Benjamin Kadoch

Thorsten Reimann

Kai Schneider

Michael Schäfer

Comparison of a spectral method with volume penalization and a finite volume method with body fitted grids for turbulent flows

Keywords: computational fluid dynamics, periodic hill, volume penalization, Fourier spectral method, finite volumes, wavelets

Comparison of a spectral method with volume penalization and a finite volume method with body fitted grids for turbulent flows

Introduction

Computational fluid dynamics in complex geometries is still a challenge especially for high Reynolds numbers. The thin boundary layers which destabilize, subsequently detach and thus generate coherent vorticies are a key feature to understand turbulence generated by walls. Two main approaches Email address: benjamin.kadoch@univ-amu.fr (B. Kadoch)

Preprint submitted to Computers & Fluids

January [START_REF] Chang | Simulations of laminar and turbulent flows over periodic hills with immersed boundary method[END_REF]2016 have been developed so far, either body fitted grids using, e.g., finite volume or finite element discretizations, or immersed boundary methods (IBM) which are becoming more and more attractive due to their high flexibility. Indeed, for the latter the complex geometry is typically embedded into a larger computational domain of simple shape, e.g., a rectangle. Hence classical discretizations, using Cartesian grids, for which efficient solvers are available, can be employed. The influence of the geometry and the no-slip boundary conditions is modeled by modifying the underlying Navier-Stokes equations.

The origin of IBM can be traced back to [START_REF] Courant | Variational methods for the solution of problems of equilibrium and vibrations[END_REF] [START_REF] Courant | Variational methods for the solution of problems of equilibrium and vibrations[END_REF] in the context of constraint optimization. A large range of immersed boundary methods can be found since then, for example Lagrangian multipliers [START_REF] Glowinski | On the solution of the Dirichlet problem for linear elliptic operators by a distributed Lagrange multiplier method[END_REF], level-set methods [START_REF] Cheny | The LS-STAG method: A new immersed boundary/level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties[END_REF], fictitious domain approaches and surface [START_REF] Peskin | Flow patterns around heart valves: a numerical method[END_REF] and volume penalization approaches [START_REF] Angot | A penalization method to take into account obstacles in viscous flows[END_REF]. For reviews we refer the reader to [START_REF] Peskin | The immersed boundary method[END_REF] and [START_REF] Mittal | Immersed boundary methods[END_REF].

Validation and benchmarking are an essential step in code development.

Comparing the results of different numerical methods enables us to know the advantages and drawbacks. Their domain of validity can thus be checked and their precision can be assessed.

In this study, we have chosen the volume penalization method [START_REF] Angot | A penalization method to take into account obstacles in viscous flows[END_REF] coupled with a Fourier pseudo-spectral method [START_REF] Canuto | Spectral Methods in Fluid Dynamics[END_REF]. Investigations on the Gibbs ocsillations which appear in Fourier based schemes can be found in [START_REF] Keetels | Fourier spectral and wavelet solvers for the incompressible Navier-Stokes equations with volume-penalization: Convergence of a dipolewall collision[END_REF],

as well as filtering techniques for removing this Gibbs phenomenon. The code has been benchmarked in two space dimensions [START_REF] Schneider | Numerical simulation of the transient flow behaviour in chemical reactors using a penalization method[END_REF] and applied to moving obstacles [START_REF] Kolomenskiy | A Fourier spectral method for the Navier-Stokes equations with volume penalization for moving solid obstacles[END_REF]. To evaluate the performance of this new method in three-dimensional, turbulent fluid flow, it is compared here to that of a wellestablished second-order finite-volume method in terms of accuracy and efficiency. The latter, called "Fastest" [START_REF]Fastest -User Manual[END_REF] is based on a finite volume discretization and uses body fitted grids. The flow configuration is a well documented benchmark "flow over periodic hills" [START_REF] Breuer | Flow over periodic hills -Numerical and experimental study in a wide range of Reynolds number[END_REF]. This benchmark is of interest, since it is a geometrically simple test case and there is no difficulty to specify inflow/outflow boundary conditions. Moreover, the physical mechanisms of separation on curved surfaces and the streamwise decorrelation enhanced, by choosing a sufficiently large distance between the two hills, are an attractive features of this test case. An other class of IBM was investigated recently using this benchmark in [START_REF] Chang | Simulations of laminar and turbulent flows over periodic hills with immersed boundary method[END_REF]. The three-dimensional Fourier pseudo-spectral code including volume penalization (denoted by "Pen4Flow") to impose nonperiodic boundary conditions is thus applied for the first time to compute fully developed turbulent flows. The aim of the paper is to show a detailed benchmark of the volume penalization method for three-dimensional turbulent flows and to compare the efficiency with respect to a classical finite volume code Fastest.

For the two solvers several grid levels are employed. As the considered flow is turbulent, statistical analysis are performed and compared with each other. In addition we also use orthogonal wavelets to decompose the flow into different scales of motion. Statistical quantities can thus be defined as a function of scale and direction.

The manuscript is organized as follows. In section 2 the "Flow over periodic hills" configuration is explained and the two numerical methods are shortly presented. Then in section 3, the results for the different codes are described and the flow statistics are compared. Scale dependent analysis of the results are also performed. Finally, the influence of the Reynolds number is investigated for the pseudo-spectral code with the volume penalization method. Some conclusions and perspectives are exposed at the end of the manuscript.

Flow configuration and Numerical methods

Flow configuration

The classical benchmark "Flow over periodic hills" [START_REF] Breuer | Flow over periodic hills -Numerical and experimental study in a wide range of Reynolds number[END_REF], is considered in detail at a Reynolds number of Re = 1400 which is based on the bulk velocity u b at x = 0 and the bump height H. To check the influence of the Reynolds number, we also considered Re = 700 and Re = 2800. The boundary conditions are periodic along the streamwise x-direction and the spanwise z-direction. The flow is confined by a hill at the bottom and by a wall plane at the top (y-direction). The flow configuration is illustrated in Fig. 1.

Pen4Flow: Fourier pseudo-spectral code with "volume penalization" method

In this subsection, we present the volume penalization method and the methodology to include the periodic hill configuration. More details on the numerical and mathematical validation of this method can be found [START_REF] Kolomenskiy | A Fourier spectral method for the Navier-Stokes equations with volume penalization for moving solid obstacles[END_REF][START_REF] Kadoch | A volume penalization method for Navier-Stokes flows and scalar advection-diffusion with moving obstacles[END_REF].

∂u ∂t + ω ω ω × ∇u + ∇Π -ν∇ 2 u -F = - 1 η χ(u), ∇ • u = 0, x ∈ Ω (1)
with

-∇ 2 Π = ∇ • ω ω ω × u + F + 1 η χ(u) (2)
where u = (u x , u y , u z ) is the velocity, ν the kinematic viscosity, Π is the modified pressure, ω ω ω the vorticity, F an external force, η the permeability (or penalization parameter) and χ the mask function which equals to 1 in Ω s (solid domain) and 0 in Ω f (fluid domain). Figure 2 (a) shows the computational domain Ω = Ω f ∪ Ω s where Ω f is the fluid domain (in white) and Ω s is the solid domain (in black). The solution of the penalized Navier-Stokes equations does converge towards the solution of the Navier-Stokes equations with no-slip boundary conditions [START_REF] Angot | A penalization method to take into account obstacles in viscous flows[END_REF]. The difference between the exact solution of the penalized equation and the exact solution of the Navier-Stokes equations, called modeling error is proportional to √ η and for sufficiently small values of η the solution is precise enough.

In space all variables are represented as truncated Fourier series, product and nonlinear terms are evaluated in physical space while derivatives are evaluated in spectral space. For time integration a second order Adams-Bashforth scheme is used for the nonlinear and the penalization term. The viscous term is integrated exactly using the semi-group formulation [START_REF] Kolomenskiy | A Fourier spectral method for the Navier-Stokes equations with volume penalization for moving solid obstacles[END_REF],

which means that the semi-group of the heat kernel is used for time integration. This corresponds to the exact integration of the linear viscous term.

The numerical study was carried out on three different grids that differed in the number of nodes employed, see Table 1. As the volume penalization method takes care of generating the boundary layer, the numerical grid has to be chosen sufficiently fine to be able to resolve these thin layers. For each grid, the size l y in the y-direction of the computational domain is chosen such that y = 3.036 coincides with a collocation point and that the solid domain on the top contains 10 points along the y-direction. In all simulations, the penalization parameter is η = 10 -3 and the CFL constant is equal to 0.1.

The choice of the penalization parameter is a compromise which works well for the resolutions considered here, more details on the influence of the penalization parameter can be found in [START_REF] Keetels | Fourier spectral and wavelet solvers for the incompressible Navier-Stokes equations with volume-penalization: Convergence of a dipolewall collision[END_REF]. For Pen4Flow, we observe that the time step limitation comes from the CFL condition and not for the stability limit (∆t < η) imposed by the penalization term.

The flow is driven by an external force F = f e x ∈ Ω. In order to keep the desired mass flow rate constant throughout the whole simulation, a control function similar to [START_REF] Benarafa | RANS/LES coupling for unsteady turbulent flow simulation at high Reynolds number on coarse meshes[END_REF], updates the forcing term on every time step:

f n+1 = f n + (u b -2ũ n + ũn-1 )/(α f ∆t) ( 3
)
where ũn is the measured bulk velocity at time level n, α f a damping factor (α f = 10 for Pen4Flow and α f = 1 for Fastest). In all computations the time-step for Pen4Flow is ∆t = 10 -3 and for Fastest ∆t = 1.5 10 -3 .

The flow is initialized with u(t = 0) = 0. To trigger the instability, an impulsion is imposed during the first time interval [0, 2], i.e., the forcing term

is extended to F = f e x + f imp e z with f imp = A sin(2x2π/l x ) * exp(1 -2(y - l y /2)
2 ) * sin(2z2π/l z ) and A = 1 being a suitable amplitude.

Fastest: Finite volume method

To solve the transient, incompressible Navier-Stokes equations with the finite volume method on body-fitted grids, we employ the solver Fastest [START_REF]Fastest -User Manual[END_REF]. This code is based on a cell-centered finite volume method on blockstructured grids. Discretization of the convective term is carried out using a multi-dimensional Taylor series expansion which preserves second-order accuracy on strongly distorted grids [START_REF] Lehnhäuser | Improved linear interpolation practice for finite-volume schemes on complex grids[END_REF], while the time is advanced by the Crank-Nicolson scheme. This numerical procedure is of second order in space and in time. The SIMPLE procedure is employed for the solution of the pressure. Velocity and pressure fields are coupled via the Rhie-Chow interpolation. The discretized system of equations is implicitly solved by Stone's method.

Two grids were generated for the finite volume computations, the first of which, subsequently referred to as fine grid, was constructed to fulfill the requirements of a DNS, while a coarse grid was derived from that by omitting every second node in each direction. A precursor RANS simulation employing the ζ-f model of Hanjalić [START_REF] Hanjalić | A robust near-wall ellipticrelaxation eddy-viscosity turbulence model for CFD[END_REF] gave an estimate of the Kolmogorov length scale η. Following Pope's recommendations [START_REF] Pope | Turbulent Flows[END_REF], a maximum ratio of ∆x/η = 2.1 was ensured over the entire domain for the construction of the fine grid. For the boundary layer resolution, the normalized wall distance of the first cell center y + 1 was kept below 0.1 everywhere. A coarse grid was derived from the fine one by omitting every second node in every index direction.

The time step size was chosen such that CFL < 1 for all time steps.

During the computation, the mass flux was kept constant utilizing (3) as external forcing. However, u(t = 0)/u b = 1.0 was chosen as initial condition, and during the initial simulation period no additional forcing was necessary to excite the flow instabilities.

Results

First we consider the periodic hill configuration at Reynolds number Re = 1400 and compare Fastest and Pen4Flow with the results obtained by Breuer et al. [START_REF] Breuer | Flow over periodic hills -Numerical and experimental study in a wide range of Reynolds number[END_REF]. Second, we focus on Pen4Flow and study the influence of the Reynolds number.

Case Re = 1400

In the following we present the results at Re = 1400 and study the influence of the resolution. Table 1 summarizes the numerical and physical parameters for both codes.

Ω f Ω s 0 1 2 3
Scaling of the CPU time with respect to the number of processors is given in Fig. 3 for both codes using coarse resolution, obtained on the HHLR supercomputer ("Hessischer Hochleistungsrechner"). Note that for Pen4Flow, the fluid domain height is constant and equal to y f = 3.036 and the solid domain height is chosen such that the solid contains 10 grid points and the boundary of the fluid coincides with a grid point. been obtained with the volume penalization method at resolution 512 3 . We observe that the flow is turbulent, characterized by the presence of vortices of different size. We can also identify a recirculation zone corresponding to an upstream (negative) velocity. In the following we will consider statistical quantities only. Table 2 shows that both codes yield similar values for the separation and reattachment point. A slight difference can however be observed for the reattachment point which might be due to the penalization method. x + ω 2 y + ω 2 z colored with the horizontal velocity u x computed with the Pen4Flow code at resolution 512 3 for Re = 1400.

N x N y N z y l /H FTT Pen4Flow
N x N y N z x sep /H

Moment statistics

Table 3 summarizes the first and second order moments together with the skewness and flatness values of the velocity fields for both codes at a given time instant in the statistically steady state. We can note that averaging the results over time does not change the results and slight differences (≤ 2%) appear only for high order moment statistics. The flatness reflects the intermittency in a turbulent flow while the skewness factor of velocity is related to the energy transfer, according to the Kolmogorov 4/5 law. The results show that all values are in good agreement, even for the higher order statistics. In all cases the difference is less than 5%.

M 1 M 2 Flatness: Skewness: M 4 /M 2 2 M 3 /M 3/

Mean velocity and Reynolds stress profiles

First we consider the volume penalization code and we study the influence of the resolution on the statistics. Figure 6 shows mean velocity and Reynolds stress profiles at four different streamwise locations, x = 0.5, 2, 4 and 6 for three different resolutions, 256 3 , 512 2 × 256 and 512 3 . Note that the profiles are all averaged over time (between 30 and 40 flow through times) and over the spanwise direction. We observe that the influence of the resolution is weak for all mean velocity profiles (Fig. 6 only shows the x-component u x )

while it is much more sensitive for the second order moments, illustrated in Fig. 6 (right) for the cross Reynolds stress u ′ x u ′ y . Hence in the following we will consider only the two higher resolutions for Pen4Flow.

In Fig. 7 and8, the coarse grid computations using Fastest show a good agreement for mean velocities with the fine grid computations, similar to what has been observed for Pen4Flow. Concerning the Reynolds stresses the differences are less pronounced than for Pen4Flow, but are still slightly more visible, in particular for the u ′ x u ′ y (see Fig. 7, bottom, right). The profiles are averaged over time and over the spanwise direction. The codes used are Pen4Flow and Fastest with fine and coarse grids. The results from [START_REF] Breuer | Flow over periodic hills -Numerical and experimental study in a wide range of Reynolds number[END_REF] are also plotted and denoted by BPRM09 for LES (adapted grid) and DNS (MGLET: non uniform spatial grid).

Wavelet analysis for the velocity fields

Wavelet decomposition of flow fields yields information on scale, position and direction. Scale-dependent statistical analysis of turbulent flows can thus be performed by considering the wavelet coefficients for example of velocity or vorticity, see, e.g., [START_REF] Schneider | Wavelet methods in computational fluid dynamics[END_REF]. Wavelet-based directional energies can be defined which capture the properties of velocity gradients. The intermittency of the flow in different directions can be quantified with scale-dependent directional flatness. The scale-dependent velocity flatness was shown to quantify the spatial variability of the energy spectrum [START_REF] Bos | Small scale intermittency in anisotropic turbulence[END_REF]. Here we will present applications of scale dependent statistics of the velocity field for the periodic hill flow computed with both methods presented above.

The velocity field, u(x) = (u x , u y , u z ), is decomposed into an orthogonal wavelet series: with

x = (x, y, z) ∈ [0, x l ] × [0, y l ] × [0, z l ] rescaled onto [0, 2π
] and given at resolution N = 2 3J , where J is the number of octaves in each spatial direction. The decomposition of u into an orthogonal wavelet series yields:

u(x) = λǫΛ ũλ ψ λ (x), (4)
where the multi-index λ = (j, i x , i y , i z , d) denotes the scale j, the position i = (i x , i y , i z ), and the directions d = 1, 2, ..., 7 of the wavelets. Indeed, the wavelet decomposition in three dimensions yields seven directions due to the tensor product construction [START_REF] Daubechies | Ten lectures on wavelets[END_REF]. The corresponding index set Λ is Λ = {λ = (j, i x , i y , i z , d), j = 0, ..., J -1; i x , i y , i z = 0, ..., 2 J -1 (5) and d = 1, ..., 7}.

Due to the orthogonality the wavelet coefficients are given by ũλ = u, ψ λ where •, • denotes the L 2 -inner product, defined by f, g = [0,2π] 3 f (x)g(x)dx, for each velocity component. The coefficients measure fluctuations of u around scale 2 -j and around position 2πi/2 j in one of the seven possible directions. The N wavelet coefficients ũλ are efficiently computed from the N grid point values of ω using the fast wavelet transform, which has linear complexity [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]. In this study, we have chosen the Coiflet 30 wavelet, which has 10 vanishing moments ( x p ψ(x)dx = 0, p = 0, ..., 9) and which is well adapted to represent the current flow simulations.

The a-order moments dependent on the scale index j and the direction d are defined as:

M a (j,d) (u i ) = ũi a (j,d)
, where i = i x , i y , i z and a ∈ N.

(j,d) is the average over all positions i of the wavelets for a given direction d and a given scale j.

At a given time instant in the statistically steady state, we applied the wavelet based multi-scale analysis. Scale-dependent moments are shown in figure 9 for Pen4Flow with 512 3 collocation grid points and for the finite volume code with a fine grid interpolated on an uniform Cartesian grid 512 3 .

Large scale corresponds to J = 0 and small scale to J = 8. The second order moments show the distribution of turbulent kinetic energy at different scales and yield similar results for both codes. The scale-dependent flatness, which is a measure for intermittency of the flow, obeys to the same behavior, but the difference between the two curves is small. The skewness (not shown here) gives for both cases similar values. Both codes show thus the same behavior for the scale dependent statistics from the largest to the smallest scale. To get further insight into the scale dependence of the Pen4Flow and the Fastest computations, we consider in Fig. 10 scale dependent moments of the velocity fields in different spatial directions, defined in eq. ( 6) using an orthogonal wavelet decomposition. In Fig. 10

Influence of the Reynolds number

In the following, we investigate the influence of the Reynolds number using the Pen4Flow code and compare the results again with those obtained by Breuer et al. [START_REF] Breuer | Flow over periodic hills -Numerical and experimental study in a wide range of Reynolds number[END_REF]. In Table 4, the separation and reattachment points are compared for three Reynolds numbers, Re = 700, 1400 and 2800. Slight differences are observed for the separation point, while the discrepancies are larger for the reattachment by comparison with those obtained with BPRM09 LESOCC code.

x As example in Fig. 11, we selected the position x/h = 4 as the compari-327 son of the results for the other positions shows the similar tendencies. Again ever for u ′ x u ′ x we observe larger differences compared to the mean velocity. for Reynolds number Re = 700, 1400, 2800. The profiles are averaged over time and over the spanwise direction. The code used is Pen4Flow with fine and coarse grids. The results from [START_REF] Breuer | Flow over periodic hills -Numerical and experimental study in a wide range of Reynolds number[END_REF] are also plotted and denoted by BPRM09 for LES (adapted grid).

Conclusion

The flow over a periodic hill configuration has been revisited using a Fourier pseudo-spectral code with volume penalization and a finite volume 20 code with body fitted grids. This classical benchmark involves complex, non-Cartesian geometries and turbulent flow conditions. The presence of a recirculation zone caused by the presence of the hill constitutes an additional difficulty. This detailed study showed that the volume penalization code and the finite volume code reproduce well the physics of this flow and produce similar results. Wavelet analysis enabled us to show that the multi-scale physics is almost the same for both codes. The volume penalization method is a powerful technique to impose Dirichlet boundary conditions in complex geometries using still a Fourier pseudo-spectral method, which has been confronted here for three different Reynolds numbers.

The pros and the cons for Pen4Flow and Fastest can be summarized as follows. Due to the volume penalization method, Pen4Flow is very flexible for changing the geometry and the convergence of the method is mathematically justified and proved in [START_REF] Angot | A penalization method to take into account obstacles in viscous flows[END_REF]. The penalization method becomes more efficient for higher Re numbers [START_REF] Nguyen Van Yen | Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint[END_REF] because the penalization boundary layer is proportional to √ ην. Moreover for this code, no linear system has to be solved (differential operators are diagonal) and most computational cost is due to the FFT, for which highly efficient parallel implementations are available. We can note also that the Gibbs oscillations are not amplified [START_REF] Keetels | Fourier spectral and wavelet solvers for the incompressible Navier-Stokes equations with volume-penalization: Convergence of a dipolewall collision[END_REF] and the method does not show numerical diffusion and dispersion. However, the penalization term models boundary conditions with first order accuracy only which is also the case for the geometrical errors due to the mask function (staircase effect). Equidistant grids are required in Pen4Flow, which is less efficient as fine grid size is required close to the boundary. Furthermore, the volume penalization technique requires more grid points as the governing equations have to be solved in the whole domain, including both solid and fluid domains. Finally, the time step is limited due to stability restrictions imposed by the penalization term in addition to the classical CFL limit. In conclusion, considering these latter points, the volume penalization method is an attractive alternative to simulate turbulent flows in complex geometries.

Fastest takes advantage of body fitted grids which are well adapted to the geometry and involve no interpolation error. As a consequence less grid points are required compared to Pen4Flow. The curvilinear grid with blockstructured grids enables to have boundary conditions fulfilled exactly. Nevertheless, Fastest adopts the SIMPLE projection scheme for imposing the incompressibility, which is not exactly fullfilled. The finite volume scheme, used in this code, is second order accurate in space and time, and an alge-braic system has to be solved in each time step. The parallelization is also less efficient, as shown in Fig. 3. Finally, Fastest is less flexible compared to Pen4Flow since the grid has to be generated for a given geometry.

Interesting perspectives of Pen4Flow are the application to compute turbulent flows in complex geometries which are simply defined by the mask function and which may even vary in time, as it is the case in fluid-structure interaction problems. The implementation of additional transport equations, e.g. for passive scalars, or the Boussinesq approximation is another promising direction. The necessary Neumann boundary conditions can be implemented using the technique proposed in [START_REF] Kadoch | A volume penalization method for Navier-Stokes flows and scalar advection-diffusion with moving obstacles[END_REF].

Figure 1 :

 1 Figure 1: Sketch of the 2d hill geometry. The dimension in spanwise z-direction is l z = 4.5H.

Figure 2 :

 2 Figure 2: Numerical grids: (top) Pen4Flow (128 3 for visualization), (bottom) Fastest for the coarsest grid (442 × 184 × 200).

Figure 3 :

 3 Figure 3: CPU time(s) for one iteration as function of the number of processors and for different resolutions using Pen4Flow and Fastest codes.

Figure 4

 4 Figure 4 shows a three-dimensional representation of isosurfaces of the vorticity norm colored with the streamwise velocity. The computation has

Figure 4 :

 4 Figure 4: Isosurface of the vorticity norm ||ω ω ω|| = ω 2x + ω 2 y + ω 2 z colored with the horizontal velocity u x computed with the Pen4Flow code at resolution 512 3 for Re = 1400.

Figure 5 (

 5 Figure 5 (top) shows the probability density functions (PDF), estimated with a histogram using 50 bins, of the velocity at a given time instant in the statistically steady state. We find that the PDFs do almost collapse except for weak negative values where we can observe a small deviation. Note that the PDFs are plotted in log-lin representation, hence the difference is negligible. Spectra of turbulent kinetic energy along the spanwise direction are shown in figure5(bottom). A small inertial range with a k -5/3 behavior can be identified for k = 3 to 10, followed by a faster decay. The spectra of both simulations collapse for low wavenumbers up to k = 60. For the Fastest code we then find a saturation, while for the spectral code energy further decays with increasing wavenumber. The saturation might be due to the linear interpolation of the finite volume computation.

Figure 5 :

 5 Figure 5: Top: PDF of the velocity (averaged over the three components) at a given time instant in the statistically steady state. Bottom: Turbulent kinetic energy spectrum versus wavenumber k in the spanwise direction. The codes used are Pen4Flow and Fastest.

  04 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0

Figure 6 :

 6 Figure 6: Mean velocity (left) and Reynolds stress (right) profiles at four different locations x = 0.5, 2, 4 and 6 for Pen4Flow using different resolutions.

Figure 8 :

 8 Figure 8: Mean velocity (top) and Reynolds stress profiles (middle and bottom) at x = 4.The profiles are averaged over time and over the spanwise direction. The codes used are Pen4Flow and Fastest with fine and coarse grids. The results from[START_REF] Breuer | Flow over periodic hills -Numerical and experimental study in a wide range of Reynolds number[END_REF] are also plotted and denoted by BPRM09 for LES (adapted grid) and DNS (MGLET: non uniform spatial grid).

Figure 9 :

 9 Figure 9: Scale-dependent second order moments (left) and scale dependent flatness (right) for the three different velocity components for Pen4Flow with 512 3 collocation grid points and for Fastest with a fine grid interpolated on an uniform Cartesian grid 512 3 .

  (top) the second order moment of the three velocity components in the longitudinal and transverse directions, i.e., for u x the longitudinal direction corresponds to x while y and z are the transverse directions, is shown. Large scale corresponds to J = 0 and small scale to J = 8. For clarity, we decided to omit the mixed directions xy, xz, yz, xyz. The scale dependent first (not shown here) and second order moments, are almost the same for both codes. The behaviors of the scale-dependent flatness are also similar; indeed the difference between the two curves is small for all scales, except for the smallest scale J = 8 where the deviation is more pronounced.

Figure 10 :

 10 Figure 10: Scale-dependent second order moments (top) and scale dependent flatness (bottom) for the three velocity components in the three main directions. The codes used are Pen4Flow with 512 3 collocation grid points and Fastest with a fine grid interpolated on a Cartesian uniform grid 512 3 .

Figure 11 :

 11 Figure 11: Mean velocity (top) and Reynolds stress profiles (middle and bottom) at x = 4for Reynolds number Re = 700, 1400, 2800. The profiles are averaged over time and over the spanwise direction. The code used is Pen4Flow with fine and coarse grids. The results from[START_REF] Breuer | Flow over periodic hills -Numerical and experimental study in a wide range of Reynolds number[END_REF] are also plotted and denoted by BPRM09 for LES (adapted grid).

Table 1 :

 1 Numerical parameters, flow domain height y l /H, flow-through times (FTT = L/u b ) used for statistical evaluation for Re = 1400.

	256 3	256 256 256 3.15941 48.88
	Pen4Flow 512 2 256 512 512 256 3.09647 14.69
	Pen4Flow 512 3	512 512 512 3.09647 29.19
	Fastest coarse	221 92 100	3.05	215
	Fastest fine	442 184 200	3.05	213

Table 2 :

 2 Numerical parameters, separation point x sep /H and reattachment point x reatt /H.

Table 3 :

 3 Moments of the velocity field (averaged over the three components) for the Pen4Flow code (512 3 ) and for the Fastest code, at a given time instant in the statistically steady state. The data of the finite volume computed on the fine grid are interpolated onto an uniform grid with 512 3 grid points.

Table 4 :

 4 Reynolds numbers, separation point x sep /H and reattachment point x reatt /H. The results from[START_REF] Breuer | Flow over periodic hills -Numerical and experimental study in a wide range of Reynolds number[END_REF] are also plotted and denoted by BPRM09 for LES (adapted grid).

		sep /H x reatt /H
	Re = 700
	Pen4Flow N = 256 3	0.30	5.30
	BPRM09 LESOCC	0.29	5.24
	Re = 1400
	Pen4Flow N = 512 3	0.26	5.26
	BPRM09 LESOCC	0.26	5.19
	Re = 2800
	Pen4Flow N = 512 3	0.24	5.44
	BPRM09 LESOCC	0.21	5.41

Nevertheless, for the worst case, Re = 2800, the differences are below 10%. 333

Acknowledgements B.K. thankfully acknowledges financial support and hospitality from the Graduate School of Excellence Computational Engineering at TU Darmstadt. B.K. and K.S. thank D. Kolomenskiy for providing the spectral code Pen4Flow. K.S. thanks the international affairs office of Aix Marseille Université for financial support.

The comparison of the statistics for the two codes is shown for the mean velocity and Reynolds stress profiles at location x = 0.5 and x = 4 in figure 7 and figure 8, respectively. The profiles are averaged over time and over the spanwise direction. We observe that, the mean velocities u x are the same, the mean velocities u y , the Reynolds stress u ′

x u ′ x and u ′ y u ′ y almost coincide, except at x = 0.5 where we observe a small deviation for u ′ y u ′ y .

The profiles of the Reynolds stress u ′ z u ′ z and u ′ x u ′ y exhibit slight differences for x = 2 and x = 4. Significant differences in u ′ z u ′ z are observed if we compare with the results from [START_REF] Breuer | Flow over periodic hills -Numerical and experimental study in a wide range of Reynolds number[END_REF], however Pen4Flow and Fastest do still match. The difference is probably due to the fact that, in [START_REF] Breuer | Flow over periodic hills -Numerical and experimental study in a wide range of Reynolds number[END_REF], Breuer et al.

plotted the turbulent kinetic energy k instead of the Reynolds stress u ′ z u ′ z .

We checked the turbulent kinetic energy computed from our data and found a good agreement with the u ′ z u ′ z obtained in [START_REF] Breuer | Flow over periodic hills -Numerical and experimental study in a wide range of Reynolds number[END_REF].

We can note also that for the mean vertical velocity u y , at x = 6 some fluctuations and an overestimation around y = 1 are observed for Pen4Flow.

However, the fluctuations tend to disappear by increasing the resolution. The higher moments are more sensitive when we change the methods, but we can conclude that the two codes yield similar results which are in reasonable agreement.

we start with the mean velocity profile for u x and u y , we observe that for


Niki Baccile
email: niki.baccile@upmc.fr

Anne-Sophie Cuvier

Claire Valotteau

Inge N A Van Bogaert

Practical methods to reduce impurities for gram-scale amounts of acidic sophorolipid biosurfactants Authors

Raw sophorolipids, a variety of bioderived glycolipid biosurfactants produced by the yeast Candida bombicola, generally present themselves as a mixture of acetylated acidic and lactonic forms mixed with residual fatty acids. To recover the acidic form alone in large amounts, e.g. tens of grams, which is interesting for various applications in cleaning technologies, material science, medicine, one must either purify the mixture through column chromatography, thus tremendously reducing the final yield, or remove the fatty acids via hot hexane extraction followed by alkaline hydrolysis and a liquid-liquid extraction process; in the latter, pentanol, used as extraction medium, is a high boiling point solvent and, consequently, difficult to remove under standard vacuum extraction conditions. Here, we compare, quantify and draw the positive and negative aspects of the typical hexane/pentanol extraction route compared to two additional ones, never reported for this system: a cold extraction from a pentanol/alcane mixture and a consecutive filtration on inverse C18modified silica phase and silica gel in, respectively, water/acetonitrile and methanol/CH2Cl2 mixtures. The first one generates highly pure products but quantitative extraction is quite long

(up to several days) while the second is an extremely handy and rapid (few hours) way but it is less quantitative and it provides a product with not less than 6% impurities.

Introduction

The emerging field of bio-based surfactants has been intensively studied in the past 20 years [START_REF] Kjellin | Surfactants from Renewable Resources[END_REF] in agreement with the 12 principles of green chemistry. Glycolipids are surface active agents composed of a hydrophilic carbohydrate-based head linked to a fatty acid or a fatty alcohol through an ester, an amide or an ether bond. [START_REF] Von Rybinski | Alkyl Polyglycosides-Properties and Applications of a new Class of Surfactants Angew[END_REF] They already replace, and/or are destined to massively replace, classical surfactant families like alkylsulfates or alkylammonium in home and personal care products. For these reasons, their physicochemical properties have been studied in the past 15 years. Effect of molecular structure (e.g., chain length, number of sugar units), concentration, use of co-solvents (short-chain alcohol, for instance), co-surfactants on the critical micelle concentration (cmc), surface tension, rheology, cleaning power but also self-assembly and mesophase formation have shown particular perspectives for many of these compounds. [START_REF] Von Rybinski | Alkyl Polyglycosides-Properties and Applications of a new Class of Surfactants Angew[END_REF][START_REF] Stubenrauch | Sugar Surfactants -Aggregation, Interfacial, and Adsorption Phenomena[END_REF][START_REF] Hoffmann | Phase and Aggregation Behaviour of Alkylglycosides[END_REF] These molecules are generally synthesized via several chemical strategies, reviewed by von Rybinski, from renewable compounds like fatty acids and sugars. In term of their cost, they are still more expensive in their pure form with respect to classical surfactants. For instance, commercial alkylpolyglucosides (APG) are actually sold as formulations presenting either a non-constant number of glucose units or mixtures of at least two different chain lengths.

In this context, various authors have shown the interest of a specific class of biosurfactants, sophorolipids (Figure 1), a family of entirely bio-derived glycolipids. These natural compounds can be obtained in large amounts by yeast culture in presence of several carbon sources, like glucose, fatty acids but even alkanes and waxes. [START_REF] Rau | Sophorolipids: a Source for Novel Compounds Ind[END_REF][START_REF] Tulloch | Structure and Reactions of Lactonic and Acidic Sophorosides of 17-Hydroxyoctadecanoic Acid Canad[END_REF][START_REF] Asmer | Microbial Production, Structure Elucidation and Bioconversion of Sophorose Lipids[END_REF][START_REF] Zhou | Supramolecular Assemblies of a Naturally Derived Sophorolipid[END_REF] Due to their reduced environmental impact biosynthesis, confirmed by a recent life cycle analysis study, [START_REF] Develter | Surfactants from Renewable Resources[END_REF] sophorolipids (SL) have attracted a fair attention in various fields: skin-care industry, [START_REF] Maingault | Use of Sophorolipids and Cosmetic and Dermatological Compositions[END_REF][START_REF] Van Bogaert | Microbial Production and Application of Sophorolipids[END_REF][START_REF] Shete | Mapping of Patents on Bioemulsifier and Biosurfactant: a Review[END_REF] anticancer, [START_REF] Fu | Sophorolipids and Their Derivatives are Lethal Against Human Pancreatic Cancer Cells[END_REF] structure-directing agents and self-assembly properties. [START_REF] Zhou | Supramolecular Assemblies of a Naturally Derived Sophorolipid[END_REF][START_REF] Baccile | Sophorolipids: a Yeast-Derived Glycolipid as Greener Structure Directing Agents for Self-Assembled Nanomaterials[END_REF]15,16,17 Raw sophorolipids are a mixture of two forms, lactonic and acidic, having different degrees of acetylation. Nevertheless, the acidic non-acetylated form of sophorolipids has gained much interest and has been exploited in most of the above-cited studies. For this reason, it becomes quite important to obtain acidic sophorolipids (SL-COOH) from the raw mixture.

Unfortunately, at the moment, the development of a full microbial approach to accomplish such a task is still a matter of ongoing studies in biotechnology-specific laboratories. A modified strain producing non-acetylated sophorolipids has been created, but the synthesized molecules are still a mixture of acidic and lactonic forms and yield is much as less as compared to the wild type. [START_REF] Saerens | One-step production of unacetylated sophorolipids by an acetyltransferase negative Candida bombicola[END_REF] Recently, a strain producing solely acidic sophorolipids was constructed and attempts to combine both features in one strain are still ongoing. The common strategy to obtain non-acetylated acidic sophorolipids exists in the application of chemical conversion and purification procedures to the raw mixture of sophorolipids, which mainly consists into: 1) hexane washing to remove residual fatty acids; 2) alkaline hydrolysis at about 100°C followed by acidification; these steps remove all ester groups and are responsible for the production of acetic acid and salt; 3) 1-pentanol extraction, which allows the recover of the SL-COOH, leaving salts and acetic acid in the water phase; 4) silica column purification, typically using methanol/dichloromethane as eluent. This procedure, even if efficient in obtaining a pure acidic SL-COOH compound, has many drawbacks, listed hereafter. 1) Alkaline hydrolysis can lead to parasite reactions that add impurities to the final compound; 2) 1-pentanol extraction is a tedious step because of its high boiling point (137-139 °C), making its removal under vacuum a delicate, potentially time-consuming, task, not exempt of parasite esterification reactions between pentanol itself and the COOH group of SL-COOH; 3) silica column purification is a complicated operation for large amounts of product and require good practical knowledge in organic chemistry.

Significant loss of product can be experienced on the column due to strong attractions between silanols and sugars.

For these reasons, we propose in this work to review the purification of acidic sophorolipids. We compare the classical "pentanol route", for which we provide more experimental details hard to find in the current litterature, to two alternative ones never proposed before for this system: a cold precipitation, referred to a as a "modified pentanol extraction", and a "double silica filtration", both more adapted to work with large amounts of product. In the latest method, hydrophilic and hydrophobic impurities are retained on, respectively, chromatographic SiO2 and inversed-SiO2 (SiO2-C18) substrates and for which no pentanol is used at all. Finally, we compare the advantages and inconvenients for each route, also providing the final yields. In the end, the goal of this work is to provide the opportunity to choose between more than one route so to optimize effort, experimental time, yields and level of impurities. Further development of any of these routes could eventually be of interest to develop a scale-up purification process.

Experimental

Sophorolipids mixture was obtained from Sopholiance (France). This product is certified with 54.6 w% of dry matter, estimated by Mass Spectroscopy and where about 10 w% of the dry compound is constituted by residual fatty acids, indicating that the sophorolipid mixture constitutes a little less than 50 w% of the overall product. [START_REF] Kjellin | Surfactants from Renewable Resources[END_REF] H solution NMR spectra were acquired on a Bruker AV300 Ultrashielded© Avance spectrometer equipped with a 5mm (1H/BBF) BBO probe with Z-axis gradient using a standard 30° flip angle.

HPLC analysis was performed on a Varian Prostar HPLC system using a Chromolith Performance RP-18e 100-4.6mm column from Merck KGaA at 30°C and Evaporative Light Scattering Detection (Alltech). A gradient of two eluents, a 0.5% acetic acid aqueous solution and acetonitrile, had to be used to separate the components. The gradient started at 5% acetonitrile and linearly increased till 95% in 40 min. The mixture was held this way for 10 min and was then brought back to 5% acetonitrile in 5 min. A flow rate of 1 mL/min was applied.

Solvent extraction was done on a Büchi Rotavapor® R-210/R-215, equipped with a digital controller V-850, heating bath B-491, vacuum pump V-700 and closed-circuit chiller F-105. Silica gel 60 (0.015-0.040 mm) for column chromatography was obtained from Merck and C18-grafted silica powder 40-60 µm (Vrac-C18-500) was obtained from AIT France.

Results and discussion

The complexity of a sophorolipid mixture is well-known and, at the moment, impossible to avoid. This is merely a result of the expected variation one tends to observe in (complex/nonspecific) biosynthetic systems. A typical mixture can contain various impurities (mainly residual fatty acids) but it is also characterized by a mixture of different sophorolipids with slightly different chain lengths (C16, C18) and saturated/unsaturated double bonds. By far, the most abundant species (60 % -80 %, according to the origin), which we refer to as SL, is the C18:1, shown in Figure 1. All purification steps performed in this work are not meant in any way to separate sophorolipids molecules with different structural features but only non-SL impurities, which will be detailed below. Figure 2 shows the typical 1 H NMR spectrum of such a product, whose main impurities are constituted by fatty acids, acetic acid and NaCl, but small fractions of additional compounds derived from side reactions can also pollute the system. We must highlight the following resonances belonging to various types of impurities, whereas all other peaks are directly related (not commented here) to the acidic form of sophorolipids:

-δ= 0.92 ppm, typical of a CH3 group in an aliphatic chain, here attributed to residual fatty acids. This impurity is easy to recognize and distinguish from SL because the chemical shift corresponding to same group in SL is slightly unshielded.

-δ= 1.96 ppm, typical of the CH3 group of free acetic acid, derived from acetate in the raw sophorolipids. This resonance is easy to attribute because of its sharpness.

-6.30 < δ < 5.70, unknown impurities N°1, derived from overheating issues -δ= 8.51 ppm, unknown impurity N° 2, derived from overheating issues

The 9-5 ppm portion of the spectrum belonging to the raw mixture before hydrolysis is shown in the same figure for comparison, indicating the absence of any signal in this region. These resonances are the typical consequence of a poorly controlled hydrolysis reaction and are presented here on purpose, mainly for didactical purposes. The nature and way to remove these impurities, which are responsible for the dark brown/black colour of the mixture, will be discussed later and can be avoided by simply keeping the reaction temperature below 90°C, as commented before. Method 1: Classic pentanol extraction. The hexane washing combined with pentanol extraction, followed by silica column chromatography purification, was described before by several authors. [START_REF] Rau | Sophorolipids: a Source for Novel Compounds Ind[END_REF]19 Unfortunately, very little specific information is available on this procedure, which has more than one inconvenience. First of all, pentanol is a difficult solvent to remove under vacuum, an operation which can take from hours to days according to the quantities and the type of apparatus employed for the task. In fact, removal of pentanol using low heating (T≤ 45°C) can be done with a cooled serpentine refrigerated below T= 6°C and vacuum pressure below 10 mbar. These operating conditions are probably common for organic-chemistry laboratories but they are definitely not generalizable for an average, non-organic related, environment. When removal is done in the absence of a cooled serpentine, one must work with a standard water supply, of which the temperature does not fall below 15-18 °C. To achieve complete solvent removal one can either raise the bath temperature up to 60-70 °C, thus inducing the risk of esterification between pentanol and the sophorolipid COOH group, or work under very low vacuum conditions (below 1 mbar) at room temperature, which requires the use of a liquid nitrogen trap to condense the pentanol vapours. Even if a combination of both approaches can provide good results, the whole approach is time-consuming followed by many safety issues (potential glassware explosion, use of cryogenic fluids, long extraction time, etc.).

The best adapted experimental details for this purification procedure are given here. The pentanol/water phase transfer can be done on the previously hydrolyzed mixture using 5 x 100 all hydrophobic compounds (fatty acids and eventual residual pentanol) marking the spectrum at δ= 0.92 but also those impurities resonating between 6.30 ppm and 5.70 ppm. Eventually, the positive point of using this route is the possibility to treat fairly large amounts of matter (here 20 g but this can definitely be increased) with good yields (above 80 w% with respect to the initial mass of raw sophorolipids). In terms of final purity, the result is more than satisfactory. All hydrophilic compounds (acetic acid, hydrophilic impurities) are efficiently removed, provided that bulk water is removed as well. The level of hydrophobic compounds can be safely kept below 5 mol%, as estimated by [START_REF] Kjellin | Surfactants from Renewable Resources[END_REF] H NMR, especially if Soxhlet treatment is performed and if the final powder is heated under strong vacuum.

mL
The main drawback for this route is the absolute need of a performing, well-designed, vacuum rotavapor with a closed cooling system and an efficient pump. Under these conditions, this approach is safely suggested.

Method 2: Modified pentanol extraction.

To overcome the drawbacks mentioned just above, it is also possible to recover the acidic sophorolipid from pentanol by decreasing the polarity of the medium. This can be easily done by adding an equi-volumetric amount of a short chain alcane (pentane, hexane or heptane) to the pentanol medium. The mixture can then be stored at -18 °C to induce precipitation of the acidic sophorolipids. The product is then recovered by a simple filtering step. The final product is particularly pure as fatty acids remain in solution but further purification, if needed, can be performed using the already-mentioned Soxhlet extraction process.

This procedure is extremely easy to perform as no specific instrument is needed and it provides a product of good purity, below the detection limit of 1 H NMR, as shown by the corresponding spectrum in Figure 4. Nevertheless, several drawbacks that may impact the overall time and energy consumption should also be pointed out. First of all, residence times at -18°C are quite long (> 12h);

secondly, extraction should be repeated several times (here, we went up to three times) to increase the yield, which are below 80 %. Finally, it is of crucial importance to remove any bulk water before storing the solution. In fact, if this is not done, the actual precipitate is a water-rich sophorolipid precipitate that melts at room temperature.

Method 3: Double silica filtration. On the basis of silica column chromatography experience, we propose here an adapted, fast and easy way to recover acidic sophorolipids. Experiments performed on a column are difficult to carry for large amounts (tens of grams) of compound.

We have found large loss in matter and relative purification efficiency. For this reason, we adapted the column principle to work in a classical beaker, much easier to manipulate. As discussed previously, the hydrolyzed sophorolipid mixture contains both hydrophilic and hydrophobic impurities, in addition to sophorolipids themselves. Removal can then be done with a double silica phase extraction in the suited solvent mixture, where both hydrophilic silica gel 60 and hydrophobic C18-grafted silica are used at different times. The use of two different solid phases is preferred to a single one to always keep sophorolipids in the liquid phase, thus reducing adsorption/desorption phenomena which would consequently reduce the final yield. In the following, Figure 5, Figure 6 and Figure 7 and related discussion show the preliminary tests that are carried out on a given amount of hydrolyzed sophorolipids in the corresponding solvent to which the desired amounts of the adsorbant is added. At the end of this section, we show an all-in-one consecutive purification procedure (C18-modified silica followed by silica gel extraction) performed on larger amounts of the same hydrolyzed sophorolipid batch. The purification tests of hydrolyzed SL sample on bulk silica gel to remove hydrophilic impurities is shown on Figure 6, where a CH2Cl2/CH3OH = 70/30 solvent mixture is used.

Sophorolipid concentration in the solvent is 50 mg/mL; to this, various amounts of silica gel have been added. Solution is stirred for 10 minutes, left at rest and analyzed both using 1 H NMR and HPLC. NMR spectra (Figure 6) show that both the intensity of the resonance at Interestingly, one can clearly see that this procedure has no effect at all on filtering out the hydrophobic species, whose NMR signals at 0.92 ppm and 6.30-5.70 ppm and HPLC retention times between 34 and 40 min, refer to. On the contrary, the elimination of these compounds can be efficiently done using a bulk amounts of C18-modified silica powder.

Sample preparation is similar for the previous experiment except for the solvent mixture. In These tests indicate the good efficiency of the double silica-based treatment. Even if the advantage of changing solvents and substrates is to always keep sophorolipids in the liquid phase so to limit their adsorption/desorption, we still do experience a loss of SL-COOH in the process, as testified by the HPLC data (Figure 6b and Figure 7b), where the SL signal in the 17.5-21.5 min region is also affected by the filtering process.

To quantify this procedure in terms of both recovered amount of sophorolipids and residual impurities, we have treated the same initial sophorolipid batch in a consecutive way starting from the hydrolyzed form of the as-provided compound. Since this is dissolved in an aqueous medium and to limit the number of overall steps, we prepared a 100 mg/mL solution from the hydrolyzed batch itself, where the solvent is constituted by a water/acetonitrile 90/10 mixture and used an overall volume of 100 mL, to which were added 5 g of C18-SiO2. This was stirred at room temperature and after about 30 min the silica phase was removed by filtration.

The aqueous liquid phase was removed under reduced pressure, being extremely careful to remove water. The solute was redissolved in about 400 mL of CH2Cl2/CH3OH -70/30, to which about 100 g of silica gel was added and filtered after the adsorption of the hydrophilic impurities. This is quite clear as for the most polluted solutions, which colours are brownish, one can clearly see an increase of transparency upon filtration, the darkest compounds being adsorbed on the solid phase. We found this part being the most critical step for this process, given the large amount of silica employed. It is advisable to rinse the collected solid phase at least once or twice with an equivalent amount of the solvent mixture. Finally, all solvent fractions are grouped together and evaporated under reduced pressure.

This method has the advantage to provide an easy and fast way to recover an acidic sophorolipid compound but two main drawbacks must be outlined: the lower yield (about 50/60 %) and higher impurity with respect to the pentanol routes. The first point depends on the large amounts of silica gel employed, which retain part of the sophorolipid in solution.

The second point depends on the efficiency in terms of impurity adsorption for large amount of matter. Both of them could be improved by a better selection of the solvent mixtures, for instance. The typical spectra obtained for the compound before and after treatment are shown in Figure 8. One can observe the drastic reduction in both the peak related to the hydrophilic and hydrophobic impurities, even if both are still observed and whose amount is estimated to be below 10 mol%. In the following table we highlight the advantages and disadvantages for each method presented above. All in all, the pentanol route is quite robust and it provides a rather pure, readily desalted, product. Nevertheless, we do not recommend using the pentanol extraction under reduced pressure in the absence of a well-equipped vacuum extraction system. The most practical and best way in terms of purity is undoubtedly the precipitation in a cold pentanol/alcane mixture, but this is a long way (several days) if one wants to obtain the highest yield possible. Finally, the double silica filtration is an interesting, very fast, method for immediate use, even if lower yields and residual impurities are higher with respect to the pentanol routes. For possible scale-up, further work must be done to develop these processes and reduce all inconvenients, thus maximizing yields and minimizing impurities. For instance, in the double silica extraction, one could try to adjust the polarity of the solvents and use optimized amounts of both silica gel and inversed silica phase. In the modified pentanol route, on the contrary, one could use a more efficient solvent combined to pentanol to reduce the precipitation times.

Conclusion

This work has shown the possibilities offered by two alternative routes in the extraction process of acidic sophorolipids in a water solution after the base-catalyzed hydrolysis of the lactonic form in a commercial formulation. We initially detailed the classical pentanol extraction route, which consists of a hexane washing (to remove residual fatty acids) previous to solvent/solvent extraction at room temperature followed by the pentanol vacuum removal. This process, almost quantitative and providing a good quality product, has several limitations and drawbacks to treat large amounts of matter (tens of grams): 1) manipulation of large amounts of solvents (pentanol, hexane); 2) need for a specific vacuum extraction setup for rapid and effective pentanol removal. In particular, the second point is the most limiting one to operate the extraction correctly and in the absence of a specific apparatus with a closed refrigerating system, the extraction can be extremely long and the product has good chances to be polluted with residual pentanol. Furthermore, esterification between pentanol and the COOH group of SL is possible if one rises the heating temperature to 60°C -70°C for several hours. To overcome these drawbacks, we have proposed two alternative routes. In the first one, one can perform a cold extraction by adding an alcane in a 1:1 v:v proportion to the SLcontaining pentanol solution. This method is very effective in terms of final purity as only acidic sophorolipids precipitate while all other hydrophobic impurities (e.g., fatty acids) remain in solution. This way should also avoid the need to pre-treat the raw sophorolipid mixture with hexane washing. Several problems are inherent to this approach: its length (up to several days for a quantitative extraction) and the presence of residual water bound to sophorolipids, which should be carefully removed before product filtration. Further development in terms of best co-solvent and careful bulk water removal could greatly help to overcome these points. In summary, this method is by far more practical and safer than the first one and it can be employed in any, non organic-chemistry specialized, environment.

Nevertheless, in the quest for a fast production, one can try the third approach that we propose, consisting in a double solid/liquid extraction on a C18-modified silica powder and normal silica gel. This method, adapted from column chromatography, can be developed here because of the little heterogeneity of the raw product after alkaline hydrolysis: hydrophilic compounds (e.g., acetic acid, minor impurities), hydrophobic molecules (fatty acids, if not removed in advance) and acidic sophorolipids of average hydrophilicity. This is a fast track approach with several drawbacks, that can probably be overcome with further process improvement. Due to undesired solid-phase adsorption of sophorolipids, this is by far the least quantitative method described so far and in spite of the designed choice of solid/liquid duo that allows to always keep sophorolipids in the liquid phase. So far, this method is also the least effective in terms of impurities removal (especially the most hydrophilic ones), even if the overall amount can be as low as 6%. On the contrary, this method employs volatile solvents easy to remove with a standard rotavapor apparatus. Finally, we also identify several points that one should consider for a successful extraction, such as the limitation in the hydrolysis temperature, which if too high it promotes the formation of hydrophilic impurities, the use of Soxhlet extraction to remove fatty acids at the end rather than at the beginning of the process, thus limiting the amount of employed hexane, the careful removal of bulk water from the pentanol mixture, use of low temperatures when working with pentanol to avoid esterification, etc… In summary, we have shown multiple ways that one can follow to recover acidic sophorolipids, proposing the positive and negative aspects for each one and indicating how these methods could be developed further to overcome the drawback and, eventually, to end up with an up-scalable, highly performing, process.

Figure 1 -

 1 Figure 1 -Alkaline hydrolysis reaction transforming the acetylated lactonic sophorolipid to the deacetylated acidic form of sophorolipids.

Figure 2 - 1 H

 21 Figure 2 -1 H NMR spectrum of the raw compound after alcaline hydrolysis. The 9-5 ppm region is specifically highlighted to show the effect of hydrolysis on the purity compound mixture before and after hydrolysis. Full black circle indicate the solvent resonances.

Figure 3 - 1 H

 31 Figure 3 -1 H NMR spectra showing the effects of the pentanol extraction process on the hydrolyzed sophorolipid. The top spectrum shows the effect of Soxhlet extraction on the pentanol-washed compound. The arrows indicate the impurities related to acetic acid and fatty acids.

Figure 4 - 1 H

 41 Figure 4 -1 H NMR spectrum of the purified sophorolipid using the modified pentanol extraction method. The 9-5 ppm and 0-2.5 ppm regions are highlighted to show the loss of all impurities. The typical spectrum before purification is given on Figure 3 (bottom spectrum).

Figure 5

 5 Figure 5 shows the typical HPLC chromatogram of hydrolized SL before silica filtration, showing three distinct regions corresponding to hydrophilic species (1.7 min), sophorolipids (17.5-21.5 min) and fatty acids (34-40 min).[START_REF] Van Bogaert | Production of new-tonature sophorolipids by cultivating the yeast Candida bombicola on unconventional hydrophobic substrates[END_REF] This specific mixture containing three families of compounds with large differences in polarity is a good candidate for selective bulk silica adsorption.

Figure 5 -

 5 Figure 5 -Typical chromatogram of hydrolyzed SL. Peak attribution is done in the text.

1.96 ppm and 8 .Figure 6 -

 86 Figure 6 -a) 1H NMR and b) HPLC data showing the effect of increasing amounts of silica gel (given on the figure) on the adsorption of the hydrophilic impurities from the hydrolyzed form of sophorolipids. Sophorolipid concentration is 50 mg/mL, while the volume of the solution is 10 mL.

Figure 7 -a) 1 H

 71 Figure 7 -a) 1 H NMR and b) HPLC data showing the effect of increasing amounts of C18-modified silica (given on the figure) on the adsorption of the hydrophilic impurities from the hydrolyzed form of sophorolipids. Sophorolipid concentration is about 90 mg/mL, while the volume of the solution is 10 mL.

Figure 8 - 1 H

 81 Figure 8 -1 H NMR spectra of the SL-COOH (top spectrum) consecutively purified with silica gel and C18-modified silica (bottom spectrum). Spectra are normalized.

  , thus allows solvent economy and efficient fatty acid removal, as well as all other organic impurities. An additional advantage of Soxhlet extraction concerns the removal of residual pentanol, which can contaminate the system. This is shown in Figure3. The

	spectrum before pentanol extraction refers to a hydrolized acidic SL product which was
	actually initially washed with hexane. Three types of impurities can be found: residual fatty
	acids (δ= 0.92), acetic acid (δ= 1.96) and the above mentioned spurious impurities between 9
	and 5 ppm. Clearly, a hexane washing prior to alkaline hydrolysis is not enough to remove

portions of pentanol in a separatory funnel. Once all fractions are collected, one should take care of removing exceeding water using a pipette or magnesium sulfate. As for pentanol extraction, one can set a 47°C water bath and a cooling system at 6°C. Vacuum pressure should be slowly lowered from 80 mbar to 8-10 mbar to initially remove residual water, then acetic acid and finally pentanol over 2-4 hours. The residual product has the aspect of a viscous brown/dark brown (large impurities) oily phase or a pale yellow/grey dried powder (low impurities). The main impurities are residual fatty acids, structural water and eventual oxidized compounds, mentioned previously. The eventual brownish colour depends on watersoluble species; so, if water is carefully removed before vacuum extraction, these should not be recovered in the end. On the contrary, fatty acids can only be removed by alcane (generally, hexane) washing. This procedure can take place either before, as previously suggested, alkaline hydrolysis on the raw product or after pentanol extraction, once a powder is obtained, using Soxhlet extraction, as we actually suggest here. Soxhlet extraction works in a closed circuitresidual hydrophobic compounds. Upon pentanol extraction, only the resonances at 8.51 ppm and 1.96 ppm are filtered out, showing that the impurity at 8.51 ppm is also related to a hydrophilic compound. At this stage, the peak at δ= 0.92 is still present. One should note that residual pentanol can also be observed at this resonance. Finally, Soxhlet extraction removes

Acknowledgements

The research leading to these results has received funding from the European Community's

Seventh

Framework Programme (FP7/2007-2013) under Grant Agreement n° Biosurfing/289219 and the French state funds managed by the ANR within the Investissement d'Avenir programme under the reference ANR-11-IDEX-0004-02 (Labex MATISSE). We kindly acknowledge Dr. Cyril Martini and Dr. François Ribot (Laboratoire de Chimie de la Matière Condensée de Paris, Collège d France/UPMC/CNRS, Paris, France) and Dr. Philippe Simon (Chimie des Processus Biologiques, Collège de France, Paris, France) for helpful discussions. Stijn Verweire (InBio, Ghent University, Ghent, Belgium) is acknowledged for technical assistance on HPLC.