S Cabanes
email: simon.cabanes@hotmail.fr

B Favier

M Le Bars

Some statistical properties of three-dimensional zonostrophic turbulence

Keywords:

. In this framework, the so-called zonostrophic regime is achieved when large topographical variations of the fluid layer combine with rapid rotation in a well developed three-dimensional (3D) turbulent flow. At steady state, strongly energetic, zonally dominated, large-scale axisymmetric structures emerge scaling with Rhines' theoretical scale. This model differs from the shallowlayer scenario where the flow is confined to a quasi-two-dimensional (2D) fluid shell and the anisotropic βeffect arises from latitudinal variation of the Coriolis force. Thus, we aim to reveal, in the specific framework of the deep-layer scenario, signatures of the zonostrophic regime and of a β-topography in statistical flow properties. To do so, we run two large-scale 3D direct numerical simulations in a cylindrical geometry of a highly turbulent and rapidly rotating flow. These two simulations use similar set of parameters but with and without topographical β-effect. We propose a comparative phenomenological description of the temporal and spatial statistics of the three components of the velocity field. Interestingly, we report that peculiar correlations occur between the vertical and radial flow components when a β-topography is imposed and show a feature possibly due to a zonostrophic dynamics in 3D frequency spectra. These results suggest the development of new tools to remotely investigate gas giants zonal winds by extracting statistical flow properties from direct observations. Ultimately our analysis may support the relevance of the deep models in the study of prevalent features of planetary dynamics.

Introduction

The strong east-west winds on the gas giants, such as Jupiter [START_REF] Bagenal | Jupiter: the planet, satellites and magnetosphere[END_REF] and Saturn [START_REF] Del Genio | Saturn atmospheric structure and dynamics[END_REF], are unmistakable markers of powerful underlying dynamics, inaccessible to direct measurements. Understanding how these intense zonal winds emerge and interact with deeper planetary envelops, is a major scientific challenge for ongoing spacecraft missions as well as for numerical and laboratory studies. While classical homogeneous three-dimensional (3D) turbulence leads to a direct cascade of the energy, turbulence theory predicts that large scale structures naturally emerge from small-scale motions in two-dimensional (2D) systems. This well-known phenomenon is called an inverse energy cascade [START_REF] Kraichnan | Inertial Ranges in Two Dimensional Turbulence[END_REF]. Consequently, macro-turbulence on Jupiter has been interpreted to result from the inverse energy cascade within the quasi-2D shallow troposphere [START_REF] Cho | The morphogenesis of bands and zonal winds in the atmospheres on the giant outer planets[END_REF] (i.e., the planetary weather layer). Similar assumptions are used in global circulation models designed to simulate large-scale circulations observed in the Earth's atmosphere and oceans. It is then common to invoke that the vertical dimension (along the planetary radius) is much smaller than any typical horizontal scale, thus reducing simulations to a quasi-2D configuration (e.g. solving shallow-water equations on a planetary spherical surface). However, recent studies have revived the possibility to generate Jupiter like jets from a 3D convective envelope, extending through the molecular layer between the surface weather layer and the deeper dynamo region [START_REF] Christensen | Zonal flow driven by deep convection in the major planets[END_REF][START_REF] Heimpel | Simulation of equatorial and high-latitude jets on Jupiter in a deep convection model[END_REF][START_REF] Kaspi | The deep wind structure of the giant planets: Results from an anelastic general circulation model[END_REF][START_REF] Heimpel | Simulation of deep-seated zonal jets and shallow vortices in gas giant atmospheres[END_REF][START_REF] Cabanes | A laboratory model for deep-seated jets on the gas giants[END_REF]. These studies invoke that rapid planetary rotation leads to flow invariance along the planetary spin axis. This dynamical confinement due to a dominant Coriolis acceleration forces the flow to be invariant along the rotation axis and sets the required conditions for the inverse cascade to feed large scale structures. This duality between 2D and 3D dynamics, inverse and direct cascades, is at the core of fundamental geophysical fluid dynamics [START_REF] Pouquet | Geophysical Turbulence and the Duality of the Energy Flow Across Scales[END_REF]. In both shallow and deep configurations, the dynamics is quasi-2D, namely depth invariant, and large scales develop on the horizontal flow components. In the shallow model, horizontal is defined by the spherical planetary surface and depth is along the planetary radius. In the deep model, the relevant horizontal and depth components are defined to be respectively normal and along the planetary spin axis. This defines a complete framework to study the specificity of the shallow and deep-layer scenarios.

In the specific context of planetary systems, the large scale structures formed by quasi-2D flows display an additional zonal anisotropy in the form of east-west dominant motions.

Such features appear in spherical geometry where the boundary curvature of planetary fluid layers preferentially channels energy into the large-scale zonal motions via the so-called β-effect [START_REF] Vallis | Generation of mean flows and jets on a beta plane and over topography[END_REF]. Large-scale drag eventually halts the inverse energy cascade that propagates up to the Rhines' length scale L Rh = π 2U/β, and sets the characteristic width of zonal jets [START_REF] Rhines | Waves and turbulence on a beta-plane[END_REF][START_REF] Sukoriansky | On the arrest of inverse energy cascade and the Rhines scale[END_REF]). Here U is the root-mean-square (rms) fluid velocity. In shallow turbulence, where quasi-bidimensionality results from spatial confinement, zonal anisotropisation arises from latitudinal variation of the Coriolis force with β = 2Ω cos θ/R, where Ω is the rotation rate, θ is the latitude and R is the radius of the planet. In deep turbulence, where quasi-bidimensionality results from strong anisotropy induced by the Coriolis force, zonal anisotropy emerges from variations of the axial fluid column height h with cylindrical radius r, with β = (2Ω/h)dh/dr.

Both scenarios gather all suitable physical ingredients and zonal jets develop, provided that the Coriolis force dominates over fluid inertia (i.e, low Rossby number Ro), which in turn, must dominate over viscous dissipative effects (i.e., large Reynolds number Re and small Ekman number E). Depending on the details of the system considered, the large-scale jets can remain weak and meandering like in the Earth's oceans [START_REF] Maximenko | Observational evidence of alternating zonal jets in the world ocean[END_REF], or ultimately reach the so-called zonostrophic regime where zonal energy dominates over turbulent fluctuations. These differing regimes of jet flow behaviors can be characterized in terms of the zonostrophy index R β [START_REF] Galperin | Geophysical flows with anisotropic turbulence and dispersive waves: flows with a β-effect[END_REF]. This index is the ratio of the Rhines scale and the transitional scale L β ≈ /β 3 1/5 above which the β-effect affects the energy cascade, with being the inverse energy transfer rate through the system. Simplistically, R β ≈ L Rh /L β may be regarded as a measure of how strongly anisotropic the turbulent flow may be. Zonostrophic flows are argued to exist at R β ≥ 2.5 [START_REF] Galperin | Anisotropic turbulence and zonal jets in rotating flows with a β-effect[END_REF][START_REF] Sukoriansky | On the arrest of inverse energy cascade and the Rhines scale[END_REF]. Jovian mid-latitude jets exceed this threshold [START_REF] Galperin | Cassini observations reveal a regime of zonostrophic macroturbulence on Jupiter[END_REF]) with a typical value R β ≈ 5, while Earth's oceans are limited to R β ≈ 1.5 [START_REF] Galperin | Anisotropic turbulence and zonal jets in rotating flows with a β-effect[END_REF]. Strongly zonostrophic regime is the focus of the present paper.

As of now, although these shallow and deep models have been systematically challenged, there has been no successful attempt to contrast their relevance in the study of the prevalent features of planetary dynamics. Today, new observations from the ongoing Juno NASA's mission pave the way to conduct in-depth analyzes of Jupiter's flow properties. The upcoming challenge will be to identify, using the measurements of the Jovian anisotropic turbulent flow, typical signatures of the shallow and deep turbulent scenarios. For that purpose, laboratory and numerical models for planetary jets are now requested to come up with a full description of turbulent flow properties in the framework of zonostrophic systems. However, planetary-like jets have been reproduced mainly in numerical models with simplifying assumptions and limitations [START_REF] Cho | The morphogenesis of bands and zonal winds in the atmospheres on the giant outer planets[END_REF]Polvani 1996, Jones and[START_REF] Jones | Compressible convection in the deep atmospheres of giant planets[END_REF]. Jets dynamics is sometimes studied in idealized numerical models solving, in the framework of β-plane approximation, for 2D quasi-geostrophic turbulence in single (Scott andDritschel 2012, 2013) or multiple [START_REF] Berloff | On latency of multiple zonal jets in the oceans[END_REF] planetary layers in vertical. Attractive alternative studies equally retrieve basic jets properties, in idealized 2D system with β-plane, by modeling kinetic energy transfer from small turbulent scales to large zonal scales using stochastic methods or closure models [START_REF] Bouchet | Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier-Stokes equations[END_REF][START_REF] Marston | Generalized quasilinear approximation: application to zonal jets[END_REF]. It is then common, when a planetary like spherical configuration is considered, that the shallow scenario is modeled solving for three-dimensional general circulation model (GCM) of a thin shell in the outer atmosphere, assuming weak depth variations along the planetary radius [START_REF] Schneider | Formation of jets and equatorial superrotation on Jupiter[END_REF]. Numerical studies designed to simulate the deep model (in which rotation leads to axial flow invariance of a 3D-turbulent system) are only considered using hyper-diffusion or stress-free boundary conditions [START_REF] Christensen | Zonal flow driven by deep convection in the major planets[END_REF][START_REF] Heimpel | Simulation of equatorial and high-latitude jets on Jupiter in a deep convection model[END_REF], 2016). Note that, contrary to shallow-layer models, where depth invariance is assumed and not produced by the dynamics, vertical motions including wave motions may play an important role in deep-layer 3D turbulent models (see e.g. [START_REF] Yarom | Experimental observation of steady inertial wave turbulence in deep rotating flows[END_REF]. Previous laboratory studies (see for instance the recent studies by Di [START_REF] Di Nitto | Simulating zonation in geophysical flows by laboratory experiments[END_REF][START_REF] Smith | Multiple Zonal Jets in a Differentially Heated Rotating Annulus[END_REF][START_REF] Zhang | Beta-plane turbulence: Experiments with altimetry[END_REF][START_REF] Read | An experimental study of multiple zonal jet formation in rotating, thermally driven convective flows on a topographic beta-plane[END_REF][START_REF] Galperin | Anisotropic macroturbulence and diffusion associated with a westward zonal jet: From laboratory to planetary atmospheres and oceans[END_REF] have reached weakly zonostrophic regimes (with R β < 2.5).

In the present paper, we extend the analysis initiated by [START_REF] Cabanes | A laboratory model for deep-seated jets on the gas giants[END_REF] of the first laboratory/numerical model generating planetary-like zonal jets in a deep 3D-layer.

In addition to a large-scale rotating experiment, [START_REF] Cabanes | A laboratory model for deep-seated jets on the gas giants[END_REF] performed a 3D turbulent direct numerical simulation (DNS) with a cylindrical β-topography designed to emulate part of a 3D convective planetary layer. Contrary to previous numerical simulations aforementioned, 3D Navier-Stokes equations have been solved in realistic experimental conditions using no-slip boundary conditions. Deep-seated jets have been observed with similar flow properties in the numerical simulation and in its laboratory realization. The strength and scale of this multiple jets system have been shown to be comparable relative to those observed on the gas planets. In the direct continuity of these first results, we propose here a phenomenological description of flow statistics in this high resolution direct numerical simulation referred to as DNS1. In order to investigate the specific role of the β-topography and its impact on turbulent flow properties, we run a second DNS using similar dynamical conditions but without layer depth variation (labeled as DNS2). Spatial and temporal spectral flow description, comparatively in the β and non-β simulations, allows us to extract the signatures of the zonostrophic turbulence in a fully 3D system with realistic dissipative effects. In particular, we evidence the peculiar correlation of the vertical and radial flow components under topographic β-effect and report, for the first time, a signature of the zonostrophic regime in frequency spectra. Our findings may open new horizons to remotely investigate the dynamics of gas giants' zonal winds using phenomenological description of turbulent flow properties.

Numerical setups

We consider the motion of an incompressible fluid with constant kinematic viscosity ν and constant density ρ. The fluid is contained inside a circular cylinder of radius R rotating with a constant rate Ω around the vertical axis. In our first model DNS1 (see figure 1(a)), the vertical extend of the fluid layer h(r) is axisymmetric and corresponds to the analytical paraboloid shape of a free surface that depends on the rotation rate and on the gravitational acceleration g given by

h(r) = h 0 - Ω 2 R 2 4g 1 -2 r 2 R 2 , ( 1
)
where h 0 is the fluid height in the absence of rotation. This upper paraboloidal surface is assumed to be stress-free whereas both side and bottom boundaries are no-slip. This appropriately models the experimental setup described in [START_REF] Cabanes | A laboratory model for deep-seated jets on the gas giants[END_REF] and also mimics topographic variations near planetary poles. The center, with minimum fluid height, corresponds to the pole, while the edge of the domain corresponds to lower latitudes. The simulation is set so that the nature of the β parameter (i.e., cylindrically-radial topographic variations)

and the large dynamical jets' aspect ratio (i.e., vertical jets' depth larger than their horizontal width L Rh ) reproduce a deep planetary layer. We have also run a second DNS (labeled as DNS2 in the following), dynamically similar to the first run DNS1 but this time with a top flat stress-free surface. All parameters characterizing the system remain the same, but the uniform layer depth is now h 0 , which corresponds to the limit case β = 0. This second case is run as a reference, as there is no topographic effect in that case. It is also close to other rotating turbulence experiments, as for instance by [START_REF] Yarom | Experimental observation of steady inertial wave turbulence in deep rotating flows[END_REF]. For the two cases, energy is injected through a small-scale steady horizontal volumic force corresponding to Taylor-Green vortices. The forcing is given by the following formula:

F =            AS(r) sin (πk f x) cos (πk f y) sin (πk f z) AS(r) cos (πk f x) sin (πk f y) sin (πk f z) 0   if 0 ≤ z ≤ 1/k f , 0 elsewhere.
(2)

The function S(r) is given by

S(r) = 1 2 tanh 60(r -0.15) -tanh(60(r -0.92) , (3)
and is used to avoid the generation of intense vortical structures close to the outer boundary and at the center of the cylindrical domain. A is the ratio of the force amplitude to the centrifugal acceleration and can be seen as a forcing Rossby number (A = 2 × 10 -3 in the following). We focused on the configuration where the forcing is at relatively small scales (i.e., with a typical horizontal wavenumber k f = 12) to ensure scale separation between the injection of the energy and the jet scale, but not too small so that the local Reynolds number is still large. Note that it is very demanding numerically to allow for a scale separation between the injection and the large-scale flows, while solving at the same time for the inertial range associated with the three dimensional direct cascade. Contrary to previous numerical simulations, designed to model deep-seated jets with negligible role of viscous dissipation, we solve for a viscous flow with realistic no-slip boundary conditions applied on the bottom and side walls. In particular, multiple banded flows are not found in the most recent, high-resolution models that couple the molecular envelope to the deeper dynamo region.

In these models, bottom dissipation damps the higher-latitude deep jets out of existence [START_REF] Heimpel | On the relationship between zonal jets and dynamo action in giant planets[END_REF][START_REF] Gastine | Zonal flow scaling in rapidly-rotating compressible convection[END_REF][START_REF] Jones | A dynamo model of Jupiterś magnetic field[END_REF]). Here, long-lived jets can persist even under conditions including viscous dissipation and no-slip conditions.

We solve the Navier-Stokes equations in the rotating frame using the spectral element solver Nek5000 [START_REF] Fischer | Simulation of high-Reynolds number vascular flows[END_REF](Fischer et al. , 2008)). Variables are decomposed into Lagrange polynomials based on Gauss-Lobatto-Legendre quadrature points inside hexahedral elements. For both simulations, we use polynomials of order 11 so that each element contains 12 3 grid points.

The original mesh corresponds to a full circular cylinder of unit radius and unit height with multiply the vertical coordinate of each element by the local required fluid depth. The mesh is also refined close to all boundaries. We use a third order explicit extrapolation scheme for the nonlinear convective terms and the linear Coriolis and forcing terms, and a third-order implicit Backward Difference scheme for the linear diffusive terms. The code is parallelized

Global properties and visualizations

The Let us first discuss the zonal spectrum computed from spectral decomposition of the axisymmetric component of the flow, here based on the time and azimuthal average of the zonal velocity. As shown in figure 3a, we recover the theoretical prediction

< u 2 r > < u 2 φ > < u 2 z > 0 0.002 0.004 0.006 0 2 0 0 4 0 0 6 0 0 8 0 0 Time (rotation units) < u 2 r > < u 2 φ > < u 2 z >
E ϕ (k ⊥ ) = C ϕ β 2 k -5 ⊥ ,
with k ⊥ the orthogonal wavenumber, perpendicular to the azimuthal component, and C ϕ = 2 the zonal constant [START_REF] Sukoriansky | Universal spectrum of two-dimensional turbulence on a rotating sphere and some basic features of atmospheric circulation on giant planets[END_REF], 2007[START_REF] Galperin | Cassini observations reveal a regime of zonostrophic macroturbulence on Jupiter[END_REF]. To distinguish between the axisymmetric and non-axisymmetric components of the flow, one defines the origine to be the rotation spin axis for spectral analysis. The axisymmetric flow is then a one dimensional profile along the radius in our cylindrical geometry. Following this approach, the orthogonal wavenumber corresponds to the radial wavenumber, k ⊥ = k r . We note here that cylindrical harmonic decomposition of the flow would have been more appropriated than time and azimuthal average of the velocity field to compute the axisymmetric flow component.

However, to facilitate comparison with spectral analysis in the laboratory reported in Cabanes et al. (2017) (which was performed in Cartesian coordinates due to the lack of a complete cylindrical field) we decided to adopt the same data processing for statistical description of the flow. We ascertain however that both approaches lead to comparable results. Thus, consistently with aforementioned studies [START_REF] Sukoriansky | Universal spectrum of two-dimensional turbulence on a rotating sphere and some basic features of atmospheric circulation on giant planets[END_REF], 2007[START_REF] Galperin | Cassini observations reveal a regime of zonostrophic macroturbulence on Jupiter[END_REF], kinetic energy distribution follows a k -5 r slope on both sides of the transitional wavenumber, it achieves maximum values at large scales and drops at wavenumbers smaller than the theoretical Rhines wavenumber. This characteristic spectrum of zonostrophic flows has been commented in details by [START_REF] Cabanes | A laboratory model for deep-seated jets on the gas giants[END_REF] and confirms that the present model is comparable with Jupiter's jets flow. In the flat geostrophic case (DNS2), we verify that energy of the time and azimuthal average of the zonal velocity is two orders of magnitude smaller than in case DNS1 and that Rhines scale is not a relevant scale in the dynamics.

DNS2 is in the rapidly-rotating but non-zonostrophic regime (Ro 1, Re 1 and β = 0).

These two simulations set the way to investigate the role of the topographic β-effect in 3D-turbulent rotating flows and to characterize the specificity of the deep-layer scenario.

Based on [START_REF] Cabanes | A laboratory model for deep-seated jets on the gas giants[END_REF], who attested the establishment of a zonostrophic regime in DNS1, we now extend the study to a full spatial and temporal statistical characterization of the three components of velocity fluctuations.

Azimuthal fluctuations

Because the velocity field is quasi-invariant with depth, we restrict our flow description to a horizontal plane normal to the rotation axis, specifically z = 0.15. averaged flow to the instantaneous velocity field in our Cartesian grid. We ascertain here that temporal and azimuthal average are equivalent in our Cartesian grid to extract the axisymmetric flow component. At initial time, energy is injected from basal forcing at scale k f (blue curves). Temporal evolution of the zonal spectra clearly shows how energy transits in time from initial injection at small-scale to the formation of large-scale structures. With topographic β-effect, narrow but clear inertial ranges develop on both sides of the forcing scale following the double cascade scenario predicted by Kolmogorov-Batchelor-Kraichnan (KBK) [START_REF] Kraichnan | Inertial Ranges in Two Dimensional Turbulence[END_REF]) in 2D-turbulence (figure 3(b)). The existence of two inviscid invariants (energy and enstrophy) of the 2D Navier-Stokes equations implies an inverse cascade of kinetic energy to large scales (k < k f ) with a power-law of E(k) ∝ k -5/3 , and a direct cascade of enstrophy to small scales (k > k f ) inducing the power-law E(k) ∝ k -3 . These features are well retrieved in the instantaneous azimuthal velocity of our 3D system, which is dynamically quasi-2D because of the effect of strong rotation. This double cascade scenario has previously been observed in 2D turbulence numerical simulations [START_REF] Boffetta | Evidence for the double cascade scenario in two-dimensional turbulence[END_REF] as well as in forced rotating flows [START_REF] Campagne | Direct and inverse energy cascades in a forced rotating turbulence experiment[END_REF]. It is now reported in the present work when topographical effects are dominant, although we are limited by the relatively small scale separation between the energy injection scale and the container's typical length scale. Indeed, we note that previous statistical analyses of the zonostrophic regime have been reported in studies solving for 2D vorticity equations [START_REF] Galperin | Geophysical flows with anisotropic turbulence and dispersive waves: flows with a β-effect[END_REF][START_REF] Sukoriansky | Rossby waves and zonons in zonostrophic turbulence[END_REF]. In these studies, the injection wavenumber k f is much larger than the transitional wavenumber

k β (namely k f k β ).
The inverse cascade consequently develops on a much wider inertial range k < k f , leading to a k -5/3 spectrum spreading on both sides of the k β wavenumber. In the present work, injection and transitional wavenumbers are approximatively identical, due to numerical constraints, leading the inverse cascade to develop on wavenumbers k < k β ≈ k f and the direct cascade to develop on wavenumbers k > k β ≈ k f . This lack of scale speration is waiting for extended numerical capacity to increase k f , while maintaining turbulence at the injection scale, and finally disclose the dynamics around the transitional wavenumber k β in 3D-turbulent zonostrophic flows. Similarly, the k -5 slope on wavenumbers k > k β observed Figure 3(a) is disputable as it occurs beyond the injection wavenumber k f .

When no topographic effect is present (figure 3(c)), the inverse cascade still transfers energy to large scale vortices. There is however no equivalent to the Rhines scale so that the cascade continuously feeds large-scale flows that finally saturate due to frictional effect on the lateral and bottom rigid no-slip boundaries. The resulting kinetic energy spectrum is apparently modified compared to the case with topography, and we observe instead a slope closer to k -3 , altough we cannot be fully conclusive since there is less than a decade available. One possible explanation is the presence of so-called condensates formed by a 2D turbulence inverse cascade in close or periodic containers [START_REF] Chertkov | Dynamics of energy condensation in twodimensional turbulence[END_REF]) that follow a steeper power-law E(k) ∝ k -3 . Note that in the context of forced rotating turbulence at

Rossby and Reynolds number similar to the ones considered here, both k -5/3 and k -3 slopes have been observed in the inverse cascade range depending on the anisotropy of the forcing [START_REF] Sen | Anisotropy and nonuniversality in scaling laws of the large-scale energy spectrum in rotating turbulence[END_REF], Pouquet et al. 2013). In addition, the large-scale friction due to the side boundary might affect differently the non-axisymmetric flow without topography compared to the dominantly axisymmetric flow in the presence of topography, thus leading to different spectral slopes at large-scales. The reason why this finite size effect does not occur on the azimuthal component of the zonostrophic regime has to be investigated, but may relate to the dynamical, non-rigid, confinement imposed by the Rhines scale [START_REF] Vallis | Generation of mean flows and jets on a beta plane and over topography[END_REF]. Accurate determination of the slope remains to be validated using larger numerical simulations (i.e. larger scale separation) and laboratory experiments. These frequency spectra can be split into two frequency ranges separated by the cutoff frequency ω = 2Ω of the inertial waves. In the range ω > 2Ω, the dynamics evolve on short time scales and the effect of rotation is moderate. Turbulent energy is nearly isotropically distributed over the three components of the flow (see figure 5). We notice that these high frequency velocities are only weakly affected by the β-topography: all spectra of both simulations are almost indistinguishable. We observe a ω -3 power law tentatively explained by a Taylor hypothesis: the high-frequency spectrum corresponds to the advection by the large-scale quasi-steady energetic patterns of the direct cascade described in figure 3.

The low frequency range ω < 2Ω, where rotation dominates (i.e., low Rossby number) and 3D inertial waves develop, shows the most striking signature of the β effect. In simulation DNS2, when no β-topography is present, rotation acts to separate the horizontal flow from the vertical flow [START_REF] Campagne | Disentangling inertial waves from eddy turbulence in a forced rotating-turbulence experiment[END_REF]. This horizontal anisotropisation is well observed in the frequency spectra figure 5 A striking feature of these frequency spectra is the energetic plateau within the range of inertial waves frequency, when a β-topography is imposed (figure 5(a)). The particularity of a β-topography in rotating system is that it injects potential vorticity and generates Rossby waves that propagate in the zonal direction. Our observations indicate that such a dynamics strongly affects turbulent statistics in a way that energy is equally distributed over a large frequency range. Similar features are observed in frequency spectrum of simplified 2D global circulation models used to simulate planetary atmospheres [START_REF] James | Wave-zonal-flow interaction and ultra-low-frequency variability in a simplified global circulation model[END_REF].

We note that the lowest frequencies in our spectra, which reflect the stationary zonal jets, correspond to a maximum of energy. In the absence of β-topography, frequency spectra follow a ω -1 power-law for vertical velocity and for the lowest frequencies of horizontal velocities (figure 5(b)). The ω -1 power-law behavior is a prevalent feature in turbulent flows related to the statistical properties of coherent structures [START_REF] Herault | Experimental observation of 1/f noise in quasi-bidimensional turbulent flows[END_REF]. The horizontal velocities at intermediate frequencies exhibit a nearly ω -5/3 power-law, in agreement with the recent laboratory studies of [START_REF] Yarom | Experimental observation of steady inertial wave turbulence in deep rotating flows[END_REF] and [START_REF] Campagne | Disentangling inertial waves from eddy turbulence in a forced rotating-turbulence experiment[END_REF].

Conclusions

Our phenomenological description of turbulent properties in the so-called zonostrophic regime, and comparatively with the geostrophic (non-zonostrophic) regime, constitutes a coherent and advanced analysis of the first numerical/laboratory model for deep-seated planetary jets presented in [START_REF] Cabanes | A laboratory model for deep-seated jets on the gas giants[END_REF]. We report spectral evidences that an inverse energy cascade occurs under rapid rotation in a 3D-turbulent cylindrical system. We This feature is probably due to the formation of 3D Rossby waves that are non-existent at constant fluid depth in DNS2. The upcoming data from current and future spacecraft missions will have to be analyzed in view of these spectral analyses that report statistical properties of a deep turbulent planetary layer. Additionally, more theoretical work is requested to explain this 3D zonostrophic regime that is now accessible to both experimental and numerical investigations.

Figure 1 .

 1 Figure 1. Sidecut and top view of DNS1 (a-b) and DNS2 (c-d) after t = 1000 rotations showing the instantaneous non-dimensional azimuthal velocity field at E = 5 × 10 -7 , Ro = 2 × 10 -3 and Re = 4 × 10 3 , based on the rms flow velocity and the averaged fluid depth. The dashed squares in (b) and (d) show the location where spectral analysis is performed (see section 4).

  time evolutions of the rms values of all three velocity cylindrical components (u r , u φ , u z ) are shown in figure 2. After a very short transient phase where all components grow rapidly (few rotation times), a secondary transient growth occurs in both cases on much longer timescales (typically several hundreds of rotation times) until complete saturation eventually occurs. In case DNS1, this slow secondary growth only affects the azimuthal component whereas radial and vertical components are similar and saturate quickly around a lower

Figure 2 .

 2 Figure 2. Time evolution of the rms velocity components for the case DNS1 with topographical effects (left) and the case DNS2 without topographical effects (right). (Colour online)

  Figure 1(b) and (d) shows horizontal cuts of the zonal velocity. At steady state, we observe large scale non-axisymmetric structures when the β-topography is absent (see DNS2 figure 1(d)). On the contrary, a paraboloidal surface preferentially channels energy into the zonal direction leading to prograde/retrograde jets formation (see DNS1 figure 1(b)). To facilitate the spectral S. Cabanes, B. Favier and M. Le Barsanalysis of the velocity fluctuations, we extrapolate them with spectral accuracy on a high resolution 512 × 512 Cartesian square grid (dashed lines in figure1). 2D spectral analyses are computed along both Cartesian coordinates (x, y) to obtain energetic distribution of the zonal u ϕ , radial u r and vertical u z flow components as a function of the total wavenumber k = k 2x + k 2 y , with k x and k y the two Cartesian wavenumbers.

Figure 3 .

 3 Figure 3. (a) Kinetic energy spectra of the time and azimuthal averaged zonal velocity field for DNS1 (red solid line) and DNS2 (black solid line). The theoretical energy spectrum Eϕ(kr) for the zonostrophic regime (Sukoriansky et al. 2007) is represented by a black straight line as a a function of the radial wavenumber kr. Theoretical values of the Rhines k Rh , the transitional k β and the injection k f wavenumbers are respectively shown as dashed grey, vertical dashed black, and solid lines. Typical value of k β is estimated using (U T rms ) 2 /2τ E , with τ E the Ekman spinup-time. (b-c) 2D zonal kinetic energy spectra Eϕ as a function of the total wavenumber k. Power spectra are given at different times t = 0, 5, 8, 31 and 1000 rotations (blue, orange, yellow, purple and red) for DNS1 (b) and DNS2 (c). We also report the injection wavenumber k f as a vertical line and the transitional wavenumber k beta as a vertical dashed line. (Colour online)

Figure 4

 4 Figure 4(a) reports 2D spectra at steady state of the vertical and radial fluctuations (note that time averaged radial and vertical flows are both negligible) in the horizontal planes displayed in figure 1. With topographic β, the amplitude of those two components is much smaller than the azimuthal one (consistent with figure 2(a)). Also, both radial and vertical velocity fluctuations are limited to scales that do not exceed Rhines wavenumber. On the contrary, the large scale vortices formed when the top surface is flat have the same signature on the radial and zonal component: we retrieve a E(k) ∝ k -3 power-law behavior (see figure 3(c) and 4(a)).

Figure 5 .

 5 Figure5. Frequency energy spectra computed from 1000 rotations time series of the zonal (thin solid lines), radial (thick solid lines) and vertical (dashed lines) velocity field components for DNS1 (a) and DNS2 (b). Note that for comparison purpose, we also report on a), the frequency spectra of the zonal component obtained from DNS2 (grey solid line). (Colour online)

  recover several statistics observed in more classical regimes of forced rotating turbulence but also show some new results specific to the case with topography. The presence of a paraboloidal surface shape induces a strong zonal anisotropy of the turbulent flow similarly to deep convective planetary layers. It results that the dominant zonal flow structures into jets and that the statistical flow properties follow the double cascade scenario reminiscent of the 2D turbulence paradigm[START_REF] Kraichnan | Inertial Ranges in Two Dimensional Turbulence[END_REF], at least for the azimuthal flow velocity. Our high resolution numerical simulations allow us to access the flow in 3D and at all times. We observe that radial and vertical flow components are correlated due to topographical deformation of the top surface. Power spectrum of the vertical velocity field is consequently more energetic in the zonostropic regime where radial motions are converted into vertical motions. This result constitutes an interesting signature of the deep-layer scenario. Ultimately, we report frequency spectra where all three components of the flow are strongly affected by the β-effect.

GGAF-2016-0069-Cabanes Statistical properties of three-dimensional zonostrophic turbulence

April 13, Geophysical and Astrophysical Fluid Dynamics GGAF-2016-0069-Cabanes

Statistical properties of three-dimensional zonostrophic turbulence

Acknowledgements

The authors thank Prof. J. Aurnou (Spinlab, UCLA) for initial suggestions and stimulating discussions, which motivated the present work. This work was supported by the French Agence Nationale pour la Recherche (Program SIMI5 JCJC 2013, Grant No. ANR-13-JS05-0004-01).

We also acknowledge support from IDRIS for computational time on Turing (Project No. 100508) and from the HPC resources of Aix-Marseille University (project Equip@Meso No. ANR-10-EQPX-29-01 of the program Investissements d'Avenir).

S. Cabanes, B. Favier and M. Le Bars that vertical and radial motions are not correlated. Under strong rotation and without topographical variations in a 3D rapidly-rotating system, the vertical velocity component actually behaves as a quasi-passive scalar. However, when a paraboloidal topography is imposed, the PDF shown in figure 4(b) shows that positive/positive and negative/negative values of the radial/vertical velocities are correlated. In other words, inward (outward) radial motions, when they impact (move away from) the topography, convert into downward (upward) vertical motions. We note that these correlations are maximum for radius where β is maximum (not shown). Joint PDF of the other velocity components are not reported in this manuscript as they do not display specific correlations. These correlations of the radial and vertical flows are a specificity of 3D-rotating turbulence under topographical β-effect, as it is the case in 3D-convective planetary layers. Similar flow measurements on gas planets' surface may highlight the dynamics underlying the jets. Note that velocity correlations are obtained from a numerical simulation where the free surface is not deformable. In the laboratory, the iso-pressure paraboloidal free-surface is non-rigid and deformations can occur (typically of a few mm for maximum radial velocities). Even if these deformations appear to be weak, it may attenuate the observed correlation. Direct measurements of velocity correlations in the laboratory will be further required to confirm our results.

Frequency spectra

We now focus on the frequency spectra and on the possible signatures associated with the observed large-scale flows. To complete the description of our system and extract all its specificities, we report in figure 5 frequency spectra of the instantaneous velocity for the three components of the flow. Frequency spectra result from time series spanning over 1000

April 13, Geophysical and Astrophysical Fluid Dynamics GGAF-2016-0069-Cabanes Statistical properties of three-dimensional zonostrophic turbulence


Nadja Gaudillière
email: nadja.gaudilliere@gmail.com

Romain Duballet

Charles Bouyssou

Alban Mallet

Philippe Roux

Mahriz Zakeri

Justin Dirrenberger

Large-Scale Additive Manufacturing of Ultra-High-Performance Concrete of Integrated Formwork for Truss-Shaped Pillars

Keywords: 3D printing, Concrete, Cementitious materials Large-scale additive manufacturing, Architecture, Design Truss-shaped pillars

In the present paper a new additive manufacturing processing route is introduced to produce ultra-high-performance concrete complex architectonic elements, by printing integrated formwork. Interdisciplinary work involving material science, computation, robotics, architecture and design resulted in the development of an innovative way of 3D printing cementitious materials. The 3D printing process involved is based on a FDM-like technique, in the sense that a material is deposited layer by layer through an extrusion printhead mounted on a 6-axis robotic arm. An architectural application is used as a case-study to demonstrate the potentialities of the technology. Along with the detailed description of the design and construction process, a description of the responsibilities and their distribution amongst the stakeholders involved in the project is given. The steps taken to include the 3D printed element in an authorized regulatory context are presented as well. The structural elements produced constitute some of the largest 3D printed concrete parts available until now. Multi-functionality was enabled for structural elements by taking advantage of the complex geometry which can be achieved using our technology for large-scale additive manufacturing. The proposed process succeeds in solving several of the current issues problems that can be found in the production of 3D printed architectural features for an AEC industrial context and therefore suggests an immediately viable route for industry assimilation.

Introduction

Until recently, additive manufacturing (AM) techniques were confined to high value adding sectors such as the aeronautical and biomedical industries, mainly due to the steep cost of primary materials used for such processes. In the last decade, the development of large-scale AM in such domains as design, construction and architecture, using various materials such as polymers, metal, and cementitious materials, has surged.

Historically, the first attempt at cement-based AM was made by Pegna [START_REF] Pegna | Exploratory investigation of solid freeform construction[END_REF] using an intermediate process between the classical powder bed and inkjet head 3D printing (3DP) [START_REF] Sachs | Three-dimensional printing techniques[END_REF] and fused deposition modelling (FDM) [START_REF] Crump | Apparatus and method for creating three-dimensional objects US Patent US5340433A[END_REF], in order to glue sand layers together with a Portland cement paste. Many groups have been involved with the development of large-scale AM for construction applications, all of which have been using processing routes derived from FDM or 3DP, although varying depending on the chosen material and targeted application.

Three pioneering technologies can be mentioned regarding cement-based AM. Contour Crafting technology [START_REF] Khoshnevis | Automated construction by contour crafting-related robotics and information technologies[END_REF] is based on a printhead mounted on a crane, extruding simultaneously two layers of material intended to be used as formwork. The main drawbacks of Contour Crafting technology are its limitation to vertical extrusion and a complex implementation of the system for industrial production. Loughborough University concrete printing technology [START_REF] Buswell | Freeform construction: mega-scale rapid manufacturing for construction[END_REF], using a similar system with a printing nozzle and a crane, uses a material with higher performances and to a small extrudate, allowing a good geometrical control. However, the printing process is relatively slow and the use of an overhead crane both discourages the printing of complex geometries and complicates the industrial development. Finally, D-Shape technology [START_REF] Cesaretti | Building components for an outpost on the lunar soil by means of a novel 3D printing technology[END_REF] consists in a large-scale sandbed locally solidified by deposition of a binding agent, layer by layer. Although designed for off-site production of prefabricated elements, D-Shape technology currently aims to demonstrate the feasibility of the process on-site. More detailed information regarding existing cement-based AM techniques can be found in the literature [START_REF] Buswell | 3D printing using concrete extrusion: a roadmap for research[END_REF][START_REF] Duballet | Classification of building systems for concrete 3D printing[END_REF].

In the past few years, several other technologies have been developed for cementbased AM techniques and applications in construction, both by university research groups and new industrial players. Landmarks of AM construction applications have been produced during this same period, such as housing printed by the Chinese company Winsun and other small-scale building printed by companies such as Apis Cor (Russia) or CyBe (Holland). Major achievements also include two bridges, one by Acciona in Madrid and the other by TU Eindhoven in the Netherlands.

Beyond the mention in the Contour Crafting patent of the use of AM as integrated formwork for construction, only few constructive experiments based on this technique have been documented. AM of formwork can be seen in the Batiprint printing process, though the material used is foam, making the formwork disposable. The case study presented in this paper is the first using method UHPC concrete to produce integrated formwork for an actual building project using the XtreeE 3D concrete printing system.

The aim of the present work is to shed a new light on the perspective of 3D concrete printing in the construction industry, specifically using the integrated formwork technique, by describing this project. With many new industrial players but not many largescale realisations yet, cement-based AM manufacturing appears to be at a critical point of development. The present paper seeks to discuss the industrial assimilation of these new construction techniques and the issues currently at stake in this matter. First, an introduction on designing structures for large-scale additive manufacturing is given. In Sect. 3, the concrete formwork 3D printing technique is presented, followed by the case study demonstrating the potential advantages of such a technique for the construction industry. Finally, conclusions are drawn regarding the results obtained in this work.

Design for Additive Manufacturing

Generating and modelling shapes for additive manufacturing follows specific rules, coming from both processing constraints, and functional objectives. According to [START_REF] Gosselin | Large-scale 3D printing of ultra-high performance concretea new processing route for architects and builders[END_REF], the concept of freeform previously used in the literature is not adequate nor sufficient for describing concrete 3D printing. For a given printing process and automation complexity, one can attain specific types of topologies within a given time-frame and performance criterion for the structure. Although out of the scope of the present paper, design conditions for large-scale additive manufacturing depend on many other parameters than just the properties of extruded cementitious materials; parameters such as the printing spatial resolution, overall size of parts to be printed, the environment, the presence of assembly steps, etc. A classification of such relationships between geometrical complexity, processing, and design is proposed in [START_REF] Duballet | Classification of building systems for concrete 3D printing[END_REF].

The processing constraints depend mostly on the fresh material properties in its viscous state, as well as early-age behaviour, in interaction with the building strategy and the stiffness of the structure being built. On the other hand, functional requirements will depend on the properties of the hardened material as well as the structural geometry for effective stiffness, and other functional properties such as thermal and sound insulation. See [START_REF] Gosselin | Large-scale 3D printing of ultra-high performance concretea new processing route for architects and builders[END_REF] for a geometrically induced thermal insulation case study. Both types of constraints must be considered at the design stage. Material properties of 3D printed concrete used by XtreeE, including in the case-study presented in this paper, are highlighted in [START_REF] Gosselin | Large-scale 3D printing of ultra-high performance concretea new processing route for architects and builders[END_REF] as well. More information on 3D printed concrete in general can be found in the very recent review published by [START_REF] Roussel | Rheological requirements for printable concrete[END_REF].

Printing path generation is a critical step to be considered in early design phase. It is indeed the prime problem, embracing together machine, material and structural requirements. 3D-to-2D slicing, which is by far the most common method adopted in the context of 3D printing, yields planar layers of equal thickness built on top of each other. This approach is not optimal from a design and structural viewpoint as it will induce shifts when two consecutive layers have different sizes and limit the attainable geometries. In the context of 3D printing. Other approaches to tool-path generation have been developed, such as the tangential continuity method introduced in [START_REF] Gosselin | Large-scale 3D printing of ultra-high performance concretea new processing route for architects and builders[END_REF] to optimise the structure being built by creating layers of varying thickness. These layers exhibit a maximized surface area of contact between each other, hence stabilizing the overall structure. However, in the present case-study, as no other reliable software allowing the use of other methods existed yet, the first method was used.

Concrete Formwork 3D Printing

The process of large-scale concrete 3D printing developed by XtreeE has been mentioned previously in [START_REF] Gosselin | Large-scale 3D printing of ultra-high performance concretea new processing route for architects and builders[END_REF], the overall process is summarized in Fig. 1. Based on this process, a construction strategy can be derived for concrete formwork 3D printing. The principle consists in 3D printing the formwork necessary for casting another structural material such as ultra-high-performance concrete for fibre-reinforced concrete, or insulation material such as foamed concrete, as shown on Fig. 2. The printed formwork is left in-place and becomes a so-called lost formwork. An optimal trade-off must be considered from the early design steps as to the ratio of printed material within the built part, which can be critical for reaching economic viability. Depending on the application considered, concrete formwork 3D printing can be more efficient than either all-3D concrete printing, or traditional building techniques, from both an economic and/or building strategy viewpoint. This assertion is demonstrated in the next section on the Aix-en-Provence post case study.

4 Pillar in Aix-en-Provence, France

Context

This 4 m-high freeform pillar is placed in the sports facilities of a school in Aix-en-Provence, France. It supports a concrete awning covering part of the playground (Fig. 3). The sports facilities project was mandated by the Aix-Marseille Metropolis. The pillar, part of this larger project, was handled by the following people: Marc Dalibard as architect (also the architect for the whole sports facilities building), Artelia as structural engineering office, AD Concept as construction company, LafargeHolcim as material supplier and Fehr Architectural as UHPC concrete caster. For the construction of the pillar, the responsibilities of the actors were divided as follow: Marc Dalibard, as architect, was also the manager of the overall project, and defined the shape and placement of the pillar. Artelia, as structural engineer, supported XtreeE both during design and construction phases, and was tasked with defining the load case on which to base the topological optimization and verifying the strength and stability of the printed pillar in accordance with the load case. LafargeHolcim supplied a specific 3D print concrete, developed with XtreeE in an earlier collaboration and Fehr Architectural casted UHPC inside the pillar, a task requiring a specific licence. Each one of the key players supported XtreeE in the definition of the fabrication strategy adopted for the pillar, by providing input regarding their field of expertise. XtreeE identified a fabrication strategy for the post and adapted the printing system developed earlier (presented in Sect. 3) according to the fabrication strategy and its requirements. During the design stages, XtreeE co-defined the load case with structural engineer Artelia and designed an exact shape for the pillar through structural design. In the fabrication stages, XtreeE coded the manufacturing files using HAL Robotics for the printing system and performed the manufacturing, before co-supervising the placing of the pillar on site with architect Marc Dalibard.

Although relying on the skills of well established players of the construction market and its various subgroups, it is interesting to note that the Aix-en-Provence pillar studied in this paper presents a new workflow between the stakeholders of the project. This new workflow is established due to the appearance of a new player, XtreeE. XtreeE plays in this case the roles of technical expert in 3D printing, giving advice for every step of the conception and manufacturing process, of designer of the final shape and of manufacturer for the 3D print elements. While the new industrial protagonists in the field of AM manufacturing for construction applications might not, in the future, play such polyvalent role in projects, the Aix-en-Provence pillar gives an example of the array of possible interventions, and therefore of the many changes that could be implemented, not only with AM, but also with various rising digital conception and manufacturing tools.

Design

In the initial project designed by architect Marc Dalibard, a complex truss-shaped pillar supporting the roof was already planned, as shown on Fig. 4. But though the idea of a complex truss-shaped pillar was featured, no viable design for the pillar existed at this stage of the project. XtreeE came in at this point and took over the design of the pillar, based on the sketches provided by the architect. The design of the pillar is based as much on the formal intention highlighted in these sketches as on the constraints fixed by the building regulations in effect at the time and by the 3D printing manufacturing method.

As no building regulation existed regarding 3D print items integrated in buildings at the time of construction, and to stick to the projected schedule, the choice was made to use the lost formwork manufacturing method, as introduced in Sect. 3. Instead of having to validate the pillar and its manufacturing method by using an ATEx (Experimental technical appreciation), a long and expensive procedure for experimental construction in France, the lost formwork made it possible to rely on existing building regulations on UHPC concrete. Furthermore, having recourse to 3D printing technology for the fabrication of architectonic elements offers the possibility to build nonstandard geometries, with fine and curved elements. However, the complexity of those shapes proscribes implementing steel reinforcements inside. This impossibility, along with the fact that the shape of the pillar as designed by the architect had for main characteristic the slenderness of its parts, also called for the use of UHPC inside the formwork.

The critical design constraint for layer by layer extrusion techniques is the maximum inclination if the geometry. In the case of the Aix-en-Provence pillar, this issue was avoided by printing supports at the same time than the geometry to enable any angle for the parts of the truss. This has been commented in [START_REF] Duballet | Classification of building systems for concrete 3D printing[END_REF].

To define the precise shape of the pillar, we relied on an optimization method, to ensure an optimal use of matter in the truss by refining the limbs and taking into account the fabrication constraints as well as the wishes of the architects regarding the shape. The entire circular volume containing the pillar is used as research space for the truss to develop, and the applied load case included the weight of the concrete awning supported by the pillar as well as site-specific constraints (wind, etc.). The resulting final shape is shown on Fig. 5. A more thorough examination of the possibilities offered by topology optimization in the context of 3D concrete printing is available in [START_REF] Duballet | Design of space truss based insulating walls for robotic fabrication in concrete[END_REF]. Given the selected approach of lost formwork printing, the pillar is made of an outer shell that is 3D printed and later filled with UHPC. The pillar is divided in three smaller parts (cf. Fig. 6), both for transportation constraints and to reduce hydrostatic pressure when self-compacting concrete is casted inside. Each part is to be filled with concrete and then assembled together to form the whole element. During casting, metallic connectors are inserted in the concrete at each end of the parts to ensure reinforcement continuity of the construction joints. Finally, chemical glue is used to seal the connection.

Manufacturing

Manufacturing of the pillar includes several stages: 3D printing the outer shell, at XtreeE headquarters in the south of Paris, France, casting the UHPC and integrating the connectors, at the Fehr Architectural production facility in Germany, and final assembly on site in Aix-en-Provence. 3D printing the outer shell inside the facility enables, like for UHPC casting, a precise control and monitoring of the environment, to ensure ideal temperature and humidity conditions for the concrete to behave as expected.

As a precaution, after running trials on smaller geometries similar to the pillar, it was decided to 3D print the formwork in four parts rather than three. The concrete formwork took 15 h to print in total, approximately 3 h and 45 min for each part of the post, with the former XtreeE printing systemadjustments made over the past year now make it faster. The printing of one of the parts is shown on Fig. 7. Setting time starts to occur after several minutes. Once the formwork was successfully 3D printed, an assembly trial, shown on Fig. 8, was conducted at our facility to ensure the results were as precise as expected before shipping the parts.

The casting of UHPC in each part of the pillar was then operated by the team of Fehr Architectural. To resist the hydrostatic pressure resulting from the casting, supports printed with the pillar were left in place until the UHPC set, as shown on Fig. 9. The supports were then cut, as shown on Fig. 10, and the parts were shipped on site to Aix-en-Provence. The definitive assembly of the parts was performed there, before installing the pillar in place, with sliding supports on top. Finally, on request of the architect, the pillar was given a smooth finish by coating it to cover the line pattern specific to 3D printed objects. The difference of surface roughness is shown on Fig. 11. The various advantages of large-scale additive manufacturing of ultra-highperformance concrete, as well as the concrete formwork 3D printing technique, have been reviewed based on the analysis of the case study of a complex pillar part of an industrial construction projects in France, performed using the XtreeE 3D printing technology. A final significant element of comparison can be given by confronting the total production price of the Aix-en-Provence pillar to the total production price of a traditionally built complex pillar: a 62,5% total price gain is obtained, based on our information for the price of the Aix-en-Provence pillar, and quotes obtained for a traditional manufacturing. One of the main reason for this difference, on top of the gain identified on time, materials, and workforce, is the absence of a specific material and shaping for the mould, hence eliminated by using the lost formwork method. The multiple aspects of potential socio-economic gain for relying on additive manufacturing are three-fold: (1) materials saving by using the right amount of matter where needed, given that a topology optimization computational framework is available; (2) time saving by reducing the number of steps in the construction process, as well as being BIM-compatible for construction-planning strategies; (3) workforce saving by limiting on-site manual building steps, therefore enhancing safety on the construction site and implementing a new workflow between project stakeholders.

Although the lost formwork strategy allowed to get around experimental technical certification, further work should have to be conducted with certification authorities for the construction industry to define a legal and regulatory framework for 3D printed structures. The technological feats presented in this work are commercially available, but a legal framework and economic market are to be developed for the 3D printing technology to transfer into the mainstream construction industry.

Studying the conditions of design and fabrication of the Aix-en-Provence pillar also provides hints on possible improvements to the developed process, to push industrial assimilation forward.

Along with the paradigm shift of additive manufacturing comes the possibilities enabled by topology optimisation, which aims at attaining the most efficient structure geometrically possible for a given set of requirements. Optimality in terms of industrial design has become more and more critical due to scarcity of material resources and the need for lightweight structures.

A driving force for additive manufacturing is its ability to produce more complex 3D shapes in comparison to casting or subtractive processes. This complexity allows to design optimal structures based on topology optimisation techniques. One of the main current challenges is to modify optimisation algorithms to account for the additive manufacturing constraints, especially with regards to the processing parameters and structural stability while printing. A possible answer to these challenges would be to consider the multiphysics phenomenon aspect of 3D printing, which involves the elastic stability of the overall structure being built, the kinetics of hydration, the evolving viscoplasticity of fresh cement, the evolution of temperature within the printing environment, etc. As a matter of fact, all these physical problems, at multiple time and space scales, can be modelled on their own, but coupling them generates complexity and uncertainty regarding the process of 3D printing. Therefore, efforts should be concentrated on understanding and modelling the printing process in its multiple physical aspects, only then optimisation will be fully integrated with the processing, which would virtually change the way 3D printed structures are conceived today.

Fig. 1 .

 1 Fig. 1. Workflow of the large-scale concrete 3D printing process.

Fig. 2 .

 2 Fig. 2. Schematic view of the concrete formwork 3D printing.

Fig. 3 .

 3 Fig. 3. Pillar in Aix-en-Provence, photo by Lisa Ricciotti.

Fig. 4 .

 4 Fig. 4. Initial sketch for the pillar.

Fig. 5 .

 5 Fig. 5. Final design of the pillar after topology optimization.

Fig. 6 .

 6 Fig. 6. Splitting and assembling principles for the 3D printed pillar system.

Fig. 7 .

 7 Fig. 7. Printing of one of the parts of the pillar's formwork.

Fig. 8 .

 8 Fig. 8. Assembly trial after printing of the four parts.

Fig. 9 .

 9 Fig.9. UHPC casting inside the envelope, with supports left in place.

Fig. 10 .

 10 Fig. 10. Cast and lost formwork assembly from which the printing supports were cut.

Fig. 11 .

 11 Fig. 11. Before (left) and after (right, photo by Lisa Ricciotti) surface smoothing through manual coating.