A Hugo Noubel

Viviana Lago

EXPERIMENTAL INVESTIGATION OF AERODYNAMIC FORCES AND SHOCK WAVES FOR A WAVERIDER IN SUPERSONIC AND SLIP REGIME CARRIED OUT IN THE MARHY FACILITY

Keywords: Waverider, Wind tunnel, Supersonic rarefied flow, Lift-to-Drag ratio, Shock wave

Hypersonic vehicles are more and more studied in the military domain but also for other missions as the interplanetary or commercial missions. These vehicles optimize the ratio between lift and drag forces to success their missions. Hypersonic vehicles studied are launched at high altitude by rockets and they realize a re-entry by changing their angle of attack in function of speed and altitude. In this study, aerodynamic forces (drag and lift) are experimentally measured for different angles of attack and for four different rarefied flows conditions representative of altitudes ranging between 50 km and 80 km. The first part focus on aerodynamic forces and the behavior of the vehicle at altitudes higher than the initial optimal one which is 50 km in this case. The second part consists to study the shock angles at same angles of attack and rarefied flows to correlate shock wave angles and aerodynamic forces.

INTRODUCTION

Hypersonic flight is increasingly sought after worldwide for a variety of missions. The design of hypersonic vehicles change according the mission but the common requirement for all missions is to obtain the greatest Lift-to-Drag ratio. The higher this parameter is, the longer the waverider will travel. Hypersonic vehicles could be used for different missions: military [START_REF] Helley | Hypersonic weapons: Background and issues for Congress[END_REF], interplanetary [START_REF] Jr | Hypersonic waveriders for planetary atmospheres[END_REF] or commercial [START_REF] Haney | A waverider derived hypersonic x-vehicle[END_REF]. They are two types of hypersonic vehicles: the Hypersonic Cruise Missiles (HCM) as the X-51A waverider and the Hypersonic Glide Vehicles (HGV) as the HTV-2 [START_REF] Walker | The darpa/af falcon program: the hypersonic technology vehicle# 2 (htv-2) flight demonstration phase[END_REF] which has no propulsion. In this study, only hypersonic glide vehicles will be considered. Several countries have already developed their own vehicles, such as United States with the HTV-2, China with the DF-ZF [START_REF] Helley | Hypersonic weapons: Background and issues for Congress[END_REF] or Russia with the Avangard [START_REF] Hwang | Research and development trends of a hypersonic glide vehicle (hgv)[END_REF].

Hypersonic gliders do not have booster and they are launched by rockets at high altitude (between 50 km and 100 km). For this type of vehicle, the re-entry trajectory is important and constitutes a real challenge. In his article, Linshu [START_REF] Linshu | Optimal trajectory analysis of hypersonic boost-glide waverider with heat load constraint[END_REF] presents a schematic trajectory with the different phases of the re-entry into the atmosphere. The trajectory and capabilities of hypersonic vehicles give them unprecedented attributes and an ability to be unpredictable. Trajectory design performance is the essential section of a spacecraft [START_REF] Dong | Robust trajectory planning for hypersonic glide vehicle with parametric uncertainties[END_REF]. All trajectory optimization strategies [START_REF] Sti Rizvi | Trajectory optimisation for a rocket-assisted hypersonic boostglide vehicle[END_REF] are based on an inviscid flow field [START_REF] Yu Jianglong | Cooperative guidance strategy for multiple hypersonic gliding vehicles system[END_REF].

However, part of the waverider trajectory takes place at high altitude and the assumption of an inviscid flow is not relevant. To design a waverider, one of the methods used is the Rasmussen method [START_REF] Rasmussen | Waverider configurations derived from inclined circular and elliptic cones[END_REF] which considers only the inviscid flow. This method is relevant for low altitudes but a realistic method must consider: viscous effects with wall shear stress, weak and strong hypersonic viscous interaction, boundary layer displacement and the effects of sliding flow at high altitudes. [START_REF] Paul | Flow of rarefied gases[END_REF]. Low density effects affect and significantly reduce the Lift-to-Drag ratio [START_REF] Anderson | Hypersonic waveriders-where do we stand?[END_REF]. To improve the performance of waveriders in the low-density regime, Anderson, Bowcut, and Corda proposed a new method of geometry design in 1986 [START_REF] Corda | Viscous optimized hypersonic waveriders designed from axisymmetric flow fields[END_REF][START_REF] Jd Anderson | Viscous optimized hypersonic waveriders[END_REF]. This method involves introducing skin friction into the geometry optimization process. This change introduces the new family of waveriders called optimized viscous hypersonic waveriders.

In the upper atmosphere and low density, the assumption of a continuous flow regime fails. Depending on the Mach number and the density of the flow, the flow regime can be considered as a slip flow or a transitional flow [START_REF] Bowcutt | A perspective on the future of aerospace vehicle design[END_REF]. In these regimes, to describe the interaction between the molecules, statistical methods are required such as the Boltzman equation. To solve this equation, the most appropriate numerical simulation is the direct Monte Carlo simulation (DSMC) proposed by Prof. Bird. Many numerical simulations have been performed to identify rarefaction effects [START_REF] Rault | An efficient dsmc algorithm applied to a delta wing[END_REF][START_REF] Kautz | Direct simulation of waveriders in hypersonic rarefied flow[END_REF]. To complete and validate the numerical simulations, experimental data are needed. A few studies have been performed in rarefied domain with delta wings [START_REF] Allègre | Angle of attack and leading edge effects on the flow about a flat plate at mach number 18[END_REF][START_REF] Allegre | Rarefied hypersonic flow characteristics of delta wings and trailing edge spoilers[END_REF]. The few experiments are due to a lack of facilities and the main researches on the waverider in wind tunnel were realized at medium altitude: 22.5 km for Nagashetty [START_REF] Nagashetty | Tomographic visualization of the hypersonic flow field over a waverider[END_REF], 30.6 km for Rolim [START_REF] Cavalcanti Rolim | Experimental results of a mach 10 conical-flow derived waverider to 14-x hypersonic aerospace vehicle[END_REF]. Cockrell [START_REF] Jr | Aerodynamic characteristics of two waverider-derived hypersonic cruise configurations[END_REF] performed aerodynamic force measurements in a Mach 4 flow with a waverider at a Reynolds number equal to 3.99 × 10 -6 . Only a few numerical studies have been performed in the rarefied regime [START_REF] Galloway | Heat transfer to a delta wing and two waverider wings in rarefied hypersonic flow[END_REF][START_REF] Mn Macrossan | Scaling parameters for hypersonic flow: correlation of sphere drag data[END_REF]. Most of the work on the waverider topic focuses on the study the aerody-namic forces in order to optimize the geometries and improve the Lift-to-Drag ratio. Regarding the study of waverider impact angles, there are few studies. Anderson [START_REF] Anderson | Hypersonic waveriders for high altitude applications[END_REF] presents a method for determining the variation of the thickness of a normal shock as a function of Mach number and altitude. The method is based on the Grad steady state to obtain the thickness of the normal shock and the method is confirmed in 1955 by Shermann. In 1979 Allegre proposed in his thesis [START_REF] Allègre | Problèmes d'interactions liées aux régimes d'écoulements supersoniques et hypersoniques raréfiés[END_REF] experimental data on the shock angles of delta wings in rarefied regime.

The main objective of this work is to experimentally study the aerodynamic forces and shock angles of a waverider in several slip regime flows at different Knudsen number. The chosen waverider is based on Rolim's waverider [START_REF] Cavalcanti Rolim | Experimental results of a mach 10 conical-flow derived waverider to 14-x hypersonic aerospace vehicle[END_REF] which was designed with the Rasmussen method for Mach 10 -50 km. In a first step, the objective is to analyze the behavior of the vehicle in off-design flow conditions and to evaluate the degradation of the Lift-to-Drag ratio for different velocities and pressures typical to altitudes ranged between 50 km and 80 km. The shock wave angles are analyzed to characterise the evolution of the upper and lower shocks angles. General equations are proposed to describe shock waves of both sides of the model as a function of Mach number and free stream density for any angle of attack. Finally, a discussion is proposed to correlate the aerodynamic forces and shock wave angles, including viscous layers.

EXPERIMENTAL CONDITION

Waverider model

The geometry of waverider investigated is based on the geometry of the Rolim waverider which contributed to the development of the Brazilian waverider named 14-X. The model of the vehicle was built with the Rasmussen method. The optimal flight conditions for this waverider are a Mach number of 10 and an altitude ranged between 40 km and 50 km. In the present study, experiments were carried out with flows at Mach numbers 2 and 4 and free stream pressure flows ranged between 2.67 Pa and 71 Pa which corresponds to a range of altitudes between 60 km and 80 km. The objective is to study the aerodynamic behavior of the waverider in off-design conditions. In this study, the waverider is confronted with slip regime and flows more viscous than those with pressures representative of the 50 km altitude.

The Rolim waverider has been scaled to match with the flow core size of the experimental conditions the MARHy wind tunnel. The test model is therefore 100 mm long, 35.7 mm wide and 6.6 mm high. The non-dimensional parameters used in this paper were determined with a characteristic of 100 mm. The top surface is slightly curved.

Wind tunnel description

The MARHy (Mach Adaptable Rarefied Hypersonic) wind tunnel can recreate supersonic and hypersonic flows at low density to simulate the rarefied flows conditions at high atmospheric altitude.

Fig. 2: Wind tunnel MARHy

The facility can be divided in two parts: the wind tunnel and the pumping unit. The wind tunnel consists of three elements.

In the flow direction, the first element is the settling chamber, then there is the nozzle and finally the experimental chamber. In supersonic configuration, the settling chamber is a tank where the gas (air) is introduced before passing through the nozzle. The pressure is controlled by a micrometric valve to obtain the stagnation pressure P 0 of the used nozzle. All conditions in the settling chamber are considered to be stagnation conditions and are indicated by the index 0 (in Table 1, Table 2, Table 3 andTable 4). The second element of the test chamber is the nozzle. In supersonic conditions contoured and a set of different nozzles are availed to create a flow with a single Mach number and a single free flow pressure. For this study, four flow conditions were studied with four different nozzles to be changed for each new experimental condition. The last element is the test chamber where the test model is located. It is connected with the pumping unit by the diffuser.

The pumping unit consists of two primary pumps, two intermediate roots pumps, and twelve high vacuum roots pumps. The operating conditions of the profiled nozzle determine the number of pumps needed to stabilize the correct static pressure inside the test chamber. The free stream pressure P ∞ is adjusted operating the motorized butterfly valve placed between the wind tunnel and the pumping group. The pumping unit allows for unlimited test time unlike other wind tunnels.

Operating conditions

Each experimental condition in supersonic rarefied configurations requires a specific profiled nozzle. In this study, four supersonic flows are studied and the associated nozzles are presented in this section. The nozzles were chosen to study the impact of Mach number variation and static pressure variation on aerodynamic forces and shock waves. Three nozzles are dedicated to the study of the impact of the pressure: Mach 4 -2.67 Pa, Mach 4 -8 Pa and Mach 4 -71 Pa. The Mach 4 -8 Pa nozzle is also used to study the impact of the Mach number with the Mach 2 -8 Pa nozzle. Force measurements with aerodynamic balance were carried out in all the experimental conditions. Regarding flow visualization and shock waves analysis, the experimental condition at P ∞ = 71.11 Pa could not be investigated because the glow discharge technique cannot be applied.

Table 1: Flow conditions of the Mach 2 -8 Pa nozzle

Stagnation conditions

Free-stream conditions gas ambient air gas ambient air P o (Pa) 62.58

P ∞ (Pa) 8 T e o (K) 293 T e ∞ (K) 162.78 ρ o (kg.m -3 ) 7.44 × 10 -4 ρ ∞ (kg.m -3 ) 1.71 × 10 -4 µ ∞ (Pa.s) 1.11 × 10 -5 U ∞ (m.s -1 ) 511.43 Re ∞ (m -1 ) 7738.6 M a ∞ 2.0 λ ∞ (m) 2.79 × 10 -4

Stagnation conditions

Free-stream conditions gas ambient air gas ambient air P o (Pa) 404.79

P ∞ (Pa) 2.67 T e o (K) 293 T e ∞ (K) 69.76 ρ o (kg.m -3 ) 4.82 × 10 -3 ρ ∞ (kg.m -3 ) 1.33 × 10 -4 µ ∞ (Pa.s) 4.78 × 10 -6 U ∞ (m.s -1 ) 669.61 Re ∞ (m -1 ) 18 049 M a ∞ 4.0 λ ∞ (m) 2.20 × 10 -4

EXPERIMENTS DESCRIPTION

Aerodynamic sting balance

A home-made aerodynamic balance has been developed and used to measure forces in supersonic and hypersonic rarefied

(kg.m -3 ) 1.44 × 10 -2 ρ ∞ (kg.m -3 ) 4.00 × 10 -4 µ ∞ (Pa.s) 4.78 × 10 -6 U ∞ (m.s -1 ) 669.61 Re ∞ (m -1 ) 54 147 M a ∞ 4.0 λ ∞ (m) 7.33 × 10 -5
(kg.m -3 ) 1.28 × 10 -1 ρ ∞ (kg.m -3 ) 3.55 × 10 -3 µ ∞ (Pa.s) 4.78 × 10 -6 U ∞ (m.s -1 ) 669.61 Re ∞ (m -1 ) 497 800 M a ∞ 4.0 λ ∞ (m) 1.2 × 10 -5
flows. It is a two-axis balance that measures drag and lift forces applied to the model.

The balance is composed of two modules that separately measure the two forces. Each module is designed with thin strips that are perpendicular to the force they measure. Strain gauges are attached to the strips and characterize the deformation by a variation of the electrical signal. Then, a calibration with a digital Newtonmeter allows to correlate the variation of the signal and the force.

Different angles of attack are studied. To measure the forces for each angle, the sting balance is positioned horizontally on a rotating device and thus avoid gravity forces. Otherwise, The drag depends on the position of the center of gravity, so a change in the center of gravity may induce a new calibration. After each rotation, the test model and balance are moved by a motorized robot to maintain the same flow conditions around the test model.

To measure the drag and lift forces, the waverider is positioned in the flow and an initial ten seconds recording is made. During this recording, one thousand points are recorded every second. Next, the parasite forces that can act on the balance are measured. To do this, a flat plate is placed in front of the model so that the flow no longer interacts with the waverider. A new recording of ten seconds is made. This second recording allows to characterize the parasite forces and particularly the one created by the pumping group. The forces due to the flow acting on the waverider are the result of the difference between the two sets of recordings. This experimental protocol is realized five times for each angle of attack to improve the accuracy of measurements. Drag and lift forces depend on the angle of attack values so trigonometric formulas are used to correct forces and decorrelate the drag and lift [START_REF] Hugo | Experimental analysis of waverider lift-to-drag ratio measurements in rarefied and supersonic regime[END_REF]. The standard deviation was determined for all four nozzles and on average the drag force has a standart deviation of 1.86 mN and 0.6 mN for the lift module. All details are represented by error bars on each value of each graphs (Figure 3, Figure 4, Figure 5 and Figure 6).

Glow discharge

To observe waverider shock waves in rarefied flows, only the glow discharge method is possible. For moderate pressures, as is the case for the Mach 4 -71 Pa nozzle, the glow discharge technique does not allow the flow to be visualized because the discharge remains confined around the electrode due to the pressure.

The glow discharge technique consists of weakly ionising the gas using a negatively polarized copper ring which is placed at the outlet of the nozzle. The ionization creates a pink glow. Ionization produces excited molecules of N 2 and N +

2 emitting photons at wavelengths 337 nm for the positive system 2 nd and 391.4 for the negative system 1 st [START_REF] Menier | Influence d'une décharge électrique continue sur un écoulement supersonique raréfié[END_REF].

Ionized flow field around the model is captured with the ICCD Kuro camera. For each angle of attack, three different images are recorded with different exposure times. To improve the quality of the images, the background flow field without the model is recorded to have a better contrast on the shock wave. The shock wave is then highlighted and can be detected with the software ImageJ. The analysis of the shock waves shows that near the nose, the shock shape is rounded because the thick boundary layer is merged. The angle of the shock wave have been determined from the middle axis of the test model to the end because the shape can be approximated by a linear evolution . This method gives angles values with an accuracy ranging from 0.048 degrees to 2.59 degrees. When applying an angle of attack to the waverider, the lower shock will become more visible, with a low accuracy of the shock wave angle value, and the presicion on this detection will be better which is the opposite with the upper angle.

RESULTS

Lift, drag forces and Lift-to-Drag ratio

For hypersonic glide booster vehicles, the most important parameter during flight is the Lift-to-Drag ratio. These vehicles have no boosters, so they must optimize their angle of attack and Lift-to-Drag ratio to glide the greatest distance possible. The Lift-to-Drag ratio is the ratio of horizontal distance traveled for each unit of vertical distance lost. For example, a ratio of two means that the waverider glides two kilometers horizontally for every vertical kilometer lost. The goal of wa-veriders is to decrease drag and increase lift to maximize the ratio of both.

For each experimental condition, the angles of attack investigated range between 0 degree and 25 degrees. For these angles, the waverider has its nose higher than its base like an airplane during take-off. The objective of these experiments is to visualize the evolution of the drag force, the lift force and the Lift-to-Drag ratio as a function of the angle of attack, the Mach number and the free stream pressure. These two last parameters will define the rarefaction parameter. Nevertheless, giving an angle of attack to the wave rider, le local flow on the upper and lower side of the waverider will be different leading to different local rarefaction parameter. Figure 3, Figure 4, Figure 5 and Figure 6 show the evolution of the two aerodynamic forces as a function of the angle of attack. The trend of the two forces is different but similar for all experimental conditions. The drag describes a parabolic curve which has a minimum near 0 degree. At this angle, the waverider has the better aerodynamic behavior because its wet section is the smallest of all configurations (angles of attack)follows. It is an advantage to have the lowest drag but it is the worst case if we consider the lift curve. Indeed, the lift curve describes a straight line as a linear function. In aerodynamics, the lift of lifting bodies has a maximum which is synonymous with stall. Thus, the angle of attack range describes only one part of the total lift curve. The maximum angle of attack studied is 25 degree because beyond the waverider blocks the central flow. The optimal angle of attack for the waverider is the angle for which the ratio of lift to drag is maximum. At this point, the waverider can travel the maximum distance. The goal during cruise is to maintain the highest possible Lift-to-Drag ratio to be sure to reach the landing point.

Figure 7 shows that the maximum Lift-to-Drag ratio is not reached for the same angle of attack for all the experimental conditions. However, the maximum of the values is differ-ent and the highest Lift-to-Drag ratio is reached for Mach 4 -71 Pa at 10 degrees angle of attack. This flow condition is the least rarefied and represents the lowest altitude (50 km) in terms of viscous effects. Thus, in correlation with the altitudes, nozzle for which flow conditions have the greatest viscous parameter is the Mach 2 -8 Pa and for which the maximum value of Lift-to-Drag ratio is the lowest . Thus, during a re-entry, the Lift-to-Drag ratio will increase and the angle of attack corresponding to the maximum value will decrease when the altitude decreases (decrease of the rarefaction degree). The angle of attack has an impact on aerodynamic performances and it is important to change it during the flight to keep the best trajectory possible. Regarding flow conditions at Mach number 4, we observed that the increase in pressure is synonymous with an increase in the Lift-to-Drag ratio. The variation of pressure has a great impact on aerodynamic performances and the waverider is more operational in low altitude. But during a re-entry into the atmosphere, the waverider does not keep the same speed. If we analyze the two nozzles at 8 pascals, we can conclude that the increase in speed increases the performance of the waverider.

In conclusion, the Lift-to-Drag ratio increases when the altitude decreases and the speed increases. Thus if this waverider is launched from the rocket at 80 km of altitude, its aerodynamic performances will be weak (low Lift-to-Drag ratio). During the first phase of the re-entry (80 km to 50 km) the speed and the pressure will increase as waverider trajectories predict it [START_REF] Dong | Robust trajectory planning for hypersonic glide vehicle with parametric uncertainties[END_REF] and the waverider will begin to improve its performances and will glide. The L/D maximum value will be reached at 50 km for a Mach number of 10 are the flight conditions for which the design has been optimised. The study shows that when a waverider is optimized for a low or medium altitude, it will have a difficult first re-entry phase. The waverider is not adapted to the rarefied domain and it can be improved by taking into account the boundary layer in the optimization part as proposed by Anderson [12]. This can improve the maximum flight range.

Shock wave angle study

A waverider geometry is designed with the shock wave obtained by a supersonic or hypersonic flow over a cone in inviscid flows . The purpose is to analyze the shock wave angles of the waverider to see if a correlation with aerodynamic forces is possible. if so, it could help to optimise the geometry of the waverider.

The Mach 2 -8 Pa, Mach 4 -2.67 Pa and Mach 4 -8 Pa flows are studied with the glow discharge method. Different angles of attack are studied: 0 degree to 25 degrees, same angles as the aerodynamic forces to compare both results.

The shock wave angle is divided into two regions: the upper shock angle and the lower shock angle. The variation of the upper shock angle is presented in Figure 8 and that of the lower shock in Figure 9. 8, the general trend is similar for all experimental conditions presenting a linear behavior that shows an opening of the shock angle as the angle of attack increases. The increase in the shock angle is well known in the continuous regime with the work of Anderson [START_REF] David | Fundamentals of aerodynamics[END_REF] but in the rarefied regime the phenomenon is amplified because of the thickening of the boundary layer. Accuracy decreases as the flow become more viscous and shock waves more diffuse, thus the error bar increases with the angle of attack, which is typical of a diffuse shock. The shock is difficult to be precisely detected and the accuracy decreases.

On the other hand, Figure 9, the shock angle decreases and approaches the waverider surface as the angle of attack increases. This trend is the same for any flow condition. A convergence appears when the angle of attack is higher than 20 degrees.

The most rarefied nozzle is Mach 2 -8 Pa, then Mach 4 -2.67 Pa and finally Mach 4 -8 Pa. In both figures, the order of rarefaction degree is respected but there is a large difference between the Mach 2 nozzle and the two Mach 4. This observation shows that the best rarefaction parameter to describe the evolution of the shock angle is a product between the density and the Mach number.

With this last observation, two formulas were found to describe the evolution of the shock wave angle (upper and lower shock wave) as a function of the angle of attack, free stream density and Mach number. Parameter ξ is defined as ξ = ρ ∞ (1 + 0.2M 2 ) 2. These formulas are based on the experimental data. On Figure 8 and Figure 9 we plotted experimental and calculated points. It can be noted that there is good agreement between the fitting equation and the experimental data suggesting that these equations could be applied to determine the shock angles for any other flow condition.

These formulas allow to complete the study and estimate the evolution of the nozzle Mach 4 -71 Pa shock wave angles.

Shock wave impact on aerodynamic forces

The aerodynamic forces, drag and lift, are here the consequence of the interaction between the waverider and the flow density distribution around the geometry. As presented in the previous section, the shock wave can be divided into two regions, the upper and lower shock wave. Together, they create an equilibrium that is reflected in the aerodynamic forces. In this section, the aerodynamic forces and the shock angles are correlated by the relationship µ -α -θ. As presented in the Figure 10, µ is the upper angle of the shock wave, α is the lower angle of the shock and θ is the angle of attack. When this relationship is expressed as a function of αω, with ω the corner angle of the waverider equal to 3.72 degrees, the Figure 11 shows that a relation between forces and shock angles can be drawn. For the four flow conditions studied, the shape of the curves is similar and has a maximum. When the figure is correlated with Figure 7, it can be seen that the curves have a similar shape and the maximum of the curves is located for the same angles of attack as the Lift-to-Drag ratio curves. This result highlights the link between the aerodynamic forces and the angles of attack. The Figure 11 can help to determine the optimal Lift-to-Drag ratio for any flow condition as long as shock angles are known. These values are obtained with the two equations presented earlier. Thus, when designing the waverider, it is possible to know the optimal angle of attack for any rarefied flow so at any moment of the re-entry when the velocity and density of the flow are known. This behavior can be also correlated with the pressure variation across the shock. However formulas are valid for this waverider and other studies like this one can allow to generalize formulas to all geometries.

Pitot measures could be made to obtain pressure profile at upper and lower shock angles to understand reasons of the maximum of Lift-to-Drag ratio.

CONCLUSION

Experimental data were obtained for aerodynamic forces and shock wave angles in slip regime in Mach 2 and 4. The model is a waverider designed by Rolim and optimized for Mach 10 and 50 km of altitude. Lift-to-Drag ratio was determined for different angle of attack ranged between 0 and 25 degrees and at several experimental condition. Results shows that the angle of attack for the best L/D value decreases when increasing Mach number and free stream pressure. The analysis of the evolution of the shock wave angles presents a strong relation with the aerodynamic behavior. This means that to improve the performances of waveriders flight at any altitude, viscous layers should be considered when designing the waverider. For this purpose, this study proposes a relation to pre-

Fig. 1 :

 1 Fig. 1: Waverider geometry

  Force measured at Mach 4 -2.67 Pa and Mach 2 -8 Pa are the lowest because flows they are the most rarefied, with the lowest density.

Fig. 3 :

 3 Fig. 3: Drag and lift at Mach 2 -8 Pascals

Fig. 4 :Fig. 5 :Fig. 6 :

 456 Fig. 4: Drag and lift at Mach 4 -2 Pascals

Fig. 7 :

 7 Fig. 7: Lift-to-Drag ratio curves of all flows conditions

Fig. 8 :

 8 Fig. 8: Upper shock

2 ( 1 )

 21 5 M µ = (3.8 × 10 -3 θ + 9.1 × 10 -1 ) 1 2ξ + 7.72 × 10 -1 θ + 13.α = (-1.31 × 10 -2 θ + 1.02) 1 0.09ξ 0.6 -4.12 × 10 -4 θ 3 + 3.20 × 10 -2 θ 2 -8.73 × 10 -1 θ + 10.7 (2)

Fig. 10 :

 10 Fig. 10: Shock wave angles

Fig. 11 :

 11 Fig. 11: Optimal angle of attack from shock wave angle analysis

Table 2 :

 2 Flow conditions of the Mach 4 -2.67 Pa nozzle

Table 3 :

 3 Flow conditions of the Mach 4 -8 Pa nozzle

	Stagnation conditions	Free-stream conditions
	gas ambient air	gas ambient air
	P o (Pa) 1214.39	P ∞ (Pa) 8
	T e o (K) 293	T e ∞ (K) 69.76
	ρ o

Table 4 :

 4 Flow conditions of the Mach 4 -71 Pa nozzle

	Stagnation conditions	Free-stream conditions
	gas ambient air	gas ambient air
	P o (Pa) 10797.3	P ∞ (Pa) 71.11
	T e o (K) 293	T e ∞ (K) 69.76
	ρ o

2nd International Conference on Flight Vehicles, Aerothermodynamics and Re-entry Missions & Engineering (FAR) 19 -23 June 2022. Heilbronn, Germany

2nd International Conference on Flight Vehicles, Aerothermodynamics and Re-entry Missions & Engineering (FAR)

ACKNOWLEDGMENTS

This work is fully financed by the Agence Nationale de la Recherche for the project APHYRA (APHYRA-19-ASTR-0014-01). PhD is supported by the French National Research Agency (ANR) as part of the Programme d'Investissements d'Avenir (LabEx CAPRYSSES; Grant No ANR-11-LABX-0006-01).


Hugo Noubel

Viviana Lago

Céline Baranger

Experimental investigation of waverider aerodynamic forces and shock waves in hypersonic slip regime in the wind tunnel MARHy

Keywords: Mach 20.2, Wind tunnel, Waverider, rarefied flows, Aerodynamic forces

This paper presents an experimental investigation focused on aerodynamic forces and shock waves of a waverider. The study is carried out in the wind tunnel MARHy located at the ICARE laboratory of CNRS-Orleans. The facility can recreate the atmospheric re-entry flight of any vehicle in terms of viscous parameter and Knudsen number. The present work was carried out at hypersonic flow condition with a Mach number 20 and low density equivalent to 100 km of altitude. The model is a waverider optimized for optimal Lift to Drag ratio at Mach 10 and 50 km of altitude. The objective of this work is to analyse the behaviour of this waverider at high Mach number and altitude. The investigation of aerodynamic forces shows the waverider falls down for angles of attack higher than 15°. To confirm and explain this stall density inside the shock wave have been analysed from flow field a visualisation. This study shows a phenomenon of separation bubble which imply a constant pressure at the waverider surface. The bubble is observed thanks to luminosity intensity profiles and showing the displacement and increase of this bubble with the local rarefaction degree. As a consequence the shear layer can not reattach due to the separation bubble so the lift force decrease and the waverider falls down.

Introduction

The aerodynamic efficiency of a vehicle's flight is rated by the maximum of Lift-to-Drag (L/D) ratio of the forces acting on the vehicle. Waveriders are designed as the class of hypersonic vehicles that generate the highest L/D value compared to generic hypersonic designs. The characteristic of this type of vehicle is to generate a shock wave which remains attached under the leading edge to create a high L/D ratio and glide.

The first waverider concept appeared in 1959 with the work of Professor Terence Nonweiler [START_REF] Nonweiler | Aerodynamic problems of manned space vehicles[END_REF] on folded delta wings which were named "Caret-wings". The design of waveriders was then improved with the work of M.L. Rasmussen [START_REF] Rasmussen | Waverider configurations derived from inclined circular and elliptic cones[END_REF] who developed a method for designing waveriders based on the shock created by a circular cone in a supersonic or hypersonic and inviscid flow fields. Waveriders are designed for a flight path point in terms of Mach number and flight altitude. Their geometry is therefore optimized for a specific flight condition, and outside of this configuration the flight skills of the waveriders will be degraded. Over decades many works were addressed to improve methods for calculating geometry waveriders for a wide Mach range of practical applications. Morphing waveriders have been proposed by Maxwell and all [START_REF] Maxwell | Morphing waveriders for atmospheric entry[END_REF]. The concept focuses on morphing bottom surface providing on-design waverider aerodynamic performance across a wide range of Mach numbers [START_REF] Phoenix | Mach 5-3.5 morphing waverider accuracy and aerodynamic performance evaluation[END_REF]. All these strategies are developed with CFD models based on perfect fluid flow fields [START_REF] Liu | Novel approach for designing a hypersonic gliding-cruising dual waverider vehicle[END_REF][START_REF] Jianglong | Cooperative guidance strategy for multiple hypersonic gliding vehicles system[END_REF]. This implies that viscous fluid effects are not integrated as a parameter of the numerical simulations.

HiSST: International Conference on High-Speed Vehicle Science & Technology However, part of the waverider trajectory takes place at high altitude where the inviscid flow assumption is irrelevant. A realistic design should also consider viscous effects with constrained wall shear, weak and strong hypersonic viscous interaction, boundary layer displacement and high altitude slip flow effects. When slip effects become large, a viscous boundary layer results with a non-zero slip velocity condition at the wall, which increases viscous drag, thus decreasing the flight efficiency.

In the literature, there are few experimental results obtained under flow conditions in slip regime and or rarefied. Those that exist relate to works carried out with flat plates and delta wings [START_REF] Coumar | Influence of mach number and static pressure on plasma flow control of supersonic and rarefied flows around a sharp flat plate[END_REF][START_REF] Allègre | Angle of attack and leading edge effects on the flow about a flat plate at mach number 18[END_REF][START_REF] Allegre | Rarefied hypersonic flow characteristics of delta wings and trailing edge spoilers[END_REF][START_REF] Lengrand | Rarefied hypersonic flow over a sharp flat plate: numerical and experimental results[END_REF].

New experimental results are therefore useful and necessary to validate and possibly improve the numerical results that seek to optimize the design and the flight path of this type of vehicle. Few research on waverider shock waves in low density flows can be found in the literature. In 1986, Anderson [START_REF] Capriotti | Viscous optimized hypersonic waveriders[END_REF][START_REF] Anderson | Viscous optimized hypersonic waveriders[END_REF] included skin friction effects in the optimization process and proposed a new generation of waveriders named optimized viscous hypersonic waveriders.

The authors studied the behaviour of a waverider in supersonic flows in the slip regime. The waverider geometry studied is inspired by the design proposed by Rolim [START_REF] Rolim | Experimental results of a mach 10 conical-flow derived waverider to 14-x hypersonic aerospace vehicle[END_REF] and has been optimised for a flow at Mach 10 and an altitude of 50 km. Our first approach is to study the behaviour of this waverider outside the optimal conditions. A first step of the study has been carried out for supersonic Machs numers.

Experimental work was carried out in the MARHy supersonic/hypersonic rarefied wind tunnel. Three flow conditions were studied: Mach 2-8 Pa, Mach 4 -2.67 Pa and Mach 4 -8 Pa giving viscous parameter values ( √ Re/M ach) of 13.91, 10.62, and 18.34 respectively indicating that the experimental flow conditions obey the slip regime. Our experimental work investigates the behaviour of the Lift-to-Drag ratio as a function of operating conditions and angle of attack [START_REF] Hugo | Experimental analysis of waverider lift-to-drag ratio measurements in rarefied and supersonic regime[END_REF]. Shock wave visualisations were analysed to establish a relationship between the evolution of lift and drag forces with shock wave angles for upper and lower shocks [START_REF] Noubel | Correlation between waverider shock waves and aerodynamic forces in supersonic rarefied flow, experimental investigation in the wind tunnel marhy[END_REF]. This experimental work presents the second step of our investigation focused on the study of the aerodynamic behaviour of the same waverider for hypersonic conditions at Mach 20.2 and a very rarefied flow regime as it corresponds to an altitude of 100 km. We also present measurements of drag and lift forces and an investigation of their behaviour based on detailed shock analysis.

Experimental condition 1.Waverider model

For this study, the investagated waverider is based on the work of Rolim [START_REF] Rolim | Experimental results of a mach 10 conical-flow derived waverider to 14-x hypersonic aerospace vehicle[END_REF]. The initial waverider is taking part of the bresilian waverider project 14-X. The waverider used is optimized thanks to the Rasmussen method which does not take into account viscous effects. This is one of the reason why the Rolim waverider is optimized for 50 km and Mach 10. In this study, the flow investigated is Mach 20 and the static pressure associated is 6.7.10 -2 Pa which is typical to altitude of 90 kilometers. The objective is to alaysed the aerodynamic behaviour of the waverider when it flies at altitudes higher than the initial optimisation and at speed higher than the optimal. The waverider has already been investigated for Mach numbers ranged between Mach 2 and Mach 4 for pressures associated to altitudes between 50 km and 70 km. The objective of this study is to analyse an extreme case where the rarefaction degree of the flow is high and the speed is maximum during a re-entry phase.

As mentioned before, the waverider is based on the Rolim waverider. Dimensions have been scaled to fit with the flow core of the experimental conditions. The maximum of length possible is 100 mm to be sure to be inside the flow core and realised a maximum of angle of attack. The test model is 100 mm long, 35.7 mm wide and 6.6 mm high, Fig. 1. The top surface named the extrados is slightly curved and the boottom part named the intrados is flat.

Wind tunnel description

To create the hyperonic rarefied flow, the wind tunnel MARHy is used. This wind tunnel have been built in 1961 at the 'Laboratoire d'Aerothermique' in Meudon. Previously named SR3, when the wind The wind tunnel is composed of three part, the settling chamber, the test chamber and the pumping group.

The settling chamber is the beginning of the wind tunnel and it is the section where the gas is introduced and feed the contoured nozzle. To produce flows of different Mach numbers, the nozzle is changed and it allow to experiment subsonic, supersonic and hypersonic rarefied flows ranged between Mach number 0.8 and 20.2.

In this paper the hypersonic configuration is quite different than supersonic configurations. While for subsonic and supersonic rarefied flows, air is used, for hypersonic flows the used gas is nitrogen. Also, the hypersonic nozzle is different than the others because the gas, nitrogen, need to be heated to avoid freeze in the divergent section Nitrogen is introduced in a vessel where a graphite resistance will heat the nitrogen by Joule effect. After the carbon heater, nitrogen is compressed through the nozzle the throat and then expands through the divergent section of the nozzle. All components of the nozzle are cooled with a water circuit.

The test chamber is the next part of the wind tunnel. In this section, the test model is positioned closed to the exit nozzle to be sure to be inside the isentropic core. Instead of supersonic nozzles, the isentropic core for hypersonic nozzle is located close to the nozzle so the test model need to be positioned with accuracy to be sure to investigate the good flow conditions. The test chamber is a cylinder of 2 meters diameter and 3.5 meters long and positioned perpendicular to the flow. The large volume allows to use HiSST: International Conference on High-Speed Vehicle Science & Technology several scientific devices inside the test chamber. In addition to intrusive instruments, optical cameras can be use thanks to two flanges equipped with quartz and fluorine optical windows.

Then the final part of the wind tunnel, the pumping unit Fig. 3, which components have been entirely renewed during the transit between Meudon and Orleans. 2 primary pumps, 2 intermediary pumps and 12 root pumps compose the pumping unit. The association of 16 pumps allow to work without interruption for several hours. A diffuser through a butterfly valve connects the pumping unit and the testing chamber, which control the static pressure inside the testing chamber.

Operating condition

As presented before, different flow configuration can be experimented in the wind tunnel MARHy. In this study the hypersonic contoured nozzle Mach 20.2 -0.068 Pa is used. This nozzle operates with nitrogen which is heated when it will cross a graphite heater. This heater is supplied with an electric generator at high intensity. The gas is warm at 827°C but the gas in the testing chamber is only 13.32 K.

Table 1. Flow conditions of the Mach 20.2 -0.068 Pa nozzle

Stagnation conditions Free-stream conditions

gas Nitrogen gas Nitrogen P o (Pa) 350000

P ∞ (Pa) 0.068 T e o (K) 1100 T e ∞ (K) 13.32 ρ o (kg.m -3 ) 1.07 ρ ∞ (kg.m -3 ) 1.73.10 -5 µ ∞ (Pa.s) 9.12.10 -7 U ∞ (m.s -1 ) 1502.14 Re ∞ (m -1 ) 28480 M a ∞ 20.2 λ ∞ (m) 1.05.10 -3

Experimental description

In this study, two different experiments were carried out: first the study of aerodynamic forces by means a home-made aerodynamic sting balance and secondly the visualization of shock waves using the glow discharge technic.

Aerodynamic sting balance

The aerodynamic sting balance used to measure waverider aerodynamic force have been designed at the ICARE laboratory by Hugo Noubel [START_REF] Hugo | Experimental analysis of waverider lift-to-drag ratio measurements in rarefied and supersonic regime[END_REF], Fig. The sting balance is a two-axis balance which can measure only two forces: drag and lift. The balance has two separated modulus, one to measure the drag force and the second one to measure lift. Both modulus are composed by lamellae equipped with semiconductor strain gages. There are four strain gages on each modulus to compose a Winston bridge to improve the quality of measures and reduce thermal effects. The sting balance is positioned horizontally in the wind tunnel in order to impose an angle of attack on the waverider without being affected by the force of gravity and thus not being impacted by a change in the position of the model's centre of gravity. For each angle of attack, the aerodynamic forces must be corrected using trigonometric referential change formulae. For each angle of attack, the waverider is centrered in the flow core to keep the same free stream condition. When the angle of attack is too high and the free stream pressure too far from the optimal conditions the angle of attack is not studied. For this reason, the maximal angle of attack investigated for the hypersonic nozzle Mach 20.2 0.068 Pa is 20°.

Thanks to semi-conductor strain gages, this balance as a high accuracy. For each angle of attack, the measurements were carried out several times to ensure repeatability and to determine a mean value with a standard deviation. With the hypersonic nozzle forces, the minimal standard deviation between forces is 0.18 mN obtained for zero degree angle of attack.

Glow discharge

Different technics are usually applied to visualize shock waves arround a test model. These technics depend on the density of the flow. For example the PIV or Schlieren technics are not relevant for the present case because the density inside the test chamber is too low. To give an example, Settles [START_REF] Settles | Schlieren and shadowgraph techniques: visualizing phenomena in transparent media[END_REF] estimates the Schlieren technic appliable for density up to 1% of the atmospheriic density, i.e. 1.20.10 -2 kg/m 3 and in the investigation flow density is about 1.73.10 -5 kg/m 3 . To fix the problematic of low density, the solution is to ionise the flow thanks to an electron beam or glow discharge.

For this investigation, the technic used is the glow discharge as it was already used in many supersonic flows configuration. giving good results n s As the Fig. 5 shows it, the color of the ionized flow is pink. The Spectroscopic analysis of the ligth emitted by the the glow dishcarge showed the presence of only emissions from the 1 st negative system (N + 2 ) at 391.4 nm and the 2 nd positive system (N 2 ) with a wavelength of 337 nm.

As described in the article of Noubel and coumar thesis [START_REF] Noubel | Correlation between waverider shock waves and aerodynamic forces in supersonic rarefied flow, experimental investigation in the wind tunnel marhy[END_REF][START_REF] Coumar | Etude des mécanismes physiques induits pas un actionneur plasma appliqué au contrôle d'écoulements raréfiés super/hypersoniques dans le cadre de rentrées atmosphériques[END_REF], the intensity of the glow discharge can be associated to the molecular concentration. The measured light intensity is correlated to density To visualize shock wave, the waverider is positioned at the center of the flow core, the glow discharge switch on and an image is record. Then the waverider is removed, the glow discharge still switches on and then a new image is recorded. This second image is a background which divided to the model/flow field image gives a better detection of the shock wave as the Fig. 6 show it. For this study, glow discharge is obtained with the copper ring biased negatively with a voltage of -2 kV for an intensity of 7 mA.

Fig 6. Original and postraited images

For the aerodynamic forces study, angles of attack investigated are ranged between 0°and 20°. The reason to not test angles of attack higher is due to experimental constraints in particular regarding the pressure inside the test section, which must remain close to the optimal value. One of the main reasons of the increase of pressure is due to the sting balance, which is too big for the flow. For the glow discharge study, the device is smaller than the sting balance and so the angle of attack 25°could be investigated in addition to the others.

HiSST: International Conference on High-Speed Vehicle Science & Technology

Results

Aerodynamic forces

Aerodynamic forces have been measured for different angles of attack. Results are summarized on the Fig. 7. Two curves are plotted representing the drag force in orange and the lift force in green. Both curves have different behave, the drag force is fitted with a quadratic curve with a minimum near 0°. For the lift force the trend curve is linear for angles of attack lower than 15°which is similar to investigations in supersonic rarefied flows presented by Noubel [START_REF] Hugo | Experimental analysis of waverider lift-to-drag ratio measurements in rarefied and supersonic regime[END_REF]. In this study with hypersonic flow conditions, the force trends are similar to those determined in rarefied supersonic flow, but for angles of attack less than 15°. For angles of attack higher, both curves are falling down which is a new observation compared to previous studies.

The lift force curve can be divided in three parts. The first one is a linear increase between 0°and 10°, then the curve seems constant and finally after 15°the curve is drastically decreasing. This trend is similar to a phenomenon well known in the aeronautic which is the stall. The stall in aeronautic is a boundary layer separation which creates a drastic decrease of the lift force. The stall depends on the angle of attack, the wing shape and the flow conditions. Comparing lift curves during a stall [START_REF] Pattermann | On stall behavior in aerobatic figures with an aerobatic glider[END_REF] with the trends obtained here it seems that for angles of attack greater than 15°a stall is observed. An instability can be observed on lift force 15°point thanks to the big the standard deviation, obtained with the repetition of several measures.

One of the major indications of this aerodynamic forces study is on the angle of attack to be adopted during flight. Regardless of the values the Lift-to-Drag ratio we can say that during a hypersonic flight, if the pressure is too low around the waverider there is a risk of stalling and it can compromise the mission. The waverider studied does not have booster to correct its trajectory so if it reaches the stall angle it could have drastic impact on the trajectory. This study is important because it shows for the first time experimentally that a stall can be reached if a non-optimized waverider reach hypersonic speeds in rarefied flow. If the stall is reached, the waverider need to reduce its angle of attack to increase its speed and try to recover positive lift force.

If we interest on the Lift to Drag ratio we can see the curve increase and reach a maximal value at 10°. This angle characterizes the optimal angle for the waverider for this flow conditions. For an angle of ten degree, the waverider can flight on the highest distance possible and here for each vertical kilometre lose at these flow conditions, the waverider glide over 610 meters. These data show the waverider has poor aerodynamic performances because the Lift to Drag ratio is less than 1. The main reason of Lift-to-Drag ratio these performances is due to the geometry, which is not optimized for these flow conditions. Indeed the boundary layer is not considered by the Rasmussen geometries design, but for this rarefied flow condition, the boundary layer is thick. Therefore, to improve aerodynamic performances the geometry need to be adapt to be able improve glide in hypersonic rarefied flows. On the Fig. 8 the instability due to the stall can be observed thanks to the standard deviation and values which decrease after 15°as the a aerodynamic forces presented before.

Shock waves study

To confirm the stall determined from the analysis of aerodynamic forces, the shock wave on the extrados of the waverider is studied. In aeronautic when a stall is reached, a turbulent flow is observed on the extrados of the wing which leading to a decrease of lift. The goal of this section is to understand what the reason of the stall for a waverider is in hypersonic rarefied flow.

When the background is removed from the original iCCD image, the shock wave is better visible. Then, it is possible to determine the shock wave angle of the extrados and intrados but here the most important thing is what happen inside the extrados shock wave. As it is explained in previous subsection, the stall is a boundary layer separation and in the continuum regime, a vortex can be observed. If we refer to the Fig. 6 there is no wortex visible on the original picture or on the prostrated image. The only way to observe a variation inside the extrados shock wave is to plot luminosity intensity profiles perpendicular to the upper surface. This method allows characterizing the variation of density inside the shock wave because luminosity intensity variations and density variations are related as explained in the section 'Glow discharge'.

Luminosity profiles are plotted along a straight line, which start at the end of the waverider surface and cross all the shock wave. As it is presented on the Fig. 9, five profiles are plotted to have the evolution of the luminosity along the surface. This analysis is repeated for all the angles of attack. The profile at the waverider nose is not considered because we know there is an attached shock wave and the variations at the nose will not give any information on the variation of the pressure inside the shock, as it can be the case for the other profiles. So if the Fig. 9 is considered, there are five profiles, starting by the left, named: profile 100 mm, profile 80 mm, profile 60 mm, profile 40 mm and finally profile 20 mm which is the closer profile to the nose.

For each profile a curve is obtained and give information on the shock density behaves. The Fig. 10 presents the different regions of the shock. There are four regions in the graph, which describe the shock wave. The x-axis represents the length of the profile so at 0 mm it is associated to the waverider

Fig 10. composition of the shock wave on a luminosity intensity profile

The first region is the boundary layer marked by the number one in red. It is characterised by an increase of the luminosity intensity and the distance between the wave rider surface (x = 0 mm) and the maximum of the luminosity intensity is considered as the boundary layer thickness. The more a flow is rarefied and the more the boundary layer will be thick. In continuum regime, for supersonic flows this region is extremely thin nevertheless it can be visible for hypersonic flows.

The second regioon of the curve is a decrease of the luminosity intensity. This decrease describes a curve and the inflection point of this curve, associated to the green dot line, is the middle shock [START_REF] Kovacs | Detection method for shock-waves in viscous flows[END_REF]. The distance between the waverider surface (x = 0 mm) and the green dot line is the thickness of the shock.

The third region represented in blue is the transition between the decrease and the constant region of HiSST: International Conference on High-Speed Vehicle Science & Technology the luminosity intensity. This region delimits the foot shock. After the foot shock, the curve is constant because the profile describes what happen outside the shock wave. Outside the shock wave the density is constant that is the reason why the luminosity intensity profile does not vary.

After considering a classical profile we will analyse all the profiles for all angles of attack to identify if anything can explain the stall observed for the aerodynamic forces studied Fig. 7. Fig. 11, presents the totally graph fo the profiles along the waverider for angles of attck up to 25 °. the x-axis is the axis length in mm from 0 mm to 60 mm and the y-axis is the normalized luminosity intensity. For each profiles the luminosity intensity is divided by the luminosity intensity outside the shock wave to compare all values to free stream conditions. Fig. 11 allowed to detect any 'anormal' behaviour.

First of all, for angles of attack of 0°and 5°density profiles of shocks waves seems 'normal' because all the region of the shock (boundary layer, middle shock and foot shock) are well visible. For the angle of attack 5°the shock wave is thicker than the angle of attack of 0°because the local rarefaction degree above the extrados due to the angle of attack is bigger for 5°. The increase of shock wave thickness is due to a boundary layer thicker which is characteristic on profiles with the maximum of luminosity intensity reached for highest values of profile length.

For the case of the angle of attack 10°we can note that profiles are more horizontal and different region the shock wave density profiles are more difficult to be identified. This could be interpreted as a diffused shock. Nevertheless, profiles seems to be similar to those obtained for 5°.

Then for higher angles of attck some differences can be pointed that can explaine the measured variations of aerodynamic forces.

For the angle of attack 15°, the slope of the profile 100 mm is not linear as the previous profiles at 100 mm. The variation of density inside the shock wave does not increase normally and a flat region appear that can explain forces instabilities observed at 15°. This change of slope is more visible for the angle of attack 20°. The penultimate line shows a profile at 100 mm with a flat region and then an increase of luminosity intensity. The situation observed for the profile 100 mm at 15°is visible for the profile 80 mm at 20°. The flat region seems to propagate along the waverider extrados surface. This hypothesis is confirmed with the last line and the angle of attack of 25°. On this line the Profile 40 mm is similar to the profile 80 mm at 20°and the profile 100 mm at 15°. Then for the three other profiles of this line, each one starts with a flat region but the longest is near the waverider base. The profile at 100 mm of 25°start with a negative slope followed with a flat region and finally a positive slope. It shows an important variation of density inside the shock wave when local rarefaction degree is important.

In conclusion, to this shock wave analysis, when the angle of attack increases, the density inside the shock varies. For high angles of attack, the density near the surface is low and remains constant over a distance. This distance increases with the angle of attack and the phenomenon propagates along the waverider surface. For 15°angle of attack it is only located at the waverider base and then for 25°a ngle of attack the phenomenon is clearly visible at 40 mm of the waverider nose.

Discussion

With the two previous studies, aerodynamic forces and shock waves, we can state there is a correlation between them. Indeed the stall observed in the aerodynamic force begin for angles higher than 15°. At the same time, the shock wave investigation showed a variation of the density inside the shock wave, which can explain the stall.

The Fig. 12 recap all the profiles plotted for each angle of attack. In light blue there is the profile at 20 mm, in orange the profile at 40 mm, in purple profile at 60 mm, in green profile at 80 mm and in dark blue the profile at 100 mm.

For angles of attack ranged between 0°and 10°the lift-to-drag ratio increases and aerodynamic forces follow the general trend: linear for the lift and quadratic for the drag. This behaviour is 'normal' as those of shock wave profiles e as explained in the Fig. 10.

Then for the angle of attack 15°an instability was observed on aerodynamic forces which correspond to the first apparition of a flat region on shock wave profiles. Indeed the profile of 100 mm at 15°has a s considered, it can be noted at the waverider surface the density is higher than density of profile 80 mm which is not observed for previous angles of attack. This observation means that the pressure on the waverider base increases so the increase of pressure creating a force toward the waverider surface. This pressure can explain why the waverider is falling down and have a negative lift force for 20°. This interpretation can be confirmed with results obtained with an angle of attack of 25°where the profile 100 mm as a surface density really high compared with the other cases and so we can suppose the lift force will still be negative. It is supposed the air stuck under the intrados move to the extrados and increase of pressure. Maxwell has already mentioned observation of flow recirculation [START_REF] Maxwell | Morphing waveriders for atmospheric entry[END_REF]. The increase of pressure on the profile at 100 mm for all angles of attack is represented by the Fig. 13. A complementary study will be necessary to measure the wall pressure to confirm a possible recirculation effect.

In 1972, Allegre and all made a study on delta wings for a flow at Mach 8 and a Reynolds number of 2200/cm. Flows conditions have lower speed and higher Re number than the present study. In their investigation, they determined aerodynamic forces and wall pressure for different angles of attack (between -20°and 20°). This study showed a stability on aerodynamic forces for all angles of attack and this stability is founded in the wall pressure results. The wall pressure for a same profile decrease and converge to a single value. It is not the case of our study where aerodynamic forces are not stable and the wall pressure seems to increase for 20°and 25°compared to the other angles. If there is no stall the lowest wall pressure for the profile 100 mm will be the same that for 25°and not 15°. The Allegre work seems to favour the hypothesis that the flat region that appears on the density profiles are indications of the creation of a phenomenon that distorts the density within the extrados shock wave.

Flat region on profiles are visible for three angles of attacks and seems to propagate in direction of the We can observe the emergence of a zone with a lower luminosity intensity (in green) for the angles of attack 15°, 20°and 25°and near the surface of the waverider. This zone seems to increase in size and move to the waverider nose, when angle of attack increases corresponding to the flat region of the density profiles. This zone seems round and it probably correspond to a separation bubble.

Gaster [START_REF] Gaster | The structure and behaviour of laminar separation bubbles[END_REF] describes this phenomenon in 1966 in the domain of stall. This phenomenon appears in laminar conditions, and create a shallow region of reverse flow at the surface of an airfoil. The fluid in the forward region of the bubble is static which create a constant pressure region. This specificity can explain flat regions observed in the density profiles so it confirm the origin of stall. In his paper, Gaster precise that if the shear layer fails to reattach, the lift force fall. It is named the thin airfoil nose stall. This last point confirms that a variation of density in the extrados initiate the stall and the fall of In conclusion, aerodynamic forces and shock waves are related because the creation of a separation bubble due to the low rarefaction degree, high angle of attack and flow conditions will change the density inside the shock wave. The density will be constant in the bubble and will imply a fall of the lift force and so a stall. The objective, if this situation appear, will be to reduce the angle of attack to reattached the shear layer and restoring lift and good aerodynamic performances.

Conclusion

The goal of this investigation was to determine aerodynamic forces of a waverider in slip regime at Mach 20.2. The waverider is based on the Rolim studies and it is designed thanks to the Rasmussen method. Aerodynamic forces are measured with an aerodynamic sting balance for angles of attack from 0°to 20°. This investigation shows that angles greater than 15°instabilities in the lift force appears.

The lift force at 20°is negative so the waverider falls down. To understand the reason of the stall, a complementary study was realized to investigate and analyse the variation of pressure inside shock waves. The technic used is the glow discharge and the extrados shock wave is analysed because in case of stall, instabilities appear in this region. The shock wave study revealed that after 15°angle of attack the pressure remains constant near the waverider surface. This zone of constant pressure increase with the angle of attack and move to the waverider nose. This phenomenon is a separation bubble which appears between the surface and the boundary layer. Thanks to 3D mapping it have been concluded

HiSST: International Conference on High-Speed Vehicle Science & Technology the reason of the stall is due to this bubble which not allow the boundary layer to reattach and so create a fall of lift. To avoid the lift fall, the waverider need to reduce its angle of attack to remove the bubble. An other solution is to design a new waverider because a hypothesis of flow recirculation from the the bottom to the top can be made. The new waverider need to avoid recirculation and complementary studies will be realised. The objective is to mesure wall pressure and confirm the separation bubble and try to understand if there is a flow recirculation.

Fig 1 .

 1 Fig 1. Waverider geometry

Fig 2 .

 2 Fig 2. Wind tunnel MARHy

Fig 3 .

 3 Fig 3. Wind tunnel MARHy pumping group

4 .

 4 The principle of intern balance and more precisely HiSST: International Conference on High-Speed Vehicle Science & Technology sting balance have been preferred to extern balance because intrusive devices can be used inside MARHy wiind tunnel and it offers more accuracy.

Fig 4 .

 4 Fig 4. Sting balance positionned in the wind tunnel MARHy

Fig 5 .

 5 Fig 5. Glow discharge visualization

Fig 7 .

 7 Fig 7. Drag and Lift forces measured for different angles of attack

Fig 8 .

 8 Fig 8. Lift-to-Drag ratio

Fig 9 .

 9 Fig 9. Waverider and profiles

Fig 11 .Fig 12 .

 1112 Fig 11. Detailed luminosity profiles plotted for each angle of attack

Fig 13 .Fig 14 .

 1314 Fig 13. Luminosity intensity profile 100 mm for all angles of attack

Fig 15 .

 15 Fig 15. 3D mapping of the luminosity intensity to visualise the separation bubble

CNRS, ICARE, UPR 3021, 1C avenue de la Recherche Scientifique, CS 500060, F-45071, Orléans, cedex

, France, hugo.noubel@cnrs-orleans.fr 2 CNRS, ICARE, UPR

3021, 1C avenue de la Recherche Scientifique, CS 500060, F-45071, Orléans, cedex 2 , France, Viviana.lago@cnrs-orleans.fr 3 CEA-CESTA, 15 avenue des Sablières, CS 60001, 33116 Le Barp Cedex, France, celine.baranger@cea.fr

Copyright © 2022 by the author(s)

Acknowledgments

This work is fully financed by the Agence Nationale de la Recherche for the project APHYRA (APHYRA-19-ASTR-0014-01). PhD is supported by the French National Research Agency (ANR) as part of the Programme d'Investissements d'Avenir (LabEx CAPRYSSES; Grant No ANR-11-LABX-0006-01).


Simon Thorpe

Universal Basic Income and the Economic Singularity

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

The members of the Economic Singularity Club (ESC) share the belief that advances in AI and automation will have profound effects on humanity in the coming decades. In particular, it seems likely that machines could replace humans in a wide range of situations, making a substantial proportion of the population unemployable. For anyone with doubts about the potential impact of state-of-the-art technologies such as GPT-3 (Generative Pre-trained Transformer 3), the recent article in the Guardian entitled "A robot wrote this entire article. Are you scared yet, human?" gives a flavour of what is already possible. GPT-3's potential for replacing entire teams of journalists (and other highly skilled and well-paid professions) makes it clear that technology is not just a threat for people earning their livings from repetitive, manual tasks.

Our book "Stories from 2045 -Artificial Intelligence and the Future of Work" published in January 2019, included a wide range of scenarios, ranging from the positively dystopian, to quite optimistic. One of the aims of ESC must be to propose strategies that help ensure that our future is more "White Mirror" than the "Black Mirror" scenarios of Charlie Brooker's popular television series.

In this presentation, I would like to argue that providing a Universal Basic Income (UBI) is the single most potent antidote to the societal threats associated with the AI revolution and that we should be pushing to implement such proposals as rapidly as possible.

It is perhaps useful to note that providing a UBI is in no way incompatible with other ideas that have emerged from the ESC group. In particular, Calum Chase has argued that we could move towards a world of what he calls "Fully Automated Luxury Capitalism", made possible when the cost of providing goods and services drops so low that people could live decently without spending money. This sort of revolution has already occurred in some areas. Here is a list of some of the areas where goods and services can be provided to people with low incomes -even if they do lose their ability to find paid employment.

• Internet. Billions of people can now access a wide range of services including radio,

TV and internet at virtually zero cost. Wikipedia, Facebook, Google, Spotify and Banking services are all available to anyone with an internet connection and a mobile phone or computer. And there are now numerous schemes for providing free smartphones to people who do not have the means to buy one themselves, see for example a recent example in Lagos, Nigeria. • Energy and Water. Every citizen could have a basic allowance of electricity and water for free, and companies only allowed to charge for consumption exceeding the basic quotas.

• Education. In virtually all countries, education is provided free of charge at least until the end of high-school, but there are many countries where even university education is either free or at a minimal cost. They include Norway, Finland, Sweden, Germany, France and Denmark. • Health. Although there are some glaring exceptions (such as the USA), the majority of countries now provide free universal health care. • Transport. There are now well over 100 cities where transportation is free, and the movement is growing. Tallinn became the first EU capital to provide free transport in 2013, and in 2020, Luxembourg became the first country with entirely free transport. But things could become even better when self-driving electric vehicles become a reality. Governments could decide to provide a Uber-like service so that anyone with a smartphone could simply request transport to go anywhere. The reduction in congestion and the number of vehicles required could easily justify the expense.

So, it seems clear that people could have many of the requirements for a modest lifestyle without changing the system completely. And while this is not Fully Automated Luxury Capitalism, this tendency is likely to accelerate. Other vital components needed to live a decent life may be somewhat harder to provide at zero cost because requirements vary so much between individuals. Food, clothing and housing can all be expensive items, even though you can reduce costs a lot by choosing to live a more frugal lifestyle. People could opt to eat less meat and dairy produce, recycle clothes and ignore the need to have the latest fashions and choose to live away from expensive and overcrowded cities.

But could people really live decently when advances in AI and robotics have restricted their ability to earn money? Before discussing the potential role for UBI, let us examine how the current systems would cope with a rise in technological unemployment. In many countries, the state would step in to help people with little or no resources using a whole range of means-tested schemes that often oblige people to prove that they are actively looking for paid work. But such schemes have many serious problems. Many people who are eligible for aid fail to take advantage of the schemes either because they are often overly complicated or because of the shame of living on welfare. And in many cases, the assistance is removed as soon as the person starts to earn any additional income, meaning that the effective marginal tax rate can be very high -80% or more. The resulting poverty traps can mean that people get stuck in poverty, even if they would like to work.

Poverty traps can be avoided if, instead of providing support in the form of means-tested benefits, the state simply provides support to everyone, using regular monthly payments with no strings attached -the very definition of a Universal Basic Income (UBI). In this article, I will look at the fundamental issue of how such a mechanism could be implemented and how it could change the nature of society. In particular, I will argue that a Universal Basic Income (UBI) will be a vital feature of any future solution to the economic singularity.

The main problem -financing a UBI

One of the objections to a Universal Basic Income is the claim that is would simply cost too much. If you simply multiply the amount of the UBI payments by the number of people, you certainly get huge numbers. But, as we will see, a radical change in the entire tax and welfare system can easily cover the costs.

Suppose that every adult receives a standard sum of money every month, with no strings attached, but then paid tax at a fixed rate on any additional income, whatever the source. Interestingly, it turns out that the resulting scheme could be very close to the current system, but without the administrative overhead. Consider the situation in France where less than half the working-age population pays income tax. You could provide €600 a month to every adult, and tax all earnings at a flat rate of 30%. 61% of the French population earns less than €2000 a month, which is the point where their tax payments (30% of €2000, or €600) cancel out the basic income. All such people would effectively get a form of negative income tax, an idea already proposed by Milton Friedman in the 1960s. Remarkably, the amount of income tax revenue paid by the 39% who earn more than €2000 a month is enough to finance the entire system. In such a case, the UBI coupled with a flat rate 30% tax on all earnings would be a purely redistributive system -taking money from the high earners and using it to pay those earning less than €2000.

Interestingly, a 30% flat-rate tax rate has already been imposed by the French government on all income from financial sources (dividends, rents, etc.). The reform would thus be a simple extension of this same flat-rate tax to all forms of income, including salaries, pensions etc. It is also worth noting that it is easy to scale the amount of the Basic Income even in this selffinancing system. Essentially, increasing the flat-rate tax by an additional 10% adds an extra €200 to the UBI. You can therefore choose between €600 a month with a 30% flat-rate tax, €800 with a 40% tax or even €1000 a month at 50%.

Further evidence that a UBI can indeed be affordable comes from a website, developed by a team based in Bordeaux, where you can simulate different ways to finance a UBI. I found that it was possible to provide a UBI of €920 per month for all adults, coupled with half that amount for those under 18 (€460). Total cost would be €645 billion, but the simulator shows that this could be fully financed. There would even be a surplus of €9 million. Much could be achieved by fusing many of the existing benefit schemes into the Basic Income payments. They include the equivalent of the job seekers allowance (€15.4 bn), housing aid (€17.4 bn), tax exemptions for low salaries (€38.7 bn), family benefits (€36 bn), state pensions (€219 bn) and unemployment benefits (38.1 MD€). But it also requires the elimination of tax niches (€34 bn) and the effects of tax individualisation (€37 bn). I also set the rate for Income Tax at 0% but replaced it by something called the CSG (General Solidarity Contribution) which is a flatrate tax with no loopholes that applies to all forms of income. I increased it from the current rate of 7.9% to 30%. I even managed to abolish VAT altogether, but increased the existing tax on wealth from 1.18% to 2% and set a carbon tax at €100 per ton. I chose the option of taxing financial transactions which would raise a further €50 billion based on a rate of just 0.012%, but this could be easily increased. In conclusion, this official government simulator, using official figures, demonstrates that financing a UBI is perfectly doable -if there is the political will.

Later on, we will look at two additional ways of financing a UBI at a global level. But these can be combined with the proposals mentioned here. So, if we can agree that implementing a UBI is possible, what would be the advantages?

Advantages 1 -Simplification

Implementing a UBI provides a vast simplification of existing schemes for people on low incomes, or who risk losing their livelihoods because of advances in AI and robotics. Some argue that it is better to target such support to avoid making payments to people who don't need the money -why pay a basic income to millionaires? But fundamentally, providing a basic income of €600 (or whatever) to someone earning €100,000 a month is functionally the same thing as giving them a tax break of €2000 a month.

Another thing comes for free with such a system. Switching to a Basic Income & flat-rate tax system would allow the suppression of hundreds of tax breaks and loopholes that make many tax systems impossible to navigate without professional help. In the US, the complexity of the tax system forces even people with relatively modest positions (such as university professors) to pay professionals to fill in their tax returns. While the claim that the tax code in the US now runs to around 70,000 pages is apparently not valid, the system is very complex. It could be that the UK wins the title for the most complex system, with 17,000 pages of tax code. Much of this complexity results from lobbying by pressure groups who argue that their sector merits specific reductions in taxation. Trying to eliminate all those loopholes one by one could be extremely arduous, so entirely scrapping income tax would be a simple way to clear the system. But bear in mind that a direct subsidy can replace any tax break -they are functionally equivalent. Of course, it is relatively easy to defend a tax break (since no-one likes paying taxes and lobbying for tax breaks is considered a fair game). In contrast, it is much harder to justify receiving taxpayers' money.

France provides a good illustration. The government is currently offering €6000 in aid to anyone buying an electric vehicle, and a further €6000 for switching from oil-fired heating to Solar panels and a heat pump. The government could have implemented such aid via tax breaks, but this would only affect people who pay tax -less than half the population. Direct subsidies are thus a much fairer way to distribute state aid because everyone can benefit. And, since they are available to all citizens, they resemble an Unconditional Basic Income. The only real difference is that the help comes as one-off payments -not regular monthly payments.

Advantage 2 -Choice of lifestyle

A second significant advantage of having a UBI is that it enables people to choose their lifestyle without the need to spend a large proportion of their time earning money. Our current system obliges many people to work 9 to 5, five days a week from the day they quit education to the day they retire forty or so years later. Those who fail to fit the pattern and currently depend on welfare payments are often treated as second-class citizens, "scroungers", or worse. And that is true even if they spend all their free time doing valuable, yet unpaid work.

There are already groups of people who can avoid paid work entirely, without being labelled scroungers for living on benefits. Retired people with comfortable pensions, people with inherited wealth, and a few people who have earned enough to retire early can all live without working for money. Another very significant group, especially historically, are "housewives" or "stay-at-home moms". At the start of the 20 th Century, close to 90% of women were housewives or homemakers. Even as recently as 2011, roughly 20% of French women described themselves as housewives. These days, the status of housewife is only possible for people whose partners are earning enough money to give them that option. Most women are now obliged to out to work to pay the bills.

So there are already several groups within society that already have the liberty to devote their time to a vast range of activities that can be very valuable, even though unpaid. These include working in the home and caring for children but also charity work and other worthy causes. They can also indulge in a wide range of creative activities that are highly rewarding, even without pay.

You might argue that people need to be paid money to motivate them to do things that are vitally important for society. But some of humanity's most impressive achievements have been done without monetary rewards. The entire Wikipedia project involves only about 280 paid staff, with the rest of the work done by unpaid volunteers. Currently, around 70,000 people make five or more edits to the English language Wikipedia per month -all for free. It is worth noting that Wikipedia has practically demolished the market for commercially produced encyclopaedias like the venerable Encyclopaedia Britannica that was probably published in print form for the last time in 2010, after 244 years. Likewise, open-source software projects like Linux produced by armies of unpaid enthusiasts have demonstrated that they can rival products from Microsoft and Apple, despite their phenomenal resources. Indeed, since 2017, Linux is now the only system used on the TOP500 supercomputers.

So, there are clearly several groups in society that can choose what they do with their time.

Why not give this option to everyone? Imagine the change if everyone could get involved in unpaid work -including those currently obliged to go out and earn a living 40 hours a week, all year round. Should such people be obliged to give up their valuable weekends and vacation time to do useful things that they would be delighted to do if they only had the choice?

Interestingly, in his 2018 book "Bullshit Jobs: A Theory", anthropologist David Graeber found that around 40% of people in paid work believe their jobs are entirely pointless and serve no useful function whatsoever. It follows that if a UBI allowed all those people to stop wasting their time, and do something useful, we would all benefit. Tragically, David Graeber died on the 2 nd September 2020.

Advantage 3 -Ecology

Yet another advantage of UBI would be its ecological impact. Even a modest UBI would allow some people to adopt a low consumption lifestyle, perhaps by moving to rural areas and growing their own food. Around 1000m 2 of land can feed a family of four, but this currently requires about 6 hours of labour per day. However, low-cost intelligent robots could significantly reduce the amount of work needed by picking fruit or removing weeds. Apart from anything else, this would dramatically reduce the need for herbicides and help protect the environment. The more people make this choice, the better our chances of saving our planet, so it should be actively encouraged.

In France, for example, there are around 28 million hectares of agricultural land. If 1% of that land were requisitioned, and distributed to people in the form of allotments that could be used for growing food, it would allow a substantial proportion of the French population (around 11 million people) to become self-sufficient in food. With the average cost of agricultural land in France at around €6000 per hectare, the land needed to feed a family of 4 would only cost about €600 -a very modest investment. At such prices, rural communes in France could simply give land to all residents, enabling them to attract city dwellers to move to the country and revitalise rural communities.

Advantage 4 -Flexibility UBI would end the current inflexible system where the length of the working day and the times when people have to work are imposed from above. With a UBI, people could freely decide how much time they spend doing paid work. Some might choose to work 100 hours in a week some of the time (it would be their choice). But others might do only 10 hours of paid work per week or even give up paid work entirely. Likewise, there would be no obligation to work all year round. People could easily decide to work just one week every month, or three months every year. With this degree of flexibility, the entire concept of being "unemployed" would cease to exist.

A UBI would also impact the many vital jobs that are currently poorly paid. The recent covid-19 pandemic has highlighted the importance of frontline health staff, including nurses and auxiliary personnel. If such workers were not obliged to work long hours, they might prefer to work part-time -maybe just one or two days a week, or for relatively short periods. To entice them to work, employers could be obliged to increase the level of pay to convince them to continue working long hours. The net effect would be that vital but gruelling jobs that are currently poorly paid would end up being among the best-paid jobs in the economy.

The desire for flexibility has been a factor in the development of the so-called Gig economy and zero-hour contracts, especially in the UK. For a business, this can be a significant advantage -especially when there is a need for seasonal workers in areas like tourism and agriculture. While current zero-hour contracts lead to worker exploitation, in a system with a UBI, such criticisms lose their pertinence.

UBI has other advantages for business by effectively subsidising local production. Imagine that you are in the business of manufacturing cars, trying to decide where to implant a new factory. A country with a UBI would have a natural advantage because some proportion of employees' living expenses are already covered, reducing overall manufacturing costs. A similar argument already applies to health provision. In most developed countries, universal health care provision reduces costs for industry. The glaring exception is the USA, where employers have to cover the full costs of health care for workers and their families. Businesses in the USA are thus seriously disadvantaged relative to practically every other country on the planet. It is odd, even incomprehensible, that the business leaders who currently back Trump's plan to eliminate "Obamacare" have failed to register this point.

Towards a Global UBI?

So far in the essay, I have talked about implementing a UBI at a local level -national or regional. Replacing the current complex benefits and taxation system by a UBI coupled with a flat-rate tax on all revenues could indeed work locally. But the challenge imposed by advances in AI and automation are ones that will affect just about everyone on the planet. Can we imagine a scenario to introduce a UBI at a global level?

Suppose that we decided to provide a UBI at 50% of the Median Per Capita Income. How much would it cost? Using data from the World Population Review produces a total of just under $10.6 trillion. That may seem a huge number. But choosing to provide a UBI at 50% of the median per capita income is quite generous. For example, in the UK it would correspond to nearly £400 a month for every man, women and child -up to £1600 for a family of four.

The figures for France would be similar -around €440 a month. The costs would be much higher in Scandinavian countries where the cost of living is much higher, and where median incomes are about 50% higher. But the cost of the equivalent deal in African countries where median income is typically well below $35 a month would be almost trivial.

In my contribution to ESC's book on "Stories from 2045", I made the radical suggestion that a very modest 0.1% tax on all financial transactions could entirely fund the $10.6 trillion needed for a universal UBI. Data compiled from the Bank for International Settlements (BIS) shows that in 2018, these transactions totalled at least $14.9 quadrillion -over one thousand times more than the sum needed to provide a universal UBI. And that sum is almost certainly massively underestimated because the BIS figures fail to include major players like the Chicago-based Options Clearing Corporation (OCC) that describes itself as "the world's largest equity derivatives clearing organisation", clearing 5 billion contracts a year. Likewise, according to BIS, foreign exchange transactions total $6.6 trillion a day so that with 250 trading days a year, this totals around $1.6 quadrillion. Daily volumes of interest rate swaps are also around the same value, so the $14.9 quadrillion figure is entirely plausible. Of course, some in the financial sector may claim that even a 0.1% financial transaction tax would have catastrophic consequences. But the fact is that the 0.1% rate is similar to the transaction charges imposed by the industry itself, so this argument is bogus. If 0.1% of the value of transactions was indeed funneled into the general world economy by simply giving it to the worlds' citizens, the boost to the entire system could be extremely beneficial.

There is another relatively simple, yet currently, unexplored way to move towards a genuinely universal UBI. I have recently proposed the idea of a universal 1% annual tax on all assetspayable wherever they are held. In jurisdictions where no asset tax was present (as is the case in tax havens like the Cayman Islands), the tax would need to be paid to a global organisation such as the United Nations. If international agreement for such a scheme was possible (admittedly a tall order), it could eliminate tax havens at a stroke because it would no longer be possible to avoid the tax by relocating assets.

How much revenue would such a tax generate? The companies in the Forbes Global 2000 list have combined assets of over $200 trillion. Clearly, the total would be a lot higher if we included assets for all the roughly 43,000 publicly traded companies in the world, rather than just the top 2000. We should also include the many privately-owned companies that are also required to declare their assets every year. While those numbers are harder to find, the annual revenue for just the top 50 exceeds $2.5 trillion, suggesting that the accumulated assets of privately-owned companies could also be enormous. We can also add in the global value of real-estate that Savill's estimates at around $280 trillion and currently increasing at over 6% a year. We should also be able to include the value of other assets such as yachts, aircraft, jewellery and works of art. A simple way to determine the value of such assets for the asset tax would involve using the insured value of the goods. Adding together the values of all these assets, whether owned by individuals, trusts or corporations could easily reach the eye-watering figure of $1 quadrillion. With a 1% annual asset tax, this would be yet another way to finance a universal UBI set at 50% of median income for the entire population of the planet.

We have thus seen that there are various options for financing a UBI. Locally, governments could reform their tax and benefits system to provide a basic income. But with international cooperation, a universal financial transaction tax of around 0.1%, or a universal asset tax of 1% per annum, would be enough to provide a UBI for everyone. By using both taxes, it would be possible to use even lower rates. It is therefore hopefully abundantly clear that providing a UBI is perfectly doable -if there is the political will.

The suggestion that a global organisation such as the United Nations could organise the distribution of a basic income to everyone on the planet is undoubtedly very bold -and perhaps not very realistic. But the advantages could be enormous. Nearly every nation on earth has signed up for the United Nations list of 17 Sustainable Development Goals. While everyone agrees on these objectives, there is little agreement about how to finance them. But it is worth noting that a global UBI could help many of them. The obvious ones are Goals 1 (No Poverty), 2 (Zero Hunger) and 3 (Good Health and Well-being) that would be directly affected, but a global UBI would help just about every one of them. Furthermore, allowing the United Nations to implement the scheme would give it enormous negotiating power, because the provision of the basic income payments could be made conditional on the governments in particularly countries respecting basic civil rights including freedom of speech.

Final comments on UBI and ESC

My final point returns to the fundamental question of how we could cope with the consequences of the AI revolution and increasing automation on the amount of paid work in the economy. I would like to argue that by introducing a modest UBI now, or in the near future, governments could easily cope with any future changes in the job market.

To make this point clear, suppose we start from the "ideal" situation of full employment with 100% of the working age population working full time -five days a week. Imagine that over some period, the amount of paid work drops by 50%. There are clearly a range of strategies.

One would be to lay-off 50% of the population, and keep those that still have jobs working full time. Another would be to reduce the working week from 5 days to 2.5 days. But who is to say that everyone would want that solution?

With a UBI in place, and set at a level where it was possible to get by with just the UBI (if someone was happy to live a low consumption lifestyle, growing their own food), you might have a situation where equal numbers of people decide to work full time (5 days a week), part time (4 days a week, 3 days week, 2 days a week and 1 day a week), or to do no paid work at all. With one-sixth of the population in each group, you would also have enough people to do all the paid work that was available. But critically, no one would have to impose this in a topdown way. And market forces would naturally adapt to ensure that all the essential paid work was done by increasing pay levels when necessary.

Suppose that, a few years later, the amount of paid work had dropped to just 20% of the original value. There would again be a wide range of possibilities without having to decree that everyone has to work a one-day week. The obvious option would have 20% of the population working full time, and 80% doing no paid work at all. But you could just as easily have two-thirds of the population opting to do no paid work, and the remaining third being divided equally between 1-, 2-, 3-, 4-and 5-day working weeks -with one in 15 still opting to continue working full time.

Of course, the two-thirds of the population doing no paid work would not necessarily be sat watching endless television series or playing video games (even though they would certainly have that option). They could also indulge in all the enjoyable activities currently reserved for the retired, for those who have inherited wealth, those lucky enough to have earned enough to retire early, or married to someone wealthy. They could also devote themselves to the myriad options for rewarding and valuable unpaid work that would improve our world.

The beauty of having a modest UBI as a baseline is that everyone would be able to choose precisely how they divide their time between paid work, recreational activities, and valuable but unpaid work.

It is also important to realise that in such a scenario, market forces will still be able to play their role. If the level of the UBI is set a level sufficient to live modestly without paid work, but not enough to provide a lavish lifestyle, money will still be effective at motivating people whose hard work is considered particularly valuable, such as chefs in Michelin-starred restaurants, skilled musicians, actors, sports stars and entrepreneurs. You could even decide the level of the optimal amount for the UBI by fixing it such that a given number of people (say 10%) choose to opt out of paid work.

A final point to make is that this future world where everyone chooses freely how they divide their time may be as close as we can get to a real utopia. It would ensure what Jeremy Bentham called "the greatest good for the greatest number". The simple fact is that people differ and there is no point in seeking a "one-size fits all" solution where everyone has to do what some bureaucrat has decided is the optimal amount of paid work. Not only will people be free to choose their particular mix, but that mix can also change continuously from day to day, from week to week, from month to month and from year to year. There would be no such thing as being "unemployed" since not doing paid work would be a question of choice, and there would be no shame in choosing such a lifestyle. Indeed, if we are serious about saving the planet, we should be actively encouraging such behaviour. Likewise, the very notion of defining an age when you are allowed to retire would be seen to be nonsensical. Effectively, some people might "retire" as soon as they left school -as the children of millionaires can already. Does that mean that they would never do anything of value?

For me, the main reason why we have ended up in the current system stems from the fact that the vast majority of people appear to believe that their "value" can be measured by the amount they earn. Hopefully, most people will be able to see that this is an illusion. The elimination of this misconception could be the most positive impact of advances in AI and automation.

But the final point is that, with a UBI in place, and the development of a situation where everyone chooses their life mix freely, there will be little need to fear the rise of AI and automation. These new technologies can potentially be enormously beneficial for humanity and for all life on our planet. UBI is the mechanism that we need to be able to see that technology progress without threatening our societies.