The objective of this paper is to propose a technique for measuring the velocity of a bubble cloud driven by a flow, in a regime where multiplescattering effects are predominant. The most popular acoustic velocitymeasurement technique is the Doppler effect. However, when applying this technique, multiplescattering could be a problem. Most of the papers dealing with Doppler effect inscattering media assume no multiplescattering occurs . In some other papers it is proposed to model the Doppler effect inMultipleScatteringregime using a Monte-Carlo simulation in acoustics  or in optics , and it is shown multiplescattering causes distorsion of the measured flow. Another paper proposes a technique using a beam-forming method , assuming that single scattering is predominant, that is to say, no multiplescattering occurs. Inmultiplescatteringregime, another technique has been studied, using a coherent wave phase conjugation process . This technique proposes a measurement based on the coherent part of the field, and may lose effectiveness when this coherent part becomes negligible. The technique developed in this paper is based on the sensitivity of the incoherent field to a slight perturbation in the medium. It uses only the incoherent part of a multiple scattered field to evaluate the bubble-cloud velocitymeasurement by studying the evolution of the incoherent field-field correlation coefficient. Our technique is closely related to Diffusive Wave Spectroscopy [18–22] in optics, Coda Wave Interferometry [23, 24] in seismology and Diffusive Acoustic Wave Spectroscopy (DAWS) in acoustics [25–29].
and H . A part of the signal extending ±0.012 s around each de- tected extremum has also been removed in order to localise the entire part of the signal corresponding to spurious peaks. Both the threshold and the time extension of ±0.012 s have been de- termined from the comparison of the spectrometer raw signal to the images obtained by shadow imaging. By inspecting the images, it is possible to detect precisely the start and the end of a bub- ble passing through the measurement volume. The corresponding part of the signal is removed and replaced by a linear interpola- tion. The linear interpolation is of course questionnable since the concentration is not defined in the region occupied by the bub- bles. Similar methods have been used for processing liquid veloc- ity measurements by hot-film anemometry inbubbly flows. Dis- cussion of their consequences on the signal statistics can be found in Serizawa et al. (1983) and Suzanne et al. (1998) . Here we have checked that the present interpolation has not significant influ- ence upon PDFs of concentration fluctuations. Concerning spec- tra, different other techniques have been used for their determi- nation for velocity fluctuations from signals that are interrupted by the passages of the bubbles, such as smoothing the disconti- nuities by a Gauss function ( Lance and Bataille, 1991 ) or consid- ering only intervals between bubbles where the signal is continu-
difficult. As a result, engineering design rules and recommendations for stirred tank applications are typically valid for turbulent or laminar flows only.
The few studies in the literature dealing with the transitional flowregimein stirred tanks address specific particularities of these flows; however, determination of the general underlying reasons for physical phenomena occurring is not generally the focus. For example, Machado and Kresta (2013) and Machado et al. (2013) studied the transition from turbulent ta transitional flowin various stirred tank geometries in order ta determine the limits of fully turbulent scaling in different zones of the tank. They showed that although fully-developed turbulent flow is generally considered for Re> 20 000 in standard stirred tank geome tries, this is only true close ta the impeller. Reynolds numbers greater than 300 000 are required in order ta attain fully-developed turbulence at heights in the tank greater than 0.9T. However, in non-conventional geometries like the confined impeller stirred tank (CIST), fully turbulent flow was observed at Reynolds numbers as low as 3000. More generally, it was found that fully turbulent flow occurs at lower Reynolds num bers as the scale of the tank decreased and the ratio of impeller-to tank diameter DIT increased. They also showed that dimensionless veloc ity profiles at a fixed Reynolds number can depend on the viscosity of the Newtonian fluid, therefore affecting the flowregimein the tank. Liné et al. (2013) analysed the flow and the dissipation rate of a shear thinning liquid in the vicinity of a Rush ton turbine. Due ta the rheology of the fluid, the flow was expected ta be in the transitional flowregime with an impeller Reynolds of 530, determined using the Metzner-Otto carrela tian. However, the validity of this method and the spatial varia tion of the flowregimein the tank was not discussed. Recently, Alberini et al. (2017) have performed 3-dimensional Particle Tracking Veloci mentry (PTV) measurements in a stirred tank in transitional flow at Re= 70 and 1000 with the objective of comparing the measurement technique with 2-dimensional 2-component Particle Image Velocime try (PIV) for Newtonian and non-Newtonian flows. Although the work does not focus on the characteristics of transitional flow, the authors note a slight difference in the PTV and PIV velocity fields in the impeller discharge with non-Newtonian flow. This is attributed ta the unstable nature of transitional flow and suggests that the tangential velocity component may be of significance in such flows.
Vial, C., Laine, R., Poncin, S., Midoux, N., Wild, G., 2001. Influence of gas distribution and regime transitions on liquid velocity and turbulence in a 3–D bubble column. Chem. Eng. Sci. 56 (3), 1085–1093 .
Wang, F., Huang, S.-D., Zhou, S.-Q., Xia, K.-Q., 2016. Laboratory simulation of the geothermal heating effects on ocean overturning circulation. J. Geophys. Res. Oceans 121, 7589–7598. https://doi.org/10.1002/2016JC012068 .
measured temperature fluctuations are limited to frequencies lower than 10 −1 Hz. At approximately 10 −2 Hz we observe a peak, beyond which there is a very steep decrease of the spectrum (the same frequency can be observed in figure 5 a). As is well known (Castaing et al. 1989 ), this peak corresponds to the large-scale circulation frequency ( fLS ≈ Vff /4H), which can be estimated from the free fall velocity Vff = √ g β1TH which is ∼6 cm s −1 for the lowest RaH and ∼11 cm s −1 for the highest Ra H . In both the single-phase and two-phase cases, a higher level of thermal power is seen for higher RaH numbers. The same trend is seen for all the measurement positions at mid-height. If we now compare the single-phase and two-phase spectra, we can see that with the bubble injection, the thermal power of the fluctuations is increased by nearly three orders of magnitude. The bubblyflow also shows fluctuations at a range of time scales, with a gradual decay of thermal power from f ' 0 .1–3 Hz. The observation that substantial power of the temperature fluctuations resides at smaller time scales, as compared to the single-phase case where the power mainly resides at the largest time scales, further confirms that the bubble-induced liquid fluctuations are the dominant contribution to the total heat transfer.
1995 ), only a measurement bias can explain the observed difference: this is a possible explanation since the slip velocity is determined from the difference between two measured velocities whose accuracy is probably not better than 2 %.
In the near-wall region, the slip velocity decreases (ﬁgure 13 ). Even if the bubble diameter decreases as well (ﬁgures 4 and 5 ) their dynamics belong to the capillaro- gravity regime for which the velocityin still liquid depends little on the bubble diameter. Why is the slip velocity much smaller than the velocityin still liquid in the wall region? A recent numerical experiment (Adoua, Legendre & Magnaudet 2009 ) has shown that the drag coefﬁcient of an oblate bubble increases with the shear rate Sr = 2d U L,y /U L if it is greater than 0.2. For Sr = 1, they found that the drag coefﬁcient can be more than twice its value for a homogeneous ﬂow. In the present experiments Sr can increase up to 1 when the distance from the wall decreases. At this stage it is impossible to check how much the shear rate can reduce the slip velocity because the process is sensitive to the bubble aspect ratio, a quantity that has not been determined in our experiments.
Vial, C., Laine, R., Poncin, S., Midoux, N., Wild, G., 2001. Influence of gas distribution and regime transitions on liquid velocity and turbulence in a 3–D bubble column. Chem. Eng. Sci. 56 (3), 1085–1093.
Wang, F., Huang, S.-D., Zhou, S.-Q., Xia, K.-Q., 2016. Laboratory simulation of the geothermal heating effects on ocean overturning circulation. J. Geophys. Res. Oceans 121, 7589–7598. https://doi.org/10.1002/2016JC012068.
gap) are tested. In order to enable the characterization of the bubble dispersion by visualisations, the global void fraction was willingly limited to a small value (α<0.23%), smaller than in Mehel 15, 17, 19 .
This paper is organized as follows. The next section is devoted to the description of the experimental facility. Characterization of the flow structure and viscous torque measurements in the single-phase flow are also shown and discussed in this section. The section ends with a description of a specific measurement technique for tracking of bubbles. Section 3 is devoted to the characterization of the two-phase flow: we present the void fraction distributions, the Eulerian velocity fields of the gas-phase measured in a meridian plane and the viscous torque measured in two-phase flow. Section 3 also develops a comparison of our results with related work, discussions about bubbles localization are made and the phase diagrams, which summarize the various types of bubbles arrangement, are built. Lastly section 4 concludes the paper and outlines further work.
normalized correlation time and length decrease strongly with α from 200 to 1 for τ ∗ and from 4 to 1 for ξ ∗ . In this regime, the increase of defects number is responsible for this strong decrease of space- time correlation in the pattern. Meanwhile, the flow has the capacity to reproduce the pattern in axial direction and in time; the correlation length and time remain larger than the space-time period of the flow ( ξ ∗ > 1 and τ ∗ > 1). As mentioned before, the flow is structured, as it keeps the memory of the pattern. For α(%) αIDC = 0.005%, the bubbly patterns have small correlations, the correlation length and time being smaller than the space-time periods ( ξ ∗ < 1 and τ ∗ < 1). This critical value of the air volumetric fraction αIDC = 0.005% correctly characterizes the transition from the structure patterns flow (SCP) to the intermittent defect chaos regime (IDC). In the DDC regime, for α > 0.02%, the normalized correlation length and time saturate. This means that the dynamics of the defects does not change anymore. Indeed, the defects number is expected to saturate, as the gap is of finite dimension. We can suppose that saturation occurs at large alpha when a connection between Taylor vortex pairs (defect) occurs over each time and axial periods. The azimuthal wave promotes the connection by making easier bubbles jumping from the crest to the trough of the upper Taylor vortex pair. Bubbles jumping (i.e. bubbles escape from the vortices) is enhanced (1) by the increase in the gravity effect (achieved by increasing the relative contribution of the terminal rising velocity of isolated bubbles Vb /V i or decreasing Vi and (2) by an important accumulation of bubbles in the Taylor vortices (achieved at large values of Qg) which induces an important rising velocity of bubbles clouds under collective effect An important axial flux of the bubbles is required to connect the vortex pairs by bubbles jumping: this is ensured at large values of the volumetric fraction α ∼ Q g /V i.
A Taylor-Couette flow can be considered as a valuable configuration to study bubble induced modifications of the viscous drag. It is a closed system and characterizing the viscous drag implies to measure the viscous torque applied on the inner cylinder. Moreover, for moderate to high Reynolds numbers, the Taylor-Couette flow has several similarities with the turbulent boundary layer flow that develops over a flat plate. In particular, in the very near wall region of inner and outer cylinders, there is an inner layer of constant shear stress and negligible curvature effect characterized by a linear evolution of the azimuthal velocity with respect to the distance from the wall 3 . Farther from the wall, the azimuthal velocity follows a logarithmic law as observed in turbulent channel flows and for high Reynolds numbers (Re>10 6 ), the slope asymptotically tends to the Von Karman constant 4 . Furthermore, in the transition regime, the occurrence of Taylor contra-rotating vortices (with associated inflow/outflow jets regions) is very similar to the energetic turbulent structures, taking place in the very near wall region over a flat plate 5 .
been measured at T=22°C: ε 1=3.7 ε 0, η l=16.5 mPa.s. Its density ρ l=1.14 10 3 kg.m -3 is close to that of the particles. Its conductivity has been raised to γ 1 =5.4 10 -8 S.m -1 upon addition of an ionic-surfactant (AOT salt, sodium dioctylsulfosuccinate, Sigma Aldrich). The particle conductivity is small enough ( γ 1 ≈ 10 -14 S.m -1 ) to consider them as insulating. Accounting for the electric characteristics of the particles and of the suspending liquid, we deduce the values of the critical field and of the Maxwell time: Ec=1800V/mm and τ M=1.6ms. The volume fraction of the solid particles is φ =5.10 -2 and the viscosity of the suspension, η s=19.3 mPa.s., has been measured with a controlled stress rheometer Carrimed CSL 100. The particle volume fraction has been chosen in such a way that the effect of Quincke rotation on the material rheology is sufficiently high, without inducing too much ultrasound multiplescattering that would have made the measurement of the velocity profile impossible.
Two-phase turbulence has been studied using a DNS of an upward turbulent bubblyflowin a so-called plane channel. Fully deformable monodispersed bubbles are tracked by the Front-Tracking algorithm implemented in TrioCFD code on the TRUST platform. Realistic fluid properties are used to represent saturated steam and water in pressurised water reactor (PWR) conditions. The large number of bubbles creates a void fraction of 10%. The Reynolds friction number is 180. After the transitional regime, the flow is simulated until convergence of statistics is achieved. Time- and space-averaging is used to compute main variables at the average scale (e.g. void fraction, phase velocities. . . ). Budget of forces and Reynolds stresses are also computed from the local fields. They provide reference profiles to improve momentum transfer closures and turbulence modelling. The velocity profile and the flow-rate are compared to a similar single-phase flow simulation. Strong buoyancy forces create a large relative velocity. Averaged surface tension forces also play a significant role in the flow equilibrium. In the prospect of assessing a single-pressure Euler-Euler two-fluid model, the macroscopic momentum jump condition is deduced from averaging DNS fields. The resulting balance shows that the classical assumption of opposite forces acting on each phase should be revised. Indeed, neither surface tension, nor pressure difference is negligible.
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The simulation of transitional flows in stirred tanks is challeng ing because hydrodynamic instabilities create unsteady flow, which needs ta be correctly captured. Turbulence models are typically not well adapted at transitional flow Reynolds numbers because the eddy viscosity hypothesis used in the models is designed for high Reynolds number turbulence and also because the wall fonctions assume a log law at the wall. Ultimately, the full resolution of the time-dependent Navier-Stokes equations on an extremely fine 3-dimensional mesh would be desirable however such simulations require excessive com puting efforts, which may not be viable for practical engineering applications. As a result, there are very few studies in the literature that deal with the simulation of transitional flowin stirred tanks and almost ail of the available literature may be attributed ta a single author ( Derksen, 2011, 2012a, 2012b, 2013; Zhang et al., 2017 ). In his studies, Derksen has used the Lattice Boltzmann method ta perform direct numerical simulations (ONS) of flows with Reynolds numbers in the range 2000-12000. These are extremely intensive computations, which require around 100 impeller revolutions on a highly-refined grid. Although these studies do not focus particularly on the underlying nature of the transitional flow, but rather specific mixing applications, the results demonstrate that moderate Reynolds numbers allow the flow ta be simulated directly, without the use of a turbulence clo sure or subgrid-scale models. Although, in their recent study, Zhang et al. (2017) have shown that even for simple Newtonian flows, ONS failed ta correctly predict the flow patterns and velocity fluctuation levels at certain transitional Reynolds numbers. Their work also shows the current state of confusion in modeling transitional flows as they attempted ta apply transition models developed for externat aerody namics ta internai flow and performed fini te volume simulations using 2nd order upwind differencing that they erroneously called finite vol ume ONS. The limits and capacities of simulation techniques (e.g. CFD, Lattice Boltzmann) for transitional flows in stirred tanks are therefore not clearly identified. Moreover, as various authors have discussed, the flowregimein a stirred vesse! is rarely constant in the entire volume ( Ducci et al. 2008; Machado et al. 2013 ); in some cases, flow may be fully turbulent in the vicinity of the impeller but in the transitional or even laminar flowregimein zones further away. It is not straightforward ta know how this variation inflowregime should be taken into account in simulations that can be performed with reasonable computational effort and therefore applicable for engineering applications.
The objective of this part is to study the e ffect of buoyancy (through varying the viscosity of the mixture and the bubble size), as well as the e ffect of the Reynolds number on the bubble arrangement in the gap. Unfortunately, it is not possible to test experimentally the influence of these variables separately. Indeed, for a given bubble size, C increases as H decreases, with the increasing of the Reynolds number. Table III reports the two-phase flow configurations, for which the Eulerian cartographies of the gas-phase are displayed. These flow conditions were selected because the gas Eulerian cartographies of each of them illustrated the various types of bubble arrangement observed in the framework of the present study. Cases 1 to 3 exhibit di fferent arrangements for same bubble size and same viscosity, when increasing the Reynolds number. Case 4 corresponds to same bubble size as the previous cases but with an increase of the Reynolds number and an increase of the rising velocity by changing the viscosity. Case 5 illustrates the influence of the bubble size for same Reynolds number and same viscosity. Case 6 corresponds to a condition of high Reynolds number and large bubbles, thus leading to large void fraction.
We have to model and render this sub-surface scattering e ffect for photorealistic
rendering of these materials. But taking it into account greatly increases the computational complexity of illumination simulation. In order to compute the outgoing light at a specific point, we now have to take into account the incoming light at all neighbouring points, from all directions, instead of just the light incoming on this point. This adds two dimensions to the sampling, increasing the computation time. We also have to store the behaviour of the material, a function that express the relationship between incoming and outgoing light. Because the relationship is both spatial and angular, this function, the BSSRDF, has 6 dimensions. With regular sampling on all directions, storage is prohibitive.
The second oscillator we presented has a smaller subcavity (concave mirrors with a smaller focal length) which creates a smaller spot size at the center of the crystal. The stability zones in this case are bended in such a way that the two stability zones are superimposed. In the meeting between the two stability zones, on the edge of the second stability zone we find the ideal point to have the laser in ML. At the ideal ML position the focal for the first oscillator is approximately 50 mm (35 µm spot size), for the second oscillator, the beam size is smaller than 10µm, in consequence the thermal lens is one order of magnitude smaller on the order of the mm. This enabled us to work in a region that is stable and at the same time with a spectral bandwidth bigger than an octave. We also verify that the point in which the cavity operates in ML the pump and beam size are similar at the central of the crystal, no other configuration of the sub-cavity as the same two characteristics. In the first cavity that was presented, the final model gives presents two points that where ML operation is feasible, one of them is coincident with the literature (see for instance [87, 112, 137]) on the other the pump and beam sizes do not match (compare Fig. (2.24) c) and the bottom graph of Fig. (2.28)), we envision that in future works an engineering parameter to define the ideal ML position would not only have into account the decrease in mode size due to the increase in power (which the ML parameter already does) but also the match between the pump and beam size inside the crystal.
As stated above, a reasonable assessment is to assume that the average flow is limited by the expulsion of hydrocarbons from source rocks (Carruthers and Ringrose, 1998). Since Bo<<1, the buoyancy force of a single oil-filled pore is much smaller than its capillary pressure, which prevents an isolated, non-wetting fluid inside a single pore from moving; however, buoyancy of connected pores is additive and clusters with many pores (of the order of 1/Bo pores long in the vertical direction) may overcome this barrier. In order to build such a cluster, one has to assume that a few discrete stringers of continuous oil phase slowly emerge from the nearby source rock (Hirsch and Thompson, 1995). Once a stringer has a sufficient buoyancy to overcome capillary forces it may migrate until it is halted by a collection of smaller pore throats.
The numerical procedure is developed with scripts already existing in the Matlab Tool- box dedicated to the resolution of partial differential equations (PDE) with finite elements method. All scripts are freely available on request.
The cross section is meshed with triangular elements with a constant mesh size ( t mesh ). Afterward, the mesh is refined only near the bed (Fig. 2 ) to reduce the compu- tational time. For smooth flow, the minimum size should ensure that the distance between a first node and a boundary allows verifying the condition 30 ≤ (z+ = u ∗ z ∕𝜈) ≤ 100 where 𝜈 the molecular viscosity. For rough flow, the mesh can be larger and a maximum value of h / 30 is used. To start the PDE resolution an initial condition is given by considering a constant velocity deduced from the Manning equa- tion with n = 0.025 . In a first step, the equation is solved with an uniform shear stress based on the hydraulic radius ( u ∗0 = √gS f R h ). The boundary condition at the free sur- face is a slip condition which leads to a maximum velocity at the free surface. Although in practical case, the secondary currents involve a dip phenomenon, this pre- sent model does not solve this issue because the applications studied here are focused on boundary layer. For wall boundaries, the velocity u k is imposed considering the shear stress u ∗0 . The Eq. 3 is numerically solved with the Partial Differential Equation Toolbox using a Finite Element Method. The velocity distribution is then used to (8) ( u k