + ŵ(t) 2 + ŵx (t) 2 (59) V 2 (t) = ŵ(0,t) 2 + ŵx (1,t) 2 + ŵxx (t) 2 (60) V 3 (t) = ũ(t) 2 + ũx (t) 2 (61) V 4 (t) = ũ(0,t) 2 + ũx (1,t) 2 + ũxx (t) 2 (62) ϒ 1 (t) =Γ 1 (t) + ŵxx (t) 2 (63) ϒ 2 (t) =Γ 2 (t) + ũxx (t) 2 (64)
Under the conditions ( 14), ( 41)-( 42), there exist η 1 , η 2 > 0 such that (58) rewrites

V (t) ≤ - η 1 (V 1 (t) +V 2 (t)) 1 +V 1 (t) -η 2 (V 3 (t) +V 4 (t)) (65)
+ (M 0 + 2 + a)(|ε 1 | + |ε 2 |)(V 1 (t) +V 2 (t) +V 3 (t) +V 4 (t))
Observing that there exist positive constants µ 1 , µ 2 , µ 3 and

µ 4 such that µ 1 (V 1 + V 2 ) ≤ ϒ 1 ≤ µ 2 (V 1 + V 2 ) and µ 3 (V 3 + V 4 ) ≤ ϒ 2 ≤ µ 4 (V 3 + V 4 ) and that ϒ i ≥ Γ i (i = 1, 2), one obtains the existence of the functions ε * 1 and ε * 2 introduced in Theorem 2 such that, for |ε 1 | < ε * 1 (Γ 1 (0), Γ 2 (0)) and |ε 2 | < ε * 2 (Γ 1 (0), Γ 2 (0)), then V (t) ≤ - η 1 (V 1 (t) +V 2 (t)) 2(1 +V 1 (t)) - η 2 2 V 3 (t) (66)
The rest of the proof of Theorem 1 remains unchanged.

V. NUMERICAL SIMULATIONS To illustrate the merits of the proposed controller, we consider a numerical example inspired from the fluid flow system considered, e.g., in [START_REF] Bresch-Pietri | Prediction-based control of moisture in a convective flow[END_REF] [START_REF] Hernandez Perez | A time-delay approach fro modeling and control of mist in a poiseuille flow[END_REF]. It consists in an horizontal tube equipped with inlet and outlet fans and a mist injector. The mist injection is fixed and the control objective is to regulate the moisture at the output of the tube, that is, the ODE state X is the moisture while the control U is the fan power. A schematic view of the set-up is given in Fig. 1.

As studied in [START_REF] Bresch-Pietri | Prediction-based control of moisture in a convective flow[END_REF] [6], a stable first-order delay equation encapsulates the main features of the dynamics under consideration. Here, we consider that diffusion also occurs inside the tube. The moisture measured at the output thus satisfies (52) with A = -1/τ 0 and B(θ ) = θ /τ 0 in which the uncertain variable is the system gain (which slowly varies with thermodynamics conditions, and is then considered as uncertain). This system hence satisfies Assumptions 1-3 for any constant K(θ ) < 0.

For simulation, we take1 a = 1/6, τ 0 = 9s, θ = 0.08 with Π = [0.05, 0.11]. We consider parameter PDE uncertainties as in (52) with ε 1 = 0.1 and ε 2 = -0.17. We choose K(θ ) = -40. We compare closed-loop performances with a (similarly tuned) PI controller. Corresponding results are depicted in Fig. 2. One can observe that, when using the proposed adaptive controller, the obtained transient is much smoother.

In particular, this controller enables to avoid oscillatory response which can be observed with a PI controller and are responsible for overshooting and actuator saturation. This is the main interest of the proposed controller.

VI. CONCLUSION

In this paper, we proposed an adaptive controller enabling convection and diffusion compensation for a diffusive PDE-ODE cascade. Future works should focus on alleviating the assumption that the ODE state is measured, by studying a normal representation of the plant. A direct extension of the proposed work could be the design of a PDE-parameter adaptive controller. Also, counter-convection (that is, a < 0) could be handled by a methodology similar to the proposed one by extending the design of [START_REF] Susto | Control of PDE-ODE cascades with neumann interconnections[END_REF]. However, as counterconvection has a potentially destabilizing effect, one cannot rely anymore on an open-loop observer for the PDE state, as done in this paper, and should assume that additional information is available for the PDE state. This is a direction of future investigation.

Fig. 1 .

 1 Fig. 1. Schematic view of the flowtube system under consideration as an illustrative example.

  Evolution of the moisture, the fan power and the parameter estimate.

Fig. 2 .

 2 Fig.2. Numerical results for the closed-loop dynamics of the moisture modeled by (52) with A = -1/τ 0 and B(θ ) = θ /τ 0 , for two different controllers. The initial condition corresponds to a moisture (relative humidity) equilibrium of 50% that one wants to regulate to a new operating condition corresponding to 60% of moisture. The proposed adaptive controller has been implemented with a feedback gain K( θ ) = -40, an update gain γ = 60 and a tuning parameter b 1 = 10. The PI controller has been implemented with a proportional gain K P = -40 and an integral gain K I = -1.

Note that the convection speed is actually time-varying, but we neglect those variations here for the sake of simplicity. Also, diffusion is likely to be negligible compared to convection for this set-up in practice. Yet, for the sake of illustration, we consider the same scale for diffusion and convection coefficients in simulation.


Silicon soul: The vain dream of electronic immortality

Our bodies set some requirements for what it would mean to upload our brains. Nicolas Rougier, Author provided In just over 30 years, humans will be able to upload their entire minds to computers and become digitally immortal. -Ray Kurzweil, Global Futures 2045International Congress (2013) Without even considering the ethical, philosophical, social or legal scope of such a statement, it's important to consider if it actually makes any sense. To try to give an educated guess, we have to move away from computer science and look at what biology and neuroscience can teach us.

The sensible world

In his book Being There: Putting Brain, Body and World Together Again, Andy Clark explains that:

The biological mind is, first and foremost, an organ for controlling the biological body. Minds make motions, and they must make them fast -before the predator catches you, or before your prey gets away from you. Minds are not disembodied logical reasoning devices.

To better understand this assertion, it's essential to know that our bodies are literally covered with sensors -chemical, mechanical, visual, thermal, proprioceptive (perception of the body), noniceptive (perception of pain). All of them inform the brain about the outside world (exteroception) and the inner world (interoception), allowing it to regulate the body. The majority of our brain is actually dedicated to the processing of sensory information, and the largest part of that is devoted to visual information, occupying the entire occipital lobe and large parts of the temporal and parietal lobes.

We are mostly visual beings who, incidentally, think.

If someone wants to "upload his entire mind to a computer," the problem of sensors must be solved. A quick and dirty solution could be to not worry about them and just pretend that all sensory neurons would remain silent forever. However, 50 years ago, Donald O. Hebb conducted a series of experiments to study the effects of sensory deprivation. He generously paid students to recline 24/7, taking care to deprive them of most of their senses using glasses, helmets, gloves and so on. The majority of the students abandoned the experiment after two or three days because they were no longer able to develop coherent thoughts and began to suffer from auditory and visual hallucinations. The experiences evoked considerable interest from the CIA (which financed the original study) and they later "improved" the process up to the point where it was transformed into an instrument of psychological torture.

The body electric

Consequently, if we want to "upload our brain" without going insane, it's imperative for the uploaded brain to be connected to an artificial body that can perceive the outside world and act on it. But what kind of artificial body do we have today? Robotic bodies where retinas are replaced by cameras and muscles by motors? To some extent, yes, but this solution would be only a pale replica, far from the complexity and intelligence of the human body, as nicely explained by Rolf Pfeiffer and Josh Bongard's book How the Body Shapes the Way We Think.

During childhood, a brain learns through experience to control its body and to leverage its specifics. For example, consider the fingertips, which are sufficiently soft and sensitive that we can easily grasp small objects. There's no need for the brain to send a precise command -the intelligence is in the body itself. Imagine trying to do the same thing with thimbles over each finger, and you will understand how your body automatically solves a number of problems all by itself. As a first step, we could therefore use those simplified robotic bodies with their reduced sensory and motor skills. Would it affect our minds? Yes. Our cognition depends on the interactions we have with the world, and this interaction is conveyed by both our perceptions and our actions. If you change them, you also change the sensory experience of the world as well as its underlying logic. Cognition is embodied.

We further explore this question in the second part of this article.

This article was originally published in French

  Diagram of the retina, in its sensory complexity. Cajal, CC BY-SA What about the artificial eyes we'd need? Even though high-resolution cameras now exist, the eyes we're born with each have approximately five million cones and 100 millions rods, plus the various "preprocessing" stages, from the horizontal, bipolar, amacrine and ganglion cells. We are indeed very far from being able to reproduce a full artificial retina, even though some amazing research in Paris has succeeded in helping the vision-impaired see again.


Why you'll never be able to upload your brain to the cloud

In the first article in this series, we saw how the mind and body literally cannot be separated, and also why robotics isn't capable of replicating either one. But let's assume that we've solved the problems of sensors and muscles and all the rest, and accept that the uploaded brain won't truly reflect our mind. Now comes the big challenge: uploading the brain. But what is a brain exactly? This term usually refers to the cortex and possibly some subcortical structures, including the amygdala, hippocampus and basal ganglia. But the central nervous system is actually made of several other structures that are no less critical, including the cerebellum, thalamus, hypothalamus, medulla and brain stem.

Making the connections

If we consider the whole central nervous system, we are facing an average of 86 billion neurons, and each of these neurons contacts an average of 10,000 other neurons, representing a grand total of approximately 860 billion connections. This is really huge.

So exactly what do we have to upload into the computer? The type, the size and the geometry of each neuron? Its current membrane potential? The size and position of the axon and its state of myelination? The complete geometry of the dendritic tree?

The location of the various ion pumps? The number, the position and the state of the different neuro-mediators? Any of these could be critical, and they can only be taken into account in state-of-the-art computer models (and for a few neurons only). The problem is that we do not know exactly what it is that makes us who we are and different from anyone else (and I'm not even talking about learning).

As a fallback -and only if we had the proper tools to record each of these parameters once -we could try to transfer everything. However, this would require potentially some thousands or even millions of pieces of information for a single neuron. If you consider just the number of neurons, we would reach a figure in the zetta domain (for your information, the order is kilo, mega, giga, tera, peta, exa and zetta, multiplying by 1,000 at each step). This number is so huge that it cannot yet be manipulated as a whole by computer science. And we are talking only about the brain's storage, because we also have to ensure that this model runs in real time, since nobody would happily accept a silicon mind that runs at reduced speed. From a purely technical perspective, we are thus very far (really very far) from making this to happen.

Worse, research indicates that Moore's Law -which suggests that computer power doubles every 18 months -is reaching its limits, suggesting that we may never attain the necessary level of technology. The Human Brain Project foresaw this problem and planned from the beginning to use only simplified models of neurons and synapses. If you're interested in more accurate models, take a look at the OpenWorm project, which doesn't pretend to simulate any more than a few neurons.

The bird in the machine

This idea of transferring one's brain into a machine is widespread in both literature and cinema. It has gained renewed interest with recent advances in artificial intelligence.

However, there may be some confusion regarding what is actually artificial intelligence (AI) and what are its goals.

When media cover artificial intelligence, they generally refer to machine learning and robotics, neither of which really seeks to understand the brain or cognition (with some notable exceptions, such as the work of Pierre-Yves Oudeyer). This confusion likely stems from the fact that new algorithms have been designed that enable excellent performance on tasks that were previously thought to be reserved for humansrecognizing images, driving a car and so on.

But if machine learning and robotics are progressing at an amazing speed, this does not tell us anything about how the biological brain works (at least not directly). If we want to know, we have to look at neuroscience and more specifically at computational neuroscience. A parallel could be drawn between aeronautics (AI) and ornithology (neuroscience). Even though the early attempts at flying were directly inspired by the flight of birds, this was abandoned long ago in favor of the design of ever more efficient aircraft (speed, payload, etc) using techniques specific to aeronautics. To better understand birds, you must turn to ornithology and biology. Hence, talking about uploading a brain to a computer because of the progress of AI makes as much sense as gluing feathers on an airplane and pretending it's an artificial bird.

No one knows if it will ever be possible to "upload a brain to a computer." But what is certain today is that in the current state of science, this statement makes no sense and will remain so without a major epistemological breakthrough in our understanding of the brain and how it works.


Damien Szubert

Ioannis Asproulias

Fernando Grossi

Yannick Hoarau

Marianna Braza

Régis Duvigneau

Numerical study of the turbulent transonic interaction and

Introduction

The present study has been carried out in the context of the European research program TFAST, "Transition location effect on shock-wave/boundary-layer interaction", project N • 265455. One of the main objectives of this research is to provide optimal laminarity in the boundary layer upstream of the shock-wave/ boundary-layer interaction (SWBLI), in order to reduce the skin friction comparing to the fully turbulent case and therefore reduce drag, in the context of greening aircraft transport (a major objective of the Horizon 2020 European programme). Due to increased aerodynamic loads and aero-engine components nowadays, supersonic flow velocities are more frequent, generating shock waves that interact with boundary layers. Laminar shock-wave/boundary-layer interaction can rapidly cause flow separation, which is highly detrimental to aircraft performance and poses a threat to safety. This situation can be improved by imposing the laminar-turbulent transition upstream of the interaction, but this should be carefully done in order to keep the aerodynamic efficiency high (lift/drag ratio).

In this context of the European research program TFAST, several ways of controlling the position of the transition is carried out. To this end, a supercritical laminar wing (the V2C) has been designed by Dassault Aviation. This profile allows the boundary layer to remain laminar up to the shock foot, even in the environment of transonic wind tunnels of the laboratories involved in the project, and up to the angle of attack of 7.0 • . Experimental results for the present configuration are not yet available in the present research project. Regarding the related literature, the transonic buffet has been studied experimentally in detail since the 70s on circular-arc airfoils [START_REF] Mcdevitt | Transonic flow about a thick circular-arc airfoil[END_REF][START_REF] Seegmiller | Steady and unsteady transonic flow[END_REF], and most recently on supercritical airfoils [START_REF] Jacquin | Experimental study of shock oscillation over a transonic supercritical profile[END_REF]. In this latest study, a fixed transition tripping was applied at 7% of the chord. The physics governing the transonic buffet is complex and several theories have been proposed, like the effect of the feedback mechanism of waves propagating from the trailing edge, or the onset of a global instability [START_REF] Jacquin | Experimental study of shock oscillation over a transonic supercritical profile[END_REF][START_REF] Lee | Oscillatory shock motion caused by transonic shock boundary-layer interaction[END_REF][START_REF] Szubert | Shock-vortex shear-layer interaction in the transonic flow around a supercritical airfoil at high reynolds number in buffet conditions[END_REF]. Comparison of numerical results by Deck [START_REF] Deck | Numerical computation of transonic buffet over a supercritical airfoil[END_REF], Grossi et al. [START_REF] Grossi | Prediction of transonic buffet by delayed detached-eddy simulation[END_REF] and Szubert et al. [START_REF] Szubert | Shock-vortex shear-layer interaction in the transonic flow around a supercritical airfoil at high reynolds number in buffet conditions[END_REF] with the experimental results by Jacquin et al. [START_REF] Jacquin | An experimental study of shock oscillation over a transonic supercritical profile[END_REF] concerning the transonic buffet around supercritical wings with fixed transition showed the predictive capability of recent CFD methods and a physical analysis of the interaction between buffet and trailing-edge instabilities. The SWBLI involving transonic buffet and laminar wing design currently highly interests the aeronautical industries (Cleansky European project, "Advanced, high aspect ratio transonic laminar wing" [START_REF]Advanced, high aspect ratio transonic laminar wing for regional aircraft with load controll & alleviation devices[END_REF]). Laminar wing design in transonic regimes has been studied in respect of transition control by means of Discrete Roughness Elements (DRE's) [START_REF] Belisle | A transonic laminar-flow wing glove flight experiment: Overview and design optimization[END_REF].

Navier-Stokes simulations of transonic buffet as well as of the shock-vortex interaction at moderate Reynolds numbers were reported by Bouhadji and Braza [START_REF] Bouhadji | Organised modes and shock-vortex interaction in unsteady viscous transonic flows around an aerofoil: Part I: Mach number effect[END_REF], as well as DNS by Bourdet et al. [START_REF] Bourdet | Direct numerical simulation of the three-dimensional transition to turbulence in the transonic flow around a wing[END_REF]. In the high Reynolds number range, typical of aerodynamic applications, the use of appropriate turbulence modelling is necessary. Concerning transonic buffet, the unsteady shock-wave/boundary-layer interaction represents a major challenge for turbulence models and the low frequencies associated with the shock-wave motion can make the simulations very expensive. Since the first simulations by Seegmiller et al. [START_REF] Seegmiller | Steady and unsteady transonic flow[END_REF] and Levy Jr. [START_REF] Levy | Experimental and computational steady and unsteady transonic flows about a thick airfoil[END_REF] for a circular-arc airfoil, Unsteady Reynolds-Averaged Navier-Stokes (URANS) computations using eddy-viscosity turbulence models have been largely used to predict the phenomenon over two-dimensional airfoils. Pure LES simulations, even combined with specific wall-models, are yet quite costly for the high Reynolds number range of real flight configurations. For this reason, hybrid RANS-LES methods have been developed in the last decade and start to be largely used in the industrial context together with adapted, advanced URANS approaches. The hybrid methods combine the robustness and near-wall physics offered by URANS in the near region, as well as LES advantages in capturing the physics of unsteady vortices and instabilities development in the detached flow regions. Among the hybrid methods, the Detached-Eddy Simulation (DES) does not need to impose the interface between the statistical and LES regions. This is provided inherently by the choice of the turbulence length scale to use in the transport equations [START_REF] Spalart | Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach[END_REF]. In order to avoid approaching the near-wall region by the LES zone, the Detached-Eddy Simulation has been improved in respect of the turbulence length scale, ensuring a quite significant statistical zone around the body, in the context of the Delayed Detached-Eddy Simulation (DDES) [START_REF] Spalart | A new version of detached-eddy simulation, resistant to ambiguous grid densities[END_REF]. Moreover, improvement of the nearwall modelling has been achieved by means of a suitable Wall-Modelled LES (WMLES) in order to allow the flow physics modelling in the very near wall region covering the viscous sublayer by means of finer grids (but more economic than the LES) in the context of the Improved Delayed Detached-Eddy Simulation (IDDES) [START_REF] Shur | A hybrid RANS-LES approach with delayed-DES and wall-modelled LES capabilities[END_REF]. Regarding the transonic buffet simulations, Deck [START_REF] Deck | Numerical computation of transonic buffet over a supercritical airfoil[END_REF] has used a successful zonal DES approach, using mostly statistical modelling in the outer regions far from the body. He provided a detailed prediction of the transonic buffet around the supercritical airfoil OAT15A. Regarding the same configuration, Grossi et al. [START_REF] Grossi | Prediction of transonic buffet by delayed detached-eddy simulation[END_REF] performed a Delayed Detached-Eddy Simulation in the context of the ATAAC (Advanced Turbulence Simulations for Aerodynamic Application Challenges) European programme. This study succeeded in the prediction of the shock-wave self-sustained motion near the critical angle of incidence for the appearance of buffet, based on experimental results by Jacquin et al. [START_REF] Jacquin | Experimental study of shock oscillation over a transonic supercritical profile[END_REF][START_REF] Jacquin | An experimental study of shock oscillation over a transonic supercritical profile[END_REF]. Moreover, Szubert et al. [START_REF] Szubert | Shock-vortex shear-layer interaction in the transonic flow around a supercritical airfoil at high reynolds number in buffet conditions[END_REF] provided a detailed analysis of the buffet dynamics by means of the Organised Eddy Simulation (OES) approach, resolving the organised coherent modes and modelling the random turbulence background and using upscale turbulence modelling through stochastic forcing in order to keep the turbulent-non-turbulent shear-layer interfaces thin. In the present paper, the transonic buffet is applied on the V2C airfoil within the TFAST program, at 7.0 • , the maximum angle of attack allowed by the design, upstream Mach number 0.70 and Reynolds number 3.245 × 10 6 . The fully turbulent case is studied by differ-ent URANS and DDES modelling in two and three dimensions respectively. The predictive capabilities of statistical and hybrid turbulence modelling approaches are discussed. A 2D study is first carried out to investigate the main flow characteristics in respect of the angle of attack as well as the influence of the transition location. The transition location effects are also studied in the buffeting regime, by imposing the laminarity at several positions. Based on these results, the main objective of the present article is to put ahead a coupling of the aforementioned CFD methods with a non-classical optimisation approach of the transition location in the steady and unsteady transonic regimes in respect of the drag reduction and lift to drag ratio maximisation.

Numerical method and turbulence modelling

Flow configuration

Concerning the design of the V2C wing, it was validated numerically by Dassault on a 0.25 m-chord length (c) profile by means of RANS computations for various angles of attack at freestream Mach numbers of 0.70 and 0.75, yielding chord-based Reynolds numbers of approximately 3.245 × 10 6 and 3.378 × 10 6 respectively. The study was performed using a compressible Navier-Stokes code adopting a two-layer kε model, with the transition location being determined from the fully-turbulent flowfield using a three-dimensional compressible boundary-layer code by means of the N-factor amplification with a parabola method. The technique employed for laminarity and an initial design in respect of the transition prediction was based on the e N method (Ref. [START_REF] Crouch | Variable N-factor method for transition prediction in three-dimensional boundary layers[END_REF] for instance). The airfoil surface was generated in such a way that the N-factor remains small for low-to-moderate turbulence intensity levels, similar to the wind tunnel turbulence levels used for the present test-case for the experimental study currently in progress in the TFAST project. At Mach number 0.70, the flow separated between α = 6 • and 7 • . The amplification factor N was shown to be smaller than 3 up to the shock wave, thus guaranteeing laminar flow. At Mach 0.75, the value of N remained smaller than 2 up to α = 7 • . For this Mach number, there were not buffeting phenomenon, whatever the angle of attack. Moreover, for incidences higher than 1 • , the shock induces a separation of the boundary layer until the trailing edge.

Numerical method

The simulations of the V2C configuration at upstream Mach number M = 0.70 and Reynolds number Re = 3.245 × 10 6 have been carried out with the Navier-Stokes Multi-Block (NSMB) solver. The NSMB solver is the fruit of a european consortium that included Airbus from the beginning of 90s, as well as main European aeronautics research Institutes like KTH, EPFL, IMFT, ICUBE, CER-FACS, Univ. of Karlsruhe, ETH-Ecole Polytechnique de Zurich, among other. This consortium is coordinated by CFS Engineering in Lausanne, Switzerland. NSMB solves the compressible Navier-Stokes equations using a finite volume formulation on multi-block structured grids. It includes a variety of efficient highorder numerical schemes and of turbulence modelling closures in the context of URANS, LES and of hybrid turbulence modelling. NSMB includes efficient fluidstructure coupling for moving and deformable structures. For the present study, the third-order of accuracy Roe upwind scheme [START_REF] Roe | Approximate Riemann solvers, parameter vectors, and difference schemes[END_REF] associated with the MUSCL flux limiter scheme of van Leer [START_REF] Van Leer | Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method[END_REF] is used for the spatial discretisation of the convective fluxes. A similar upwind scheme (AUSM) was used by Deck [START_REF] Deck | Numerical computation of transonic buffet over a supercritical airfoil[END_REF]. For the diffusion terms, second-order central differencing has been used. The temporal discretisation has been done by means of dual-time stepping and of second order accuracy. A physical time step of 5 µs has been adopted for 2D simulations. For the 3D simulations, the time step has been reduced to 0.1 µs after detailed numerical tests. A typical number of inner iterations of 30 was necessary for the convergence in each time step.

The 2D grid has a C -H topology, and is of size 163,584 cells. The downstream distance of the computational domain is located at a mean distance of 80 chords from the obstacle. A grid refinement study has been carried out, by means of steady-state computations by using local time stepping, for the flow at M ∞ = 0.70 and α = 4.0 • using the kω SST model [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] and assuming fullyturbulent boundary layer, with two other grids: one 50% coarser, and another 30% finer. Detailed results of this convergence study can be found in [START_REF] Grossi | Prediction of transonic buffet by delayed detached-eddy simulation[END_REF]. The grid retained for the present study gave a maximum value of non-dimensional wall distance y + of about 0.55 with respect to the turbulence modelling. Fig. 1 shows the grid and the computational domain. For the 3D computations, the planar grid has been extruded to 59 cells uniformly distributed in the spanwise direction over a distance of 0.33 × c. The 3D grid contains about 9.65 M cells.

Boundary and initial conditions

On the solid wall, impermeability and no-slip conditions are employed. The far-field conditions are the characteristic variables extrapolated in time: the total pressure (P 0 = 10 5 Pa) and total temperature (T 0 = 290 K), as well as the upstream Reynolds number of 3.245 million and Mach number of 0.70. The upstream turbulence intensity is Tu = 0.08%.

The initial conditions are those of a steady-state generated field in each case.

Turbulence modelling

In the context of URANS and hybrid turbulence modelling, the following models have been used respectively: the two-equation k -ω SST model of Menter [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] as well as the OES-kε [START_REF] Szubert | Shock-vortex shear-layer interaction in the transonic flow around a supercritical airfoil at high reynolds number in buffet conditions[END_REF][START_REF] Bourguet | Anisotropic organised eddy simulation for the prediction of non-equilibrium turbulent flows around bodies[END_REF] and the DDES-kω SST models have been used with turbulence-sustaining ambiant terms to prevent the free decay of the transported turbulence variables [START_REF] Spalart | Effective inflow conditions for turbulence models in aerodynamic calculations[END_REF].

Optimisation method

In the context of the transition location study detailed in this paper, an optimisation of its location is proposed by employing a non-classical statistical learning approach. The principle consists in gathering a set of performance values, observed for different parameters, and construct a statistical model (Gaussian Process) on this basis, that reflects the knowledge and uncertainties related to the performance function. Then, this model is employed to determine the most interesting simulations to carry out, in a statistical sense. This approach is repeated until convergence [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF].

More precisely, the statistical model for the performance function f is constructed on the basis of a set of observed values

F N = { f 1 , f 2 , . . . , f N } at some points X N = {x 1 , x 2 , . . . , x N } ∈ R d (here d = 1
). F N is assumed to be one real-ization of a multivariate Gaussian Process which has a joint Gaussian distribution [START_REF] Sacks | Design and analysis of computer experiments[END_REF]:

p(F N |X N ) = exp -1 2 F N C -1 N F N (2π) N det(C N ) , (1)
for any collection of inputs X N . C N is the N × N covariance matrix, whose elements C mn give the correlation between the function values f m and f n obtained at points x m and x n . This is expressed in terms of a correlation function k, i.e.,

C mn = cov( f m , f n ) = k(x m ,
x n ; Θ) with Θ a set of hyper-parameters, calibrated on the basis of known points (likelihood maximisation principle). The Matérn class of covariance stationary kernels, which gives a family of correlation functions of different smoothness [START_REF] Sacks | Design and analysis of computer experiments[END_REF], is used for k.

After calculations based on conditional probabilities, the probability density for the function value f N+1 at any new point x N+1 is:

p( f N+1 |X N , F N ) ∝ exp - ( f N+1 -fN+1 ) 2 2 σ 2 f N+1 , (2)
where fN+1 = k N+1 C -1 N F N , (3)
σ 2 f N+1 = κ -k N+1 C -1 N k N+1 , (4)
with κ = k(x N+1 , x N+1 ; Θ) and k N+1 = [k(x 1 , x N+1 ; Θ), . . . , k(x N , x N+1 ; Θ)] . Thus, the probability density for the function value at the new point x N+1 is also Gaussian with mean fN+1 and standard deviation σ f N+1 . Therefore, the most likely value at the new point x N+1 is fN+1 . This value will be considered as the prediction of the Gaussian Process model. The variance σ 2 f N+1 can be interpreted as a measure of uncertainty in the value prediction. If the evaluation is known to be noisy, the model can account for the observation noise by modifying the diagonal terms of the covariance matrix, on the basis of the noise variance estimated for each database point [START_REF] Picheny | Noisy expected improvement and on-line computation time allocation for the optimization of simulators with tunable fidelity[END_REF].

At each step of the optimisation procedure, this Gaussian Process model is exploited to determine new points to be simulated. The most popular strategy is the maximisation of the Expected Improvement (EI) criterion [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF]. The maximisation of this criterion is numerically reached, by solving an internal optimisation problem using an evolution strategy.

Results

Two-dimensional study: angle of attack effects

This study has been carried out in URANS with the two-equation kω SST turbulence model [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] for an upstream Mach number M ∞ = 0.70. The angle of attack has been varied from 1.0 • up to 7.0 • , which is the maximum angle of attack for which the boundary layer is supposed to remain laminar from the leading edge to the shock wave. Initially, the computations adopt local time stepping. If convergence is not reached (i.e., a relative reduction of 10 -6 in the residual), time-accurate simulations with a time step of 5 × 10 -6 s are then carried out. Near the critical angle regarding the buffet, the angle of attack has been varied by an increment of 0.5 • in order to refine the critical buffet range.

Fig. 2 shows the final distributions of the pressure coefficient for the full range of incidences and the skin-friction coefficient for the steady cases. For angles of attack up to 5.0 • , the flow is steady and rear separation is always present. The shock wave can be distinguished at 2.0 • . As the angle of attack is further increased, the shock initially moves downstream, then it goes upstream for α > 3 • . From α = 4.0 • , a separation bubble appears and develops. The amount of the rear separation steadily increases with the angle of attack (Fig. 2(b)). The buffet onset, characterized by an oscillating shock wave, has been detected from 5.5 • . The main frequency increases with incidence in the range of 80-82Hz. A detailed comparison of the results obtained in the present study by the NSMB code has been carried out by using the Edge code, an unstructured compressible finite volume CFD code developed by the FOI since 1997 in collaboration with industrial and academic partners. The wall pressure distribution is plotted in Fig. 3 for angles of attack between 1.0 • and 7.0 • . This comparison showed small differences close to the critical angle, but the results were very similar at lower and higher angles of attack. This ensures about the validity of the present simulations, in absence of finalised experimental results within the TFAST programme.

Transition location effect

Two flow conditions have been selected for a numerical investigation of the transition location effect on the SWBLI, due to their interesting flow physics. First, the steady interaction arising at α = 4.0 • is addressed, featuring a reasonably strong shock just below the critical angle of attack for buffet onset. The second flow condition is the fully-established buffet regime at α = 7.0 • , which presents a large shock-wave motion region.

The transition is forced at the position x t by imposing the turbulent viscosity ν t = 0 for x < x t . Its location x t is varied from the leading edge up to as close as possible to the shock wave. The influence of the tripping point over the selected steady and unsteady transonic flow-fields is presented in the following two subsections.

Pre-buffet condition -Steady case

Results presented in the previous sections showed that, at α = 4.0 • and M ∞ = 0.70, the fully turbulent flow over the V2C airfoil is near critical with respect to transonic buffet. At that incidence, the shock wave is strong enough to induce a small separation bubble and the adverse pressure gradient over the rear part of the airfoil causes rear separation at about x/c = 0.91. The same flow condition has been recomputed considering different transition locations x t from the leading edge up to the mid-chord, remaining steady in all cases. The pressure and friction coefficients distributions over the upper surface are plotted in figure 4 for some chosen values of x t . The pressure coefficient indicates an increase of the suction effect as the transition position moves donwstream, while the shock position moves downstream. This facts yield an increase of lift. The trailing-edge pressure decreases, as well as the Cp on both sides of the rear airfoil's part. The x/C = 0.10 case can be qualitatively compared with the case of the OAT15A airfoil with fixed transition at x/C = 0.07, numerically studied by Grossi et al. [START_REF] Grossi | Prediction of transonic buffet by delayed detached-eddy simulation[END_REF] (Fig. 9 in this reference) and compared with the experimental data of Jacquin et al [START_REF] Jacquin | An experimental study of shock oscillation over a transonic supercritical profile[END_REF], where the same order of magnitude for the upstream and downstream pressure plateau is observed. A quite good comparison with the experiment is obtained. Therefore, despite the lack of experimental results up to now for the V2C airfoil, a fairly good agreement can be expected between the present CFD and experiments under way in the TFAST project. Moreover, the DDES results of Grossi et al. [START_REF] Grossi | Prediction of transonic buffet by delayed detached-eddy simulation[END_REF] provide a higher trailing-edge suction than URANS, associated with more intense separation. This feature is a similar tendency to the DDES behaviour of the present study, discussed in section 3.4, as well as with the zonal DES (ZDES) of Deck [START_REF] Deck | Numerical computation of transonic buffet over a supercritical airfoil[END_REF] (Fig. 6 in this reference). The effect of the transition location on the shock-wave position x s , on the location x b and length l b of the separation bubble as well as on the rear separation position x r are detailed in Table 1 for the complete set of simulations.

The tripping points can be easily identified on the friction coefficient by the sudden and high increase in the wall shear when the boundary layer becomes turbulent. They can also be distinguished on the pressure coefficient in the form of slight pressure disturbances in the supersonic region. As the transition location is shifted downstream, which induced a reduction in the boundary layer displacement thickness, the shock wave moves downstream, which can be noted in the figures. This produces higher Mach number levels in the supersonic pocket associated with lower pressures in that region, resulting in a stronger compression through the shock. As the laminar region increases, the progressively stronger shock wave makes the separation bubble grow continuously as indicated in Table 1 and by means of the C f distribution. On the contrary, the rear separation gets smaller, yielding a larger pressure recovery and eventually vanishing for x t /c ≈ 0.5. Table 1 provides also the force coefficients as the tripping point is varied. As the length of the laminar region, and thus the shock wave position move downstream, the lift increases due to a higher pressure difference between the upper and lower surfaces. The lift-to-drag ratio L/D is also provided. An optimal value is found near x t /c = 0.3. However, this position of transition does not give the minimum value of the global drag coefficient, which is obtained for a transition located near x t /c = 0.10, with a short laminar boundary layer region. This drag coefficient then increases with a longer laminar region, while the friction drag always diminishes as the laminar region becomes longer.

Unsteady regime

This study has been carried out to assess the influence of the transition point on the properties of the well-developed buffeting flow at 7.0 • . Besides the fullyturbulent case, three tripping locations have been considered: x t /c = 0.09, 0.16 and 0.24. For the latter, the most upstream position of the shock wave during buffet has been of about x t /c = 0.25. This limits the displacement of the tripping point, because imposing ν t = 0 inside the shock-motion region would not be an acceptable approximation. Fig. 5(a) presents the statistical pressure distributions obtained for each boundary layer tripping position. While the most upstream limit of the shock-motion range is not much sensitive to the transition location, its most downstream limit is strongly affected by the boundary layer state. As seen for the case α = 4.0 • , a larger extent of laminar boundary layer tends to move the shock wave further downstream by altering the displacement thickness distribution around the airfoil. In fact, this effect can also be observed in the unsteady case regarding the mean shock-wave position, which roughly corresponds to the point of maximum pressure unsteadiness in Fig. 5(b). As the tripping point is placed downstream, the amplitude of shock motion becomes wider, increasing the fluctuation levels in the shock-wave region as well as the trailing edge unsteadiness. This can be observed in the series presented in Fig. 6, in terms of statistical pressure fluctuation fields. Comparing the fully-turbulent simulation with the case of the most downstream transition location (x t /c = 0.24), the pressure unsteadiness increases by approximately 20% in the shock region and gets nearly two times larger near the trailing edge. The development of the shock-motion area as a function of the transition location is clearly visible in Fig. 6. In the last section of the article, an optimisation of the transition location effect has been carried out in respect of increasing aerodynamic performance.

Table 2 gives the average lift, drag and pitching moment coefficients for the three transition cases as well as for the fully-turbulent computation. The stan- dard deviation σ of the aerodynamic forces is also presented. As for the steady flow at 4.0 • , the values of the mean lift and of the moment magnitude inscrease as the triggering location moves towards the trailing edge. A slight augmentation in the mean drag is also noticed. As a result of the increasing shock-motion amplitude and of the overall flow unsteadiness, the standard deviations of the lift and drag coefficients also become larger as the extent of laminar boundary layer gets longer. Therefore the mean lift over mean drag ratio doesn't show much improvement whereas the laminar region is inscreased. Indeed, as the transition is located closer to the shock wave/boundary layer interaction, the boundary layer downstream detaches more easily than the fully-turbulent case, which gave here the higher lift-to-drag ratio. Moreover, due to the high angle of attack, the most upstream shock location is near 25% of the chord, which limits the flexibility on the transition position.

Optimisation of the tripping location 3.3.1. Problem description

As observed in the results above, the location of the transition point may have a significant impact on the airfoil performance, in particular when unsteady boundary-layer/shock interactions occur. From designer point of view, it would be interesting to quantify this influence for the different cases (steady and unsteady) and determine the best tripping location, which maximises the airfoil performance. In this perspective, a study is presented for the optimisation problem formulated as:

Maximise

f (x) = C L /C D for x ∈ I, (5)
where x is the stripping location and I the allowed search interval. This is a PDEconstrained optimisation problem including a single parameter. The major difficulty arises from the computational cost related to the unsteady flow simulations and the possible noisy prediction of the performance due to the presence of numerical errors (discretization, time integration). The use of a classical descent optimisation method is tedious, due to the unsteady functional gradient estimation. Alternatively, stochastic approaches like genetic algorithms or evolution strategies require too many evaluations to be practically tractable.

Results for the steady case

A steady flow problem is first considered, corresponding to the 2D case described above for an incidence α = 4.0 • . In this context, the performance is simply the lift-to-drag ratio computed at convergence. The tripping location can vary in the interval I = [0.1c, 0.5c]. Five configurations, corresponding to x 1 = 0.1c, x 2 = 0.2c, x 3 = 0.3c, x 4 = 0.4c and x 5 = 0.5c, are achieved independently to construct a first database. A Gaussian Process model for the lift-to-drag ratio function is then constructed according to the previous section and illustrated by Fig. 7. On this figure, one can see the model itself, its associated standard deviation and the expected improvement (EI) criterion used to drive the search and select the next point to simulate. As can be seen, after two additional simulations, the standard deviation is strongly reduced and the expected improvement almost zero. Moreover, the next point to simulate, as proposed by the EI criterion, is very close to a know point and the mesh accuracy for the tripping point location is reached. As consequence, the optimisation process is stopped. Finally, two conclusions can be drawn from this optimisation exercise: a large area, from x = 0.25c to x = 0.35c corresponds to a very high lift-to-drag ratio, and the best performance is obtained for a tripping location close to x = 0.3c.

Results for the unsteady case

We consider then the more challenging case corresponding to unsteady flows, for which shock-wave/boundary-layer interactions generate buffets. This study has been carried at an incidence of α = 5.8 • . In this case, the shock-motion amplitude is limited and allows for a wider range of transition locations than at higher incidence. Here, the performance function is the time-averaged lift-to-drag ratio, computed once a quasi-periodic flow is obtained. The admissible interval for the tripping point is moved upstream I = [0, 0.31c], to avoid the shock to be located in the laminar area. Five configurations, corresponding to the fully turbulent case x 1 = 0, then x 2 = 0.0825c, x 3 = 0.165c, x 4 = 0.2475c and x 5 = 0.31c, are achieved independently to construct a first database. Note that the configuration x 5 = 0.31c exhibits instabilities after a long time integration. For this case, the time-averaging process has been shortened to avoid these phenomena. Fig. 8 represents the Gaussian Process model for the time-averaged lift-todrag ratio, at iterations 0 and 1. The initial model (iteration 0) yields an Expected Improvement criterion localized around a maximum at x 6 = 0.2665c. This configuration is simulated and added to the database, yielding an updated model (iteration 1). Since the lift-to-drag ratio computed by simulation is very close to the one predicted by the model, the variance of the model is strongly reduced, as well as the Expected Improvement criterion, as soon as the first iteration. Therefore, the optimum tripping value should be close to x 6 .

To validate this result, three additional test points (T P) are simulated a posteriori, corresponding to x T P 1 = 0.12c, x T P 2 = 0.2c, x T P 3 = 0.25c and the results are compared to the model prediction. It appears that the performance value for x T P 3 slightly differs from the model prediction, due to the fact that the unsteady flow exhibits some low frequency oscillations, which make the estimation of the time-averaged lift-to drag ratio more difficult. To account for this uncertainty in the performance estimation, a variance estimate of the time-averaged lift-to-drag ratio is computed for all configurations, by using a classical moving average procedure. This variance is introduced into the Gaussian Process model as an observation noise. Fig. 9 shows the resulting model, that does not interpolate database points anymore, against additional test points. As can be observed, the uncertainty in the performance estimation is not negligible in this context, especially when the tripping point is close to the shock wave location, which corresponds to the best performance area (x between 0.2c and 0.3c). Nevertheless, the statistical model allows having a better analysis of the problem. In particular, one can underline that the confidence interval of the model is smaller than the standard deviation of the observations, in the zone where several points have been computed. In conclusion, for this unsteady case, the airfoil performance is better for a tripping point x between 0.2c and 0.3c, but the corresponding flows exhibit additionnal unsteadiness because of interaction with the existing buffet instability, that could be dommageable in real conditions.

Three-dimensional simulation of the fully-turbulent case

The DDES-kω SST model has been applied, using the same numerical scheme as the 2D computations and time step ∆t = 10 -7 s), in order to examine the 3D dynamics of the fully developed transonic buffet occurring over the V2C airfoil at M ∞ = 0.70 and α = 7.0 • . The turbulence length scale provided by the RANS part is computed using local turbulence properties and is given by √ k/(β * ω). A comparison of the DDES results with the URANS kω SST [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] as well as the 2D and 3D OES-kε [START_REF] Szubert | Shock-vortex shear-layer interaction in the transonic flow around a supercritical airfoil at high reynolds number in buffet conditions[END_REF][START_REF] Bourguet | Anisotropic organised eddy simulation for the prediction of non-equilibrium turbulent flows around bodies[END_REF] is provided. Concerning the grid spacing, which has to be nearly isotropic in the three dimensions, in respect of the DDES choice of the turbulence length scale, 59 cells have been distributed over a 0.33c span with a constant spacing, resulting in a final grid of about 9.65 M cells. The computations have been carried out in the SGI Altix supercomputer at CINES (Centre informatique national de l'enseignement supérieur), by using 1024 parallel processors in MPI.

Flowfield dynamics

The time-dependent lift coefficient according to the aforementioned models is presented in Fig. 10 for the fully established regimes, beyond transient phases. While in URANS kω SST the lift coefficient oscillates quasi-harmonically at a frequency of approximately 82 Hz, the DDES produces sharp-like and much stronger lift fluctuations. The high slope of the curve indicates that the shockwave speed is relatively high, especially during the lift fall when the flow separates and the shock moves upstream. This may explain, at least partially, the somewhat higher mean buffet frequency found in the DDES case (approximately 108 Hz). The large amplitude of the fluctuations suggests also that the shock-wave motion range is wider than in case of the kω SST model. The OES-kε model provides an almost sinusoidal behaviour of the oscillations at 107 Hz and a bit higher Turb. model. amplitudes than the kω SST. This behaviour is in-between the URANS and DDES evolutions. The spectral analysis of the lift coefficient is shown in Fig. 11, where St = f U O /c is the non-dimensionalised frequency, with f the frequency in Hz, U O = 228 m.s -1 the upstream velocity and c = 0.25 the chord of the airfoil. These spectra are similar to the experiments by Jacquin et al [START_REF] Jacquin | An experimental study of shock oscillation over a transonic supercritical profile[END_REF][START_REF] Jacquin | Experimental study of shock oscillation over a transonic supercritical profile[END_REF] concerning the buffet mode identification for the OAT15A supercritical airfoil configuration in the same Mach and Reynolds number range. Moreover, the OES modelling sensitised to reduce the turbulent diffusion and enhance coherent structure appearance, provides spectra of a similar shape to the study of Szubert et al [START_REF] Szubert | Shock-vortex shear-layer interaction in the transonic flow around a supercritical airfoil at high reynolds number in buffet conditions[END_REF] carried out for the OAT15A, showing the buffet frequency as well as a spectral bump related to the von Kármán mode associated with alternating vortices past the trailing edge, as shown in Fig. 17 for the V2C profile. This two-mode interaction sustains a feedback loop including also Kutta waves as shown in this figure, in qualitative comparison with experiments (Fig. 18). This aero-acoustic feedback mechanism was schematically presented in Lee [START_REF] Lee | Self-sustained shock oscillations on airfoils at transonic speeds[END_REF]. Table 3 shows the values of the buffet frequency as well as the corresponding Strouhal numbers and mean and RMS values of lift coefficient per turbulence model.

k -ω SST OES-k -ε 2D OES-k -ε 3D DDES- k -ω SST C D ×
In the spectra, DDES-the kω SST provides the highest continuous spectral level, indicating a high turbulence diffusion rate generated by this model. The kω SST and OES-kε provide in addition to the main frequency bump corresponding to the buffet instability, bumps beyond 2000 Hz, which are related to the von Kármán instability and other vortex interactions past the trailing edge as discussed in [START_REF] Szubert | Shock-vortex shear-layer interaction in the transonic flow around a supercritical airfoil at high reynolds number in buffet conditions[END_REF]. In all the spectra, the presence of the buffet mode is illustrated by a frequency bump instead of a sharp peak, because of the non-linear interactions of the buffet mode with the von Kármán, shear-layer and other more random mo- tions downstream. Therefore, the present simulations capture the dynamics of the buffet and near trailing-edge instability modes and of their interaction producing a multitude of frequencies between these modes, which sustain a feedback loop among the shock oscillation region, the separated shear layer and the near wake. These interactions and feedback loop, schematically reported in [START_REF] Lee | Self-sustained shock oscillations on airfoils at transonic speeds[END_REF], have been analysed in detail by [START_REF] Szubert | Shock-vortex shear-layer interaction in the transonic flow around a supercritical airfoil at high reynolds number in buffet conditions[END_REF], using time-frequency analysis by means of wavelets and Proper Orthogonal Decomposition in addition to a spectral analysis. Fig. 12 shows the mean surface pressure coefficient. All models are in agreement in the suction side, with a slight increase of lift near the trailing edge in case of OES-k -ε, which corresponds to a higher averaged lift coefficient (Table 3). The kω SST model produces the shortest inclination of the C p within the shock region and therefore the less developed shock oscillation amplitudes. The largest ones correspond to the DDES-kω SST, as expected from the 'sharp' lift coefficient oscillations. This feature, accompanied by a higher trailing-edge pressure plateau, is similar to a thicker airfoil's C p , as for example in the experimental study of McDevitt et al for a circular-arc airfoil in transonic regime [START_REF] Mcdevitt | Transonic flow about a thick circular-arc airfoil[END_REF], as well as in the ZDES of Deck [START_REF] Deck | Numerical computation of transonic buffet over a supercritical airfoil[END_REF] (Fig. 6 in this reference) and in the DDES-Spalart-Allmaras study of Grossi et al. [START_REF] Grossi | Prediction of transonic buffet by delayed detached-eddy simulation[END_REF] (Fig. 9 of this reference). This common tendency occurs among these approaches using different numerical schemes (the AUSM in Deck's study, the 3 rd order Roe upwind scheme in Grossi's study as well as in the current study). Therefore, it seems that the hybrid RANS-LES models provide a higher level of suction and flow detachment in the present family of supercritical airfoils. This behaviour can be explained by means of turbulent viscosity levels of the three modelling approaches used in the present study and by considering the frontier between the URANS and LES regions, commented in a dedicated discussion at the end of this section. The OES-kε produces an in-between behaviour, similar to the flow simulations around the supercritical OAT15A airfoil with fixed transition at 7% (Szubert et al [START_REF] Szubert | Shock-vortex shear-layer interaction in the transonic flow around a supercritical airfoil at high reynolds number in buffet conditions[END_REF]) which compare quite well to experimental results by Jacquin et al [START_REF] Jacquin | An experimental study of shock oscillation over a transonic supercritical profile[END_REF][START_REF] Jacquin | Experimental study of shock oscillation over a transonic supercritical profile[END_REF]. Therefore, it can be reasonably supposed that in the V2C case, a fairly good comparison of the present URANS studies (better than the DDES behaviour) is expected from the ongoing experimental campaign in the TFAST project. Fig. 13 shows the mean pressure fields superimposed with streamlines according to the previous models. The DDES illustrates the largest separation area and the OES indicates a higher circulation intensity, corresponding to the lift increase. The kω SST and OES-kε provide qualitatively comparable recirculation regions. The same feature stands for the mean velocity profiles shown in Fig. 14, in the near-wall region concerning the locations x/c = 0.2, 0.4, 0.6 and 0.8. In the In order to understand the DDES behaviour which provided such a large separation, the distribution of the RANS and LES regions has been monitored allowing assessment of the present DDES ability to switch between the two modes (URANS and LES) during buffet and of the size of the two regions. The instantaneous distributions of the delaying function 1f d of the DDES at four phases of buffet are given in Fig. 19. The irregular black areas over the upper surface indicate large regions of separation, even when the shock is at its most downstream position (Fig. 19(d)), where a large amount of rear separation exists on the upper surface. This analysis shows the existence of a RANS-mode layer covering the near-wall region around the V2C airfoil. The overall height of this layer seems to be relatively small. This might cause some degree of modelled-stress depletion (MSD) [START_REF] Spalart | A new version of detached-eddy simulation, resistant to ambiguous grid densities[END_REF] due to the erroneous penetration of the LES mode into attached boundary layers, which facilitates separation. This behaviour was also observed in the DDES studies by Deck [START_REF] Deck | Recent improvements in the zonal detached eddy simulation (zdes) formulation[END_REF] improved by a zonal DES approach. In this article, Fig. 19, the development of the shear layer instabilities appear at a considerable distance past the separation point, whereas in our case they appear earlier (Fig. 15). The article also by Uzun et al. [START_REF] Uzun | An application of delayed detached eddy simulation to tandem cylinder flow field prediction[END_REF] has been referenced thanks to a clear representation of the f d function delimiting the RANS region in the boundary layer around the body (Fig. 7 in their study), which is similar to the behaviour of this function in the present study and the fact that the shear layers past the cylinder are treated by LES in that study. In addition, these shear layers (Fig. 8 in that paper) display the instability development at a considerable distance downstream of the separation point in respect of the appearance of Kelvin-Helmholtz vortices.

The same behaviour was reported by Mockett et al. [START_REF] Mockett | Two non-zonal approaches to accelerate rans to les transition of free shear layers in des[END_REF] concerning the acceleration of the transition between RANS and LES in a free shear layer by various DES approaches. The present DDES behaviour illustrated by the previous flow visualisations, the mean C p and lift coefficients can be explained as follows: the pressure near the trailing edge (Fig. 12) is underestimated in the case of DDES, displaying a significant suction comparing with the URANS cases. The DDES provides a higher shock's excursion from the leading edge up to more than half of the chord yielding a pressure increase in this area. Therefore, the resulting lift is lower than in URANS and consequently, the corresponding circulation is lower. In this case, the pressure aspiration effect on the suction side and the overall separated region seems to be more intense than in other cases. The related instabilities are more pronounced and start more upstream in the shear layers than in cases where the excursion of the shock has a shorter amplitude. These results are not linked to a strong delay in the formation of instabilities in the shear layer and in the overall suction region but on a too early onset of instabilities. This is viewed in the 3D plots of Fig. 15 where a strong and rich statistical content of vortices are developed in the suction area from practically the leading edge. Indeed, the difference between the "peaky" shape and the more "sinusoidal" one indicates that at the same instant, the lift is lower in the DDES case, upstream and downstream of the sharp peak. This behaviour is in accordance with the "peaky" shape of the lift coefficient displayed by the DDES-SST, comparing to the OES simulations (Fig. 10). This is in accordance with the aforementioned elements concerning the pressure distribution and the mean lift. The reasons for this can be as follows. In Fig. 19, the frontier between RANS and LES regions are shown. It can be seen that a significant part of the shear layer is handled by RANS computation (see dark zone past the separation point), but this does not inhibit the development of instabilities which are quite displayed in the upstream region. Moreover, the dark region surrounding the airfoil near the wall is associated with RANS computation within the boundary layer. Therefore, the LES approaches drastically the wall region. This would need a finer grid in this area. Moreover, the reason for the DDES behaviour, also depicted by another partner (URMLS) within the TFAST European program (M. Bernardini, S. Pirozzoli, private communication), by using DDES-SA and a different numerical code, may be due to the turbulent viscosity produced by the model in association with the grid. In order to illustrate the effect of the turbulent viscosity produced by the turbulence model, the ratio ν t /ν is plotted in Fig. 20. It can be seen that the DDES produces a much lower turbulence viscosity (order of 200 in the separated regions) than the URANS-OES (order of 1800), leading to a lower dissipation level which excessively amplifies smallerscale structures in the separated area and a more intense separation. In the OES case, the higher ν t level improves this feature. The use of the Spalart-Allmaras model instead of the kω SST in the DDES provided even higher shock amplitude oscillations because the maximum ratio ν t /ν was of order 250 [START_REF] Grossi | Physics and modeling of unsteady shock wave/boundary layer interactions over transonic airfoils by numerical simulation[END_REF], Fig. 6.15 in this reference. This behaviour was shown for the lift oscillation in [START_REF] Szubert | Hybrid RANS-LES and URANS simulations of a laminar transonic airfoil[END_REF], Fig. 13. As can be shown in the lift oscillations, the 'peaky' behaviour disappears on the benefit of a more sinusoidal shape with a higher pressure plateau up to 30% of the chord, a shorter excursion of the shock as well as an improved effect on the pressure 'plateau' near the trailing edge with less suction (see Fig. 12).

In a study in progress, the OES-kε model results will be analysed in detail, in order to take benefit from the more regular buffet oscillations and simultaneously from the formation of the additional frequency bumps shown in Fig. 11, as in the study by Szubert et al. [START_REF] Szubert | Shock-vortex shear-layer interaction in the transonic flow around a supercritical airfoil at high reynolds number in buffet conditions[END_REF].

Conclusion

The present study analysed the SWBLI in the case of the transonic flow around the V2C-Dassault Aviation profile in two and three dimensions by means of statistical and hybrid RAND-LES turbulence modelling in the high Reynolds number regime of 3.245 million. The critical range of angle of attack for the buffet appearance has been investigated by means of 2D URANS computations and found near 5.5 • . The different flow phenomena occurring around the airfoil for various angles of attack at Mach number 0.70 have been analysed. The pressure and skin friction distributions have shown the angle of attack effect on the shock wave position, as well as on the state of the boundary layer interaction with the shock foot. The influence of a fixed transition location on the flow physics has been studied in the steady and unsteady cases and particularly on the buffet dynamics. Based on these results, a major outcome is a non-classical optimisation procedure coupling the CFD results with a Kriging method, applied to the transition location regarding the averaged aerodynamic coefficients. In the steady case, an optimal position of the fixed transition has been found near x t /c = 0.30 regarding the averaged lift/drag ratio. Particularly, the transition location effect on the unsteady case with buffeting conditions (angle of attack of 5.8 • ) has been analysed with the same method and yields an optimum position at x t /c = 0.2665. These elements contribute to the improvement of laminar wing design for future generation of aircraft's wings, in respect of the greening requirements of the Horizon 2020 objectives. Furthermore, the flow dynamics of a fully developed buffet case at angle of incidence of 7.0 • have been investigated in respect of the predictive abilities of statistical and hybrid turbulence modelling. The DDES simulations displayed a rich content of resolved flow structures and provided a strongly detached flow and a large shock amplitude, extended from the leading to the trailing edge. This behaviour has been analysed and discussed in respect the MSD and eddy-viscosity levels induced by this modelling associated to the present grid and numerical parameters. The URANS simulations based on the kω SST model have indicated a high turbulence diffusion level and a decrease in the appearance of instabilities pas the trailing edge, as well as a short shock amplitude. The OES approach provided an intermediate behaviour between the two mentioned with a reasonably extended shock amplitude and capturing of the von Kármán and shearlayer vortices donwstream of the SWBLI and of the trailing edge. In a study in progress, the association of DDES with OES will be examined in order to take relative benefits from both approaches.

Figure 1 :

 1 Figure 1: Multiblock domain

Figure 2 :

 2 Figure 2: Effect of the angle of attack on the steady and mean surface distributions and on the friction coefficient

Figure 3 :

 3 Figure 3: Comparison of the wall pressure coefficient between NSMB end Edge codes, for angles of attack between 1.0 • and 7.0 •

  Pressure coefficient (b) Friction coefficient

Figure 4 :

 4 Figure 4: Steady surface distributions for selected transition locations

  Mean pressure coefficient (b) RMS value of pressure on the upper surface

Figure 5 :

 5 Figure 5: Transition location effect on the statistical wall pressure at α = 7.0 •

Figure 6 :

 6 Figure 6: RMS pressure fields for different transition locations at α = 7.0 •

Figure 8 :

 8 Figure 8: Statistical model for the lift-to-drag ratio regarding the tripping location (unsteady case), for iteration 0 (left) and iteration 1 (right)

Figure 9 :

 9 Figure 9: Statistical model for the lift-to-drag ratio as a function of the tripping location (unsteady case), accounting for the observation variance

Figure 10 :

 10 Figure 10: Comparison of the time-dependent evolution of the lift coefficients between URANS kω SST, 2D and 3D OES-kε and DDES-kω SST

Figure 11 :

 11 Figure 11: Comparison of the power spectra density of the lift coefficients time-dependent evolution between URANS kω SST, 2D and 3D OES-kε and DDES-kω SST

Figure 12 :Figure 13 :Figure 14 :

 121314 Figure 12: Comparison of the wall pressure coefficient between URANS kω SST, OES-kε and DDES-kω SST

Figure 15 :

 15 Figure 15: Instantaneous Q-criterion isosurfaces for Q(c/U) 2 = 75

Figure 16 :

 16 Figure 16: Divergence of velocity field -DDES-kω SST

  (a) t * = 0 (b) t * = 0.83 (c) t * = 3.28 (d) t * = 4.92

Figure 17 :Figure 18 :

 1718 Figure 17: Divergence of velocity field -OES-kε

  (a) Maximum lift (t * = 0) (b) Shock upstream (t * = 1.46) (c) Minimum lift (t * = 2.96) (d) Shock downstream (t * = 6.02)

Figure 19 :Figure 20 :

 1920 Figure 19: RANS and LES regions around the V2C airfoil according to the DDES-kω SST

Table 1 :

 1 Transition location effect on the shock position, on separation and on the global aerodynamic coefficients

	x t /c	fully	0.10	0.20	0.30	0.40	0.50
		turb.
	x s /c	0.523	0.532	0.541	0.552	0.564	0.574
	x b /c	0.533	0.541	0.547	0.556	0.566	0.575
	l b /c (%)	1.1	2.4	4.7	6.8	8.5	9.4
	x r /c	0.911	0.925	0.946	0.965	0.981	-
	C L	0.8873	0.9174	0.9556	0.9919	1.029	1.061
	C D f × 10 2	0.610	0.574	0.510	0.460	0.396	0.334
	C D × 10 2	2.080	2.069	2.102	2.171	2.268	2.365
	L/D	42.7	44.3	45.5	45.7	45.4	44.9

Table 2 :

 2 Transition location effect on the mean global coefficients, lift, drag and moment, for the unsteady

	x t /c	Fully turb.	0.09	0.16	0.24
	C D × 10 2	6.163	6.501	6.604	6.715
	σ (C D ) × 10 2	0.9419	1.250	1.384	1.533
	C L	0.9423	0.9718	0.9927	1.018
	σ (C L )	0.0854	0.1047	0.1132	0.1204
	C m × 10 2	-4.223	-4.932	-5.267	-5.676
	C L /C D	15.3	14.9	15.0	15.2

Table 3 :

 3 Comparison of the aerodynamic coefficients and buffet frequency between turbulence modelling methods for 9 periods of buffeting

Acknowledgements

The present work was supported by the TFAST European project N • 265455 -Transition location effect on shock-wave/boundary-layer interaction. The research team thankfully acknowledges the French computing centers CINES (Centre informatique national de l'enseignement supérieur) and CALMIP (Calcul en Midi-Pyrénées) for the allocated resources as well as for their availability. They thank Matteo Bernardini and Sergio Pirozzoli for their valuable discussions about the turbulence modelling behaviour in the context of the TFAST collaboration.