text:
Mario Ostieri
email: mariostieri@gmail.com

Renato Tognaccini
email: renato.tognaccini@unina.it

Didier Bailly
email: didier.bailly@onera.fr

Daniel Destarac
email: daniel.destarac@onera.fr

Aerodynamic

Ph D Student

Aerodynamic force and Lamb vector field in compressible unsteady flows

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. Introduction

Unsteady flow phenomena are increasingly becoming important in the present industrial framework. Not only trendy applications like drones or MAVs, but also rotorcraft applications, noise reduction, or, more generally, physical insight into natural flight like birds or insects one, require accurate prediction of unsteady flow vortical structures, directly linked to force generation. During last decades, many theories have been developed to physically link aerodynamic force generation to local flow properties, like entropy or vorticity. Betz [START_REF] Betz | A method for the direct determination of wing-section drag[END_REF] introduced a so-called farfield method with which irreversible drag production could be computed by a wake survey of the flow, instead of stress analysis on the body. Oswatitsch [START_REF] Oswatitsch | Gas Dynamics[END_REF] introduced the entropy drag concept, with which irreversible drag could be associated to local entropy production. Maskell [START_REF] Maskell | Progress towards a method for the measurement of the components of the drag of a wing of finite span[END_REF] proposed a method to compute reversible drag effects, in particular the lift-induced drag, in incompressible flow, linking it to the vorticity on the Trefftz plane. Paparone and Tognaccini [START_REF] Paparone | Computational Fluid Dynamics-based drag prediction and decomposition[END_REF] and Van der Voreen and Destarac [START_REF] Van Der Vooren | Drag/thrust analysis of jet-propelled transonic transport aircraft; definition of physical drag components[END_REF] proposed drag breakdown methods for steady flows, with which total drag could be separated in lift-induced, viscous and wave components, directly linking these components to local entropy production. Tognaccini also discussed the case of "powered" configurations [START_REF] Tognaccini | Drag computation and breakdown in power-on conditions[END_REF]. These applications, however, were only limited to steady flows.

Gariepy et al. [START_REF] Gariepy | Generalization of the far-field drag decomposition method to unsteady flows[END_REF] proposed a first extension of Van der Vooren and Destarac method to unsteady flows. Toubin and Bailly [START_REF] Toubin | Development and application of a new unsteady far-field drag decomposition method[END_REF] developed another extension of the steady breakdown theory, with application to complex two-and three-dimensional configurations. Finally Toubin et al. [START_REF] Toubin | Improved Unsteady Far-Field Drag Breakdown Method and Application to Complex Cases[END_REF] further developed the method, introducing a drag component that accounts for acoustic propagation effects, with an application to a detached-eddy simulation of buffet phenomenon around an airfoil.

A new exact theory for the incompressible aerodynamic force computation was introduced by Wu et al [START_REF] Wu | Integral force acting on a body due to local flow structures[END_REF]. It relies on the local Lamb vector field (the cross-product of vorticity and velocity), non negligible in a limited region of the flow only (mid field property). Marongiu and Tognaccini [START_REF] Marongiu | Far-field analysis of the Aerodynamic force by Lamb vector integrals[END_REF] proposed an extension to turbulent flows of this theory, with applications to steady and unsteady two-dimensional flows. Marongiu et al. [START_REF] Marongiu | Lift and lift-induced drag calculation by Lamb vector integration[END_REF] succeeded in giving a rigorous definition of reversible steady force effects (lift and lift-induced force components) in terms of vortex force.

A first extension of this theory to compressible flows was already proposed by Wu et al. [START_REF] Wu | Vorticity and Vortex Dynamics[END_REF], and further developed by Liu et al. [START_REF] Liu | A dynamic counterpart of Lamb vector in viscous compressible aerodynamics[END_REF], [START_REF] Liu | Longitudinal-transverse aerodynamic force in viscous compressible complex flow[END_REF]. Mele and Tognaccini [START_REF] Mele | Aerodynamic force by Lamb vector integrals in compressible flow[END_REF] retrieved the mid field property, proposing an alternative exact formula, with applications to two-dimensional transonic and supersonic flows. Mele et al. [START_REF] Mele | Vorticity based breakdown of the aerodynamic force in three-dimensional compressible flows[END_REF] successfully extended definition of lift-induced drag to compressible regime, adding to the vortex force a compressibility correction term. Finally, Ostieri et al. [START_REF] Mele | Aircraft lift and drag decomposition in transonic flows[END_REF] applied this method to the analysis of a transonic aircraft configuration, and proposed a new definition of viscous and wave drag components.

Very recently, Ostieri et al. [START_REF] Ostieri | Exact aerodynamic force decomposition and dynamic force derivatives in non linear flows[END_REF], [START_REF] Ostieri | Linear and nonlinear decomposition of the aerodynamic force acting on an oscillating plate[END_REF] focused on the application of Wu's theory to unsteady flows. They introduced an alternative aerodynamic force expression, more accurate when dealing with moving rigid grid simulations, and introduced a new definition of dynamic force derivatives, thanks to a direct link with classical inviscid theories.

In present paper, an extension of Ostieri et al. [START_REF] Ostieri | Exact aerodynamic force decomposition and dynamic force derivatives in non linear flows[END_REF], [START_REF] Ostieri | Linear and nonlinear decomposition of the aerodynamic force acting on an oscillating plate[END_REF] theory to compressible flows is proposed. Numerical applications have been performend on a pitching airfoil in subsonic flow and an airfoil buffet test case, already proposed by Toubin and Bailly [START_REF] Toubin | Development and application of a new unsteady far-field drag decomposition method[END_REF]. A sensitivity analysis of the Lamb-vector based theory to the integration domain is firstly presented for both test cases. The vortical decomposition then has been compared with drag breakdown computations performed by Toubin and Bailly [START_REF] Toubin | Development and application of a new unsteady far-field drag decomposition method[END_REF], showing some analogies between reversible and irreversible effects found by these two methods.

II. Aerodynamic drag breakdown and local irreversible velocity

A brief summary of the thermodynamic drag breakdown theory [START_REF] Toubin | Development and application of a new unsteady far-field drag decomposition method[END_REF][START_REF] Toubin | Improved Unsteady Far-Field Drag Breakdown Method and Application to Complex Cases[END_REF] is here reported. Considering a compressible unsteady viscous flow around a moving rigid body, the aerodynamic drag D can be so defined:

D = D v + D w + D ui + D m (1)
where

D v = ∫ S v [-ρ (u irr -u ∞ ) V + τ v ] • n dS - ∫ V v ∂ ρ (u -u ∞ ) ∂t dV (2)
D w = ∫ S w [-ρ (u irr -u ∞ ) V + τ v ] • n dS- ∫ V w ∂ ρ (u -u ∞ ) ∂t dV+ - ∫ V w d ∂ ρ (u -u irr ) ∂t + 1 u irr ∂p ∂t dV (3)
D ui = ∫ S f ar [-ρ (u -u irr ) V -(p -p ∞ ) i x ] • n dS - ∫ V c ∂ ρ (u -u irr ) ∂t + 1 u irr ∂p ∂t dV (4)
D m = - ∫ S b ρ (u -u ∞ ) V • n dS (5)
In these equations V = [u, v, w] T is the velocity, u ∞ is the freestream velocity, ρ is the density, p is the pressure, τ v is the viscous stress tensor, n is the normal versor, i x is the x-axis versor, aligned with the free-stream velocity. D ui is the so-called unsteady-induced drag, D v is the viscous drag, D w is the wave drag and D m is a component that takes into account for the body motion, and is usually negligible. u irr is the irreversible velocity [START_REF] Van Der Vooren | Drag/thrust analysis of jet-propelled transonic transport aircraft; definition of physical drag components[END_REF], [START_REF] Toubin | Development and application of a new unsteady far-field drag decomposition method[END_REF], so defined

u irr = u ∞ 1 + 2∆H u 2 ∞ - 2 K pv -1 M 2 ∞ e ((Kpv-1)∆s)/Kpv R -1 ( 6
)
where R is the gas constant, K pv is the heat capacity ratio of the gas, H = h + 0.5V 2 is the total enthalpy (h is the enthalpy) and s is the entropy.

The viscous, wave and shock wave regions V v , V w , V wd and their boundary surfaces can be selected by proper sensors [START_REF] Toubin | Development and application of a new unsteady far-field drag decomposition method[END_REF] in the whole integration domain V. The complementary domain V c is defined as

V \ (V w V wd V v ).
Toubin et al. [START_REF] Toubin | Improved Unsteady Far-Field Drag Breakdown Method and Application to Complex Cases[END_REF] also proposed a decomposition of the unsteady drag D i :

D i = ∫ S f ar [-ρ (u -u irr ) V -(p -p ∞ ) i x ] • n dS - ∫ V c ρ (u -u irr ) 1 a - 1 u irr ∂u ∂t dV (7)
D pa = ∫ V c ρ u -u irr a ∂R + ∂t dV ( 8
)
where a is the local speed of sound, R + = u + 2a/(K pv -1) is the Riemann invariant. D pa is a drag component that takes into account for acoustic propagation effects, D i is what they call lift-induced drag, and

D ui = D i + D pa .
The breakdown is based on thermodynamic assumptions to define the irreversible velocity. It is numerically robust since it doesn't rely on gradients computation. It can also take into account for spurious drag effects due to numerical discretization.

III. Aerodynamic force and Lamb vector field A. Incompressible regime

Considering a viscous unsteady incompressible flow around a moving body, Wu et al. [START_REF] Wu | Integral force acting on a body due to local flow structures[END_REF] derived an alternative exact expression of the aerodynamic force in terms of the vortex force F :

F a = F + F t + F S f ar + F S b + F t b + ∆ µ . (9)
where

F = - ∫ V ρ dV , F t = - ∫ V r × ρ ∂ω ∂t dV ( 10
)
F S f ar = - ∫ S f ar r × (n × ρ )dS , F S b = - ∫ S b r × (n × ρ )dS , F t B = ∫ S b r × (n × ρa)dS , ∆ µ = ∫ S f ar r × n × ∇ • τ v dS + ∫ S f ar τ v • n dS ,
with r the position vector with respect to an arbitrary pole. F t and F S b are related with I = ρ ∫ V r × ωdV, the hydrodynamic impulse in V. Indeed, taking into account for Reynolds transport theorem with S f ar fixed in time, we have:

F t + F S b = - dI dt ≡ F τ . (11)
Equation ( 9) also holds in three-dimensional flow, provided r is replaced by r = r/2.

B. Compressible regime

Mele and Tognaccini [START_REF] Mele | Aerodynamic force by Lamb vector integrals in compressible flow[END_REF] derived an exact expression of aerodynamic force in compressible flows, again in terms of the vortex force, that is

F b = F + F m ρ + F tρ + F S f ar + F S b + F t b + ∆ µ . (12)
where

F m ρ = - ∫ V r × ∇ × ρ∇ V 2 2 dV = - ∫ V m ρ dV , F tρ = - ∫ V r × ∇ × ρ ∂V ∂t dV.
F m ρ is an additional compressibility correction for the vortex force, while F tρ is an unsteady term that, differently from F t takes also into account of compressibility effects. ∆ µ is already negligible for moderately high Reynolds number flows. In steady flows a force breakdown by means of this formula has been introduced [START_REF] Mele | Vorticity based breakdown of the aerodynamic force in three-dimensional compressible flows[END_REF]. Specifying with i x and i z the unit versors aligned and orthogonal to the freestream, we have:

D p = -i x • ∫ S f ar r × (n × ρ ) dS , (13)
L = -i z • ∫ V ρ + m ρ dV , (14)
D i = -i x • ∫ V ρ + m ρ dV . ( 15
)
where D p is the profile drag (including wave contribution), L is the aerodynamic lift and D i is the lift-induced drag.

C. Link with inviscid theories

1. Incompressible regime Very recently, Ostieri et al. [START_REF] Ostieri | Exact aerodynamic force decomposition and dynamic force derivatives in non linear flows[END_REF], [START_REF] Ostieri | Linear and nonlinear decomposition of the aerodynamic force acting on an oscillating plate[END_REF] found the link between present aerodynamic force theory and inviscid results. Considering a two dimensional incompressible inviscid flow around a flat plate, of length 2b, centered at x = 0, and integrating the pressure loading ∆p from -b to x w , where x w > b is a wake station, and taking into account that ∆p is 0 on the wake:

l = - ∫ x w -b ∆p(x, t)dx = ρV ∞ ∫ x w -b γ(ξ, t)dx + ρ d dt ∫ x w -b (x w -ξ) γ(ξ, t)dξ ( 16
)
where γ is the concentrated vorticity distribution over the flat plate, and on the wake. Similarly, for a viscous flow, taking into account for identity [START_REF] Marongiu | Far-field analysis of the Aerodynamic force by Lamb vector integrals[END_REF], valid for incompressible flows, equation [START_REF] Toubin | Improved Unsteady Far-Field Drag Breakdown Method and Application to Complex Cases[END_REF] gives:

F = -ρ ∫ V dV - d dt ∫ V r × ωdV -ρ ∫ S f ar r × n × dS. ( 17
)
This formula is valid for an arbitrary choice of the pole. Considering S f ar composed by a wake plane S w orthogonal to the wake and adopting as pole for the moment calculations the wake center on S w (r w ), we have

F = -ρ ∫ V dV - d dt ∫ V (r -r w ) × ωdV -ρ ∫ S f ar (r -r w ) × n × dS. ( 18
)
The connection of the first two terms of equation ( 18) with the inviscid formula ( 16) is evident. The last term of equation ( 18) is only a drag contribution [START_REF] Mele | Aerodynamic force by Lamb vector integrals in compressible flow[END_REF]. Furthermore, integration by part of this term gives the momentum defect on the wake [START_REF] Wu | Theory for Aerodynamic force and moment in viscous flows[END_REF], therefore it takes into account for viscous effects, and is the irreversible contribution to the aerodynamic force:

F irr = -ρ ∫ S f ar (r -r w ) × n × dS. (19)
The reversible part of the aerodynamic force is given by the remaining part, responsible for the lift generation:

F r ev = -ρ ∫ V dV + ρ d dt ∫ V (r w -r) × ωdV = = -ρ ∫ V dV + ρ ∫ V (r w -r) × ∂ω ∂t dV + ρ ∫ S b (r w -r) × n × dS . (20)
Equation ( 20) is the perfect counterpart of equation ( 16) in viscous flows. With some algebra, we can obtain another expression of F irr :

F irr = -ρ ∫ S f ar r × n × dS -ρ ∫ V r w × ∂ω ∂t dV + ρ ∫ S b r w × n × dS (21)
The definition of F r ev and F irr is rigorous as far as the definition of r w is clear. Indeed, while x w in an inviscid flow can be chosen arbitrarily, in a real viscous flow x w has to be placed at the end of the near wake, where for near wake here we intend the flow zone just after the body TE where the effect of the body vorticity (bound vorticity) is still relevant. As opposed to the thermodynamic method, the Lamb vector formula can also compute and decompose lift.

An extension to compressible flows

An extension of the definition of reversible and irreversible force components in the compressible regime is here proposed. Proposed formulae are consistent with the incompressible breakdown. We have:

F rev = F + F m ρ + F t * (22)
F irr = - ∫ S f ar r × n × ρ dS + ∫ V r w × ∂ ρω ∂t dV + ∫ S b r w × n × ρ dS ( 23
)
where F t * :

F t * = - ∫ V r × ∇ × ρ ∂V ∂t dV + ∫ V r w × ∂ ρω ∂t dV + ∫ S b (r w -r) × n × ρ dS (24)

IV. Numerical applications A. Pitching airfoil

The numerical solution around a pitching NACA 0012 here analyzed is the same performed by Toubin and Bailly [START_REF] Toubin | Development and application of a new unsteady far-field drag decomposition method[END_REF]. Simulations were performed with ONERA elsA code [START_REF] Cambier | ElsA: An efficient object-oriented solution to CFD complexity[END_REF]. The imposed pitching motion is sinusoidal, with an oscillation between -5 • and +5 • . The freestream Mach number is M ∞ = 0.3, the Reynolds number Re ∞ = 6.6 • 10 6 , the reduced frequency k = 0.1. 9000 time steps for each time period have been set. Spalart-Allmaras turbulence model and Jameson numerical scheme have been adopted. The grid is composed of about 530000 cells, with 568 body points. Grid near the body is shown in figure 1. The farfield is placed at 20 chords distance in the downstream direction, and 25 chords in the others. Vorticity contours at different timesteps are shown in figure 2.

In figure 3 computed lift coefficient has been analyzed. Equation ( 12) and near field computation are in good agreement. The same result is also obtained by C l r ev . For this test case a weak sensitivity to the choice of the integration domain was found for the lift coefficient. In figure 4 the computed drag coefficients are proposed as obtained by eqs. ( 1) and ( 12). Both formulae are in very good agreement with the near field computation. Again, as for the lift coefficient, total drag is not sensitive to the chosen integration domain.

Drag breakdown

To obtain the drag breakdown in reversible and irreversible contributions by Lamb vector method, a right choice of x w is required. Indeed, the strongest limitation of the vortical method is that drag breakdown changes significantly with the value of x w . Here a strategy is proposed for a correct x w definition. The vortex drag, as obtained by varying the integration domain is displayed at two different time steps in figure 5. The abscissa x w represents the position of the wake plain chosen for the integration. In this test, vortex drag, directly proportional to the local vorticity, has a sudden increase just after the TE, then shows a periodical-like behavior, similar to the periodical wake solution predicted in inviscid incompressible flows. This behavior starts at x w ≈ 0.4. For this test this x w had only very small variations with time. All results proposed have been obtained with x w = 0.4.

The two drag breakdowns are compared in figure 6. The decomposition performed by equation ( 1) was already presented by Toubin and Bailly [START_REF] Toubin | Development and application of a new unsteady far-field drag decomposition method[END_REF]. C d v and C d ir r values are similar: differences are within 10 drag counts. In particular, the agreement between C d ui and C d r ev is more satisfactory. Please note that, while the results of the thermodynamic method are weakly sensitive to the chosen integration domain, the proposed vorticity based reversible and irreversible breakdown is much more dependent on the choice of the wake plane.

C d ir r C d v C d r ev C d u i

B. Airfoil buffet

The unsteady RANS simulation of the buffet around an OAT15A airfoil is the same reported in [START_REF] Toubin | Development and application of a new unsteady far-field drag decomposition method[END_REF]. Reynolds number is Re ∞ = 3 • 10 6 ; freestream Mach number is M ∞ = 0.73, and angle of attack α = 4.5 • . κω Wilcox turbulence model and Roe numerical scheme have been used. The grid domain extends about 40 airfoil chords downstream, upstream and laterally. Mesh is built up of about 150000 cells, with 679 body points. A detail of the grid near the body is shown in figure 7. Each oscillation period is composed of 1000 time steps, with a frequency of 60 Hz. 5 sub-iterations only were adopted in the dual time stepping procedure. Vorticity contours at different timesteps are shown in figure 8.

Numerical issues affected the analysis of the aerodynamic force by equation [START_REF] Marongiu | Lift and lift-induced drag calculation by Lamb vector integration[END_REF]. Indeed, differently from equation (1), vortical method requires the computation of space derivatives and volume integrals in the zones where accuracy is poorer, such as the shock region and the wake, where the grid is coarser. The dual-time stepping used was good enough to obtain a converged near-field force. On the contrary, the analysis with the vortical method requires a deeper convergence; indeed sufficient agreement with the near field prediction was only possible choosing the boundary of the integration domain very near the airfoil as shown in figures 9 and 10. In this figures, x s and y s specify the size of the integration domain, x s is the position of the wake plane, whereas y s the lateral distance.

Drag breakdown

The same strategy applied to the pitching airfoil test case is here proposed. Vortex drag variation while changing wake plane position at different timesteps is shown in figure 11. In this case was not possible to find a value for x w . The analysis was carried out varying x w and comparing the vortical breakdown with the thermodynamic one.

In figure 12 is shown the breakdown proposed by Toubin and Bailly [START_REF] Toubin | Development and application of a new unsteady far-field drag decomposition method[END_REF]. The decomposition between viscous and wave components is proposed. The proposed induced drag has a positive small mean value, while D pa is larger. Wave drag varies significantly, following the behavior of the shock during the buffet [START_REF] Toubin | Development and application of a new unsteady far-field drag decomposition method[END_REF]. The drag breakdown by vortical method is proposed in figure with a comparison with the thermodynamic decomposition. The value choosen for x w is 4. There is a qualitative agreement between proposed breakdown in magnitude, but not in phase with the time. At higher values of x w the phase tends to become the same of the thermodynamic breakdown, but the magnitude is affected by the numerical errors which affects the total drag computation.

V. Conclusions

Two different aerodynamic force breakdown methods, valid in unsteady and compressible viscous flow, have been here compared. The first technique, proposed by Toubin and Bailly [START_REF] Toubin | Development and application of a new unsteady far-field drag decomposition method[END_REF] is based on a local irreversible velocity definition that relies on local entropy production, here referred as thermodynamic method. This method cannot compute aerodynamic lift, but can accurately compute aerodynamic drag as shown by Toubin and Bailly [START_REF] Toubin | Development and application of a new unsteady far-field drag decomposition method[END_REF], Toubin et al. [START_REF] Toubin | Improved Unsteady Far-Field Drag Breakdown Method and Application to Complex Cases[END_REF]. The second method is based on a Lamb-vector based theory [START_REF] Ostieri | Exact aerodynamic force decomposition and dynamic force derivatives in non linear flows[END_REF][START_REF] Ostieri | Linear and nonlinear decomposition of the aerodynamic force acting on an oscillating plate[END_REF], here referred as vortical method. This method, that can compute not only aerodynamic drag, but also the aerodynamic lift, permitted for a new definition of reversible and irreversible contributions to the aerodynamic force, where the first takes into account for local entropy production effects, and the second is connected to the lift generation.

Numerical applications have been proposed for two physically completely different test-cases: a pitching NACA 0012 in subsonic flow and an airfoil natural buffet. Toubin and Bailly drag breakdown results have been compared with the proposed vortical force breakdown. As concerning the pitching airfoil test case, total aerodynamic lift is correctly computed by the vortical method, and the reversible contribution to the force correctly computes all the aerodynamic lift. Total aerodynamic drag computed by both methods is in perfect agreement with near field computations. The numerical results revealed a satisfactory quantitative agreement between proposed drag decompositions. In the second test case, total force computed by the vortical method is again in satisfactory agreement with the near field results. The comparison between thermodynamic and vortical breakdown revealed not sufficiently satisfactory, with a qualitative agreement in magnitude but not in phase. Further improvements are necessary for the vortical decomposition, in order to obtain a breakdown formulation not dependent on the integration domain.

  C l b = lift coefficient by F b C l r ev = reversible lift coefficient C d = drag coefficient C d b = drag coefficient by F b C d i = lift-induced drag coefficient C d ir r = irreversible drag coefficient C d r ev = reversible drag coefficient C d u i = unsteady lift-induced drag coefficient C d v = viscous drag coefficient C d w = wave drag coefficient C d = vortex force drag coefficient D i = lift-induced drag D m = motion drag D pa = acoustic propagation drag D ui = unsteady lift-induced drag D v = viscous drag D w = wave drag F = aerodynamic force F a = Wu's full aerodynamic force expression F b = Mele and Tognaccini full aerodynamic force expression F irr = irreversible part of the aerodynamic force F r ev = reversible part of the aerodynamic force F t = unsteady contribution to aerodynamic forces F t * = modified F t F tb = contribution to aerodynamic force F τ = F t + F S b h = enthalpy k = ωc/2V∞ reduced frequency H = total enthalpy K pv = gas heat capacity ratio I = hydrodynamic impulse = Lamb vector m ρ = compressible correction term M ∞ = freestream Mach number n = [n x , n y , n z ] T = unit normal vector p = pressure r = position vector R = gas constant R + = u + 2/(K pv -1) Riemmann invariant Re ∞ = freestream Reynolds number S f ar , S b = far field and body surfaces T = oscillation period u irr = irreversible velocity V = (V ∞ + u, v, w) T = velocity vector V ∞ = freestream velocity x, y, z = cartesian coordinates x s , y s = max x and y values that define intgration domain x w = wake station x w = wake station vector α, ᾱ = pitching law motion and amplitude µ = dynamic viscosity ∆ µ = contribution to aerodynamic force τ R = Reynolds stress tensor τ v = viscous stress tensor V = computational domain V c = complementary volume V v = viscous volume V w = shock wave volume V wd = shock wake volume γ =

Fig. 1

 1 Fig. 1 Detail of the computational mesh used for the pitching airfoil test case.

Fig. 2 Fig. 3 Fig. 4 Fig. 5

 2345 Fig. 2 Pitching NACA 0012, Re ∞ = 6.6 • 10 6 , M ∞ = 0.3, ᾱ = 5 • , k = 0.1. Vorticity contours at different timesteps.

Fig. 6

 6 Fig. 6 Pitching NACA 0012, Re ∞ = 6 • 10 6 , M ∞ = 0.3, t/T = 0.125, k = 0.1. Comparison between proposed drag components.

Fig. 7

 7 Fig. 7 Detail of the computational mesh used for the airfoil buffet test case.

Fig. 8 10 Fig. 9 10 Fig. 10 Fig. 11 Fig. 12 CFig. 13

 81091010111213 Fig. 8 Airfoil OAT15A, Re ∞ = 3 • 10 6 , M ∞ = 0.73, ᾱ = 4.5 • . Vorticity contours at different timesteps.


Emmanuel Hache

Do renewable energies improve energy security in the long run?

Keywords: Energy transition, energy security, critical materials, patents, energy technology JEL Classification: Q48, Q58, Q34, O33, O34

The aim of this article is to analyze the geopolitical consequences of the spread of renewable energies worldwide. From a macroeconomic point of view, it would be tempting to conclude that the transition to renewables (solar, wind…) will gradually end today's geopolitics of fossil fuels based on historical relationships between energy producers and consumers. The new challenges induced by energy transition policies could paradoxically turn out being as complex as today's geopolitics of energy. Local and decentralized relations could add a new geopolitical layer to current traditional actors. Technical, economic, sociological, behavioural, spatial and legal dimensions could also complicate the emerging puzzle. A massive diffusion of renewables into the world's energy mix could also lead to new, unexpected interdependencies such as dependencies to critical materials, a new geopolitics of patents and the implementation of a renewable diplomacy.

Historically, the highlight on the strategic nature of crude oil has always been associated to World War I. This period remains a true catalyst for the importance to dispose of oil resources and to ensure their securing. Thus was reinforced the idea that the search for energy sources constituted a major diplomacy component. The 1930s and World War II further strengthened this dynamic and made the structuration of international relations possible. Indeed, the 1928 Achnacarry agreement4 , just like the 1931 Red Line agreement or the 1945 Quincy agreement, brought forward structural elements to the relations between countries consuming and producing oil. Later, in 1960, this "harmony" was ruptured by the creation of the Organization of the Petroleum Exporting Countries (OPEC), which symbolized the idea that the possession of a power source could be used as a political weapon and a power instrument.

The OPEC thus unveiled a new face of international relations, one in which the geological geopolitics of certain countries carry a too heavy weight -disproportioned when compared to their population and their gross domestic product (GDP) 5 .

The European construction was especially representative of the impact energy has on interstate relationships. Indeed, it's around this very factor that were structured the first integration attempts, in particular within the European Coal and Steel Community (ECSC), in 1952, or the European Atomic Energy Community (EAEC or Euratom), in 1957. In 1973, the OPEC's power takeover on oil markets found its match within consuming countries, through the creation of the International Energy Agency (IEA) and the establishment of mandatory strategic petroleum reserves.

Since the 1970s, questions regarding the access to and the reliance on natural resources have also been structuring elements for the establishment of energy policies in different importing countries, especially Europe and the United States of America (USA). At the European level, as well as at the French one, energy security constitutes one of the main objectives of the energy policy. This concept has been subject to a new revival during the last decade even if its definition can appear vague. IEA 6 defines energy security as "the uninterrupted availability of energy sources at an affordable price. Energy security has many aspects: long-term energy security mainly deals with timely investments to supply energy in line with economic developments and environmental needs. On the other hand, short-term energy security focuses on the ability of the energy system to react promptly to sudden changes in the supply-demand balance." By a simple logic of supply in terms of volume, the energy security policies progressively revealed their multiform nature, i.e. a necessary evolution depending on time, space and market conditions. Chester (2010) defines energy security as a polysemic concept considering that it contains various dimensions at the same time that also depends and evolves according to the economic paradigms observed in the energy markets.

Today, the distinguishing features of these energy policies are made up of four main components7 described as the 4A's: availability8 , accessibility9 , affordability10 and acceptability11 . However some authors (Cherp and Jewell, 2011, 2014) considered that these four factors do not address security questions and need to be completed. Hughes (2009) introduced the four "R's" concept. In order to clarify the energy security concept, he explains that a new methodology should be introduced based on "review (understanding the problem), reduce (using less energy), replace (shifting to secure sources), and restrict (limiting new demand to secure sources)".

For the past two decades, a strong interest has emerged in favor of the integration of renewable energies in the energy and electric mix in order to ensure security within the framework of energy transition policies, but also in order to fight against climate change.

Implementing these renewable energies is all the more relevant because they allow the state to earn double dividends, as their diffusion de facto reduces the volume of imported fossil energy12 . For instance, in France, the 2015 law on energy transition imposes a 30% diminution objective in the consumption of fossil energy, which would in turn reduce by about 30% the country's energy dependence (given that France imports 99.9% of its fossil energy), induce a decrease in commercial deficit (and thus, of their financing) and could disrupt certain geopolitical balances or relationships with other importing countries.

In 2014, renewable energies represented roughly 23% of the world's electric production -27.7% of the electric production capacity -and close to 59% of the newly set up capacities. This evolution is occurring in an environment where their promotion goes through the affirmation that the geopolitical tensions associated to their development will decrease. Thus, the transition to a consumption of renewable energies would bring about less, if any, conflicts or rivalries linked to the use of resources. The geopolitics of renewable energies, and of energy transition in general, would therefore be "softer" and less conflictual than that of carbonated energy sources. But the new challenges generated by energy transition policies could, paradoxically, reveal themselves to be just as complex as the current energy geopolitics. Hence, there is a risk of tensions in those new relations, which will be more local and less centralized, being created on top of the traditional actors (producers, consumers). It would be equally tempting to conclude that a transition towards renewable energies will progressively put an end to fossil energy geopolitics. Yet, instead of just disappearing, it seems more likely that it will shift interstate relations. Finally, the massive diffusion of renewable energies in the global energy mix could also generate new dependences.

1-Defining the energy transition concept

The notion of energy transition is rather imprecise. Transitioning from a finite energy source model to a flow energy source model, decarbonizing the energy mix… in reality, the simplest definition seems to be the following: the progressive replacement of the main primary source of energy consumption (US Department of Energy, Energy in Brief, 2001). This definition is especially useful to integrate energy transition within a historical framework. Humanity has known numerous energy transitions, including the use of fire, the manufacturing of tools and handles, wind domestication (mills), water domestication (mills and reservoirs), and, eventually, the discovery of fossil energies. Each has progressively transformed the global economic and energy environment. In the USA, more than thirty-five years were necessary for coal to replace wood in the energy mix (1885) and practically a century for oil to become the main consumed energy source (1950) (Figure 1). At the present time, the primary energy consumption in the world remains dominated by fossil energy sources (87%, of which 32.9% for oil, 29.2% for coal and 23.8% for gas), with the hydraulic (6.8%), the nuclear (4.4%), and other renewable energies (2.7%) completing the global landscape (BP Statistical Review, 2016). In this regard, the French energy mix is quite distant from the global energy mix, with the nuclear representing 41% of its primary energy consumption, oil 34%, gas 15%, coal and hydraulic only 2%, and other renewables 2% (Ministry of Ecology, Sustainable and Energy Development, 2015 key numbers). In Germany, in 2013, energy consumption was divided between oil (32.4%), coal (25.7%), gas (23%), renewable energies (11.5%) and nuclear energy (8%) (Eurostat data, 2015). All these elements might establish a new line of approach on the link between the massive diffusion of renewable energies and energy security. On the one hand, it is necessary to understand that energy transition is a long process, especially because existing energy systems are inert and historical. The consequences of energy policies thus need to be evaluated in the long term. On the other hand, it is important to ensure temporal coherence between all the elements of these policies. Finally, these energy policies integrate a national dimension which needs to be accounted for, given that each mix reflects a temporal, technical and economic construction based on the constraints and preferences of national policy makers.

2-A complex relationship with energy security

In France, the law on energy transition covers three main goals: fighting against climate change 13 , mastering the demand for energy and the research for energy efficiency 14 , diversifying provision supplies and reaching energy independence 15 . While those simultaneous goals will generate double dividends for environment and security, the massive diffusion of renewable energies forces one to rethink the relationships between producers, consumers, and transit countries. Indeed, renewable energies, unlike fossil energies, are not finite, and their geographic concentration is rather low. Moreover, they bring an important degree of diversification to energy systems. A contrario, their dependence on energy flows 13 Reducing greenhouse gas effects by 40% between 1990 and 2030, and divide them by four between 1990 and 2050. 14 Reducing the final consumed energy by 50% by 2050 (as compared to 2012), while targeting an intermediary goal of 20% in 2030. 15 A 30% reduction in the primary energy consumption of fossil energies by 2030, against a 2012 reference and an increase in the share of renewable energies at 23% in the crude final energy consumption in 2020 and at 32% in 2030.

Oil 33%

Natural Gas 24%

Coal 29%

Nuclear Energy 4% Hydro electric 7%

Renewables 3%

(wind, sunshine), the poor development of efficient storage techniques and the exacerbated rivalry with terrestrial resources, especially for their installation 16 , generates other dependences.

In national energy transition scenarios, the question of the development of renewable energies invites reflection regarding new territory geopolitics. Indeed, in numerous cases, the envisaged solutions rely on decentralized energy systems, in which case local communities will in large part be responsible of their energy needs. Questions of governance (expertise division between local communities, synergy with other local policies on housing and transportation), of global coherence (investment planning, risk of resources for renewable energies piling up), and of rivalry between different actors (citizens, companies, etc.) will undeniably pop up. In this regard, the example of Germany is interesting to observe: between 2000 and 2012, the renewable energy share in electricity consumption went up from 7% to 23%, and more than 50% of the new renewable energy capacities were invested in by citizens (private parties, cooperatives and farmers), against only 7% by large energy companies (E.ON, EnBW, RWE and Vattenfall). The success in the economic diffusion of renewable energies was the result of the establishment of a triple incentive regime, comprised of judicial, contractual (creation and management of cooperatives) and financial (fiscal advantages and buyback rates in the medium term) frameworks 17 . We could also wonder about the relevance of transposing this model in other European countries, where the markets' oligopolistic structures would render its implementation more difficult.

For scenarios relying on an externalization of electricity production structures 18 , a geopolitical appreciation of renewable energy diffusion requires studying the risks and sensitivity of substituting an actor producing a fossil resource by another producing a transformed resource.

Localizing large-scale projects can turn out to be a fundamental question in order to evaluate the impact that the diffusion of renewable energies can have on international geopolitics.

Prioritizing the objectives of energy policies for renewable energies will be just as decisive as it is in the absence of renewable energies in the energy mix. These objectives should allow a certain arbitration between the supply of resources at a lower cost and the potential risk of national supply deficits in the case of hostile relations with the capacity-hosting countries. 16 When developing biofuels, the potential impact on food insecurity will also depend on many variables, such as future agricultural yields and a change in consumption habits. 17 Noémie Poize and Andreas Rüdinger, « Projets citoyens pour la production d'énergie renouvelable : une comparaison France-Allemagne », IDDRI Working Paper, n°1, January 2014. 18 For instance, the Desertec project studies the establishment of a massive renewable energy diffusion in North Africa, in particular to feed the European grid in electricity.

3-Substituting resource geopolitics with one another: the example of critical materials

While technology is usually put forward in the dynamics of energy transition, it could also represent an impediment to the massive diffusion of innovations in the medium term. The first identified risk is that of the technology itself, of its cost, of its accessibility, and of its acceptance by the different stakeholders. It is accompanied by a second factor, which is linked to the decentralization and automation of systems, i.e. the risk of cyber-terrorism or of longdistance takeover of electric production units. The third identified risk tends to minimize the impacts involved with reducing geopolitical dependence when introducing renewable energies in the energy mix, i.e. the question of critical or strategic metals.

Found in numerous decarbonization technologies, critical metals are essential to the energy transition, whether it be directly -integration within technologies -or indirectly -component linked but independent of the technology, such as batteries for electric vehicles. Hence, whether it be for the sectors of electric vehicles and plug-in hybrids (cobalt, lanthanum, lithium, etc.), of catalysts or fuel cells (platinum, palladium, rhodium, etc.), the wind-powered (neodymium, dysprosium, terbium, etc.), the civil aeronautical (titanium), or even solar PV (cadmium, indium, gallium, etc.) sectors, all the innovations involved in the decarbonization process rely in fine on the availability of strategic minerals. Yet, the large-scale diffusion of energy transition technologies could exacerbate tensions on these metals' markets, for several reasons.

Most of these markets are rather small in size compared to that of non-ferrous metals 19 ; they are unorganized 20 , poorly transparent, and the majority of their transactions are done by mutual agreement. Thus, a "strategic" resource, i.e. essential for the industry but potentially risky in terms of supply, would rapidly risk becoming "critical" in the case of a massive deployment of these technologies. Moreover, the metals used for innovations in the energy transition process are, for the most part, coproducts generated from mining activities. Hence, 19 The markets for non-ferrous metals (copper, aluminum, nickel, etc.) produce in millions of tons, while smaller metals usually produce in tons, hundreds of tons, or, less frequently, in thousands of tons. 20 Like the London Metal Exchange for the non-ferrous metals there is no global trading platforms and organized market for strategic markets such as lithium, platinum, rhodium… their extraction and production are geologically and economically reliant on other metals 21 . In this context, supply elasticity according to price movement remains low, which doesn't help in reducing tensions in the short term.

Let us also be reminded that there are very close links between metal production and energy production. Both sectors are intimately tied, given that not only is around 8% to 10% of the world's primary energy used to extract and refine metallic resources, but the mining industry itself represents 20% of the energy used across the industrial sector at a global scale. Any increase in the global metal demand will induce an increase in the energy demand. In this context, the question of the sensitivity of renewable energies -in terms of energy or metal consumption -is bound to be asked. Finally, resource localization and actors' strategy (industrial structures, embargo policies, etc.) can render critical the use of raw materials.

Lithium, a strategic metal for the production of batteries, is quite representative of what is now at stake in the process of energy transition : the electrification potential of vehicles at a global scale, the concentration of reserves and production within a small number of countries 22 compared to crude oil for example (Figure 3, Figure 4), and the market's oligopolistic structure might give countries' material reliance a new face (Figure 5).

4-A new geopolitics of patents

In parallel to criticality, the question of industrial property rights is essential in order to understand the evolution of energy politics, and especially that of renewable energies. Indeed, the industrial property of the most reliable technologies in the decarbonization sector will necessarily influence their diffusion cost. The four technological decarbonization families (biofuels, wind-powered, solar PV and solar thermal) were at the receiving end of a 280 billion dollar investment in 2014, close to the number recorded in 2011, with solar power remaining the most promising field at the moment and wind-powered the most mature. The World Intellectual Property Organization (WIPO) 23 thus compared, in 2014, the number of patents filed for these four categories between 2006 and 2011: over this period, their volume surpassed the one registered between 1975 and 2005 (Table 1). Moreover, their recent annual growth rate exceeded the one observed for patents filed across all technology sectorsbetween 13% and 27% for these four technological families, against 6% globally. Japan and Germany are the only countries still resisting in a sector where China, with its large inland and its capacity to carry out large scale solar projects, should remain dominant in the following years. It would also be relevant to study the solar PV sector. Indeed, during the early 2000s, the only countries producing panels of this type were the USA, Germany and Japan; since 2010, China manufactures more than half of them. Moreover, China moved upmarket, widely exceeding both in conception and innovation in the traditional silicon-based PV field. This dynamic is further reinforced by the delocalization movement of numerous research and development (R&D) centers from developed countries to China.

While the link between patent-innovation and innovation-deployment isn't necessarily straightforward or easy to establish24 , recent developments in these four technological families inform us on the potential weight carried by different countries or zones in terms of power on the technological markets for the years to come. China, a key player in energy geopolitics, could strengthen its role thanks to the rise of renewable energies, but as a supplier of technological solutions.

The patent issue is usually analyzed with a normative finality, in order to create a relevant framework regarding the transfer of decarbonization technology from North countries to South countries. Yet, a more geopolitical approach would also mean questioning the industry dimension, especially through the search for a financial and economic market power for these technologies. The USA, Asia (China, Japan), and Europe -especially Germany -should remain major R&D zones for the decarbonization sector, while China should assert its position as a potential future leader. This card redistribution brings forward the question of new forms of cooperation to develop within the great environmental challenges to come.

Developing cooperation and drawing win-win agreements for as many countries as possible (whether it be with a balance between private and public R&D or within a transnational cooperation framework) proves necessary if we are to avoid generating new exacerbated dependences. Not accounting for this dimension when favoring renewable energy policies would only replace a dependence on resources with a dependence on technology.

5-Conclusion : Towards a diplomacy of renewable energies?

The diffusion of renewable energies in the global energy mix would also affect countries producing fossil resources. On the one hand, it would be legitimate to think that these policies will have profound consequences on fossil energy markets, with oil and coal being the first targeted. This diminution -or rather, slowing down at first -of the import volume would therefore affect producing countries' demand security25 and would have large macroeconomic implications for them. The decrease in prices, and thus in export and budgetary revenues in the long term, could ultimately modify the national and regional balance.

On the other hand, this movement would have a significant impact on the economic structures of producing countries and, by contagion effect, on the economic and international financing circuits. Indeed, the diversification movement towards renewable energies, already observable in many producing countries (Saudi Arabia, United Arab Emirates, etc.), could incite them to strongly diminish the extraction rhythm of fossil resources, which would in turn allow them to divide up their revenue in the long term. The regional balance and relative power of states could thus be transformed, contributing to a distinct risk increase in certain geographic zones.

Finally, the decrease in oil revenues would have significant consequences on developed countries, especially those, USA being the main one concerned, which benefit from the recycling of petrodollars for their debt emissions. Hence, by contagion effect, the change in the model of oil producing countries would not be without consequence for the global economy and financial balance. The policies regarding an energy transition towards renewables should not completely iron out traditional energy geopolitics, but rather open a new chapter in international relations. While they could decrease the reliance on different fossil energy producers and allow the energy mix to be less carbonated, they are not exempt of generating new dependences themselves. Indeed, a dependence on resources such as fossil energy could be replaced with a dependence on other resources, such as strategic metals, to which a major technological component would add up (patents), especially for the diffusion of the most efficient decarbonization technologies in countries in the global South. For this reason, the question of international cooperation is thus fundamental and will also forge the geopolitics of renewables. In line with the COP 21, it seems necessary for this to happen quickly in order to pave the way for a new energy world order. With the diffusion of renewable energies, global energy geopolitics will undeniably become more complex.

Figure 1 :

 1 Figure 1 : Energy consumption in the United States

Figure 3 :

 3 Figure 3 : Oil reserves by country (left figure), lithium reserves by country (right figure)

Figure 4 :Figure 5 :

 45 Figure 4 : Oil production by country (left figure), lithium production by country (right figure)

Figure 6 :

 6 Figure 6 : New Investment in Renewable energy by region in 2015 (in billion $)

Figure 2 : World Primary Energy : consumption by fuel

	45				Petroleum									Coal
	40				Natural Gas								Hydroelectric Power
	35				Nuclear Electric Power Other Renewable Energy*				Wood
	30
	25
	20
	15
	10
	5
	0
	1775	1805	1835	1855	1870	1885	1900	1915	1930	1945	1951	1954	1957	1960	1963	1966	1969	1972	1975	1978	1981	1984	1987	1990	1993	1996	1999	2002	2005	2008	2011

Source : BP Statistical Review, 2016

Table 1 : Global Patent Filing Rates

 1 China's sensational debut on numerous segments of renewable energy technologies. Hence, in the biofuel sector, China appears alongside Japan as an innovative pole. Indeed, 11 companies or institutions out of the 20 new incomers in the ranking since 2006 are localized on Chinese territory (Sinopec, Nanjing University, etc.). In the solar thermal technologies sector, Japan,

	Middle East &	India, 10.2	Brazil, 7.1
	Africa, 12.5
	America (excl.
	Brazil & US),
	12.8
			China, 102.9
	US, 44.1
	Asia (excl.
	China &
	India), 47.6	Europe, 48.8

IFP Énergies Nouvelles, 1-4 av. de Bois Préau, F-92852 Rueil-Malmaison, France.

The French Institute for International and Strategic Affairs, (IRIS), France.

I'm grateful to Théo Fayolle, Eva Lancelin and Hugo Liagre for insightful comments and suggestions. Of course, any remaining errors are mine. The views expressed herein are strictly those of the authors and are not to be construed as representing those of IFP Énergies Nouvelles or IRIS.

Standard Oil of New Jersey, Anglo-Iranian and Royal Dutch Shell signed, on September 17 th 1928, the Achnacarry agreement, which openly stipulates an equal division of markets and production information, as well as new modalities determining prices. This agreement, considered to be the official birth certificate of the Seven Sisters, was then signed by Mobil Oil, the Standard Oil of California, Gulf Oil, Texaco and later by the Standard Oil of New York, Gulf Oil, Texaco and the Compagnie française des pétroles (CFP). Its main objective was to discipline the different key market actors and avoid any price rivalry.

Saudi Arabia and Norway make interesting examples when it comes to their disproportionate positioning with regards to their demographic or economic weight, through the sole presence of oil resources on their territory.

IEA website: http://www.iea.org/topics/energysecurity/

This is in reference to the four As policy. For further reference, see Asia Pacific Energy Research Centre, A Quest for Energy Security in the 21st century, Tokyo, August 2007. For more academic reading on the topic, see Jessica Jewell, Aleh Cherp et Keywan Riahi, « Energy security under de-carbonization scenarios: An assessment framework and evaluation under different technology and policy choices », EnergyPolicy, vol. 65, February 2014, pp. 743-760.

Here, « availability » refers to crude availability, i.e. an observable excess on the market.

A resource may be available but not accessible for several reasons: the absence of commercial relations, contractual divergences, or intestate conflicts. Accessibility somewhat measures the time it takes to access a resource.

This notion comprises a significant economic dimension, i.e. the relation between the resource's cost and the buyer's revenue (or their short-term capacity to pay for the resource).

The notion of acceptability refers to the environmental questions and how they will be received by the local population. At a more global scale, it integrates questions of environmental sustainability.

Patrick Criqui et Silvana Mima, « European climate-energy security nexus: A model based scenario analysis »,Energy Policy, vol. 41, February 2012, pp. 827-842.

Indeed, energy price structuring within countries, the complementarity between private and public R&D, or the sector's global governance are all sectors which can impede the links between patent, innovation, and largescale technology deployment.

Demand security is a concept developed by the OPEC, and is meant to be the counterpart to supply security from consuming countries to producing countries. It symbolizes specifically the necessity for producing countries to have a stable and foreseeable trajectory.

The bioenergies development: the role of biofuels and the CO2 price. December 2011 # 83. E. LE CADRE, F. LANTZ, A. FARNOOSH Bioenergies usages in electricity generation utility means through a modelling approach: application to the French case. December 2011 # 84. I. ABADA, V. BRIAT, S. GABRIEL, O. MASSOL A generalized Nash-Cournot model for the north-western European natural gas markets with a fuel substitution demand function: the GaMMES model. December 2011 # 85. O. MASSOL, A. BANAL-ESTAÑOL Export diversification and resource-based industrialization: the case of natural gas. December 2011

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