Eric Bonnet
email: bonnet@ganil.fr

R Chbihi

J D Dayras

E Frankland

F Galichet

Gagnon-Moisan

B Borderie

N Le Neindre

M F Rivet

R Bougault

A Chbihi

R Dayras

J D Frankland

E Galichet

F Gagnon-Moisan

D Guinet

P Lautesse

J Lukasik

D Mercier

M Pârlog

E Rosato

R Roy

C Sfienti

M Vigilante

J P Wieleczko

B Zwieglinski

Aladin Indra

Collaborations

Fragment properties of fragmenting heavy nuclei produced in central and semi-peripheral collisions

Keywords: Intermediate energy heavy-ion reactions, central and semi-peripheral collisions, multifragmentation, fragment partitions, collective energy PACS: 25.70.-z, 25.70.Pq, 24.10.-i

Fragment properties of fragmenting heavy nuclei produced in central and semi-peripheral collisions.

Introduction

Heavy-ion collisions at intermediate energies offer various possibilities to produce hot nuclei which undergo a break-up into smaller pieces, which is called multifragmentation. This phenomenon is expected to bring information and constraints on the phase diagram of nuclear matter through measured fragment properties [START_REF]Dynamics and Thermodynamics with nuclear degrees of freedom[END_REF]. In particular by comparing in detail the properties of fragments emitted by hot nuclei formed in central and semi-peripheral collisions (i.e. with different dynamical conditions for their formation) one can expect to reveal features which characterize where those hot nuclei break in the phase diagram [2,3,4,5]. It is the final goal of this article; the first question being: do we see any different features in fragment properties ? For reactions at small impact parameters in the Fermi energy domain, one can select the collisions where the two nuclei merge into a quasi-fused system (QF) after full stopping [6]. At larger impact parameters and for higher incident energies, only a fraction of each nucleus interacts producing in the outgoing channel quasitarget (QT) and quasi-projectile (QP) sources. For both cases large energy dissipations occur but some constraints applied to nuclei are different with a compression-expansion cycle for central collisions and a friction-abrasion process which can produce dynamical emissions in the contact region (mid rapidity emissions) for peripheral collisions [7,8,9,10]. Therefore, to make a meaningful comparison of fragment properties which can be related to the phase diagram, hot nuclei showing to a certain extent statistical emission features must be selected. It is done for central collisions by selecting compact events in velocity space (flow angle selection). For peripheral collisions a selection method is proposed and applied to quasi-projectiles. Hot nuclei with A around 150-200 were produced at GANIL in 129 Xe+ nat Sn central collisions at five bombarding energies in the range 25-50 A MeV and at GSI in semiperipheral Au+Au collisions at 80 A MeV incident energy.

The paper is organized as follows. In section 2 we briefly present the experimental set-up and conditions. We recall the criteria allowing to select experimental events corresponding to fused systems. Then the method employed to extract the excited QPs is described. Global properties of the different selected sources are finally discussed using the excitation energy as control parameter. A comparison of fragment charge partitions associated with the different sources is shown in section 3. Section 4 presents a detailed study of kinetic fragment properties produced in central and peripheral collisions. Radial collective energies are compared by means of fragment relative velocities. Section 5 is devoted to discussion and conclusions.

Experimental selection

Experimental procedure

Beams of 129 Xe, accelerated at five incident energies: 25, 32, 39, 45 and 50 A MeV by the GANIL facility, bombarded a thin target of natural tin (350 µg/cm 2 ). Hot nuclei with A around 150-200 were produced in central collisions. Nuclei of similar masses were obtained at GSI in semi-peripheral Au+Au collisions at 80 A MeV incident energy. This energy appears to be a good compromise between high energy detection limit for H isotopes introduced by the detectors, a grazing angle within the measured angular range and a good characterization for QPs which are well separated from QTs in velocity space. For this experiment the 197 Au beam was impinging on a 2 mg/cm 2 197 Au thick target.

The data were collected with the 4π multidetector INDRA which is described in detail in [11,12]. INDRA consists of 336 telescopes covering about 90% of the 4π solid angle. The configuration used at GSI differed in the composition of the first ring (2 • -3 • ): the phoswich scintillators were replaced by 12 telescopes, each composed of a 300 µm Si detector followed by a 14 cm long CsI(Tl) scintillator. Accurate fragment identification and energy calibration were achieved with INDRA; the energy of the detected products is obtained with an accuracy of 4%. Further details can be found in [13,14,15,16]. A two step procedure was used to select QF sources. First of all "complete experimental events" were selected by requiring that at least 80% of the total charge of the system was measured; the corresponding event distributions are displayed in the left upper part of fig. 1. That minimum percentage of the total charge induces a lower limit on the total pseudo linear momentum (P tot /P proj -see eq. 1) around 70% of the entrance channel value (see right upper part of fig. 1) and selects central collisions only.

P tot = | Mtot i=1 β i γ i Z i | (1)
M tot is the total measured charged product multiplicity in the event. Then, compact single sources in velocity space were selected by imposing the constraint of flow angle ≥ 60 • [17,18,19]. To calculate θ f low the kinetic energy tensor was built with fragments (Z≥5) and starting from fragment multiplicity M f rag ≥ 1. Indeed it was shown in previous studies that while events present the topology of emission from two sources at small flow angles, they evolve towards a single-source configuration above 60 • (see figure 9 in ref [18] and figure 1 in ref [6]). The rather flat cos θ f low distribution (variation of about 35% between θ f low =60 • and 90 • ) observed for each incident energy in the selected range (see left bottom part of fig. 1) indicates a fragment emission which can be associated with a strong degree of equilibration. Note that such QF sources are produced at high excitation energy (above 3.0-3.5 AMeV) and their deexcitation through fission is suppressed [4].

The charged particles (Z = 1-4) to be associated with the single sources were determined from angular and energetic criteria. Figure 1 (right bottom part) shows the centre of mass average energy of particles in coincidence with the single sources as a function of their centre of mass emission angles. Large energies are measured forward-backward indicating preequilibrium emissions whereas rather constant average values are observed over a 2π solid angle (angular range 60 • -120 • ). So, assuming isotropic emission, twice the charged particles on that angular range were associated with the sources event by event [19].

Finally the calorimetric method [START_REF]Dynamics and Thermodynamics with nuclear degrees of freedom[END_REF] was used to evaluate the source excitation energy. The following hypotheses have been made: a level density parameter equal to A/10, the average kinetic energy of neutrons equal to their emitting source temperature and the Evaporation Attractor Line formula (A=Z(2.072+2.32×10 -3 Z)) [21] applied for fragment mass determinations. EAL is especially well adapted when heavy fragments (Z > 20) result from the deexcitation of the neutron deficient sources produced. More details on calorimetry are presented in the appendix. For all incident energies Gaussian distributions are obtained for the excitation energies of the source.

Selection of quasi-projectile sources

For peripheral collisions at 80 A MeV incident energy, the relativistic fragment (Z ≥ 5) kinetic energy tensor (events with at least one fragment, M f rag ≥ 1eq. 2) was computed in the centre of mass of the reaction to define the ellipsoid frame of each event, comprising M tot f rag fragments.

T uv = M tot f rag i=1   u,v=1,3 P (u) i P (v) i (1 + γ i )m i   (2)
QP sources were selected through a completeness criterion on the total detected charge (eq. 3) and pseudo-momentum (eq. 4) on the forward part of each event -defined as all reaction products with a positive rapidity, M f wd , in the ellipsoid frame.

Z f wd = M f wd i=1 Z i Z f wd /Z proj ∈ [0.80, 1.10] (3)
P f wd = | M f wd i=1 β i γ i Z i | P f wd /P proj ∈ [0.60, 1.10] (4)
Z proj and P proj refer to projectile charge and momentum in the laboratory.

These preliminary selections lead to an ensemble of sub-events called complete QP events. Then the velocity of each reconstructed QP source is determined using fragments only.

To perform a meaningful comparison between the fragment properties of single sources defined in the previous section and QP sources, we must minimize the contribution of dynamical emissions for fragments. Indeed, in this energy regime and for peripheral and mid-peripheral collisions it is well known that a large part of the collisions end-up in two remnants coming from projectile and target and some particles and fragments with velocities intermediate between those of the remnants. They are called mid-rapidity products. They may have several origins: direct preequilibrium emission from the overlap region between the incident partners or a neck of matter between them which may finally separate from QP or QT, or from both, as well as emissions from QP/QT partially deformed and locally highly excited [9,10,22,23,24]. On the other hand, with higher dissipation associated with the decrease of the impact parameter, the velocities of the outgoing partners (QPs and QTs) are much closer to the centre of mass velocity, which makes difficult the definition of QP and QT sources. Therefore a compactness criterion in velocity space was defined and applied to select compact QP events among complete QP events which comprise at least two fragments. With this aim, we investigate further the global energetic properties of fragments. We define the VarDyn observable as follows:

VarDyn = β QP β rel with β QP = | p (i) |/ E (i)
and p (i) , β (i) and E (i) , which represents the total energy of fragment (i), are defined in the reaction centre of mass. M f rag is the number of fragments among the M f wd products. β QP is related to the dissipated energy whereas β rel gives a hint of the dispersion of fragments in velocity space. Compact QP sources should have small values of β rel and thus large values of VarDyn.

β rel = 2 M f rag (M f rag -1) i<j | β (ij) | β (ij) = β (i) -β (j) . (5)
With this ratio we compare the average position and distance between fragments with the reconstructed position of the QP in the velocity space to evidence compact configurations corresponding to events with fragments localized around the projectile velocity (larger values of VarDyn). Let us focus first on events with M f rag = 2 ( 27% of complete events). The distribution of VarDyn is plotted in the left upper part of figure 2: it exhibits a peak for VarDyn=2.4 and a pronounced shoulder for values around 1-1.5. It has long been known that excited Au nuclei will undergo symmetric fission. Fis-sion of Au QPs was characterized by using the criterion Z 1 × Z 2 ≥ 900 [25] (6.3% of complete events). The corresponding distribution of VarDyn (open circles) has a Gaussian shape centered around VarDyn=2.5. Removing the fission contribution, the complementary part still presents a Gaussian shape centered around VarDyn=1.8 with a shoulder for VarDyn<1.5. The right upper part of figure 2 shows the final decomposition with two components (after removing fission): compact QP events (black triangles) and QP events with mid-rapidity fragments (stars), associated with small values of VarDyn. That figure shows that VarDyn>1.5 is a good criterion to select compact QP events with two fragments.

For events with M f rag > 2, the direct dependence between impact parameter, dissipation, fragment production and velocity of the QPs makes the separation between the two classes of events impossible just starting from the observed VarDyn distributions (see left lower part of figure 2). To test the lower limit previously deduced for M f rag =2, we introduce the minimum, β

(ij)
min , and maximum, β (ij) max , relative velocity between pairs of fragments calculated for each event. The right lower plot of figure 2 shows the evolution of their mean values < β

(ij) min > and < β (ij)
max > and the evolution of < β QP > with VarDyn. < β QP > increases rapidly up to VarDyn around 1.5 and then evolves much more gently, as it is limited by the projectile velocity β proj = 0.2. For VarDyn values above 1.5, < β (ij) max > becomes lower than < β QP >, which indicates source velocities sufficiently different from the reaction centre of mass velocity. On the other hand the difference between < β (ij) min > and < β (ij) max > remains small above VarDyn around 1.5 whereas it increases for lower VarDyn values. That ensemble of observations confirms that VarDyn=1.5 is a good minimum condition to select compact QP events irrespective of the fragment multiplicity. Let us come now to a comparison between selected and rejected events. Figure 3 shows, for M f rag ≥ 2 and Z 1 × Z 2 < 900 events, the fragment angular distributions in the QP frame. One notes the much flatter distribution associated with compact events, which indicates a fragment emission closer to isotropy. Information on fragment size distributions is also displayed in the right part of the figure. The rejected events comprise more light fragments (Z < 20) than the compact ones.

To summarize, the second and third rows of Table 1 indicate the proportion of the different events at this stage. Compact events, without fission events, correspond to 61.2% of complete events. 1 Summary of the percentage of different QP events. The second, third and fourth columns refer to events with fragment multiplicities M f rag = 1, M f rag ≥ 2 and Z 1 × Z 2 < 900 and to fission events (Z 1 × Z 2 >= 900). At each selection step the last column gives the percentages of kept events relative to the number of complete events.

M f rag = 1 M f rag ≥ 2 Fission
The light charged particles (Z < 5) with positive rapidity in the ellipsoid frame have different origins: mid-rapidity dynamical emissions, pre-equilibrium emissions due to the limited overlap between projectile and target and statistical emissions from QPs and QTs. The evolution of their mean kinetic energy as a function of their emission angle in the QP frame is illustrated in fig. 4 (left panel). The right panel of the figure shows, averaged over all charged particles, the same trend for different excitation energies of QP sources (for the calorimetry procedure -see just after). They clearly show a flatter behaviour for forward angles in the QP source frame. To estimate the contribution of the QP statistical emission we have adopted the method already used in [25] which consists in keeping only the particles emitted forward in the QP frame, and doubling their contribution (charge, mass and energy) assuming a forwardbackward symmetric emission.

Note that a large part of the emitted particles (a factor of 3 in multiplicity) are localized in the backward part of the sources. As a consequence, starting from a QP detected charge Z f wd constrained by the completeness criterion, this asymmetry backward/forward leads to large distributions for the sizes of the reconstructed QP sources. To overcome this drawback we have finally selected QP source sizes, Z s , which correspond to Z s /Z f wd ratios comprised between 0.9 and 1.0; it is worth noting that the total (rejected) mid-rapidity charge is on average equal to four for the selected events. This last size selection withholds 31.7% of complete QP events (bottom row of table 1). The following step consists in the evaluation of the excitation energy of the sources. An event by event calorimetry follows the procedure used above for QF sources. The hypotheses are identical except for charged particle contribution (doubling the forward part) and for fragment masses for which we use the formula (A=Z(2.045+3.57×10 -3 Z)) [21], better adapted for excited nuclei close to the beta stability valley. Note that, compared to the EAL formula, differences for masses appear only for Z greater than 40. The derived excitation energy range for sources was divided in bins of 0.5 AMeV width for comparisons with QF sources.

In the following, we will only use the excitation energy per nucleon as sorting parameter to compare fragment properties from both types of sources.

Whereas the relative values of the excitation energies for each type of source should be reliable, one can wonder about the comparison between the excitation energy scales between QF and QP sources. In the next subsection we will present arguments indicating that the two scales are in agreement within 10%.

Global properties of selected QF and QP sources

The global properties of selected sources are displayed in figure 5 energy range for the two types of sources is [5,10] AMeV and their variations in charge on that excitation energy range are around ten units of charge; the ratio between the two types of sources is about 1.2-1.3, very close to the Z ratio between Xe+Sn (104) and Au (79). The average proportion of charges bound in fragments as a function of the excitation energy per nucleon is shown in the upper right hand side of figure 5. The quantitative evolution of that observable, normalized to the sizes of the sources, < Z (N ) f rag >, is the same for both QF and QP: a linear decrease of charge bound in fragments when the excitation energy per nucleon increases. This behaviour shows that, for a given excitation energy, the sharing among particles and fragments is the same for central and peripheral collisions, which confirms that multifragmentation is mainly driven by the energy deposited into the sources [START_REF]Dynamics and Thermodynamics with nuclear degrees of freedom[END_REF]. Thus, the knowledge, for a source, of the proportion of charges bound in fragments (or the complementary knowledge of the total Z found in particles) provides a good estimate of its excitation energy. Note that the proportion varies from about 0.8 to 0.5 in the common excitation energy range. Mean fragment multiplicities normalized to the size of the source are shown in the lower left part of figure 5. Much higher multiplicities are obtained for QF sources than for QP ones but they both present a maximum in the same excitation energy range (8-10 AMeV). On the other hand the variances of the normalized multiplicity distributions, which are not presented here, are very similar. Finally in the right bottom panel of the figure are displayed the normalized multiplicities of particles < M (N ) part >, which appear completely similar. As shown previously the percentage of charge bound in fragments is the same for QF and QP sources with the same excitation energy. How is this bound charge shared among fragments? A first answer comes from the properties of the charge of the biggest fragment (Z 1 ), because several studies point out its specific behaviour [27,28,25] and because it is a good candidate for the order parameter of a phase transition in hot nuclei [29,30,31,32,33]. In figure 6 (upper left part), the evolutions, with the excitation energy, of its mean value and of the associated fluctuations are plotted. The mean value appears as mainly governed by excitation energy and is largely independent of system sizes and of production modes. This effect was already observed in [34,35] for two QF sources with charges in the ratio 1.5; its occurrence when comparing QF and QP sources would indicate that their excitation energy scales do agree, within 10%. The fluctuations, on the contrary, exhibit sizeable differences. In the common energy range, the standard deviations of Z 1 decrease when the excitation energy increases but they are larger for QP sources. In this latter case they show a maximum value around 4.5 AMeV which is in good agreement with systematics reported for QP sources in [START_REF]Dynamics and Thermodynamics with nuclear degrees of freedom[END_REF]37] and seems to correspond to the centre of the coexitence region of phase transition [35,38]. Normalized variances σ 2 Z 1 / < Z 1 > are also reported in figure 6 (upper right part); it was proposed in [39] that the maximum of that observable indicates the critical region of the phase transition. We note that, as compared to standard deviations, normalized variances exhibit a maximum for both QP and QF sources in the excitation energy region 6-7 AMeV, which seems to correspond to the gas-like border of the transition and consequently possibly to the critical region [40,38]. A surprising result coming from that comparison between QP and QF sources is the difference between the behaviours of the charge bound in fragments, Z f rag -fig 5 and of Z 1 at a given excitation energy. In the first case one observes a scaling with the size of the sources whereas the second exhibits an independent mean value. How is Z f rag partitioned into fragments? An overview of all information related to fragment charge partition can be obtained with a new generalized charge asymmetry variable calculated event by event. For two fragment events (M f rag =2), the usual proposed observable is [41]. To take into account distributions of fragment multiplicities which differ for the two sources, the generalized asymmetry (A Z ) reads:

a 12 = Z 1 -Z 2 Z 1 +Z 2
A Z = 1 M f rag -1 σ Z < Z > (6)
This observable evolves from 1 for asymmetric partitions to 0 for equal size fragment partitions (symmetric). For the one fragment events, mainly present for QP sources, we compute the A Z observable by taking the first particle in size hierarchy included in calorimetry. In the left bottom part of figure 6, the mean evolution with excitation energy of the generalized asymmetry is shown. Differences are observed which well illustrate how different are the repartitions of Z f rag between fragments for QF and QP multifragmenting sources. QP partitions are more asymmetric in the entire common excitation energy range. To be sure that this observation does not simply reflect the peculiar behaviour of the biggest fragment, the generalized asymmetry is re-calculated for partitions M f rag > 1, and noted A Z \{Z 1 }, by removing Z 1 from partitions (bottom right panel of figure 6). The difference between the asymmetry values for the two source types persists. A possible explanation of those experimental results can be found by looking at kinematic properties of fragments [2].

4 Mean fragment relative velocities and radial collective energy.

Radial expansion energy following a compression phase is predicted to be present in semi-classical simulations of central collisions in the Fermi energy domain [42,43,44,45]. In experiments it was obtained, in most of the cases, from comparisons of kinetic properties of fragments with statistical models. For QF sources the centre of mass of the reaction is used as the reference frame to derive kinetic properties of fragments. For QP sources produced in peripheral and semi-peripheral collisions, the definition of the QP frame is correlated to the fragment kinetic energies. Conversely the mean relative velocity between fragments (β rel -eq.5 for M f rag >1), is independent of the reference frame, and can provide information about possible radial collective energy.

In the left part of figure 7, the mean evolution of this observable with the excitation energy is plotted for the two types of sources. For QF sources β rel exhibits a linear increase with excitation energy. For QP sources β rel remains almost constant along the E * range. Fragment velocities are the results of the composition of at most three components: a thermal kinetic part, determined at freeze-out, mainly related to the energy deposit in sources; a Coulomb contribution dependent on the source sizes and an eventual radial extra energy. The effect of the Coulomb contribution can be removed by using a simple normalization (eq.7) which takes into account, event by event, the Coulomb influence in the velocity space of the mean fragment charge (< Z >) on the complement of the source charge (Z s -< Z >).

β (N ) rel = β rel < Z > (Z s -< Z >) (7)
The mean behaviour of the β

(N )
rel observable so defined is shown in the right part of figure 7. At an excitation energy of about 5 AMeV, β

(N )
rel values corresponding to QF and QP sources are similar. Above that excitation energy, the values for QF sources exhibit a strong linear increase. For QP sources β (N ) rel slightly increases up to 9 AMeV excitation energy and saturates above. That saturation can be attributed to the compact source selection. Indeed if one performs the same analysis of QP's (i.e. with the same limit on VarDyn) from Au+Au collisions at 100 AMeV, β

That fast divergence between the values of β (N )

rel for the two types of sources signals the well known onset of radial collective expansion for central collisions. Indeed in [40], estimates of radial collective energy (from 0.5 to 2.2 AMeV) for QF sources produced by Xe+Sn collisions are reported for four incident energies: 32, 39, 45 and 50 AMeV. Those estimates were extracted from comparisons with the statistical model SMM assuming a self similar expansion energy. The four estimated values of the radial collective energy (E R ) for the QF sources can be used to calibrate the β (N ) rel observable. The correspondence between β

(N )

rel and E R is deduced from a second order polynomial adjustment (dotted line in figure 8). From this function, we extract firstly a radial energy estimate for QF sources formed at 25 AMeV incident energy: 0.1±0.1 AMeV; the error bar being determined from the grey zone in figure 8. In the same way we also deduce two sets (80 an 100 AMeV collisions) of E R values with their error bars for QP sources using the appropriate mean values of β (N ) rel . For that calibration we take intervals of E * centered around the mean value, with a width equal to ±3σ, of the corresponding QF source excitation energy distributions. All the quantitative information concerning the evolution of radial energy with excitation energy for both types of sources is presented in figure 9. We have also added the E R values published by the ISIS collaboration [46] corresponding to the π -+Au reactions which provide sources equivalent to the QP ones in terms of excitation energy range and size. The observed evolution of E R for such sources is almost the same as for QP sources. For hadron induced reactions the thermal pressure is the only origin of radial expansion, which indicates that it is the same for QP sources. To be fully convincing, an estimate of the part of the radial collective energy due to thermal pressure calculated with the EES model [47] for an excited nucleus identical to QF sources produced at 50 A MeV incident energy is also reported (open square) in the figure [START_REF]Proc. Int. Symposium on large-scale collective motions of atomic nuclei[END_REF]. To conclude on radial collective energy we have shown that it is essentially produced by thermal pressure in semi-peripheral heavy-ion collisions as it is in hadron induced reactions. For QF sources produced in central heavy-ion collisions the contribution from the compression-expansion cycle becomes more and more important as the incident energy increases. rel and E R .

> A MeV

Discussion and conclusions

We have compared the properties of QF and QP multifragmenting sources in the excitation energy range 5-10 AMeV. They present similarities and differences. Similarities concern: the division of charge among particles and fragments, the particle multiplicities (both normalized to the source size), the average charge of the heaviest fragment of the partitions and the fluctuations of the normalized fragment multiplicities, all the mentioned as a function of excitation energy (including radial collective energy). Differences are first relative to the fluctuations in charge/size of the heaviest fragment, that are larger for QP than for QF sources. That observation was very recently mentioned [37] for normalized quantities σ(Z 1 /Z s ) and a possible explanation, by comparison E* A MeV with statistical model (SMM) calculations, was related to different freeze-out volumes. Note that a detailed study using the recently developed theory of universal ∆ scaling laws [START_REF] Botet | [END_REF]50,25] should also be of the prime interest to make progress on the understanding of the different fluctuations observed.

Secondly the asymmetry of the fragment partitions, A Z \{Z 1 }, is also larger for QP sources; that difference was observed at 5 AMeV excitation energy and above, which corresponds to the onset of collective energy. Thus, the lower asymmetry for QF sources seems to be related to the presence of radial collective expansion coming from the compression-expansion cycle for central collisions. Finally normalized fragment multiplicities for QF sources are also significantly larger above 5 AMeV. Clearly the degree of fragmentation of the system increases with the radial collective energy and partitions are also affected. What is the influence of the radial collective energy on the region where QF and QT sources break in the phase diagram (plane freeze-out volumeexcitation energy)? Estimates of freeze-out volumes for QF sources produced in Xe+Sn collisions for incident energies between 32 and 50 AMeV were very recently obtained (details can be found in [51]). They evolve from 3.9 to 5.7 V /V 0 , where V 0 would correspond to the volume of the source at normal density. These results were extracted from simulations using experimental data and agree rather well with those deduced from the statistical model MMM [52,53]. In a first attempt to calibrate the freeze-out volumes for other sources, we use the charge of the heaviest fragment < Z (N ) 1

> o < V/V 1 2 3
> or the fragment multiplicity < M (N )

f rag >, normalized to the size of the source, as representative of the volume or density at break-up. From the four points for QF sources and the additional constraint that Z

(N ) 1 = M f rag =1 at V /V 0 =1, we obtain two relations V /V 0 = f 1 (Z (N ) 1 ) and V /V 0 = f 2 (M (N )
f rag ), from which we calculate the volumes for QF sources at 25 AMeV and for QP sources. The results are plotted in fig. 10, with error bars coming from the difference between the two estimates using f 1 and f 2 ; note that error bars for the QP volumes are small up to 7 AMeV, and can not be estimated above, due to the fall of < M (N ) f rag > at high energy (see fig. 5).

The volumes of QP sources are smaller than those of QF sources (about 1.2 × V /V 0 for E * =10 AMeV). This supports the observation made previously starting from fluctuations of the charge of the heaviest fragment in a partition (see figures 13 and 18 in [37]). It is worth noting that a freeze-out volume significantly larger than that of a QP source (2.75±0.25V /V 0 compared to 1.9 V /V 0 ) at the same excitation energy per nucleon is found for QF sources at 25 AMeV. This could indicate that as soon as compression-expansion occurs, larger volumes are involved.

To conclude, we have compared in detail the static properties of hot fragmenting sources produced in central (QF) and semi-peripheral (QP) collisions. From kinematical properties of fragments, information on the different radial collective energies involved was deduced using mean fragment relative velocities, a comparison with hadron-nucleus results and an estimate of the part of the radial collective energy due to thermal pressure. The major results are the following. The weak radial collective energy observed for QP sources is shown to originate from thermal pressure only; it reaches about 0.7 AMeV at an excitation energy of 10 AMeV. The larger fragment multiplicities observed for QF sources and their more symmetric fragmentation must be related to the extra radial collective energy produced by the compression-expansion cycle occurring in central collisions. Such a cycle seems to lead fused systems to break at lower density.

Appendix

The values reported in this paper for the excitation energies of the hot sources sometimes differ from those published by the INDRA collaboration for the same systems. In this appendix we will examine the influence of the parameters entering the calorimetric equation on the excitation energy per nucleon of the sources.

The excitation energy, E * , of a hot source is calculated event by event with the relation

E * s = Mcp E cp + Mn E n -Q. ( 8
)
M cp , E cp and M n , E n are respectively the multiplicities and kinetic energies of charged products and neutrons belonging to the source; Q is the mass difference between the source and all final products. Energies are expressed in the source reference frame.

We consider Xe+Sn QF sources, for which the reference frame is the well defined reaction centre of mass, and discuss only the average values (we know that the procedure used for calorimetry broadens the distributions). We have firstly verified that the definition of a fragment (Z≥3 or Z≥5) has no influence (less than 0.05 AMeV) on the excitation energy of the source.

As for all results obtained with the INDRA array, neutrons are not detected. Their multiplicity is equal to the difference between the mass of the source and the sum of those of the final products. The source is assumed to have the same N/Z ratio as the initial system (projectile + target). The fragment masses are derived in the present paper from the EAL estimate. In other papers, the mass of the β-stability valley is chosen [37]. These two mass formulae differ for large fragments Z > 20. However due to rounding up compensations for the light fragment masses, it appears that, when using the β-stability mass, the neutron number is 2 units (7%) larger at 32 AMeV and 5 units (20%) larger at 50 AMeV. The Q-value also depends on the neutron multiplicity, at the level of a few percents.

The kinetic energy of neutrons is calculated by firstly assuming that the source has a temperature T given by: E * s = (A s /k)T 2 , which, introduced in the former equation gives:

E * s = Mcp E cp + M n f T -Q. (9)
If the neutrons are all emitted at freeze-out, the factor f is equal to 1.5, whereas it is equal to 1 if all neutrons are evaporated along a long chain [54]. Obviously, the final calculated excitation energy will be higher when f is larger. The level density parameter plays also a role in the determination of E * , the value of which slightly increasing with k.

Finally we give in the following table the values of the different terms of eq. 9 with the two sets of hypotheses used in the present paper (second and fourth rows) and in ref. [37] (third and fifth rows).

If we use all sets of data obtained by varying the fragment mass (EAL, βstability), the factor f (1.0 and 1.5) and the level density parameter (k=8,10), we end-up with values of the excitation energies per nucleon which differ at most by 1.2 MeV at 50 AMeV and 0.8 MeV at 32 AMeV. This helps putting systematic error bars on the excitation energy, ±6.5% at 32 AMeV and ±6% at 50 AMeV. These systematic error bars are in agreement with the estimates of [55,[START_REF] Vient | Habilitation à diriger des recherches[END_REF]. Note that the relative values of E * /A at the different bombarding energies are little modified. The events considered have a total detected charge equal to at least 80% of the system charge.

E inc Z min f rag A f k f E cp Q T M n E * 50

2. 2 Fig. 1 .

 21 Fig. 1. Results for Xe+Sn central collisions at 5 energies: 25, 32, 39, 45 and 50 AMeV. Top -Normalized detected charge (left) and pseudo linear momentum (right) distributions of complete events. Bottom left panel: normalized angular distributions of the CM flow angle, for complete events; bottom right panel: mean kinetic energy vs c.m. angle for light charged particles (Z ≤ 4) for the compact events (cosθ f low ≤ 0.5).

Fig. 2 .

 2 Fig.2. Top -Left panel: for QP complete events with M f rag = 2, distribution of VarDyn (eq. 5 -full squares) decomposed into fission events (Z 1 × Z 2 ≥ 900open circles) and the complementary part (triangles). Right panel: same, with the further decomposition into compact events (dotted line) and non-compact events (stars). Bottom -Left panel: distribution of VarDyn for events with 3 and more than 3 fragments. Right panel: for M f rag > 2 QP complete events, mean evolution with VarDyn of the minimum and maximum relative velocity between pairs of fragments in an event, and reconstructed QP velocity; error bars indicate the standard deviations of two-dimensional distributions.

Fig. 3 .

 3 Fig. 3. Fragment angular (left panel) and charge (right panel) distributions for compact (triangles) and rejected non-compact (open circles) events with M f rag ≥ 2 and fission events removed.

Fig. 4 .

 4 Fig. 4. Left: average kinetic energy vs emission angle in the QP frame of the different light charged particles associated with compact QP sources with < E * >=6.3 AMeV. Right: same but averaged over all charged particles for QP sources with < E * >=4.8 (lower curve) and 7.8 AMeV (upper curve).

Fig. 5 .

 5 Fig. 5. Full squares and open circles stand respectively for QF and QP sources. Top: average charge of the sources (left) and of the charge bound in fragments normalized to the source charge (right) vs the excitation energy per nucleon. Bottom: average values of the normalized fragment (left) and particle (right) multiplicities.

Fig. 6 .

 6 Fig. 6. Full squares and open circles stand respectively for QF and QP sources. Top left: average values and standard deviations (multiplied by 10 -smaller symbols joined by lines) of the charge of the biggest fragment vs the excitation energy per nucleon. Top right: normalized variances of the charge of the biggest fragment vs the excitation energy per nucleon. Bottom: evolution of the charge asymmetrywith (left) and without (right) the biggest fragment -as a function of the excitation energy per nucleon.

Fig. 7 .

 7 Fig. 7. Full squares and open circles stand respectively for QF and QP sources. Full circles correspond to QP sources produced in 100 AMeV collisions. Evolution of the mean relative velocity of fragments, β rel , (left panel) and of that normalized observable, β (N ) rel , (right panel) with the excitation energy per nucleon.

Fig. 8 .

 8 Fig. 8. Radial collective energy calibration. Correspondence between the normalized mean relative velocity of fragments (β (N ) rel ) and the radial collective energy E R ; full squares (Xe+Sn QF sources from 32 to 50 AMeV incident energies) with error bars are used to establish the correspondence (dotted line) between β (N )

Fig. 9 .

 9 Fig.9. Radial collective energy. See fig.7for symbols for QF and QP sources. Evolution of the radial collective energy with the excitation energy per nucleon for different sources; full triangles correspond to π -+ Au reactions[46] and the open square to an estimate of the thermal part of the radial collective energy for Xe+Sn sources produced at 50 AMeV incident energy (see text).

Fig. 10 .

 10 Fig. 10. Fragmentation position in the excitation energy-freeze-out volume plane. The four full squares are taken from [51]. The open rectangle gives the estimated position (with error bar) for QF source at 25 AMeV, and the open circles those for QP sources.

  Calorimetry results for central Xe+Sn reactions with two different sets of hypotheses. The left part lists the hypotheses, Z min f rag is the minimum fragment charge. The right part gives the results. All values are in MeV per nucleon, except T (in MeV).

		5	EAL 10 1.0	6.10	2.34 9.92 26.6 9.88
	50	3	β	8	1.5	6.24	2.46 9.38 31.9 11.04
	32	5	EAL 10 1.0	3.62	1.39 7.73	26	6.01
	32	3	β	8	1.5	3.72	1.40 7.27 27.9 6.64
	Table 2


Susana Quijano-Roy

Jana Haberlova

Claudia Castiglioni

John Vissing

Francina Munell

Francois Rivier

Tanya Stojkovic

Edoardo Malfatti

Marta Gómez

García De

La Banda

Giorgio Tasca

Diagnostic interest of whole-body MRI in early-and late-onset LAMA2 muscular dystrophies: a large international cohort

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


Martin Obligado
email: martin.obligado@univ-grenoble-alpes.fr

C Brun

J H Silvestrini

E B C Schettini

• C Brun

Dissipation Scalings in the Turbulent Boundary Layer at Moderate Reθ

Keywords: Turbulent boundary layer, Turbulence energy cascade, Dissipation constant, Experimental fluid dynamics

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

Introduction

The turbulent boundary layer (TBL) is a flow that has received widespread attention from the turbulence community over the last decades, mainly due to its fundamental role in aeronautics, drag reduction and environmental applications [START_REF] Smits | High-reynolds number wall turbulence[END_REF]Marusic 2009). It is a canonical flow that combines both fundamental physics and geophysical and industrial applications. Its simple geometry makes it particularly suitable for its study via experiments in wind tunnels and computational fluid dynamics.

Despite this intense activity, many open questions remain, as evidenced by the recent discovery of the outer peak in the mean squared fluctuating streamwise velocity and the new theoretical models developed in consequence [START_REF] Vassilicos | The streamwise turbulence intensity in the intermediate layer of turbulent pipe flow[END_REF][START_REF] Hultmark | Turbulent pipe flow at extreme Reynolds numbers[END_REF]. Nedić et al. have also recently reported the presence of non-equilibrium turbulence [START_REF] Nedić | Dissipation scaling in constant-pressure turbulent boundary layers[END_REF]) in both experimental and direct numerical simulation (DNS) data.

The non-equilibrium energy cascade has been reported in several flows during the last decade: forced periodic turbulence and decaying periodic turbulence (Goto and Vassilicos 2016a, b;[START_REF] Valente | Origin of the imbalance between energy cascade and dissipation in turbulence[END_REF], various types of grid-generated turbulence [START_REF] Vassilicos | Dissipation in turbulent flows[END_REF][START_REF] Mora | Energy cascades in active-gridgenerated turbulent flows[END_REF][START_REF] Hearst | Philippe: effects of multi-scale and regular grid geometries on decaying turbulence[END_REF][START_REF] Nagata | Effects of grid geometry on non-equilibrium dissipation in grid turbulence[END_REF], turbulent free-shear flows [START_REF] Vassilicos | Dissipation in turbulent flows[END_REF][START_REF] Obligado | Nonequilibrium scalings of turbulent wakes[END_REF][START_REF] Cafiero | Non-equilibrium turbulence scalings and self-similarity in turbulent planar jets[END_REF], and other shear flows [START_REF] Nedić | Energy dissipation scaling in uniformly sheared turbulence[END_REF][START_REF] Takamure | Dissipation scaling in the transition region of turbulent mixing layer[END_REF]. The Richardson-Kolmogorov cascade predicts that the turbulent dissipation rate evolves as = C u �3 ∕L , with C being a constant, L the longitudinal integral length scale, and u ′ the root mean square (rms) of the longitudinal velocity fluctuations. On the other hand, within the non-equilibrium cascade C is not constant, but instead it goes as C ∼ Re G ∕Re L , for large enough Re (defined as Re = u � ∕ , with the Taylor microscale and the kinematic viscosity of the flow). Re G is a global Reynolds number that depends on the inlet conditions (not present in the Richardson-Kolmogorov cascade) and Re L a local, streamwise position dependent param- eter (usually defined with the integral lengthscale and a local rms velocity). It can also be shown that an equivalent expression for C is C ∼ √ Re G ∕Re [START_REF] Vassilicos | Dissipation in turbulent flows[END_REF]. The modelling of C via the Richardson-Kolmogorov closure makes part implicitly and explicitly in theoretical and numerical modelling of turbulent flows [START_REF] Vassilicos | Dissipation in turbulent flows[END_REF][START_REF] Pope | Turbulent flows[END_REF][START_REF] Lesieur | Turbulence in fluids[END_REF]. For instance, it is used to estimate the number of degrees of freedom of a turbulent flow, eddy viscosity models, large-eddy simulations, etc... its role is so preeminent that some authors have defined the scaling for as the zeroth-law of turbulence [START_REF] Lumley | Some comments on turbulence[END_REF]. In fact, it can be deduced from the Kolmogorov's 4/5 law [START_REF] Monin | Statistical fluid mechanics, volume II: mechanics of turbulence[END_REF], and therefore it is closely linked to the Richardson-Kolmogorov modelling of the direct energy cascade in homogeneous isotropic turbulence. The presence of a different scaling for therefore invalidates some of the assumptions from the standard model stated above. Nevertheless, the dissipation constant is one of the hardest turbulence parameters to measure experimentally, as it requires exceptional spatial and/or temporal (when it is obtained via the Taylor hypothesis) resolution, and still remains beyond our current experimental capabilities for very large Reynolds numbers.

While the energy cascade in the TBL is a very complex process [START_REF] Castillo | Similarity analysis for turbulent boundary layer with pressure gradient: outer flow[END_REF][START_REF] George | Zero-pressure-gradient turbulent boundary layer[END_REF]) that depends on many factors as the distance from the wall, the local value of Re , among others, recent works have focused on the outer region of spatially evolving turbulent boundary layers [START_REF] Nedić | Dissipation scaling in constant-pressure turbulent boundary layers[END_REF][START_REF] Kamruzzaman | Behaviour of the energy dissipation coefficient in a rough wall turbulent boundary layer[END_REF]. The turbulent flow within this region is developed, and therefore can be approximated, to some extent, as homogeneous and isotropic on the small scales (we will discuss this point in detail in the next section). A recent work [START_REF] Nedić | Dissipation scaling in constant-pressure turbulent boundary layers[END_REF] found the presence of anomalous dissipation scalings on the TBL for Re 𝜃 < 10 4 , with Re = U ∞ ∕ , a Reynolds number based on the freestream velocity U ∞ and the momentum thickness (defined on the next section). They studied DNS and hot-wire anemometry (HW) in the outer region of a spatially evolving turbulent boundary layer in the range 0.3 < y∕𝛿<0.7 , with y the vertical coordinate and the boundary layer thickness. While there is no consensus on the exact value of y∕ for which the outer layer begins, the range explored in this work corresponds to values generally accepted to be part of such region [START_REF] Phillips | The outer region of a turbulent boundary layer[END_REF][START_REF] Sreenivasan | The turbulent boundary layer[END_REF]. Whereas recent works confirm these findings [START_REF] Liu | Non-equilibrium turbulent phenomena in transitional flat plate boundary-layer flows[END_REF], there still remains many open questions about the structure and properties of the energy cascade that we will address in the present work.

First, we compare DNS of the turbulent boundary layer with experimental data for Re in the range 1000 < Re 𝜃 < 3500 , where C is expected not to be constant but to follow a Re -1 power law. Experiments are performed with both a single hot wire (HW) and a Cobra probe. The first device allows to resolve small-scale turbulence quantities in 1D, such as while the second can resolve the mean and rms values of the velocity vector in 3D within its limited spatial and temporal resolution, discussed in detail in the next section. To the authors best knowledge, no such complementary study involving DNS and HW and Cobra probes has been made before. We remark that the Cobra probe is not adapted for a detailed study of the TBL, as its low spatial and moderate temporal resolution are insufficient for resolving the small-scale properties of the flow. We therefore have used it only to characterise the large-scale anisotropy of the flow.

We can therefore address, using DNS data (that covers, in our case, the range Re 𝜃 < 1400 ), which is the vertical range where can be approximated using the local homogeneity and isotropy assumptions for the flow velocity field. This assumption is needed to deduce the value of C experimentally with a single hot wire. The relevance of this problem has already been raised in previous works in the TBL [START_REF] Kamruzzaman | Behaviour of the energy dissipation coefficient in a rough wall turbulent boundary layer[END_REF][START_REF] Pumir | Small-scale anisotropy in turbulent boundary layers[END_REF], as an incorrect estimation of the turbulent dissipation rate could affect the conclusions obtained in terms of the energy cascade and the quantification of C , needed for turbulence modelling. After this point is clarified, the HW and the Cobra probes can be used to study the dissipation scalings for larger values of Re . Our combined approach allows to quantify the large scale 3D properties of the flow (via the mean and rms values of the velocity vector obtained with the Cobra probe) and the small scale ones from the streamwise components (as we discuss in the next section, the HW resolved via the dissipation spectra and therefore could be used to quantify all small-scale single point parameters such as the Kolmogorov lengthscale , , Re ,...).

Furthermore, the comparison between HW, Cobra and DNS allows to address other issues on the characterisation of the energy cascade via the dissipation scalings. In fact, it has been found in other inhomogeneous/anisotropic flows (like the axisymmetric turbulent wake [START_REF] Dairay | Non-equilibrium scaling laws in axisymmetric turbulent wakes[END_REF][START_REF] Obligado | Nonequilibrium scalings of turbulent wakes[END_REF]) and the planar jet (Cafiero and Vassilicos 2019)) that a better estimation of C is obtained using the kinetic energy instead of u ′ : C = L∕K 3∕2 . Our experimental setup allows to compare both definitions. Therefore, we propose a study that can characterise the energy cascade considering the anisotropic nature of both large and small scales of the flow.

We also discuss the role of Re G in the non-equilibrium energy cascade in the TBL. This is not trivial to explore, as it is even unclear how to define an inlet Reynolds number on this flow. It is usually defined via a characteristic length and velocity at the inlet of the flow (for example, the freestream velocity and frontal characteristic length for a wake). While the definition of an inlet length-scale for the TBL is unclear, Re G should be a linear function of U ∞ . We have then performed measurements at several different incoming freestream veloc- ities but at fixed streamwise positions. To the authors knowledge, no previous study of the dependency of C with U ∞ has been reported before in this flow.

To summarise, we present a numerical/experimental approach, where DNS, 1D HW and 3D Cobra techniques are complemented to characterise the TBL at moderate values of Re ( Re 𝜃 < 3400 and 50 < Re 𝜆 < 140 ). While the characteristics of the wind tunnel used in this work do not allow us to explore larger values of this parameter, this is a range frequently tested in several numerical and experimental studies. Within this work, we will first show that such complementary approach can be used to obtain all one point statistics of the turbulent flow (via the Taylor hypothesis), including C , , K, L,...we will then use the DNS results to show that can be approximated using the standard formulae that assume small scales remain homogeneous and isotropic (and therefore can be estimated with a single HW). We will then use our data to evaluate the properties of the energy cascade (as modelled by C ), and we will confirm the trends from [START_REF] Nedić | Dissipation scaling in constant-pressure turbulent boundary layers[END_REF]) that suggest the presence of a non-equilibrium cascade. Furthermore, we will study the role of Re G on it. Finally, we will confirm by means of the Cobra probe measurements, that considering the large-scale anisotropy of the flow do not modify these conclusions.

Numerical methods

The numerical results in this article are based on a turbulent boundary layer computed using DNS in a computational domain of dimensions 360 0 × 40 0 × 15 0 (where 0 is the inlet boundary layer thickness) with 3073 × 513 × 256 mesh nodes. The simulations are performed with the high-order flow solver Incompact3d (available at www. incom pact3d. com), which is based on 6th order compact finite difference schemes to discrete the incompressible Navier-Stokes equations on a Cartesian mesh stretched in the wall normal direction and a 3rd order Adams-Bashforth scheme for time advancement. The incompressibility condition is treated with a pseudo-spectral approach to solve the Poisson equation for the pressure. Additional information about the numerical code can be found in [START_REF] Laizet | High-order compact schemes for incompressible flows: a simple and efficient method with the quasi-spectral accuracy[END_REF]; [START_REF] Li | Incompact3d, a powerful tool to tackle turbulence problems with up to 0(10 5 ) computational cores[END_REF]. For the inlet boundary condition a Blasius boundary layer is prescribed with local Reynolds number equal to Re 0 = 2000 corresponding to Re = 270 . A by-pass procedure using a tripping method proposed in [START_REF] Schlatter | Turbulent boundary layers at moderate Reynolds numbers: inflow length and tripping effects[END_REF] is used to reach turbulent conditions. At the outlet, where the Reynolds number reaches Re = 1640 , an advection equation is solved for the boundary condition. Periodic boundary conditions are imposed in the spanwise direction, while classical no-slip and free-slip boundary conditions are set, respectively, at the wall and top of the computational domain. Typical values for the spatial and time resolution in wall units for Re = 1400 (or Re = 452 , based on the skin friction velocity u and the boundary layer thickness ) are x + = 10.4 , y + = 0.8 (at the wall, where y + = yu ∕ ), z + = 5.2 and t + = 0.028 (corresponding to t = 0.007 0 ∕U ∞ ). The domain configuration and numerical parameters used for this article are close to those defined in [START_REF] Diaz-Daniel | Wall shear stress fluctuations: Mixed scaling and their effects on velocity fluctuations in a turbulent boundary layer[END_REF] and the results have been validated with the KTH's database [START_REF] Schlatter | Assessment of direct numerical simulation data of turbulent boundary layers[END_REF].

In the present DNS the Reynolds number reached at the outlet of the domain is in the order of the smaller value of Re reported in experiments (see next section). The relevance of this part of the study is twofold. First, it allows to study the influence of small-scale anisotropy and inhomogeneity on the estimation of C (not accessible with HW and Cobra probe measurements). It will also be used to validate the range of wall distances in which HW and Cobra experimental results are valid, as the relatively small values of Re and generated ( is below 5 cm for all cases) imply that the log-law layer is extremely thin (of a few mm) and could be affected by the HW and Cobra probe and their holders. We can therefore verify, by comparing DNS, Cobra and HW profiles, that the experimental data remains valid in the outer layer. As we detail in the next section, we also use the DNS scalings to estimate u for our experiments (via a direct measurement of Re ). These parameters will be used to normalise different quantities with wall units and match DNS and experimental results.

Figure 1a shows the evolution of the friction coefficient C f with Re obtained with the DNS. It can be observed that our DNS collapse very well with the fit proposed by [START_REF] Schlatter | Assessment of direct numerical simulation data of turbulent boundary layers[END_REF]

, C f = 0.024 Re -1∕4
. On the other hand, Fig. 1b shows three different estimations for as a function of the vertical coordinate y (both parameters are normalised with wall units, and therefore + = ∕u 4 ). is estimated following the definitions from [START_REF] Mi | Approach to local axisymmetry in a turbulent cylinder wake[END_REF]. The exact form of is, This expression, can be simplified using different assumptions. First, if the flow is assumed to be locally homogeneous, the expression can be rewritten as, where u, v and w correspond to the streamwise (x coordinate), the wall normal (y) and the lateral (z) fluctuating velocities, respectively. Finally, for locally homogeneous isotropic turbulence, takes the form, that can potentially be estimated experimentally using hot-wire anemometry. While this last expression is usually employed to obtain for experimental data, it is not clear if it is valid or not, and if it is, in which regions of the TBL it can be employed. To cater this question, in Fig. 1b can be observed that for y + larger than 150, all three definitions give similar results. In particular, the ratio iso ∕ full presents a plateau in the outer layer (with a value of ∼ 0.8 in the range 0.4 < y∕𝛿<0.75 , see Fig. 1c). The existence of such plateau validates the approach proposed by previous works [START_REF] Nedić | Dissipation scaling in constant-pressure turbulent boundary layers[END_REF][START_REF] Kamruzzaman | Behaviour of the energy dissipation coefficient in a rough wall turbulent boundary layer[END_REF]). Nevertheless, it is remarkable that the difference between iso and full is larger than in other inhomogenous flows (see for instance [START_REF] Dairay | Non-equilibrium scaling laws in axisymmetric turbulent wakes[END_REF], where for an axisymmetric turbulent wake this difference remains below 6%). These results, that suggest that anisotropy is still relevant at small scales, should be addressed in further DNS studies.

(1) full = ⟨ u i x j + u j x i u j x i ⟩ .

Experimental Setup

Experiments were conducted in the Lespinard wind tunnel at LEGI-UGA: a large wind tunnel with a measurement section of 4 m long and a square cross-section of 0.75 × 0.75 m 2 . The TBL is generated with a PMMA wall of 3 m long, 720 mm wide and a thickness of 20 mm. It was placed horizontally on the floor of the wind tunnel. A blunt finishing, with a parabolical profile was added to allow the laminar BL to develop and avoid BL detachment. A thin piece of rough tape, placed transversally to the inlet of the wall (immediately downstream the parabolic end) was used as a tripper. Nevertheless, we did not confirm that the transition occurred always at that point, so the streamwise origin of the turbulent boundary layer could have some dependency with the freestream velocity. For all cases, outside the TBL, the turbulent intensity (for the tunnel with the PMMA wall inserted) was below 0.2% and the pressure gradient negligible. All HW measurements were made by means of a single hot wire, using a Dantec Dynamics 55P01 hot-wire probe, driven by a Dantec StreamLine CTA system. The Pt-W wires were 5 μm in diameter, 3 mm long with a sensing length of l w = 1.25 mm . Acquisi- tions were made for 60 s at sampling frequencies between 20 kHz and 30 kHz (a low-pas filter was always active at 30 kHz to counteract for aliasing). This correspond to boundary layer turnover times (defined via TU ∞ ∕ , with T the acquisition time), between 4.3 × 10 3 and 17 × 10 3 . We performed some tests at longer acquisition times and higher sampling frequencies to discard the presence on any convergence problems. It was checked that for all the datasets where C results are reported we have at least k =

2 U f = 1 (with U the (2) hom = ⟨ u x 2 ⟩ + ⟨ u y 2 ⟩ + ⟨ u z 2 ⟩ + ⟨ v x 2 ⟩ + ⟨ v y 2 ⟩ + ⟨ v z 2 ⟩ + ⟨ w x 2 ⟩ + ⟨ w y 2 ⟩ + ⟨ w z 2 ⟩ , (3) iso = 15 ⟨ u x 2 ⟩ ,
local streamwise mean velocity, k = 2 f ∕U is the respective wave number, f the largest fre- quency resolved in our data (before the onset of noise) and the Kolmogorov lengthscale, = 3 ∕ 1∕4 ). On the same range, the hot wire has a spatial resolution, l w ∕ between 4 (lower value of U ∞ ) and 9.5 (higher value), as it can also be deduced from results reported later in Fig. 5. The HW was calibrated with a standard pitot tube, and both devices were placed at the inlet of the wind tunnel for each calibration. The pitot tube was then removed from the wind tunnel during measurements.

Large scale isotropy was quantified with a Cobra probe manufactured by TFI [START_REF] Watkins | Measurement of fluctuating flows using multi-hole probes[END_REF], which is able to compute the three fluctuating velocities (u, v, w) plus their mean (U, V, W) and rms ( (u � , v � , w � ) , where u ′ actually correspond to the standard deviation of u) values, with a temporal resolution of 1250 Hz. The acquisition time was set to 120 s at the maximum sampling frequency (and consequently between 8.6 × 10 3 and 34 × 10 3 boundary layer turnover times). We remark that the probe has a 4 mm 2 sensitive area, and therefore is prone to have size effects. Their relevance will be further discussed in the next section, particularly as this area represent a characteristic length (of around 2 mm) larger than and similar to for our data. To the authors knowledge, this effect cannot be corrected, as it is a consequence from the spatial filtering due to the sensor head size.

Vertical profiles were measured at x = 2.25 m , with x the distance between the probe and the blunt end of the plate. Four different incoming velocities where tested: U ∞ = 3.5, 6.2 , 9.7 and 11.9 m/s. Another profile, at x = 2m and U ∞ = 6.9 m/s was also measured. For each U ∞ and x-position, a vertical profile was performed between y = 1 mm and y = 100 mm that comprised 26 points. HW and Cobra measurements were used alter- natively for the same vertical points and experimental conditions.

Table 1 shows the boundary layer parameters for all experimental conditions explored. The momentum thickness is defined as

= ∫ ∞ 0 u(y) U ∞ (1 -u(y) U ∞
)dy and is the corresponding 99% boundary layer thickness. The associated Reynolds number is Re = U ∞ ∕ and the friction coefficient C f has been estimated using the experimental value of and the fit from Fig. 1a. The friction velocity becomes u = √ 1 2 C f U 2 ∞ (the reasons are detailed below). Figures 2a andb show a comparison of the normalised mean streamwise velocity and its standard deviation, U + and u + (note that we define the latter as u + and not u �+ , while it still corresponds to the rms value of the fluctuating velocity, respectively, while all velocities in wall units reported here have been normalised using u ) between HW and DNS data. It can be observed that indeed, for y + > 200 the lower value of U ∞ and the DNS (with a similar x stands for the distance from the probe to the edge of the plate, U ∞ is the inlet velocity, is defined as the corresponding 99% TBL thickness.

The table also includes the momentum thickness and its associated Reynolds number is Re = U ∞ ∕ . The boundary layer thickness-based Reynolds number is defined as Re = U ∞ ∕ . C f and u , the friction coefficient and the skin friction velocity respectively, have been estimated from the DNS. The value of Re is also included 2017)). Figure 2 also shows that the log-law region is too thin in our wind tunnel, and therefore the value of u has not been resolved experimentally. We have nevertheless verified that is accurately estimated with the HW (with an error of 2% at most), and used its value to deduce u from the fit of Fig. 1a.

x(m) U ∞ (m/s) (mm) (mm) Re Re C f u (m/
We then compare the measurements from HW and Cobra probes (Fig. 3a). In this case, we expect to resolve only mean and rms quantities with the Cobra probe, that will be used to evaluate the kinetic energy on the computation of C . In the figure it can be observed that it is indeed the case for our data: differences between both curves are below our velocity error, of 2%, discussed below. Also, the freestream velocity for both cases may not be identical, as there may be a small variation in this parameter when setting the motors at a given power (caused by differences in the relative humidity, temperature and absolute pressure of the air). The figure does not include error bars, as they would mask the dependency of both curves with y + . The absolute error for the y coordinate position was of 0.5 mm, and the temperature's is of 0.1 • C . While the error from the Cobra probe is below 0.2 m/s, we estimate the error from the HW in 0.1 m/s. This last value, has been chosen in a conservative way to consider variations of temperature with respect to the value on the calibration. While we monitored the temperature during measurements, we tolerated variations of up to 0.5 • C between calibration and measurements. Otherwise, time-series from the Cobra and the HW are long enough to converge all statistics (at least 10 3 integral time scales for a given time series, Fig. 3b). With this values, and assuming the error from DNS values is negligible compared to experiments, the absolute error of C f and normalised velocities is always below 1% and for u below 3% . The error for y + remains below 4% for y∕𝛿>0.3 . Finally, the error on and Re is always below 2%. Figure 3c shows the power spectral density for all Re at y = 30 mm for all values of Re , therefore always on the outer region of the TBL. It can be observed that the temporal power spectral density measured with both probes is very close for frequencies up to 100 Hz (a good agreement is also observed throughout the autocorrelations shown in Fig. 3b). This, added to the good agreement between the mean velocity profiles at y + > 100 implies that the Cobra probe can be used to complement HW data in terms on mean and rms values of the velocity vector. The presence and extent of a 5/3 power law in the power spectral densities is debatable, particularly for the lower values of Re . Nevertheless, their shape is similar to those previously reported in the literature at similar values of Re (see for instance [START_REF] Solak | Large-scale motions from a direct numerical simulation of a turbulent boundary layer[END_REF][START_REF] Liu | Non-equilibrium turbulent phenomena in transitional flat plate boundary-layer flows[END_REF]. This is a range studied by many numerical and experimental studies, and therefore it is important to address the properties of the dissipation scalings on it. Furthermore, the shape of power spectral densities still remains very similar to regular static grid spectra previously reported for similar values of Re [START_REF] Antonia | Collapse of the turbulent dissipative range on kolmogorov scales[END_REF][START_REF] Mora | Energy cascades in active-gridgenerated turbulent flows[END_REF][START_REF] Larssen | On the generation of large-scale homogeneous turbulence[END_REF].

Results

Following the comparison between numerical and experimental data, Fig. 4a-c show the rms of all fluctuating velocity components for Cobra and DNS (for u + we also added the HW data). This figure confirms that these experimental quantities have good agreement to the DNS for y + > 300 (that gets increasingly better for larger y + ). While our DNS only the smallest value of Re from experiments, the trends of the vertical profiles with y + are consistent with other DNS studies (Schlatter et al. 2010). Also, both the HWA (reported in Fig. 2b) and Cobra probe values for y + > 300 at Re = 3370 present similar values to the DNS from [START_REF] Schlatter | Assessment of direct numerical simulation data of turbulent boundary layers[END_REF] at Re = 4064.

Figure 4d shows a comparison for only the Cobra and the DNS of the Reynolds stress ( uv + , not needed to estimate the kinetic energy), that can be in principle be resolved with the multi-hole Pitot probe. While a good agreement is observed for y + > 400 , this parameter shows a significantly worse collapse with the experimental data. In consequence, the values of the minima, that could have been used to determine u experimentally, are underestimated by the Cobra probe by a factor 3. This confirms the necessity of using the DNS to properly estimate both u and C f .

We then study the turbulent flow properties for the range 0.4 < y∕𝛿<0.75 , where the mean and rms values of all three velocity components for DNS, HW and Cobra measurements are consistent (all experimental data reported from Fig. 5 onwards corresponds to this range). The turbulent dissipation rate (that in this case corresponds to iso from Sect. 2) was estimated via the dissipation spectrum of the HW signal. It was calculated as = ∫ 15 k 2 1 E 11 dk 1 , where E 11 (k 1 ) is the 1D power spectrum. As for the DNS, this def- inition involves assuming local, small-scale, isotropy and homogeneity and in this case, the Taylor hypothesis. The noise at high frequencies has been removed and modelled as a power law, fitted for each time signal [following the protocol from [START_REF] Mora | Energy cascades in active-gridgenerated turbulent flows[END_REF]]. The Taylor micro-scale has been obtained from as = √ 15 u �2 ∕ . Finally, the integral length scale is defined via the autocorrelation function (Fig. 3b) as L = ∫ 0 0 R uu ( )d , where R uu ( )=⟨u(x)u(x + )⟩∕u �2 , and 0 corresponds to the first zero crossing, R uu ( 0 )=0 . The Taylor hypothesis is used here too, to convert from the time autocorrelation (defined in a temporal variable ) to the spatial one as = U .

While we already discussed the presence of a ∼-5∕3 power law on the power spec- tral density for Re 𝜃 > 1240 , our measurements suggest that the flow remain turbulent, with Re 𝜆 > 50 (Fig. 5c). We also see that this parameter varies significantly, a condition to disentangle equilibrium from non-equilibrium turbulence (that has to be complemented with the variation of √ Re G ∕Re , as we will discuss below). Other quantities, such as , and (Fig. 5a, b ande) also show clear trends with U ∞ , x and y + . Remarkably, L remains almost constant with these two parameters (5d), remaining at L ∼ 2 cm for all conditions considered. We also observe that remains always below 350 m therefore 5 times smaller than the Cobra characteristic length (taken as 2 mm, the square root of the sensitive area). This last parameter is also of the same order as , and we therefore confirm that this probe cannot resolve the small-scale parameters from our experimental setup. We will therefore only use this probe to estimate large-scale averaged quantities such as (U, V, W) and (u � , v � , w � ). Figure 6 shows a comparison between HW and DNS turbulence parameters, normalised with wall units. As the DNS do not resolve the value of L, C cannot be estimated. Large deviations observed for the DNS for y + > 400 correspond to y∕𝛿>1 and therefore outside the TBL (Fig. 3a), where the DNS values of u ′ and are close to zero and therefore more affected by numerical noise (Fig. 6b shows that the value of Re obtained from the DNS still approaches to zero outside the TBL). While parameters present similar values, we can see that the turbulence quantities estimated with DNS do not match the HW results. This has been indeed reported before, as even DNS of the TBL, within the range of Re studied here, happens to be very sensitive to inflow condition, domain size, etc... [START_REF] Schlatter | Assessment of direct numerical simulation data of turbulent boundary layers[END_REF].

We can then compute C from the HW data from = C u �3 ∕L (Fig. 5f). A very conservative estimation of the relative error of this parameter gives error bars below 20%. The main source of error comes from , that can be estimated at around 10%, considering Fig. 6 Turbulence parameters obtained with the HW and DNS: Taylor microscale (a), Re (b), Kolmogorov lengthscale (c) and turbulent dissipation rate (d). All dimensional parameters have been normalised with wall units. For a better comparison with the HW, DNS results have been computed using iso the resolution of our wire and the method we use to model the large wavenumbers on the dissipation spectrum, the error of u ′ could be equated to the error of the mean velocity, that is also the main source of error on the estimation of L (due to the use of the Taylor hypothesis).

To identify the properties of the cascade via the dissipation scalings, as detailed in the introduction, we have first to check if C remains constant for varying Re (Fig. 7a, while we will discuss the role of Re G below). We indeed find the same trends as Nedić et al (the data from [START_REF] Nedić | Dissipation scaling in constant-pressure turbulent boundary layers[END_REF] is also shown in Fig. 7): a constant value of around 0.6 at large Re but a dependency C ∼ Re -1 for low values of Re . While our values of Re remain always below the proposed threshold of 10000, we do find C = cst already at Re ∼ 2500 (consistently with previous findings on a rough TBL [START_REF] Kamruzzaman | Behaviour of the energy dissipation coefficient in a rough wall turbulent boundary layer[END_REF]. We remind that results at Re = 1240 do not present a clear 5/3 power law, but they still seem to be in good agreement with previously reported values. Nevertheless, all data from Fig. 7 has a gap on the range Re ∈ [150-200], making it difficult to conclude pre- cisely where the transition occurs.

These results are confirmed by the evolution of ∕L with Re (Fig. 7b). While the dependency ∕L ∼ Re -1 is expected for the Richardson-Kolmogorov cascade, ∕L ∼ Re -1∕2 G is compatible with non-equilibrium turbulence. Results are consistent with Fig. 7a, as the ratio is constant, but surprisingly no trace of Re G dependency is found in this figures.

We remark that while results from [START_REF] Nedić | Dissipation scaling in constant-pressure turbulent boundary layers[END_REF] are discriminated also by the value of y∕ , due to the large amount of experimental conditions presented here, we decided to plot our results discriminated by this parameter separately on Fig. 9. Our data presents similar trends and values as previous results, like the DNS from [START_REF] Wu | Boundary layer bypass transition[END_REF]: at fixed Re we see that C slightly decreases for increasing y∕ . At larger values of Re , C ∼ 0.6 seems also to be in good agreement with our data and previously reported values for smooth [START_REF] Marusic | Evolution of zero-pressure-gradient boundary layers from different tripping conditions[END_REF] and rough TBLs [START_REF] Kamruzzaman | Behaviour of the energy dissipation coefficient in a rough wall turbulent boundary layer[END_REF].

Nevertheless, there still remains the open question about the influence of the large-scale anisotropy on the estimation of C and the validity of the conclusions stated above. We 2014) and squares to HW data reported by [START_REF] Marusic | Evolution of zero-pressure-gradient boundary layers from different tripping conditions[END_REF]. The red dashed lines in (a) are different Re -1 laws for comparison have discussed the role of small scale anisotropy in the estimation of C on section 2. We now show on Fig. 8a a comparison of the y + profiles of u +2 and the kinetic energy K + , defined as K + = 1 2 (u �2 + v �2 + w �2 )∕u 2 . Low values of Re have similar trends and values when compared with previous DNS [START_REF] Pope | Turbulent flows[END_REF] at Re = 1410 , as K + ∼ u +2 in the outer layer. Furthermore, the anisotropy of the flow seems to increase significantly with Re . Nevertheless, when the value of u ′2 or K are used to estimate C , the curve remains almost unchanged (Fig. 8b, where both L and are still estimated with the HW, we remind again that the Cobra probe was only used for the value of K on the estimation of C ). We therefore conclude that, for values as large as Re = 100 and for Re 𝜃 < 2500 the classical assumption C = cst is no longer valid. We remark that we show results using K and not 2 3 K , and therefore the large scales remain anisotropic, but the anisotropy do not modify the trends observed.

In Fig. 9 we further study the behaviour of C and ∕L . Figures 9a-c confirm that our trends with this parameter are consistent with those found by [START_REF] Nedić | Dissipation scaling in constant-pressure turbulent boundary layers[END_REF]. In Fig. 9d we study the dependency of C with √ Re G ∕Re . As discussed in the introduc- tion, this relation takes into account not only variations with Re but it also assesses the role of Re G in the non equilibrium energy cascade. A non-equilibrium energy cascade implies, for the same flow, that C should be a linear function of √ Re G ∕Re (collaps- ing for all Re G and Re within this regime). It is not clear how such parameter will be defined in the TBL, and in Fig. 9d we use Re G = U ∞ (and therefore Re G = Re ). As both and L are relatively constant, our definition of Re G ultimately quantifies variations of U ∞ , that should be present on all definitions of such parameter. In grid turbulence and some free-shear flows, this parameter remains constant at fix streamwise position x and different Re G (i.e., variations of √ Re G and Re compensate each other at fixed position x when the freestream velocity is changed), and therefore the only way to study the nature of the cascade is via streamwise profiles, where indeed, at fixed value of U ∞ , the parameter √ Re G ∕Re changes. Interestingly, this is not the case for the outer region of the TBL, and even profiles at x = cst present important variations. We have neverthe- less left the value of x in Fig. 7 for reference. For comparison, the HW results from the Therefore, a requirement to verify the presence of this energy cascade is that the parameter √ Re G ∕Re varies throughout our datasets (as it indeed happens). While in Fig. 7 and 8 no trend was easily observed, in Fig. 9 a trend is indeed present, for low values of Re (resp. large values of √ Re G ∕Re ), consistent with a linear relation, for all values of y∕ and Re . This result requires further study as, to the authors knowledge, has not been reported before. A larger wind tunnel that allows to explore wider ranges of , L and Re could allow to better characterise the transition between both cascades and if the definition of Re G pro- posed in this work remains valid. Results discussed above from previous experiments at larger values of [START_REF] Nedić | Dissipation scaling in constant-pressure turbulent boundary layers[END_REF][START_REF] Marusic | Evolution of zero-pressure-gradient boundary layers from different tripping conditions[END_REF] suggest that studies on the nature of dissipation scalings should not only cover different vertical profiles at fixed streamwise positions, but also different values of U ∞ . While we cannot conclusively confirm the presence of a large Reynolds non-equilibrium cascade, the relations observed in Fig. 9 tends to point towards that direction. We therefore confirm the observation reported by [START_REF] Nedić | Dissipation scaling in constant-pressure turbulent boundary layers[END_REF], and the approximation C = cst is invalid already at Re 𝜃 < 2500.

Conclusions

In this work we used hot-wire anemometry and a Cobra probe to estimate both the full kinetic energy and the turbulence energy dissipation rate experimentally on a TBL. A DNS for the lower value of Re was used to quantify the influence of small scale anisotropy and inhomogeneities on the estimation of . It also validated the mean and fluctuating velocities profiles obtained experimentally in the range 0.4 < y∕𝛿<0.75 . Our results and the novelty of this work can be summarised as, -Our combined approach of DNS, HW and Cobra allowed us to complement the advantages of each technique to counteract resolution problems from each one: numerically expensive estimation (thus not achieved in our case) of L and therefore C with the DNS, bad resolution with the Cobra at small scales (impeding the calculation of small-scale quantities such as , C and ) and absence of 3D information with the HW. We show that the Cobra and HW together are indeed capable of providing information about the energy cascade, while the DNS allowed us to validate our statistics at small Re . -The DNS also showed that can be estimated using the homogeneous isotropy assumption (3) in the outer layer. While our experimental setup did not allow us to fully resolve the log law region, this confirms that experiments in larger tunnels can indeed obtain all relevant information by combining Cobra and HW probes. -We confirm evidence discussed previously [START_REF] Nedić | Dissipation scaling in constant-pressure turbulent boundary layers[END_REF]) about the presence of dissipation scalings consistent with non-equilibrium turbulence at small values of Re .

In this point the contribution of this work is twofold. First, we show that while the flow presents large-scale anisotropy, it does not affect the trends of , when it is estimated as = C u �3 ∕L . Second, we not only confirm the presence of the regime C ∼ Re -1 at low Re , but in our experiment we also changed the global Reynolds number and find that data collapses reasonably well with √ Re G ∕Re . We therefore confirm and expand the validity of the findings from [START_REF] Nedić | Dissipation scaling in constant-pressure turbulent boundary layers[END_REF]. While it is beyond the possibilities of our experimental setup, a systematic study on this scaling would require to disentangle the dependency of C with Re 0.5 G and Re separately (as done for instance in the grid experiments from [START_REF] Valente | Universal dissipation scaling for nonequilibrium turbulence[END_REF], via experiments at constant Re and different Re G , and vice-versa.

While our results suggest some form of non-equilibrium turbulence is present, and trends reported previously are not contaminated by inhomogeneities and the anisotropy of the flow, further studies are needed to address conclusively the presence of such cascade. Particularly, as this regime occurs at low values of Re , in some cases where the 5/3 power-law is not very clear, it also remains a possibility that some form of viscosity effects are still present. Nevertheless, our results confirm that the standard assumption of the constancy of C is not valid in TBLs with Reynolds numbers as large as Re = 100 and Re ∼ 2500 .

Our results also suggest that the parameter better suited to characterise this transition is √ Re G ∕Re and not Re .

In this work we have proposed to define C and Re G with L and , respectively, but these two parameters do not vary significantly in our dataset. Our aim in this point is to stimulate discussion on the role of these parameters. Nevertheless, as several length scales can be defined in the TBL, further experiments in larger wind tunnels, exploring broader ranges and larger values of and Re , would contribute to understand the role and the definition of different length scales in the energy cascade in the TBL. They would also allow to explore the range Re ∈[100 -300] , and better characterise the transition observed for HW and DNS studied here in terms of Re and Re .

Funding JHS and EBCS thank the founding of Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001. A part of this work was granted access to the HPC resources of IDRIS under the allocation 2019-A0060107567 made by GENCI. This study was funded by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001. A part of this work was granted access to the HPC resources of IDRIS (Grant Number 2019-A0060107567) made by GENCI.

Declarations

Conflict of interest The authors declare that they have no conflict of interest.

Fig. 1

 1 Fig. 1 Friction coefficient C f as a function of Re a. The black dashed line correspond to the fit from the DNS reported in Schlatter and Örlü (2010). Different definitions of , as defined in the text b. The black dashed line corresponds to the estimation of full from Schlatter and Örlü (2010). Ratio between iso and full c. The vertical lines correspond to y∕ = 0.4 and y∕ = 0.75 . Figures b and c correspond to Re = 1400

  value of Re ) are in good agreement, both for the mean and rms values of u. y + = 200 cor- respond, for our TBL, to y < 10mm , and the depart at lower y + from the DNS of the TBL is probably caused by the bad resolution of the traverse system in the vertical direction (see below). Other cause could be an interference between the flow and the HW support close to the wall. Nevertheless, the range resolved in our dataset corresponds, for the worse case (lower Re ) to y∕𝛿>0.5 , and therefore the HW gets reliable measurements for almost the whole the range of interest for our study of C (higher values of Re always fall within the range studied inNedić et al. (

Fig. 2

 2 Fig. 2 Normalised mean (a) streamwise velocity and its standard deviation ( U + and u + , respectively) vs the vertical coordinate y + (all in wall units) for HW measurements at x = 225 cm and DNS data

Fig. 3

 3 Fig. 3 Comparison of mean velocity profiles U + (y + ) obtained with the Cobra probe and the HW at x = 225 cm and Re = 2850 (a). Normalised autocorrelation functions R uu ( ) (b) and power spectral densities (c) at y = 30 mm (that correspond to values of y∕ between 0.6 and 0.75) obtained with the cobra probe (stars) and the HW (solid lines). The black dashed line corresponds to a -5∕3 power law

Fig. 4

 4 Fig. 4 Normalised fluctuating velocities evolution with y + obtained with the Cobra probe (circles), the HW (blue solid line in (a)) and DNS (magenta and black solid lines)

Fig. 5

 5 Fig. 5 Turbulence parameters obtained with the HW. Kolmogorov lengthscale (a), Taylor microscale (b), Re (c), integral lengthscale L (d), turbulent dissipation rate (e) and dissipation constant C (f)

Fig. 7

 7 Fig. 7 Comparison of the dependency with Re of C (a) and ∕L (b) obtained via HW in the present experimental setup with the data reported in Nedić et al. (2017) (black and white symbols). Circles correspond to the DNS from Wu et al. (2014) and squares to HW data reported by[START_REF] Marusic | Evolution of zero-pressure-gradient boundary layers from different tripping conditions[END_REF]. The red dashed lines in (a) are different Re -1 laws for comparison

Fig. 8

 8 Fig. 8 Comparison between the values of u +2 and K + (a) obtained with the Cobra probe. Different definitions of C (b). Circles stand for the values of u +2 and triangles for K + . In figure (b) star markers correspond to HW data. The colours refer to the same datasets as in Fig. 7

Fig. 9

 9 Fig. 9 Different parameters obtained with the HW for our data alone. ∕L vs Re (a). C as a function of Re (b), Re (c) and √ Re G ∕Re (d). In all figures the colormap quantifies the position y∕ where data was taken. Circles stand for Re = 1240 , stars for Re = 1900 , squares for Re = 2300 , triangles for Re = 2900 and diamonds for Re = 3400

Table 1

 1 Experimental parameters.