smaller than the mesh size $\eta \ll d \ll \Delta$:

$$d_t \mathbf{u}_p = - \frac{\mathbf{u}_p - \overline{\mathbf{u}_f}}{\tau_{p,eff}} + \beta \overline{D_t \mathbf{u}_f} + a_* \mathbf{e}_* \quad (3.62)$$

In this equation we have kept $\beta$ rather than $\beta_{eff}$ introduced earlier, since the particle is smaller than the mesh it does not operate any additional filtering than the fluid acceleration already filtered at the mesh scale $\overline{D_t \mathbf{u}_f}$. The norm of the subgrid contribution is considered to be given by the relation (3.45):

$$a_* = \frac{3}{2} C_D \left( \frac{\rho_p}{\rho_f} + C_M \right)^{-1} \varepsilon_*^{2/3} d^{-1/3} . \quad (3.63)$$

In this equation, $\varepsilon_*$ is the dissipation rate at the particle scale. To model this quantity we use a stochastic equation similar to (3.54) but whose exponent is adapted from $\varepsilon_*^{3/4}$ to $\varepsilon_*^{2/3}$. Another difference concerns the parameter $\sigma^2$. This one is always expressed as the logarithm of the ratio between the time of the largest non-resolved vortices ($\tau_\Delta$) and the characteristic time of the particle. In the case of large particles, we estimate the characteristic time from $\tau_{p,eff}$ introduced previously:

$$\sigma^2 = \frac{1}{2} \ln \frac{\tau_\Delta}{\langle \tau_{p,eff} \rangle} = \frac{1}{2} \ln \frac{18 \tau_\Delta \langle \varepsilon \rangle^{1/3}}{(\rho_p / \rho_f + C_M) d^{2/3}} \sim \frac{1}{3} \ln \Delta / d . \quad (3.64)$$

The model for orientation remains identical to (3.55)-(3.56) presented above.

Variances and PDFs for different particle sizes between 5 and 30 $\eta$ and density of $\rho_p/\rho_f = 50, 100$ and $1000$ are presented in Figure 3.15. The LES results with the model are compared with the experimental results of Ref. [210, 263] and the standard LES approach, i.e. LES without model for unresolved fluctuations, and using either the linear drag law or the Schiller and Naumann drag law. The simulations are performed for a 64³ mesh size. Note that we only consider relatively heavy particles $\rho_p/\rho_f > 50$, as the added-mass effects are not taken into account here. We see that both versions of the LES without model predict a strong reduction in the variance of the acceleration with increasing $d/\eta$. We also see that, as expected, in the linear drag case, the decrease is much more pronounced than with the case using the drag law proposed by Schiller and Naumann. On the other hand, with LES with the model, the decrease of the variance of the acceleration with $d/\eta$ is much less important and presents a behavior close to $d^{-2/3}$.

We find that LES without model predicts a Gaussian distribution for the acceleration when Stokes drag is used. This deviates considerably from the experimental results, for which the tails of the PDF remain very broad even for a large value of $d/\eta$. With the nonlinear drag the LES without model predicts a wider distribution, which seems to be independent of the particle diameter. Nevertheless, the large fluctuations remain underestimated compared to the experiments. With the LES supplemented with the stochastic model, the PDF tails are in excellent agreement with the experiments of Ref. [263].

From (3.63), and considering that $C_D = 0(1)$, we estimate the variance of the acceleration of large particles:

$$\langle a_*^2 \rangle \sim \frac{\langle \varepsilon_*^{4/3} \rangle}{d^{2/3}} \quad (3.65)$$

From the log-normal model for $\varepsilon_*$ we can compute $\langle \varepsilon_*^{4/3} \rangle$ as $\langle \varepsilon_*^{4/3} \rangle = \langle \varepsilon \rangle^{4/3} \exp 2\sigma^2/9$. Consequently, with the expression of $\sigma^2$ (3.64), we then estimate that $\langle a_*^2 \rangle \sim d^{-2/3-2/27} \sim$
<figure><img src="image_38.png" /></figure>

Figure 3.15: (Left) Variance of the large particle ($\Delta > d > \eta$) acceleration normalized by the Kolmogorov acceleration ($a_{\eta}^2 = \langle\epsilon\rangle^{3/2}\nu^{-1/2}$) in logarithm scale. Crosses: LES with the proposed model; Circles: LES without the model with Stokes drag; Triangles: LES without the model and with the drag correction of Schiller and Naumann. For three density ratios: $\rho_p/\rho_f = 50$ (black), 100 (dark gray) and 1000 (light gray). For all cases the mesh size is 64³. Comparison with the experiments of Ref. [209] with the particle density ratio around 50 in black squares and the power laws $<a_p^2> \sim d^{-2/3}$ in gray dashed line and $<a_p^2> \sim d^{-0.8}$ in gray dotted line. (Right) PDF of the large particle ($\Delta > d > \eta$) acceleration normalized by its variance for $d/\eta = 5, 10, 20$ and $26$ (respectively shifted upward by one decade from each other for clarity). light gray: LES with the proposed model; black: LES without the model with Stokes drag; dark gray: LES without the model and with the drag correction of Schiller and Naumann. In each case the density ratio is $\rho_p/\rho_f = 50$ and the mesh size 64³. Comparison with the Gaussian distribution gray dashed line, with the experimental data of Ref. [263] (with $d/\eta = 1.6, 8.2, 13.4$ and $23.6$) in black dashed line and with the fit from Ref. [209] in dotted line.

$d^{-0.74}$. It is interesting to note that this scaling law is relatively close to the $d^{-0.81}$ law which is shown in Ref. [263] to better describe the experimental measurements than the $d^{-2/3}$ law.

These comparisons between LES with the model and experiments show the interest of the proposed approach. Nevertheless it would be necessary to continue the comparison with experiments and particle-resolved DNS to refine the formulation of the models.

## 3.4 Turbulence induced by a swarm of rising bubbles

In this part, we are interested in the flow induced by the rise of a cloud of bubbles. In this system, the only source of momentum is the buoyancy acting on the bubbles, and without the bubbles the liquid would remain at rest. It is a complex system in which the movements of the bubbles and the liquid are coupled, leading to the emergence of collective
phenomena and original properties of the flow. We consider the case of a homogeneous swarm of large deformed bubbles, with a Reynolds number based on the bubble size and terminal velocity $v_0$ is of a few hundreds, so that each bubble generates an intense wake. A first manifestation of collective effects is the decrease in the average bubble rising speed as the gas volume fraction $\alpha$ increases [220]. On the other hand, the main cause of bubble velocity fluctuations is attributed to wake instabilities. Indeed, when the deformation of a bubble and its Reynolds number are large enough, the wake becomes unstable and the bubbles exhibit path oscillations [186, 278, 72] and this seems to remain the case even for high volume fractions, of the order of 30% [60]. These wake-induced fluctuations are probably the reason why bubbly flows can remain homogeneous, and be generated in laboratory bubble columns [265]. However, the stability of homogeneous bubble columns remains an open problem and is limited to reasonably small geometries (of the order of one meter) with well-controlled uniform bubble injection. In most industrial applications, the gas volume fraction is not homogeneous throughout the flow and large-scale buoyancy-induced motions develop [187].

Fluid fluctuations exhibit very specific properties that have been identified experimentally [137, 277, 85, 224, 213, 161, 220, 167, 206, 5]. Several contributions to the fluid fluctuations can be distinguished [223]. For a homogeneous swarm of bubbles, there are, on the one hand, the localized perturbations around the bubbles (due to both potential effects and their direct wake) and the turbulence induced by the bubbles. The latter is essentially driven by the interactions between the bubble wakes [220, 219, 6, 223]. The mean kinetic energy varies approximately as $K \sim \alpha v_0^2$. The velocity fluctuations are strongly anisotropic, with the variance of the vertical velocity being more intense than that of the horizontal velocity. Their PDFs are non-Gaussian, with exponential tails and a strong dissymmetry between the upward and downward directions.

The structure of this flow is also characteristic, and the velocity spectrum exhibits a rapid $k^{-3}$ decay in a wavenumber range extending around the bubble diameters [137, 220, 5]. The origin of such a scaling law as well as its precise limits in the spectral domain remain poorly understood. From a dimensional point of view, we can write that the energy spectrum must be written as

$$E(k) = f^2 k^{-3} \qquad (3.66)$$

where $f$ is associated with the inverse of a timescale. Lance and Bataille [137] have proposed that the $k^{-3}$ regime is associated with an equilibrium between production and dissipation and that this frequency results from the characteristic shear rate of the wakes. Other flows also present a $k^{-3}$ spectrum. This is the case, for example, of two-dimensional turbulence at scales smaller than the energy injection scale. In this flow, the flow timescale is imposed by the conservation of the entrophy [130, 18]. Decaying turbulence subjected to intense rotation also develops a $k^{-3}$ spectrum with the timescale imposed by the rotation rate [24]. Another example concerns the turbulence under the wave surface and this time the timescale results from the frequency imposed by the swell [154, 257].

### 3.4.1 Simulation of the bubble swarm

Although the equations describing precisely this type of flow are relatively well known, their numerical simulation remains out of reach, due to the large spectrum of temporal and spatial scales involved. The smallest scales are a priori associated with the interfacial dynamics and the development of a very thin boundary layer around the bubbles, while
<figure><img src="image_39.png" /><figcaption>Figure 3.16: Cross-section of the vertical component of the liquid velocity field for $\alpha = 1\%$ and $\alpha = 7.5\%$. The blue points represent the position of the bubbles.</figcaption></figure>

the largest scales are related to the length of the wakes and the evolution of the collective dynamics of the flow that takes place. In order to simulate these flows, and to study the structure of the fluctuations of the liquid velocity field and the stability of the bubble swarm we proposed in Ref. [144] to abandon the precise description of the flow around the bubbles as well as the capillary effects while keeping a realistic dynamic of the downstream part of each wake. This modeling, based on the Euler-Lagrange approach, allows us to simulate flows with a large number of bubbles and to focus on the interactions between wakes. As we will detail below, the main difficulty of this type of calculation comes from the self-interaction of a bubble with its own wake. We have proposed a method enabling taking into account this effect and to calculate precisely the trajectory of each bubble. This method allowed us to obtain numerical simulations of the turbulence induced by a swarm of bubbles in agreement with the experiment, as illustrated in figure 3.16.

## Modeling

The use of the Euler-Lagrange approach amounts to considering a filtering of the field near the bubbles, as discussed at the beginning of the chapter. In this approach, the action of the dispersed phase on the flow is introduced as a volume source of momentum localized around the bubbles. The liquid velocity is given by the Navier-Stokes equations:

$$D_t \mathbf{u}_f = -\frac{1}{\rho_f} \nabla P_f + \nu \Delta \mathbf{u}_f + \frac{\mathbf{f}}{\rho_f} ; \quad \nabla \cdot \mathbf{u}_f = 0. \qquad (3.67)$$

where $\mathbf{u}_f$ represents a filtered velocity field around the bubbles. The volume forcing of the liquid phase is given by

$$\mathbf{f}(\mathbf{x}, t) = -\sum_{b=1}^{N_b} \mathbf{F}_{f \to b}(t) G_{\sigma}(\mathbf{x} - \mathbf{x}_b(t)) \qquad (3.68)$$

where $\mathbf{F}_{f \to b}$ is the momentum exchange rate between the fluid and the bubble $b$, and $N_b$ is the number of bubbles. $G_{\sigma}$ is the Gaussian kernel of the projection and $\sigma$ is its
characteristic size which is of the order of the diameter of the bubble. The latter is thus much larger than the mesh size: $\sigma \approx d > \Delta x$. Indeed, although the details close to the bubbles are filtered, the flow presents scales much smaller than the bubble size. These small-scale fluctuations result from the evolution of the turbulent wakes and their interactions.

The trajectory of the bubbles is obtained by solving Newton's equation (3.1) for each bubble. It involves the hydrodynamic force which depends on the velocity of the liquid (and its derivatives). This is called “two-way coupling” in the sense that the flow influences the bubbles and the bubbles influence the flow.

The use of such an approach raises some questions. (i) Which models should be adopted for the calculation of the hydrodynamic force on a bubble? (ii) Is such an approach able to reproduce the wakes? (iii) How to correct the self-induced force to obtain an adequate value of the hydrodynamic force on a bubble?

**Hydrodynamic force on the bubbles** As the flow is described on a smaller scale than the bubbles ($\ell \sim \Delta x < d$), the fluid motion directly induced by a bubble will be partly solved. However, the expression of the hydrodynamic forces already takes into account the fluid motion that the bubble induces in its vicinity (deviation of the streamlines, presence of a boundary layer and its possible detachment...). We assume that the usual expression of these hydrodynamic forces remains relevant, provided that their calculation is based on a fluid field in which the flow disturbance due to this bubble has been removed. We will note hereafter $\tilde{u}_{f,b}$ such a field. We will see below that the definition of $\tilde{u}_{f,b}$ is not trivial, and that its introduction remains an artifact for the calculation of hydrodynamic forces. Therefore rather than (3.3) giving the hydrodynamic force from the velocity described at a scale $\ell \gg d$ at the position of the bubble and its derivatives, we must consider:

$$F_{\text{fluid},b} = f[\tilde{u}_{f,b}(x=x_p(t),t) - u_p(t), \nabla \tilde{u}_{f,b}|_{x_p}, D_t \tilde{u}_{f,b}|_{x_p}, d_t u_p(t)], \quad (3.69)$$

involving the field $\tilde{u}_{f,b}$.

To illustrate the importance of this correction, let us specify that since the momentum exchanged with the bubble is injected over a volume of the order of $\sigma^3 \sim d^3$, the velocity of the liquid close to the bubble is of the order of the bubble velocity. Thus a naive calculation (i.e. without correction, $\tilde{u}_{f,b} = u_f$) would lead to a very low drag force. As this force does not oppose the buoyancy force, the bubble would present a non-physical acceleration which can be interpreted as a spurious self-induced force.

Let us return to the question (i). We have considered in Ref. [144] that the bubble is subject to the drag force, the added-mass force, the inertia force of the fluid, as well as the buoyancy. We have not retained the history force because it has a priori a negligible effect for large Reynolds numbers. On the other hand, when the velocity gradient is large at the scale of the bubble, the lift forces can certainly play a role. Similarly, the anisotropic effects of drag and added mass related to a non-spherical bubble are also important. We aim to reproduce the experiments performed at IMFT for millimetric air bubbles in water. Given the Reynolds number of the bubbles and the Morton number, the bubbles clearly adopt a non-spherical shape [165]. However, to simplify the modeling of the problem, we consider that the bubbles are spherical, assuming that in the case of the homogeneous cloud, the anisotropic aspects are not a priori essential. Consistently, we have as well not
retained the lift force<fnref n="8" />. Finally, considering that the density of the gas is very low we obtain for the dynamics of the bubble:

$$C_M \frac{dv_b}{dt} = \frac{3C_D}{4d} (v_b - \tilde{u}_{f,b}) |v_b - \tilde{u}_{f,b}| + (1 + C_M) \frac{D\tilde{u}_{f,b}}{Dt} - g + F_{1,b} \quad (3.70)$$

The drag coefficient is chosen, in agreement with the experiment of an isolated bubble, at $C_D = 0.35$ and the added-mass coefficient at $C_M = 0.5$ in coherence with the spherical bubble hypothesis. In this equation the bubble force is calculated from the corrected liquid velocity $\tilde{u}_{f,b}$.

Finally, the term $F_{1,b}$ is a repulsive force between bubbles. It is introduced to prevent the bubbles from overlapping and to ensure that the distance between bubbles remains greater than the characteristic size $\sigma$ of the momentum source.

The momentum exchanged between the bubble and the fluid via the volume force $f$ in (3.68) is given by the sum of the drag force and the added-mass force, the contributions of the Tchen and Archimedes forces being already taken into account in the pressure term in a way consistent with the zero divergence of the flow [59, 144].

**Correction of the self-induced force** The fluid velocity, corrected for the influence of the bubble $b$, is defined by introducing the perturbation due to the bubble:

$$\tilde{u}_{f,b}(x,t) = u_f - u_{f,b}^* \quad (3.71)$$

Because of the non-linearity of the system, this immediately raises the question of the definition of the perturbation $u_{f,b}^*$. It is indeed non-trivial to isolate the influence of a bubble among the fluctuations of the flow which include the effect of all the other bubbles. We propose here to define the perturbed field $u_{f,b}^*$ as the flow generated by an isolated “imaginary” bubble, in a liquid at rest, which would have followed the same trajectory and exchanged as much momentum with the liquid phase as the actual bubble $b$. Accordingly, $u_{f,b}^*$ is formally obtained by solving the following Navier-Stokes equation:

$$\nabla \cdot \mathbf{u}_{f,b}^* = 0, \quad (3.72)$$

$$\partial_t \mathbf{u}_{f,b}^* + \mathbf{u}_{f,b}^* \cdot \nabla \mathbf{u}_{f,b}^* = -\nabla p^* + \nu \Delta \mathbf{u}_{f,b}^* + \frac{1}{\rho_f} \mathbf{F}_{f \to b(t)} G_{\sigma}(x-x_b(t)), \quad (3.73)$$

where the last term on the right-hand side is the forcing calculated by taking into account the actual position of the bubble and the momentum actually exchanged between the $b$-bubble and the liquid phase. This definition describes only the perturbation directly generated by the $b$-bubble, but ignores indirect perturbations resulting from the fact that the $b$-bubble may also have affected other bubbles and nonlinear interactions within the liquid flow. Consequently, other definitions of $u_{f,b}^*$ can be proposed. For example, instead of isolating the trajectory of the $b$-bubble, one could define $\tilde{u}_{f,b}(x,t)$ as the flow generated by the motion of all bubbles except the $b$-bubble. Insofar as the Reynolds number $Re_b$ of the bubble and the volume fraction are finite, these two definitions of $\tilde{u}_{f,b}(x,t)$ are not equivalent. We believe that there is no exact, or unambiguous, way to define $\tilde{u}_{f,b}^*$, and this poses a fundamental problem for Euler-Lagrange approaches. At best the definition

<footnote n="8">⁸The value of the lift coefficient, and even its sign, being very dependent on the shape of the object, it would be very delicate to choose its value anyway</footnote>
of the hydrodynamic force should be consistent with the expression of $u_{f,b}^*$. But this raises the question of the experimental determination of the force, in order to establish a model of the force, in situations more complex than an isolated bubble in a well-controlled flow. In any case, computing $u_{f,b}^*$ directly from (3.72)-(3.73) implies solving one additional Navier-Stokes equation per bubble, which is not actually practicable and we must therefore rely on approximate models.

We refer to Ref. [144] for a review of the different approaches proposed in the literature. In general, the approaches proposed previously consider particles subjected only to the viscous drag force and assume that their Reynolds number is small, in order to express the perturbation of the velocity from the Green function of the Stokes equation (Stokeslet) [231]. In Ref. [144] we propose an integral model to calculate $u_{f,b}^*$ and its derivatives, which can be used to obtain the correct value of all the components of the hydrodynamic force and which is valid for the case of bubbles with large Reynolds number.

For the calculation of the hydrodynamic force, we only need to know $u_{f,b}^*$ at the position of the bubble $b$ and in its neighborhood (to calculate its derivatives). The main assumptions to obtain $u_{f,b}^*$ are that in the vicinity of the bubble (i) given the importance of the Reynolds number, one can neglect the viscous term in (3.73) and (ii) one considers that the flow is quasi-parallel. The details of the derivation can be found in Ref. [144], but after a few steps we obtain the following integral expression for $u_{f,b}^*$.

$$u_{f,b}^*(x,t) = \frac{1}{\rho_f} \int_0^t \mathbf{F}_{f \to b}(s) G_\sigma(x-x_b(s) + \ell_{adv}(t,s)) ds, \quad (3.74)$$

where $\ell_{adv}(t,s) = \int_s^t \tilde{u}_{f,b}(x=x_b(s'),s')ds'$. The velocity perturbation at a given position and time is obtained by integrating, over all previous instants, the momentum supplied by the bubble at the material point of the liquid at that specific position. The material point corresponding to the injection of momentum at an instant $s$ can be advected by the undisturbed flow, and will be found at a distance $\ell_{adv}(t,s)$ at the instant $t > s$. This $\ell_{adv}$ term is essential to guarantee the Galilean invariance of the model. This is illustrated in figure 3.17 showing a space-time diagram of the evolution of the perturbation. By substituting (3.71) and (3.74) in the dynamics of the bubble (3.70) we notice that the expression of the hydrodynamic force on a bubble depends on the history of the whole velocity field. This amounts to say, as we wrote at the beginning of the chapter (see Eq. (3.2)), that the hydrodynamic force must in general be written as a functional of the velocity field of the fluid described at a scale $\ell$ ($\ell$ can be greater or smaller than $d$). In fact, with the simple model (3.74) taken for the perturbation, one can write that the force depends only on the history of the velocity field at the position of the particle, and on its derivatives, which corresponds to the functional (3.3).

We do not detail here the discretization of (3.74). The details of its numerical implementation can be found in Ref. [144]. We just mention that we have developed an algorithm similar to the one used in the first chapter to efficiently compute the history integral involved for non-markovian stochastic process, by optimizing the amount of information to be stored and the computation time. With this method, the extra cost of computing the correction term is negligible.

**Wake of an isolated bubble** To validate this approach of modeling of the perturbation, we considered the case of an isolated bubble rising in a liquid at rest. In this simple case,
<figure><img src="image_40.png" /><figcaption>Figure 3.17: Spatio-temporal diagram of the evolution of the position of the bubble (continuous black line) and that of the material point where the momentum was injected at $t = t_1$ (dashed black lines). On the left: in the case of a zero carrier flow $\tilde{u}_{f,b}(x=x_b(t),t) = 0$; on the right: in the case of uniform advection by a uniform carrier flow $\tilde{u}_{f,b}(x=x_b(t),t) = -v_0 e_z$ with $e_z$ the vertical direction. The Galilean invariance imposes the preservation of the length as shown in both cases by the distance $\ell(t_1, t_2)$. The shaded area represents the region of the flow influenced by the momentum deposited at time $t = t_1$.</figcaption></figure>

we must have $\tilde{u}_{f,b} = 0$. This allows us to compare the wake generated by the bubble with the experimental results and to determine the free parameters of the simulation ($\sigma/d$ and a parameter $c_0$ involved in the discretization of (3.74)).

Without going into details (which can again be found in Ref. [144]), we have determined that for a range of resolution of the simulation $\Delta x/d < 1$ we have an optimal value for the ratio $\sigma/d$ around $\sigma/d \approx 0.25$ allowing reproducing the wake accurately with nevertheless a clear underestimation of the velocity near the bubble. Larger values of $\sigma$ further increase the filtering of the velocity near the bubble, while smaller values lead to a destabilization of the simulation due to a quantity of motion injected in a very concentrated area (the fluid velocity can thus become higher than that of the bubble).

We were able to verify that the model for the calculation of the perturbation works correctly by giving $\tilde{u}_{f,b} = 0$ as well as for its material derivative $D_t \tilde{u}_{f,b} = 0$. This allows us to calculate correctly the transient dynamics of the bubble and to obtain a correct rising speed and thus also to well predict the rate of momentum exchange with the liquid.

We could mention that the correction of the self-induced effect can be forgotten when $\ell \gg d$ whereas when $\ell < d$ this correction is essential. To be more precise we say that the correction is negligible if $|\tilde{u}_{f,b} - v_b| \gg |\mathbf{u}_{f,b}^*|$ and $|D_t \tilde{u}_{f,b} - d_t v_b| \gg |D_t \mathbf{u}_{f,b}^*|$. This can be tested a posteriori (once the simulation is done) by analyzing the trajectories. But we can also obtain a priori, in the simple case of the isolated rising bubble, that :

$$ \frac{|\mathbf{u}_{f,b}^*|}{|\tilde{u}_{f,b} - v_b|} \sim C_D \frac{d^2}{\sigma^2}. \qquad (3.75) $$

We conclude that it is essential to correct the fluid velocity for bubbles at high Reynolds number, since in this case $C_D = O(1)$ and $\sigma/d = O(1)$. Note that for particles moving at

low Reynolds number, an estimate obtained by the Stokeslet gives $\frac{|\mathbf{u}_{f,b}^*|}{|\tilde{u}_{f,b} - v_b|} \sim O(d/\Delta x)$
[231, 32], which means that the correction can be forgotten for small enough objects and a sufficiently coarse description of the velocity field.

It is interesting to note that although the coupling between the two phases conserves momentum, it does not conserve its energy [269, 247]. Indeed, the power $P_b$ of the hydrodynamic force working at the bubble velocity is a priori greater than the power $P_f$ of the diffuse force working at the fluid velocity:

$$P_b = \sum_b \mathbf{F}_{f \to b} \cdot \mathbf{v}_b > P_f = \int dx^3 \mathbf{f} \cdot \mathbf{u}_f. \qquad (3.76)$$

From a physical perspective, it is acceptable that energy is dissipated during the coupling. We consider a coarse description of the continuous phase, in which the strong velocity gradients in the close vicinity of the bubbles are not described. The dissipation of kinetic energy into heat that occurs in the region surrounding the bubble at scales smaller than $\sigma$ cannot be calculated from the resolved velocity $\mathbf{u}_f$. In the case of the isolated bubble, we can estimate analytically that

$$\frac{P_f}{P_b} = \frac{C_D}{64} \left( \frac{d_b}{\sigma} \right)^2. \qquad (3.77)$$

Consequently an important part of the mechanical energy is dissipated around the bubbles, in the boundary layer and the near wake.

### 3.4.2 Analysis of the bubble swarm

#### Presentation of the simulations

With this method we simulated the flow of the rising bubble swarm. The parameters correspond to a 2.5 mm air bubble in water. The Reynolds number based on the terminal velocity of an isolated bubble is $Re_0 = v_0d/\nu = 760$. A cubic domain of dimension $L/d = 70$ with tri-periodic boundary conditions is used. The characteristic size of the force projection kernel is $\sigma/d = 0.28$ and the resolution of the mesh is $\Delta x/d = 1/15$ which corresponds to 1024 points in each direction. We can notice that the number of mesh points per bubble can seem important for a method which does not try to solve precisely the dynamics around the bubbles. However, one must keep in mind that (i) the resolution with interface tracking methods (level-set or VOF type) for such a Reynolds number of bubbles, requires about 100 meshes per bubble (or even more) to capture the boundary layer which develops on the bubble [67] and (ii) the resolution is chosen here to capture the small scales which develop in the wakes, far from the bubbles, as we will see below. We have simulated this flow for volume fractions of $\alpha = 1\%, 2\%, 5\%, 7.5\%$ and 10% corresponding to a number of bubbles ranging from 6500 to 65000. See the visualization of this flow for $\alpha = 1\%$ and 7.5% in the figure 3.16. In this figure, we can see the wakes generated by the passage of each bubble, and their interactions giving birth to the agitation induced by the rise of a bubble swarm.

#### Comparison with the experiment

We show in figure 3.18, the PDF of the horizontal and vertical components of the liquid velocity, obtained by simulations for the different $\alpha$ and by the experiments of Ref. [220]. It is found that for both components the PDF presents an exponential decay and that
<figure><img src="image_41.png" /><figcaption>Figure 3.18: PDF liquid velocity for the horizontal velocity component (left) and the vertical velocity component (right) from the simulation for various $\alpha$ and comparison with the experiments of Ref. [220].</figcaption></figure>

the PDF of the vertical component is clearly asymmetric. We further observe that the normalized PDFs are nearly invariant with $\alpha$ as the central part of the experimental PDFs. This behavior is the signature of the turbulence induced by the interactions between wakes. For large fluctuations, the experimental PDFs show a second region characterized by a less steep exponential decay. This behavior has been attributed to the large localized fluctuations very close to the bubbles, as well as to the self-sustained oscillations of the bubble trajectories [222]. As this region is not described in our modeling, we indeed find that the second exponential part of the PDFs is not reproduced by the simulations.

For the same reason, the mean velocity of the bubbles decreases only very slightly with $\alpha$ according to the simulations, whereas experimentally, it is observed to decrease as $\langle v \rangle/v_0 \approx 0.6\alpha^{-0.1}$ [220]. Also the kinetic energies of the liquid and the bubbles are underestimated compared to the experiment.

Figure 3.19 compares the longitudinal spectra<fnref n="9" /> of the vertical and horizontal velocity $E_z(k_z)$ and $E_x(k_x)$ obtained experimentally and by simulations<fnref n="10" />. We can see that the spectra of the vertical and horizontal components are in fairly good agreement with the experiment. In particular, the simulations seem to reproduce a $k^{-3}$ evolution of the spectra as experimentally observed on small scales (large wavenumbers) and a $k^{-1}$ decay at large scales.

However, we note, on the one hand, an underestimation of the kinetic energy at small scales for the simulations. We attribute this underestimation to the lack of near-bubble resolution which leads to an underestimation of the power injected at the bubble scale (as discussed in the previous section). On the other hand, we also notice that the largest

<footnote n="9">⁹That is to say $E_x(k_x) = \int 1/2\phi_{xx}(\mathbf{k}')\delta(\mathbf{k}'.e_x - k_x)d^3\mathbf{k}'$ and $E_z(k_z) = \int 1/2\phi_{zz}(\mathbf{k}')\delta(\mathbf{k}'.e_z - k_z)d^3\mathbf{k}'$ with $\phi_{ij}(\mathbf{k}) = \sum_{\mathbf{k}'} \langle \hat{u}_i(\mathbf{k}')\hat{u}_i^*(\mathbf{k}') \rangle \delta(\mathbf{k} - (\mathbf{k}'))$ and $\hat{u}$ the coefficients of the Fourier series of the velocity field $\mathbf{u}_f(\mathbf{x}) = \sum_{\mathbf{k}} e^{i\mathbf{k}\cdot\mathbf{x}} \hat{u}(\mathbf{k})$.</footnote>

<footnote n="10">¹⁰With this approach, the spectra of the velocity field are very easy to obtain because the $\mathbf{u}_f$ field is smooth, whereas with a DNS type approach, or experimentally, the velocity of the liquid is not defined everywhere which poses some problems to make a spectral analysis. Here the approximations are made prior to the simulation, at the modeling phase, and there is no particular precaution to take for the calculation of the spectra.</footnote>
<figure><img src="image_42.png" /><figcaption>Figure 3.19: (Left) Longitudinal velocity spectra of the vertical component in the vertical direction ($E_z(k_z)$) (continuous lines) and of the horizontal component in the horizontal direction (dashed lines) from the simulations for various $\alpha$ ($E_x(k_x)$). Comparison, with the experiments of Ref. [220] for $\alpha = 2.5\%$ and $d = 2.5\,\text{mm}$ in blue, and with the power law $k^{-1}$ (gray dot-dashed line), $k^{-3}$ (gray dashed line) and $k^{-5/3}$ (gray dotted line) (Right) Frequency spectra of the liquid velocity at the bubble position from the numerical simulation and comparison with the experimental spectra of the liquid velocity of the flow past a random array of fixed spheres [6] in blue and with the $\omega^{-1}$ and $\omega^{-3}$ power laws.</figcaption></figure>

scale of the horizontal component is also underestimated, due to the absence of bubble trajectory oscillations which limit the redistribution of the fluid energy between these vertical and horizontal components.

It has also been reported that experimentally the spectra are invariant with the volume fraction and the bubble diameter. When the spectra of the numerical simulations are normalized by the injected power and by the viscosity, we observe the same invariance of the spectra of the numerical simulation with $\alpha$.

In figure 3.19 we also present the frequency spectra of the vertical liquid velocity measured at the position of the bubbles (the true velocity not the corrected velocity). These spectra is compared with the frequency spectrum of the velocity of the flow passing through an array of spheres held at a fixed position obtained in the experiments of Ref. [6] for a Reynolds number, based on the sphere diameter, of 600. Although these two flows are different, we can consider that this characterizes the fluctuations of the liquid in the reference frame of the bubble cloud. It can be seen that the simulations and the experiment also show a good agreement. For frequencies lower than $d/v_0$ the spectrum shows a $\omega^{-1}$ behavior while at high frequencies, the cutoff is much stronger with a slope close to $\omega^{-3}$. It is interesting to note that this behavior seems invariant with $\alpha$ in the simulations and that from the experiments these temporal spectra had been reported as invariant with the Reynolds number, provided that $Re > 200$ [6].

In conclusion, we have a simulation approach which, although imperfect, allows obtaining the main characteristics of the turbulence induced by the bubble swarm as it accounts for the main ingredient: the interactions between wakes.

The spherically averaged spectra of the velocity<fnref n="11" /> are shown in figure 3.20. Contrary to

<footnote n="11">¹¹i.e., $E(k) = \int 1/2\phi_{ii}(\mathbf{k}')\delta(|\mathbf{k}'| - k)d^3\mathbf{k}'$.</footnote>
<figure><img src="image_43.png" /><figcaption>Figure 3.20: Spherically averaged spectra of the velocity field E from the numerical simulation at various $\alpha$. Comparison with the power law $k^{-3}$ in gray dashed line.</figcaption></figure>

the longitudinal spectra presented in figure 3.19, the 3D spectra show a more complicated evolution with $k$ as well as a rather clear dependence with $\alpha$ at large scales. Several regions can be distinguished. The local maximum, located around $k\eta \approx 2$ on the figure, coincides, as we will see, with the scale of the bubbles which gives the cut-off scale of the energy injection. We see that for larger wavenumbers, a region in $k^{-3}$ clearly develops as $\alpha$ increases. The local minimum, located at large scales, corresponds to the wake scale. Between these two scales, the energy spectrum grows as $k^{2/3}$ for the 3 largest $\alpha$ and corresponds to the scales directly influenced by the wakes. The fact that the spherically averaged spectra E show such a qualitative difference at large scales with the spectra averaged over planes obviously indicates that the flow has a strong anisotropy at these scales. We will come back to the characterization of the anisotropy below.

## Characteristic scales

The flow being dominated by the wakes of the bubbles and their interactions, an essential scale of the flow is the characteristic length of the wakes. To determine the latter, we consider the mean field conditioned on the position of a bubble (equivalent to a spatial phase average):

$$ \langle \mathbf{u}_f \rangle_b(\mathbf{x}) = \frac{1}{T} \int dt \frac{1}{N_b} \sum_{b=1}^{N_b} \mathbf{u}_f(\mathbf{x} - \mathbf{x}_b(t), t) . \qquad (3.78) $$

This mean field is illustrated in figure 3.21 for the case $\alpha = 5\%$. This figure also shows the evolution of the vertical velocity along the vertical axis passing through the bubble for the different volume fractions, as well as for an isolated bubble. The first observation is that the wakes are much shorter in the case of the bubble swarm than the wake of an isolated bubble. By plotting the logarithmic derivative of the wakes (also on the figure 3.21) we see that the wakes present a self-similar evolution for all $\alpha$ and that the velocity presents an exponential decrease with $z$. This remarkable feature is in agreement with the experimental results presented in Ref. [225]. This exponential decay of the wakes is likely due to the diffusion of the vorticity between the wakes, leading to the cancellation of the vorticity as proposed by Ref. [108].
<figure><img src="image_44.png" /><figcaption>Figure 3.21: Left: cross-section of the vertical velocity conditionally averaged to the bubble position: $\langle u_{f,z} \rangle_b$ for $\alpha = 5\%$. Middle: evolution of the $\langle u_{f,z} \rangle_b$ along the vertical axis passing through the origin for the various $\alpha$ and for an isolated bubble (in gray dashed line) comparison with the exponential decay with a rate $L_w(\alpha)$ (in gray dotted line). Right: logarithmic derivative of the former quantity, $\langle u_{f,z} \rangle_b / d_z \langle u_{f,z} \rangle_b$, normalized by the characteristic wake length $L_w$.</figcaption></figure>

We choose the relaxation length of the exponential as the characteristic scale of the wakes $L_w$. The evolution of the ratio $L_w/d$ as a function of $\alpha$ is presented in figure 3.22. We can see that the length of the wakes shows evolution in $L_w \sim d\alpha^{-1/3}$. One can interpret this evolution as a simple geometrical relation, considering that the wakes tend to screen each other. It should be noted, however, that this is quite a notable difference compared to the experiments of Ref. [225] in which the characteristic length of the wakes is observed to be independent of $\alpha$. This certainly reflects that there is additionally a dependence of $L_w$ on $C_D$ since in the experiments the average speed of the bubbles decreases with $\alpha$.

From this characteristic length of the wakes, we define an inverse timescale $f = v_0/L_w$. This frequency $f$ can be considered as imposing a shear-rate scale to small scales $k > 1/d$. This assumption allows us to estimate the average dissipation rate in the simulations as

$$ \langle \varepsilon \rangle = \nu f^2. \qquad (3.79) $$

Equivalently, we can interpret $L_w$ as the Taylor length scale based on the velocity $v_0$, $\lambda = \sqrt{\nu v_0^2 / \langle \varepsilon \rangle}$. This is confirmed in figure 3.22 which shows that the evolution of $\lambda/d$ varies as $L_w/d$ in $\alpha^{-1/3}$.

The volume averaged power injected in the system corresponds to $\langle P_{tot} \rangle = n_b \langle P_b \rangle$ with $n_b$ the average number of bubbles per unit of volume and $P_b = F_{f \to b} \cdot v_b$ the power given by the bubble $b$. In the steady regime, this quantity is approximately given by $\langle P_{tot} \rangle = \alpha g v_0$ and as we have discussed above, it is larger than the power effectively received by the fluid in our simulations: $\langle P_{tot} \rangle > \langle P_f \rangle = \langle \varepsilon \rangle$. We will thus interpret $\langle P_f \rangle$ as the mechanical energy effectively injected in the wakes. Combining the previous relations, we find $\langle P_{tot} \rangle / \langle \varepsilon \rangle \sim \alpha R e_0 C_D (L_w/d)^2$ with $C_D = 4gd/3v_0^2$. Therefore at $C_D$ and $Re_0$ constant, the proportion of energy injected in the wakes decreases as $\alpha^{-1/3}$.

From the estimate (3.79) of the mean dissipation rate, we compute the dissipative scale $\eta = \nu^{3/4} \langle \varepsilon \rangle^{-1/4} = \sqrt{\nu/f} \sim d R e_0^{-1/2} \alpha^{-1/6}$. It is this scale that is used to normalize the spectra presented in figures 3.19 and 3.20.

For $\alpha \ge 5\%$ we observe that the kinetic energy of the liquid is invariant with $\alpha$ and is commensurate with $v_0^2$. Consequently we estimate the integral scale $L_{int} = \langle K \rangle^{3/2} / \langle \varepsilon \rangle$
<figure><img src="image_45.png" /><figcaption>Figure 3.22: Evolution of the characteristic scales $L_w$, $\lambda$, $\eta$ and $L_{int}$ in the simulations of the bubble swarm with $\alpha$. Comparison with the power law $\alpha^{-1/3}$ in dashed lines, $\alpha^{-1/6}$ in dotted lines and $\alpha^{-2/3}$ in dot-dashed lines.</figcaption></figure>

to varies as $L_{int} \sim d Re_0 \alpha^{-2/3}$. This behavior is observed for $\alpha \ge 5\%$ in the figure 3.22. Note that in the experiments of Ref. [220] it is observed that the liquid kinetic energy varies roughly as $\alpha v_0^2$. The discrepancy of the numerical simulations with the experiments is once again attributed to the absence of the fluctuations localized in the vicinity of the bubbles, which scales with $\alpha$.

## Spectral analysis of the bubble-induced turbulence

In order to identify the different regions of the spectra and to explain the observed scaling laws, we are interested in the spectral decomposition of the energy balance:

$$ \frac{d}{dt} E(k) = T(k) - D(k) + P(k). \qquad (3.80) $$

The terms of the right-hand side correspond respectively to the inter-scale energy transfer from a scale $k$ ($T$), the kinetic energy dissipation at a scale $k$ ($D$) and the energy injection ($P$). The expressions of these different terms are obtained from the Navier-Stokes equation (3.67). The transfer term $T$ is the contribution of the non-linear terms$^{12}$, $D(k) = 2\nu k^2 E(k)$ and $P(k)$ is the integral over the wave numbers $|k| = k$ of the real part of $\hat{f}_i \hat{u}_i^*$. At steady state, the left-hand term of (3.80) is zero, so $T = D - P$. We present in figure 3.23 the terms $P(k)$ and $D(k)$ for the various $\alpha$. We can see that the production term presents a cutoff for $k > 1/\sigma$ (we recall that $\sigma/d = 0.28$), and that on large scales it grows as $k^2$ for the largest $\alpha$, while it is roughly constant for small $\alpha$. Concerning the dissipation term, we notice that it also presents a peak around $k \sim 1/\sigma$. At large scales, the production dominates compared to the dissipation, which implies that $P(k) \approx -T(k)$. On the other hand, the dissipation dominates on small scales ($D(k) \approx T(k)$). From the absence of scale separation between the peaks of production and dissipation, we can conclude that this flow does not present an inertial zone. These budgets also show that there is no range of scales in which there is an equilibrium between $P$ and $D$. This contradicts the hypothesis

<footnote>$^{12}$ $T(k) = \int \sum_{k'} [-ik'_j(\widehat{u}_i \widehat{u}_j \widehat{u}_i^* + \widehat{P} \widehat{u}_j^*)] \delta(\mathbf{k}' - \mathbf{k}) \delta(|\mathbf{k}'| - k) d^3\mathbf{k} + C.C.$, where $+C.C.$ denotes the complex conjugate terms</footnote>
<figure><img src="image_46.png" /><figcaption>Figure 3.23: Evolution of P in continuous line and of −D in dashed lines for the various α. Comparison with αgv₀√2/π(kσ)² exp(−k²σ²) in dotted line. Left: the power density and wavenumber are normalized by αgv₀d and σ respectively. Right: the power density and wavenumber are normalized by the dissipative scales εη and η respectively.</figcaption></figure>

<figure><img src="image_47.png" /><figcaption>Figure 3.24: Spectra of the force E<sub>f</sub> for the various α and comparisons with (3.81) in dotted line.</figcaption></figure>

made in [137] to explain the presence of a k⁻³ zone in the velocity spectra. Furthermore, we notice that the k⁻¹ region of D, which corresponds to the k⁻³ region of the velocity spectra, is observed in the crossover between the production dominated scales and the dissipation dominated scales.

To interpret the behavior of the production term P, we study the spectra of the force applied to the flow, E<sub>f</sub>(k), corresponding to the spherical averaged of ˆf<sub>i</sub> ˆf<sub>i</sub>*. From the expression of the coupling force between the phases (3.68), we can obtain the following analytical expression for E<sub>f</sub>:

$$
E_f(k)/\alpha\sigma g^2 = \frac{1}{12\pi} (d/\sigma)^3 (k\sigma)^2 e^{-(k\sigma)^2}. \quad (3.81)
$$

For this calculation we have assumed that (i) the positions of the bubbles are independent
from each other and that (ii) the fluctuations of the rate of momentum exchanged between
the bubble and the liquid are small: ⟨F<sup>2</sup><sub>f→b</sub>⟩ ≈ ⟨F<sub>f→b</sub>⟩<sup>2</sup> = (ρgπd<sup>3</sup>/6)<sup>2</sup>. The spectra of the
force is presented in figure 3.24. We note that for all volume fractions the agreement with the proposed expression is relatively good. We distinguish two regions: a region which grows as $k^2$ which reflects the equipartition of the fluctuations of the forces at large scales ($k < 1/\sigma$) and an exponential cut-off imposed by the Gaussian projection kernel for $k > 1/\sigma$. We notice that for high volume fractions, the positions of the bubbles are not really independent anymore, because they cannot overlap, which explains the discrepancy with (3.81).

Similarly, considering that the fluid velocity at the position of the bubbles is independent between the different bubbles and of the order of the bubble velocity we can estimate that $P$ follows the same behavior as $E_f$. Namely, we can have:

$$P/\alpha g v_0 \sigma \sim (k\sigma)^2 e^{-(k\sigma)^2} \qquad (3.82)$$

Indeed $P$ presents an exponential damping for $k > 1/\sigma$, that overlaps for all $\alpha$ when normalized by $\alpha g v_0 \sigma$ as can be seen in the figure 3.23 (left). For $k < 1/\sigma$, $P$ increase roughly as $k^2$ in agreement with the previous relation for $\alpha > 2\%$. The underestimation of $P$ at large scales is attributed to the fact that the fluctuations of the velocity at the bubble position are important as can be seen on the frequency spectrum if figure 3.19 (right): $\langle \mathbf{u}_f(x = x_b)^2 \rangle > \langle \mathbf{u}_f(x = x_b) \rangle^2$. For small $\alpha$, $P$ no longer follows the proposed relation on large scales due to the presence of large structures in the flow leading to a significant correlation of the liquid velocity between distant bubbles.

For completeness, we present in figure 3.23 (right) the production and dissipation terms normalized by the dissipative scales $\langle \varepsilon \rangle$ and $\eta$ for the various $\alpha$. As noticed previously, with this normalization, we observed an overlap of the high wavenumber (typically $k > 1/2\eta$) part of $D$ for all $\alpha$. From the estimates of the characteristic scales of the previous section, we have $d/\eta = Re_0^{1/2} \alpha^{1/6}$, indicating that the gap between the production dominated scales and the dissipation dominated scales increases, slowly, with $\alpha$. It seems that the $k^{-1}$ subrange of $D$ is observed in this gap of scales, provided that $\alpha$ is large enough.

In conclusion, we consider that for $k < 1/d$ the flow structure is driven by the interactions between wakes, while in the range $1/d < k < 1/\eta$ the strong damping of the wakes imposes a shear scale.

This assumption of constant shear rate $f$ across scales allows us to explain the presence of a power law in $k^{-3}$ for the flow, thanks to a matching argument similar to that proposed by Kolmogorov in 1941. It is assumed that at scales that are small compared to the length of the wakes ($k \gg 1/L_w$) the structure of the flow depends only on the diameter of the bubbles $d$, the viscosity, and the shear rate $f$:

$$E = E(k; d, \nu, f). \qquad (3.83)$$

It is assumed that at scales larger than $\eta = \sqrt{\nu/f}$, one can neglect the effect of viscosity. Therefore, in this limit we can write:

$$E = E_I(k; d, f) = d^3 f^2 \Phi_I(kd) \qquad (3.84)$$

where $\Phi_I$ is a dimensionless function. Conversely, at scales much smaller than the bubble size, we will suppose that this diameter does not play a role anymore, and we will make the hypothesis that

$$E = E_S(k; \nu, f) = \nu^{3/2} f^{1/2} \Phi_S(k\eta) \qquad (3.85)$$
where $\Phi_S$ is another dimensionless function. Finally, if we assume that for a range of intermediate scales ($1/d \ll k \ll 1/\eta$), the two previous relations remain valid, we have $E_S(k) = E_I(k)$. Since $kd$ and $k\eta$ can vary independently, the previous equality can hold only if the following expressions are constant:

$$(kd)^3 \Phi_I(kd) = (k\eta)^3 \Phi_S(k\eta) = c \qquad (3.86)$$

This gives us for the velocity spectra:

$$E(k) = c f^2 k^{-3} \qquad (3.87)$$

in a range of scales where the shear rate can be considered constant.

The temporal spectra of the velocity seen by the bubbles (presented in figure 3.19 right) is influenced by the fact that the bubbles cut the wake of other bubbles. Thus the high frequencies of the temporal spectrum are dominated by the Doppler shift due to the high-speed crossing, of the order of $v_0$, of the dissipative structures of the flow. So using (3.87) and taking the argument of Ref. [255], with $\omega \sim v_0 k$, we can estimate the high frequency behavior of the frequency spectra:

$$E(\omega) = E(k) \frac{k}{\omega} = f^2 v_0^2 \omega^{-3}. \qquad (3.88)$$

The temporal spectra from both the experiments and the simulations present a $\omega^{-3}$ zone at high frequency.

We have seen that at large scales ($k \ll 1/d$), where the flow is dominated by wake interactions, there is a balance between production and inertia. At these scales we notice that the 1D spectra of the velocity present a $k^{-1}$ dependence. This means that each decade contains an equal amount of energy. This behavior can be explained by the intermittence of the wake passages, giving rise to an alternation between periods of activity and calm [158]. At these scales, the characteristic velocity no longer depends on a specific length scale and corresponds to the typical velocity of the bubbles $v_0$. These periods of activity (the wakes) are characterized by their self-similar character [11, 160] and present a variable intensity and duration, whereas the calm periods follow a Poissonian distribution reflecting the quasi-uniform distribution of the bubbles. The absence of a characteristic length leads directly to the absence of a characteristic time for the fluctuations. Hence the frequency spectrum of the velocity also shows a decay close to $\omega^{-1}$ at low frequency. As pointed out by Mandelbrot [155], these behaviors in $\omega^{-1}$ and $k^{-1}$ must also be connected to the non-Gaussianity of the velocity distributions as well as to a long-range correlation of the velocity.

To extend the discussion on the large scales, we need to take into account the anisotropy of the flow. For the characterization of the anisotropy, it is necessary to distinguish the fact that the energy can be carried mainly by one component of the velocity vector (anisotropy between components) from the fact that the fluctuations in certain directions can carry more energy (directional anisotropy) [233]. To characterize the latter, we consider a spherical coordinate system of the wave vector space, as schematized in the figure 3.25. The angle $\theta$ characterizes the orientation of the wave vector with respect to the vertical direction: $\sin\theta = k_z/|k|$ ($\sin\theta = 0$ corresponds to fluctuations in the horizontal direction
and $\sin \theta = \pm 1$ to fluctuations in the vertical direction). We then consider the directional spectra $E(k, \theta)$ which allows decomposing the energy of the fluctuations according to the wavelength and the orientation with respect to the vertical direction$^{13}$.

We show in the figure 3.25 the directional spectra of the velocity for $\alpha = 10\%$. It can be seen that at large scales ($k < 1/\sigma$) the energy is concentrated in the horizontal direction. This concentration is characteristic of vertically aligned tubular structures [47], which can be seen on the flow visualization in figure 3.16. At smaller scales ($k \gg 1/\sigma$), one can see that the directional spectra become invariant with $\theta$ indicating that the directional anisotropy tends to vanish.

To characterize the anisotropy between components, we present as well in the figure 3.25 the directional spectrum for the vertical component of the velocity and for the horizontal component$^{14}$. At large scales, we notice that the spectra of the vertical kinetic energy is very similar to that of the total kinetic energy, which indicates that at these scales the vertical component carries almost all the kinetic energy. This can be explained simply by the fact that the forcing due to the bubbles is essentially vertical. We can also note that the horizontal component of the velocity presents a very weak directional anisotropy. Finally, at small scales, we notice that the flow tends to become much more isotropic and presents both a decrease of the difference between the components and between the directions. This indicates that at scales where the energy injection is zero, there is a redistribution between the components that ensures a return to isotropy as $k$ increases.

It should be noted that when considering the spectra with angular dependence for the horizontal and vertical components, no $k^{-3}$ zone is distinguished. Thus, as noticed by Ref. [24] for the decaying turbulence under strong rotation, the $k^{-3}$ region for the spherically averaged spectra results from an average between the different directions and the different components, raising the question of the universality of the $k^{-3}$ power-law subrange.

If the structure of the flow becomes locally isotropic at scales much smaller than $\sigma$, one should expect the appearance of a $k^{-5/3}$ inertial range for the velocity spectra, if the local Reynolds is large enough, since at these scales there is no more energy injection. Such inertial range is not present in the numerical simulations reported here, but seems visible in the spectra obtained from the experiments of Ref. [220] (see figure 3.19 left). In case the velocity spectra first present evolution as $E(k) \sim f^2 k^{-3}$ followed by an inertial range $E(k) \sim \langle\epsilon\rangle^{2/3}k^{-5/3}$, the characteristic length of the crossover between these two regimes would be given by $\ell_I = \sqrt{\langle\epsilon\rangle/f^3}$. It is interesting to note that this length scale corresponds to the classical estimates of the scale from which a turbulent flow subject to mean shear can be considered as locally isotropic [53, 201]. Taking the usual Kolmogorov scale $\eta = \nu^{3/4}\langle\epsilon\rangle^{3/4}$, the extension of the inertial regime is given by $\ell_I/\eta = (\langle\epsilon\rangle/\nu f^2)^{3/4}$. In the simulations, as mentioned previously, no inertial range is present ($\ell_I = \eta$) and we have indeed $\langle\epsilon\rangle = \nu f^2$. It is likely that a larger power injection in the simulations (e.g. increasing $C_D$) would allow to obtain a separation between the scales of return to isotropy and the dissipative scales and thus to obtain a $-5/3$ range in agreement with the experiments. Nevertheless, even in absence of the $k^{-5/3}$ regime, this indicates that the

<footnote>$^{13}$In a more formal way $E(k, \theta)$ is defined by integration on all “longitudes” for a fixed “latitude” and modulus of the wave vector: $E(k, \theta) = \int 1/2\phi_{ii}(\mathbf{k}')\delta(|\mathbf{k}'| - k)\delta(k_z'/|\mathbf{k}'| - \sin\theta)d^3\mathbf{k}'/2\pi \cos\theta = \int_0^{2\pi} 1/2\phi_{ii}(k, \theta, \phi)k^2 \sin\theta d\phi/2\pi \cos\theta$. The normalization factor $2\pi \sin\theta$ is introduced to correct the geometrical effect due to the fact that a band near the poles covers a less important surface than a band near the equator.</footnote>

<footnote>$^{14}$Because of the axisymmetry of the flow we have $E(k, \theta) = E_z(k, \theta) + 2E_x(k, \theta)$.</footnote>
<figure><img src="image_48.png" /><figcaption>Figure 3.25: (a) Scheme of spherical coordinates in the Fourier space. (b) (c) and (d) Directional spectra of, respectively, the velocity vector, the vertical velocity component and a horizontal velocity component, for $\alpha = 10\%$. Note that the spherical spectra corresponds to an integral along a gray dashed line, while the longitudinal spectra in the vertical direction are obtained by integration along a black dashed line.</figcaption></figure>

rate of shear imposed by the bubble wakes control the relaxation to small-scale isotropy.

### 3.4.3 Summary

We summarize the results presented in this section. Firstly, regarding the coarse-grained numerical approach, we have shown that it is possible to use the Euler-Lagrange framework for simulating the dynamics of large objects, that is to say much larger than the grid size. For this, it is mandatory to correct the spurious self-induced effect in the dynamics of the bubbles. For that purpose we proposed a model that allows determining the undisturbed fluid velocity for objects moving with quite a large Reynolds number, and then to obtain a relative velocity of the objects used for the computation of the hydrodynamic forces. Although this approach has some limitations, since the fine scales in the vicinity of the bubbles cannot be reproduced, it is nevertheless a nice tool to study the collective phenomena at play in the bubble column flow. We have shown that the interacting wakes lead to a screening mechanism of bubble wakes, as observed experimentally. We have also shown that the $k^{-3}$ subrange of the energy spectra results from a rate of shear imposed by the bubble wakes. On larger scales the energy spectra evolves as $k^{-1}$ indicating a Poissonian
spatial distribution of the wakes along with long-range correlation.

## 3.5 Perspectives

Concerning the statistical analysis of the dynamics of particles in turbulence, presented in §3.2, we can mention the following tracks for future works. Firstly, we would like to complete the panorama of the effect of the various forces by accounting for the effect of gravity and lift forces. We have done some preliminary work on the subject during the PhD of Zhentong Zhang, where we observed a significant anti-correlation between the fluid inertia force and the lift force for bubbles in a turbulent flow with Stokes numbers larger than one. To explain this observation we proposed to express the particle dynamics equation as $a_{p,i} = -R_{ij}(u_{p,j} - u_{f,j}) + B_j$, where the acceleration of the particle dynamics is given by a relaxation tensor $R$, with a diagonal part that accounts for the drag and a skew-symmetric part due to the lift force, and a forcing term $B$ that accounts for fluid inertia effects and gravity. Qualitative explanation of these correlations were obtained by estimating the order of magnitude of the invariant of $R$. But a further analysis of this problem would be interesting as complex interplays between the various forces are present. It would also be useful to study the statistics of the particle acceleration and hydrodynamic force contributions conditional on the velocity of the particle, or of the fluid, similarly to the work presented in the first chapter for a fluid particles (see §2.3), and more specifically to analyze the effect of the Stokes number and density ratio on the correlation between velocity and acceleration.

Regarding the second part of this chapter dealing with the stochastic modeling for LES simulations (see §3.3), we would like to develop further stochastic model for the particles accounting for the, probable, dependence on the velocity, as discussed in the point just above. Based on the approach proposed in §2.4, we could propose an alternative framework to the decomposition between the norm/orientation presented in §3.3. Beside being simpler, we think that it can offer a more accurate and general description of the particle dynamics. A first step in this direction would be to consider the coupling of the model proposed in the first chapter for fluid particles with LES. Also of interest would be to consider the coupling with Lagrangian models for the velocity gradient tensor, as proposed for example in Refs. [89, 168, 114, 195]. With such an approach, we could obtain the evolution of the fluid velocity, acceleration and spatial derivatives along the path of particles. That opens the way to propose stochastic modeling of the particle dynamics accounting for further forces (e.g. lift forces), but also more complex sub-grid scale phenomena such as the fragmentation. Finally, let us mention that the goal of LES is to be able to predict flows in complex geometries, and in those conditions the local isotropy assumption at sub-grid scale may not hold, due for example to strong means shears. In that respect, it would be interesting to account for anisotropy and non-homogeneity of the flows, on the subgrid scale motion of the particles. Recently, Ref. [17] proposed stochastic modeling for the subgrid scale contribution with a preferred direction for the orientation imposed by the mean shear. Alternatively, we could extend the vectorial stochastic model proposed in the first chapter (2.4) to introduce dependence on the mean rate of shear tensor in the drift and diffusion terms.

Finally, let us mention some perspective concerning the simulations of the bubble swarm
presented in §3.4. First, regarding the approach, although we obtained a very good agree-
ment for the energy spectra with the experimental measurements, the simulation will cer-
tainly be more accurate if one adopts a more detailed description of the bubble dynamics
accounting for example for lift force, and non-spherical bubble shape (implying a tensorial
drag and added-mass coefficients). It will also be interesting to consider the model for the
large particles presented in §3.2.2 and 3.3.2. Although this model was derived for particles
with size lying in the inertial range of locally isotropic turbulence, it could be an efficient
way to account for fluid fluctuations at bubble scale in this flow. We could also improve
the self-induced force correction. The model presented above can be seen as a first-order
model, and one can consider a perturbative approach to account for correction terms at
the higher orders. Since the bubble Reynolds number remains moderate, it is possible
that only the second-order terms may be enough to account for the non-linear effect and
reproduce, at least qualitatively, the wake induced instability of the bubble path. Note
that, to our knowledge, no simulation in the Euler-Lagrange framework has reproduced
the zigzagging or spiraling motions of a rising bubble. Moreover, as discussed in §3.4.2 we
think that the absence of this intrinsic instability is one of the main missing features of the
simulations. Finally, we think that the model for the correction of the self-induced force
can also present some interest outside the framework of Euler-Lagrange approach. As such
model enables to define the fluid velocity at the particle position and thus the particle rel-
ative velocity, it can be of some help to propose a model for the hydrodynamic forces from
measurements of the particles acceleration and total forces obtained from experiments and
detailed DNS resolving the bubble interfaces.

We think that one of the interests of the kind of simulation presented in §3.4, is to provide
a way of studying the stability condition of a bubble swarm. In the simulation presented
above, the bubble cloud was observed to remain homogenous, but usually large-scale bub-
ble columns tend to develop some instabilities that eventually lead to non-homogenous
bubble distribution and large-scale recirculations. One way of studying the question could
be to consider the simulations subject to various initial distribution of bubbles and bound-
ary conditions to first analyze the sensitivity to injection conditions, and the presence of
shear. We can also focus on setting with a large scale forcing of the flow, as considered for
example in Ref. [5] or by Gabriel Ramirez (who his preparing his PhD at CEA). We could
then study under which conditions in terms of length and energy scales the peculiar dy-
namics induced by a cloud of rising bubbles could be predominant, or how the turbulence
will be modulated by the presence of the bubbles.
# 4 Magnetic fields, particles and turbulence

## Contents

<table><tr><td>4.1</td><td>Convection of a liquid metal subject to alternating magnetic field</td><td>97</td></tr><tr><td>4.1.1</td><td>Characteristic scales</td><td>99</td></tr><tr><td>4.1.2</td><td>Modeling</td><td>100</td></tr><tr><td>4.1.3</td><td>Characterization of the magnetic forcing</td><td>101</td></tr><tr><td>4.1.4</td><td>Experimental setup</td><td>102</td></tr><tr><td>4.1.5</td><td>Flow structures and heat flux</td><td>104</td></tr><tr><td>4.2</td><td>Perturbation of eddy-currents by inclusions in liquid metal</td><td>106</td></tr><tr><td>4.2.1</td><td>Model for the perturbation of the magnetic field due to an inclusion</td><td>107</td></tr><tr><td>4.2.2</td><td>Experimental setup</td><td>110</td></tr><tr><td>4.3</td><td>Perspectives</td><td>113</td></tr></table>

The coupling between electromagnetism and fluid mechanics, or MHD for magnetohydrodynamics, finds applications in very varied contexts ranging from liquid metal or ionized gas at a laboratory scale to conducting matter constituting the planets and the stars. In this chapter, we mainly discuss two projects, in which we have worked in collaboration with the CEA. The first one is about the magneto-thermoconvection of a liquid metal, in a context of prevention of nuclear power plant accidents. It has been the PhD subject of Sébastien Renaudière de Vaux and of Julien Guillou supervised with Wladimir Bergez and Philippe Tordjeman. The second one is related to bubble detection in a liquid metal, in the context of fast neutron reactors cooled by liquid sodium, and was studied during the PhD of Rafael Guichou, and Antoine Afflard and the PhD of Youssef Nasro Allah also supervised with Wladimir Bergez and Philippe Tordjeman. The common point of these two projects, and their main originality, is to involve flows subjected to an alternating magnetic field.

## 4.1 Convection of a liquid metal subject to alternating magnetic field

When an alternating electromagnetic field is applied to an electrically conductive fluid, typically a metallic liquid, electric currents are induced in it, they are called eddy currents or Foucault's currents. The induced currents generate both a heating of the fluid by Joule effect, and a volume force in the fluid resulting from their interactions with the applied magnetic field. These effects are used for induction heating or electromagnetic stirring applications. Additionally, when temperature heterogeneities are important, the buoyancy differences in the liquid can also lead to natural convection.
The objective of this project is to identify the different regimes that can be encountered when a metallic liquid is subjected to an alternating magnetic field and a different temperature on the boundary. In particular, we are interested in the influence of these regimes on the heat transfers. For this purpose, we consider the configuration of a liquid metal in a cylindrical container whose upper and lower walls are controlled in temperature (like in the Rayleigh-Benard convection, RBC) and surrounded by a coil through which an alternating current flows as depicted in figure 4.1. This device allows controlling the importance of the Laplace force and of the buoyancy force as well as the relative importance of the volume heating compared to the wall heating. During the thesis of Julien Guillou, we studied this flow by direct numerical simulations and experimentally, thanks to an experimental setup developed during the thesis. This study was motivated to simulate inductive heating experiments for studying severe accidents in the nuclear industry [116].

<figure><img src="image_49.png" /><figcaption>Figure 4.1: Diagram of the system studied with the cylinder containing the galinstan placed in the center of a coil. Several magnetic field lines are represented for a given frequency and allow, for this configuration, to visualize the skin effect.</figcaption></figure>

The stirring effects of an isothermal liquid metal by an alternating magnetic field depend on three dimensionless numbers, the magnetic Reynolds number, $Re_m = \frac{UL}{\eta}$, the shielding parameter, $S_\omega = \frac{2L^2}{\delta^2}$, and the Hartman number, $Ha = B_0L\sqrt{\frac{\sigma}{\rho\nu}}$ where L is the characteristic size of the cell (say its diameter), $B_0$ is the magnitude of the magnetic field, $\sigma$ and $\rho$ are the electrical conductivity and mass density of the fluid, $\eta = 1/\mu_0\sigma$ is the magnetic diffusion coefficient, $\mu_0$ being the permeability of vacuum, and $\delta$ is the skin depth. It has been shown that for a similar configuration and for $Re_m \ll 1$, the kinetic energy of the fluid presents a maximum value for $S_\omega \approx 20$ and decreases for higher $S_\omega$ [177, 248]. The Lorentz force, responsible for the flow forcing, is distributed in the skin depth, which is a function of the AC pulsation $\omega$: $\delta = \sqrt{\frac{2\eta}{\omega}}$. The presence of this optimal forcing is due to the fact that the intensity of the Laplace force increases with the frequency as the intensity of the induced currents, while the volume in which the Laplace force is significant is of the order of $\delta \times L^2$ and thus decreases with the frequency.

On the other hand, the natural convection in RBC configuration is controlled by two dimensionless numbers, the Rayleigh number $Ra = \frac{g\beta\Delta T_0 L^3}{\kappa\nu}$, and the Prandtl number, $Pr = \frac{\nu}{\kappa}$, where $\Delta T_0$ is the characteristic temperature difference, $\nu$, $\kappa$ and $\beta$ are the liquid viscous and thermal diffusion coefficients and the liquid thermal expansion coefficient, and $g$ is the gravitational acceleration. Grossmann and Lohse [96], have established the existence of turbulent flow regimes characterized by scaling laws of the Reynolds number
$Re$ and the Nusselt number $Nu$, as functions of $Ra$ and $Pr$, based on the kinetic and thermal dissipation rates in the bulk and in the boundary layer. Here $Re$ and $Nu$ are defined as $Re = \frac{UL}{\nu}$ and $Nu = \frac{hL}{\kappa}$ where $U$ is the referenced velocity, $h$ is the heat transfer coefficient and $\kappa$ is the thermal conductivity of the fluid.

For steady magnetic fields (or magnetic varying with very low frequencies), it is known that the Lorentz force damps the flow as it is mainly opposite to the component of the velocity perpendicular to the direction of the magnetic field. By accounting for the Hartmann layer thickness, the scaling relation of Grossmann and Lohse for standard RBC has been extended to the magnetoconvection subject to steady vertical magnetic field [281]. The recent simulation of Ref. [2] of the RBC with a constant and uniform external magnetic field show that flow pattern becomes quasi-two-dimensional for large Hartmann number $Ha$. We show that beyond a certain frequency this situation changes radically. The induced currents force the flow resulting in a very significant increase in the heat flux as well as a qualitative change in the flow structure compared to the Rayleigh-Benard configuration.

### 4.1.1 Characteristic scales

We consider here the flow of liquid metal in a cylindrical cell under both the effects of an imposed temperature difference and an ac magnetic field. Two velocity scales can be defined, derived from the buoyancy force and the Lorentz force, $U_B = \sqrt{g\beta\Delta TH}$ and $U_L = B_0/\sqrt{\mu_0\rho}$, respectively. In these expressions $H$ is the height of the cell, $\rho$ is the mass density of the fluid and $\mu_0$ is the vacuum magnetic permeability.

As well, two scales of temperature can be introduced, one given by the temperature difference imposed at the bottom and top walls of the cylindrical cell, $\Delta T_0$, and one corresponding to the characteristic temperature difference due to the Joule heating and obtained by an energy balance, $\Delta T_Q = \frac{\langle q_J \rangle}{\rho c_p} \frac{H}{U}$. Here, $\langle q_J \rangle$ is the time-volume average power density due to the Joule effect, $c_p$ is the specific heat of the liquid metal and $U$ is the characteristic liquid velocity, which is either $U_L$ or $U_B$. Depending on the choice of $U$ two definitions of $\Delta T_Q$ follow with different scalings at high and low frequencies:

* for $U = U_B$: $\Delta T_Q \sim S_\omega^2 Ha^2$ at $S_\omega < 1$, and $\Delta T_Q \sim S_\omega Ha^2$ at $S_\omega > 1$;

* for $U = U_L$: $\Delta T_Q \sim S_\omega^2 Ha$ at $S_\omega < 1$, and $\Delta T_Q \sim S_\omega Ha$ at $S_\omega > 1$.

Then one expects four main regimes depending on $U_L/U_B$ and $\Delta T_Q/\Delta T_0$:

* $\Delta T_Q < \Delta T_0$ and $U_L < U_B$: Flow is driven by the buoyancy force corresponding to RBC;

* $\Delta T_Q > \Delta T_0$ and $U_L < U_B$: dynamics are controlled by the buoyancy force and the volume heating by Joule effect;

* $\Delta T_Q < \Delta T_0$ and $U_L > U_B$: the Lorentz force governs the dynamics, and the heat flux is due to the imposed temperature difference $\Delta T_0$;

* $\Delta T_Q > \Delta T_0$ and $U_L > U_B$: the Lorentz and the heating by Joule effect dominate.
### 4.1.2 Modeling

The magnetic induction in a liquid metal results from the time variation of $B(r, t)$ and the velocity of the liquid. Owing to the linearity of the Maxwell equation, the magnetic field in the liquid can be expressed as the sum of the magnetic field in the absence of motion, $B_0(r, t)$, and the perturbation due to the velocity field, $b(r, t)$:

$$B = B_0 + b. \qquad (4.1)$$

The induction equation for $b$ is obtained from the Maxwell equations and the Ohm law assuming that the electrical conductivity is constant in the liquid and that the displacement currents are negligible:

$$\partial_t b = \eta \nabla^2 b + \nabla \times (u \times b) + \nabla \times (u \times B_0). \qquad (4.2)$$

The last two terms represent the induction of the magnetic field due to the motion of the liquid metal. The flow of the liquid metal is given by the Navier-Stokes equations, including the additional buoyancy force (expressed with the Boussinesq approximation) and the Lorentz force and is supplemented by the temperature equation with Joule heating:

$$\partial_t u + u \cdot \nabla u = -\frac{1}{\rho} \nabla P + \nu \Delta u + \beta T g + \frac{1}{\rho \mu_0} B \cdot \nabla B, \qquad (4.3a)$$

$$\frac{\partial T}{\partial t} + u \cdot \nabla T = \kappa \Delta T + \frac{\mathbf{J}^2}{\rho c_p \sigma}, \qquad (4.3b)$$

$$\nabla \cdot u = 0. \qquad (4.3c)$$

In the momentum equation (4.3a), $P$ is the total pressure including the magnetic pressure $\nabla B^2/2\mu_0^1$. The boundary conditions are a no-slip condition on the walls for the velocity, a Dirichlet conditions on top and bottom walls for the temperature with a difference $\Delta T_0$ on the two sides, and an adiabatic condition on the vertical wall. For the induction equation (4.2), the boundary conditions are integral relations ensuring the matching with the solution outside the metal $\nabla^2 b = 0$, with continuity of $b$ and its derivatives (in the absence of surface currents) at the boundary [112, 99].

The second term on the right-hand side of the induction equation (4.2), $\nabla \times (u \times b)$, can be neglected when $Re_m \ll 1$. At small frequency, i.e. $S_\omega \ll 1$, and $Re_m \ll 1$, eq. (4.2) reduces to the quasi-static diffusion equation ($\eta \nabla^2 b = -\nabla \times (u \times B_0)$). In the quasi-static approximation, the eddy currents are obtained from $B_0$ explicitly, $J = \sigma(\nabla\Phi + u \times B_0)$, where the electrical potential is obtained by a Poisson equation ensuring that $\nabla \cdot J = 0$. At high frequency, i.e. $S_\omega \gg 1$, the quasi-static assumption is not valid. We can decompose the current density in the liquid as $J = J_0 + j$, where $J_0$ denotes the current density in absence of flow induced by the time variation of the magnetic flux and $j$ is, according to the Ohm law, $j = \sigma(e + u \times B_0 + u \times b)$, with $e$ the perturbation of the electrical field due to the flow. For large enough frequency and small magnetic Reynolds number, $j$ can be neglected compared to $J_0$, as we can estimate that $O(|J_0|/|j|) = \sqrt{S_\omega/R_m}$. Therefore for $Re_m \ll 1$ and $S_\omega \gg R_m^2$ it is possible to completely decouple the resolution of the flow from the resolution of the magnetic field. The magnetic field provides sources of

<footnote n="1">¹The Lorentz force can be expressed as $F_L = J \times B = 1/\mu_0 B \cdot \nabla B - \nabla B^2/2\mu_0$, assuming the liquid metal is incompressible, the second term contributes only to the hydrostatic pressure.</footnote>
momentum and heat for the liquid with no influence of the liquid motion on these sources. Therefore as in the quasi-static approximation it is not required to solve the induction equation (4.2).

The magnetic and electrical fields, $B_0$ and $E_0$, applied to the system, at rest, are solutions of the set of equations:

$$ \nabla \cdot B_0 = 0, \qquad (4.4a) $$

$$ \partial_t B_0 = -\nabla \times E_0, \qquad (4.4b) $$

$$ \nabla \times B_0 = \mu_0(J_0 + J^e(t)), \qquad (4.4c) $$

$$ J_0 = \sigma E_0, \qquad (4.4d) $$

where $J_0$ and $J^e(t)$ are respectively the current density in the liquid metal and in the coil. For the later we impose its amplitude $J_e$ and pulsation $\omega$ which fix $Ha$ and $S_\omega$. The magnetic field $B_0$ within the cell and outside is computed with Comsol for the various values of current intensity and frequency. The Maxwell equations (4.4) have been solved in 2D axisymmetric geometry for $H = D = 0.1 \text{ m}$ and a coil of rectangular section $5 \times 5 \text{ mm}^2$ having 19 turns and 3 layers.

The equations (4.3a)-(4.3c) are solved with the finite volume code Jadim. The magnetic field computed with Comsol, with a resolution much finer than the DNS grid, is sampled at the vertices of the DNS grid. The Comsol simulations provide the magnitude and phase of the magnetic field inside the liquid. Consequently, the magnetic field and thus the momentum and heat sources can be computed at each instant in the DNS simulations. The details of the numerical simulations can be found in Refs. [100, 99].

### 4.1.3 Characterization of the magnetic forcing

In figure 4.2 we present the spatial distribution of the norm of $1/\mu_0 B_0 \cdot \nabla B_0$ for $S_\omega = 15$.

This figure shows that the force is dominant close to the two edges, top and bottom, of the cylinder. The right part of Fig. 4.2 presents the amplitude of the Joule power density

$q_J = \frac{J_0^2}{\sigma} = \frac{\eta}{\mu_0}(\nabla \times B_0)^2$ due to Joule effect also for $S_\omega = 15$. It is seen that the Joule heating is located in the skin depth close to the vertical wall.

<figure><img src="image_50.png" /><figcaption>Figure 4.2: Distribution of $|\frac{1}{\mu_0} B_0 \cdot \nabla B_0|$ responsible for the fluid motion (left part of the cell cross-section) and $q_J$ (right part) for $I_0 = 1 \text{ A}$ and $S_\omega = 15$.</figcaption></figure>
<figure><img src="image_51.png" /><figcaption>Figure 4.3: Left: Joule heating total power $Q_J$ in the liquid metal volume normalized by $Q_c$ versus the shielding parameter $S_\omega$. Right: Volume-average Lorentz force normalized by $F_B$ versus the shielding parameter $S_\omega$. Each curve is computed at $I_0 = 1$ A; the power laws in $S_\omega$ are mentioned on each curve. The colored domain indicates the range of variation of $S_\omega$ for the DNS.</figcaption></figure>

The total power deposited by Joule effect in the liquid metal, $Q_J = \int q_J dV$, normalized by the heat transferred by pure conduction, $Q_c = k\Delta T_0 \frac{\pi D^2}{4H}$, is plotted as a function of $S_\omega$ in figure 4.3. This curve shows that there are two asymptotic behaviors: the power scales as $S_\omega^2$ for $S_\omega < 1$, and $\sqrt{S_\omega}$ for $S_\omega > 100$. Additionally, the spatially averaged amplitude of the Lorentz force, normalized by the buoyancy force, $F_B = \rho g \beta \Delta T_0 \frac{\pi D^2 H}{4}$, scales as $S_\omega$, for $S_\omega < 1$, and is constant for $S_\omega > 100$. We find that these scaling relations are the same as in the case of an infinite cylinder. In this work, all the simulations are realized in the intermediate region, i.e. $1 < S_\omega < 100$.

### 4.1.4 Experimental setup

During the PhD of Julien Guillou, we develop an experimental setup that allows us to study the dynamics of a metal liquid in a Rayleigh-Bénard cell subjected to an ac magnetic field combined with a thermal gradient (see schematic representation in figure 4.4).

The convection cell is a cylinder with a unit aspect ratio and internal diameter $D = 10$cm. The top and bottom walls are two silicon carbide (SiC) disks and the lateral wall is made in polyether ether ketone (PEEK). Both these materials are good electrical insulators, and are therefore not affecting the magnetic field. SiC has a high thermal conductivity contrary to PEEK, so we can consider that the lateral wall is adiabatic whereas the top and bottom plates are at constant temperature. The two plate temperatures are set by circulation of water with different imposed temperatures.

A copper coil with an internal cooling (made by SEF) with a height of 10.2 cm and an external diameter of 17.4 cm, surrounds the convection cell. The coil is made of 57 turns with four insulated layers. The coil power supply is obtained by transforming the three-phase electrical power of the laboratory in continuous current and then to an alternating current with specified amplitude and frequency thanks to a H-bridge coupled with a Pulse With Modulated controller (made by Arcel). The coil with the cylinder fill with galinstan behaves as a low-pass filter with a cut-off frequency at about 11 Hz. To
<figure><img src="image_52.png" /><figcaption>Figure 4.4: Diagram of the experimental device setup. (a) : Thermocouples, UDV transducer and flowmeter data acquisition system. (b) : Thermal regulation of the bottom side of the RBC cell. (c) : Rayleigh-Bénard cylinder heated by induction and position of thermocouples and the UDV transducer (green). A water circulation fixes the temperature difference between the top and bottom walls; flowmeters record the water flow rate in the disk. (d) : coil power supply, capacitors, oscilloscope and function generator. (e) : coil cooling device.</figcaption></figure>

perform experiments at higher frequency, we connected a rack of capacitors in series with the coil. We have used up to three 70 µF capacitors mounted in parallel in the rack. The number of capacitors is variable to adapt the resonance close to the current frequency. We were then able to drive the intensity of the current $I_0$ between 2 and 67 A and the frequency $f$ between 15 and 1000 Hz. Note that the experiment is designed to reach current intensity up to 500 A, but the capacitor used in the first experiments presented here does not allow too high currents.

The heat flux on both bottom and top plates are obtained by measuring the flow rate of the circulating water and its temperature difference between inlet and outlet. To measure the temperature distribution in the liquid, three rings of 11 type T thermocouples are placed in the cell close to the wall and at three different heights: $H/4$, $H/2$ and $3H/4$. 3 thermocouples are also positioned along the axis of the cylinder at the same heights. Finally, an acoustic transducer (UDV) with a diameter of 8 mm and placed at 1 cm from the wall, allows measuring the vertical distribution of the convection velocity close to the cylinder wall.

The details of the setup and the experimental protocol are presented in Ref. [99].

Experiments have been realized for $15 \le f \le 1000$ Hz, $2 \le I_0 \le 67$ A and $2 \le \Delta T_0 \le 11$ K. Under these conditions, we have $6 \le Ha \le 200$, $1 \le S_\omega \le 70$, $2.3 \times 10^5 \le Ra \le 3.1 \times 10^6$ for galinstan, and $3 \times 10^7 \le Ra \le 2 \times 10^8$ for water.
<figure><img src="image_53.png" /><figcaption>Figure 4.5: Left: Streamlines in vertical cross section of the time and azimuth averaged velocity for $I_0 = 100$Å and $S_\omega = 34$ computed by DNS. Right: standard deviation of the vertical velocity in vertical cross-section for $I_0 = 50$Å and $S_\omega = 10$, obtained from time and azimuthal averaging.</figcaption></figure>

### 4.1.5 Flow structures and heat flux

We focus on the regime dominated by the Lorentz force ($U_L > U_B$), and heat transfer controlled by the wall temperature difference $\Delta T_0 > \Delta T_Q$. Hence we study the dynamics of the liquid metal under forced convection by Lorentz force, perturbed by Joule heating and buoyancy force.

When imposing an alternating magnetic field, the global flow patterns are drastically modified compared to the RBC flow. From the numerical simulation, we show that in the Laplace force forced convection regime, the flow is organized in two toroidal recirculation cells in agreement with the literature [185, 63]. This is illustrated in the figure 4.5, showing the average vertical velocity for $Ha = 301$ and $S_\omega = 34$ in a vertical plane of the vessel. It is seen that both cells have an opposite azimuthal vorticity. For $z < H/2$ the liquid metal goes upward close to the wall and goes downward in the bulk, and conversely, for $z > H/2$. This double-torus flow structure appears very robust for all intensity and frequency. Only, for the lowest intensity consider in our DNS (10Å), the torus appears and disappears alternatively, showing that the flow is characterized by a mixture between free convection and forced convection.

The toroidal cells are separated by a mixing zone located at mid-height plane. In this region, the flow presents on average the structure of a plane jet pointed radially inward, and subject to mean vertical temperature gradient. As the mean vertical velocity is zero in this region, the momentum and heat transfer between both half of the vessel, are controlled by excursion or bursting of the recirculation cells across the midplane, as seen in figure 4.5 showing the standard deviation of the vertical velocity for $Ha = 150$ and $S_\omega = 10$. This figure points out that the velocity fluctuations are maximum in the mid plane and close to the wall, where the Joule heating is the largest. In this region, the characteristic value of the RMS is of the order of $U_L/10$. We mentioned that the normalized variance map remains similar whatever the $Ha$ value, at high frequency ($S_\omega \ge 10$).

We present in Fig. 4.6 the average temperature of the three rings of thermocouples at various $Ha$ and $S_\omega$ as well as the profile of the temperature averaged over the horizontal plane from the DNS. We observe that DNS and experiments present the same trends in
<figure><img src="image_54.png" /><figcaption>Figure 4.6: Vertical profile of the average temperature. The solid lines represent the average of the dimensionless liquid temperature from the DNS for different values of Ha and S<sub>ω</sub>: Ha = 150 and S<sub>ω</sub> = 10 (orange), Ha = 150 and S<sub>ω</sub> = 51 (green), Ha = 301 and S<sub>ω</sub> = 68 (blue) and RBC (red). The points represent the measurements of the experimental average dimensionless temperature of the three rings of thermocouples for Ha = 56.4 and S<sub>ω</sub> = 34.1 (square), Ha = 65.3 and S<sub>ω</sub> = 68 (star), Ha = 113 and S<sub>ω</sub> = 50.9 (cross) and RBC (red dots).</figcaption></figure>

Ha and S<sub>ω</sub> in presence of the Lorentz force, the average temperature display a two-plateau profile. Each toroidal recirculation cells present different average temperatures, showing that within a recirculation cell the temperature is rapidly homogenized whereas the heat transfers between the two cells is not as efficient. When the Joule heating becomes significant, the average temperatures of the two cells shift to higher values and the difference between the two plateau temperatures decreases.

We consider the evolution of the average heat flux at the bottom and top walls obtained from DNS and the experiments. Based on the standard definition, Nu is equal to the ratio of these fluxes to the flux in absence of fluid motion and Joule heating, Q<sub>c</sub> = kΔT<sub>0</sub> πD<sub>c</sub><sup>2</sup> / 4H. From the mean thermal budget, the difference between the bottom and top Nusselt numbers is given by:

$$Nu^{+} - Nu^{-} = Q_{J}/Q_{c}, \qquad (4.5)$$

where the evolution of Q<sub>J</sub>/Q<sub>c</sub> with S<sub>ω</sub> is given in the left panel of figure 4.3, Note that Q<sub>J</sub>/Q<sub>c</sub> also varies as Ha<sup>2</sup>.

We observed that the heat flux increases significantly with respect to RBC when the Lorentz force is applied. This is visible even for small frequency and intensity. For example, at I<sub>0</sub> = 10A and S<sub>ω</sub> = 1, the Nu is increased by a factor ≈ 1.5.

To characterize the forced convection we introduced a Péclet number defined with a characteristic velocity U<sub>ω</sub> = U<sub>L</sub>f(S<sub>ω</sub>), Pe<sub>ω</sub> = U<sub>ω</sub>H/κ. The function f(S<sub>ω</sub>) characterizes the influence of the frequency on the Lorentz force. It behaves as f(S<sub>ω</sub>) ~ S<sub>ω</sub> for S<sub>ω</sub> ≪ 1 and for S<sub>ω</sub> ≫ 1 it tends to a constant. The evolution of the function f(S<sub>ω</sub>) is obtained by solving the Maxwell equations and is shown in the right panel of figure 4.3.

Figure 4.7 displays the Nu variations with Pe at the top wall for the experimental and numerical data. We first observe that all the results merge in a unique curve which presents two regimes: for S<sub>ω</sub> < 10 and Ha < 30, which corresponds to Q<sub>J</sub>/Q<sub>c</sub> < 1, Nu is constant. This result can be understood by considering that U<sub>L</sub> ≲ U<sub>B</sub> and the heat transfer is mainly driven by ΔT<sub>0</sub>. For S<sub>ω</sub> ≥ 10 and Ha ≥ 30, which corresponds to Q<sub>J</sub>/Q<sub>c</sub> ≥ 1, Nu scales
<figure><img src="image_55.png" /></figure>

Figure 4.7: Evolution of $Nu^+$ vs. $Pe_\omega$ in Lorentz forced convection. The red points correspond to experimental data for $Q_J/Q_c > 1$. The blue points correspond to experimental data for $Q_J/Q_c < 1$. The square gray points correspond to DNS. For $Q_J/Q_c > 1$, $Nu \sim Pe^{2/3}$.

as $Pe^{2/3}$. Under this condition, $U_L > U_B$ and the Lorentz force controls the heat transfer by producing two toroidal recirculation cells. Note that the heat flux on the bottom wall can be obtained from (4.5). As long as the Joule effect can be neglected ($Q_J/Q_c \ll 1$), $Nu^+ \approx Nu^-$. When the Joule heating becomes significant, for high frequency and current intensity, the difference between the heat flux at the bottom and top rise and is directly given by eq. (4.5).

## 4.2 Perturbation of eddy-currents by inclusions in liquid metal

In this project we focus on the modifications of the magnetic field caused by the presence of a dispersed phase in a liquid metal. More specifically we are interested in electromagnetic measurements of the dispersed phase properties such as the void fraction, the characteristic size of particles or bubbles, and their velocity. Such measurements, even for a single inclusion, remain challenging because of the opacity of liquid metal. In this goal, various technics have been developed, based and acoustic method [70], X-ray imaging [4, 121, 122], or perturbation of electromagnetic fields [151]. Here we consider the use of an Eddy Current Flow Meter (ECFM) to detect and characterize the voids in a liquid metal. ECFM is currently used for measuring the flow rate of liquid metal flows in pipes [244]. In this device, an alternating magnetic field is generated in the flow by a primary coil. The fluid motion induces small eddy currents that perturb the magnetic field. These perturbations induced a variation of the magnetic flux that can be measured on the variation of electromotive force (emf) on a secondary coil.

More specifically, we consider the design depicted in figure 4.8, which consists of three external coils surrounded the pipe. A primary coil generates an alternating magnetic field in the flow, and the two secondary coils, located on each side of the primary coil, measure the difference of magnetic flux in two cross-sections of the flow. This configuration present the advantage of being easy to set up and not perturbing the flow. In the thesis of Mithlesh Kumar [132, 133] advised by Wladimir Bergez and Philippe Tordjman, it was shown that that the electrical potential difference between the two secondary coils of the ECFM, can be decomposed, at first order, as two independent contributions, one related to the global
<figure><img src="image_56.png" /><figcaption>Figure 4.8: The ECFM is composed of three coils surrounding the pipe. P is the primary coil, S₁ and S₂ are the secondary coils; J is the eddy current field generated by P which is perturbed by the presence of the non-conducting inclusion (in blue).</figcaption></figure>

velocity of the conductor, characterized by a magnetic Reynolds number $Re_m$, and the second due to the variation of the void fraction $\alpha$. In the thesis of Rafaël Guichou [97, 98] the perturbation of the magnetic flux by the presence of an electrically non-conducting spherical particle in the liquid metal has been studied theoretically and experimentally. For low pulsations of current in the primary coil, it was shown the perturbation of the magnetic vector potential due to the inclusion satisfies a Poisson equation. Solving this equation, one can model the voltage in the secondary coil as a function of the particle diameter, its radial position in the pipe and the primary current frequency. This model is in good agreement with experimental results, for a small shielding parameter, $S_\omega \ll (R/R_b)^2$, where the shielding parameter is defined as $S_\omega = \omega R^2/\eta = 2(R/\delta)^2$, with R the radius of the pipe, $\delta = \sqrt{2\eta/\omega}$ the skin depth, and $R_b$ it the particle radius. In the post-doctorate of Youssef Nasro Allah and the PhD of Antoine Afflard, we are developing this approach to account for inductive phenomena and extend the validity to higher frequency. We also performed numerical solutions and experiments to confirm the model. These results are presented below. We are also addressing the inverse problem of finding the size and position of the bubbles from the voltage signal of the secondary coils.

### 4.2.1 Model for the perturbation of the magnetic field due to an inclusion

We consider, a non-conducting particle of diameter d traveling along an infinite cylindrical pipe filled with a liquid metal and passing through an ECFM, as shown in figure 4.8. The effect of the traveling void inclusion is modeled with the vector and scalar potentials $\mathbf{A}$ and $\phi$. Their equations in dimensionless form, satisfying the gauge condition $\nabla \cdot \mathbf{A} = 0$ reads:

$$ \nabla^2 \mathbf{A} = nS_\omega \frac{\partial \mathbf{A}}{\partial t} - nRe_m \mathbf{u} \times \nabla \times \mathbf{A} + n\nabla\phi - \mathbf{j}_e, \quad (4.6) $$

$$ \nabla^2 \phi = nRe_m \nabla \cdot (\mathbf{u} \times \nabla \times \mathbf{A}), \quad (4.7) $$
<figure><img src="image_57.png" /><figcaption>Figure 4.9: Distribution of $A_{0,\theta}(r, z)$. Right: real part. Left: imaginary part.</figcaption></figure>

where $n$ is equal to 1 in the liquid metal and 0 elsewhere, $\mathbf{u}$ is the velocity field in the liquid metal and $\mathbf{j}_e$ is the external electrical current density applied in the primary coil. Here the magnetic Reynolds number is defined as $Re_m = UR/\eta$, with $U$ the average velocity of the liquid metal in the absence of a particle. At infinity we have the condition $|\mathbf{A}| = 0$. The source term $\mathbf{J}_e$ is modeled as a harmonic in time, unit azimuthal current density uniformly distributed in the volume corresponding to the primary coil. The advantage of considering the vector potential is that the emf on the secondary coils is directly obtained by computing the circulation of $\mathbf{A}$ on these coils. The emf (in dimensional form) induced by one turn of a coil is

$$ \frac{emf}{\omega\mu_0 J_e R^3} = d_t \oint_c \mathbf{A} \cdot d\mathbf{l} \qquad (4.8) $$

where $c$ is the contour of the turn of the coil.

The main hypothesis is to consider that the velocity and void position are quasi-static compared to the frequency of the current. We will further assume that $Re_m$ and the void fraction $\alpha$ are small. With these approximations, we can obtain the linear response of the ECFM. For that we decompose the potential vector in $\mathbf{A} = \mathbf{A}_0 + \mathbf{A}_u + \mathbf{A}_\alpha + O(Re_m^2, \alpha^2, \alpha Re_m)$ where $\mathbf{A}_0$ is the vector potential in the absence of flow and particle, $\mathbf{A}_u$ is the perturbation of the vector potential due to the liquid metal flow without any particle, and $\mathbf{A}_\alpha$ is the perturbation due to the presence of the particle<fnref n="2" />. According to [132], we have $O(|\mathbf{A}_u|/|\mathbf{A}_0|) = Re_m$ and $O(|\mathbf{A}_\alpha|/|\mathbf{A}_0|) = \alpha \sim (R_b/R)^3$. With the previous approximations, the scalar potential is zero and the equations for the Fourier transform in time of $\mathbf{A}_0$, $\mathbf{A}_u$ and $\mathbf{A}_\alpha$ are

$$ \nabla^2 \mathbf{A}_0 - in_0 S_\omega \mathbf{A}_0 = \mathbf{j}^e, \qquad (4.9) $$

$$ \nabla^2 \mathbf{A}_u - in_0 S_\omega \mathbf{A}_u = -n_0 Re_m \mathbf{u} \times \nabla \times \mathbf{A}_0 + O(Re_m^2), \qquad (4.10) $$

$$ \nabla^2 \mathbf{A}_\alpha - in_0 S_\omega \mathbf{A}_\alpha = -in_b S_\omega \mathbf{A}_0 + O(Re_m^2, \alpha^2, \alpha Re_m), \qquad (4.11) $$

where $i^2 = -1$, $n_0$ equals 1 in the pipe (liquid metal without any particle) and 0 elsewhere, and $n_b$ equals 1 in the particle and 0 elsewhere (note that $n + n_b = n_0$). According to the quasistatic hypothesis, when the particle moves, one has to consider that $n$ and $n_b$ are two slowly varying functions of time, i.e. time plays the role of a parameter in Eq. (4.11). Exact solution of (4.9) has been obtained for an infinite cylinder [98]. In this case $\mathbf{A}_0$ is azimuthal ($\mathbf{A}_0 = A_0(r, z)e_\theta$), and an example of its real and imaginary part is shown in figure 4.9.

<footnote n="2">²In this decomposition their is no effect of liquid metal flow perturbation due to the particle motion. this could be introduced as higher order terms.</footnote>
<figure><img src="image_58.png" /><figcaption>Figure 4.10: Real part (left) and imaginary part (right) of the green function for the perturbation of the vector potential $\mathbf{A}_\alpha$ as defined in (4.13) as a function of $r$ for various values of $z$ for a fixed value $\theta = \theta' = 0$, $r' = 1/2$ and $z' = 0$, corresponding here to the position of the source.</figcaption></figure>

Equation (4.11) is a Helmholtz equation with a source term depending on $\mathbf{A}_0$. Accordingly the perturbation of the vector potential is given by

$$ \mathbf{A}_{\alpha}(\mathbf{r}) = -iS_{\omega} \int n_{b}(\mathbf{r}') \mathbf{G}(\mathbf{r}, \mathbf{r}') \cdot \mathbf{A}_{0}(\mathbf{r}') \, d\mathbf{r}'^3, \quad (4.12) $$

where $\mathbf{G}(\mathbf{r}, \mathbf{r}')$ is the tonsorial Green function solution of

$$ \nabla^2 \mathbf{G}(\mathbf{r}, \mathbf{r}') - in_0 S_{\omega} \mathbf{G}(\mathbf{r}, \mathbf{r}') = -\delta(\mathbf{r} - \mathbf{r}') \cdot (\mathbf{e}_r \otimes \mathbf{e}_r + \mathbf{e}_{\theta} \otimes \mathbf{e}_{\theta} + \mathbf{e}_z \otimes \mathbf{e}_z). \quad (4.13) $$

To obtain this Green function, we apply the Fourier transform in $z$ and $\theta$. The resulting Fourier coefficients are solutions of a Bessel equation, and can thus be expressed as a combination of Bessel functions. Finally, the Green function in physical space is obtained from an inverse Fourier transform performed numerically. An example of the resulting $\mathbf{G}(\mathbf{r}, \mathbf{r}')$ is shown in figure 4.10.

For a small particle volume, taking the pointwise limit for the source term, $n_b(\mathbf{r}')$ in (4.12) becomes $v_b\delta(\mathbf{r}_c - \mathbf{r}')$, where $v_b$ is the particle volume normalized by the pipe radius and $\mathbf{r}_c$ the position of its center, also normalized by the pipe radius. This leads to an algebraic equation for the vector potential $\mathbf{A}_\alpha$:

$$ \mathbf{A}_{\alpha}(\mathbf{r}) = -iv_b S_{\omega} \mathbf{G}(\mathbf{r}, \mathbf{r}_c) \cdot \mathbf{A}_{0}(\mathbf{r}_c). \quad (4.14) $$

The condition of the validity of this solution ($v_b S_\omega \ll 1$) means that the particle must be small enough when the gradients of the eddy currents become large. Equation (4.14) shows that, in the linear approximation, the perturbation of the vector potential due to the presence of the particle, varies linearly with its volume as it has been already found for small particles at low frequencies [132, 98].

With this model, we can compute the time evolution of the amplitude of the tension difference between the two secondary coils $\Delta V_\alpha$ generated by a traveling inclusion. In order to validate this approach, we compare the results of this model with numerical simulations, performed with Comsol, of (4.7) with $Re_m = 0$ and the quasi-static approximation. Such
<figure><img src="image_59.png" /></figure>

Figure 4.11: Theoretical ECFM response. Solid line: linear model; broken lines: numerical solution. Left: modulus of the complex amplitude of the ECFM response; right: its phase. Blue: $R_b = 1$ mm, $R_c = 9$ mm; brown: $R_b = 3$ mm, $R_c = 4$ mm; in both cases, frequency 1000 Hz. Abscissa Z corresponds to the axial position of the particle. The red and green areas show the positions inside the secondary and primary coils, respectively.

comparisons are shown in figure 4.11 for two non-conducting beads of different diameters, $R_b = 1$ mm and $R_b = 3$ mm, traveling in the pipe of radius $R = 13$ mm with two constant radial positions $R_c = 9$ mm and $R_c = 4$ mm, respectively. In both cases the frequency of the current in the primary coil is 1000 Hz corresponding to $v_b S_\omega = 8.8 \times 10^{-3}$ and $0.24$ respectively for the two bead sizes. In figure 4.11 we present the evolution of the modulus $|\Delta V_\alpha|$ of the generated signal as well as its phase $\arg \Delta V_\alpha = \phi_\alpha$ with respect to the current in the primary coil. We observed a very good agreement between the Comsol simulations and the model for both quantities. As seen in this figure, as the bead is traveling through the ECFM, the signal experience an oscillation. We note $\Delta V^* = \Delta V_\alpha(z_{max})$, where $z_{max}$ is the position of the bead giving the maximum of $|\Delta V_\alpha|$.

Based on the inversion of the mapping $(R_b, R_c) \rightarrow (|\Delta V_\alpha^*|, \phi_\alpha^*)$ we proposed in ref. [1] a method to measure the size of an inclusion in a liquid metal.

### 4.2.2 Experimental setup

The capability of an ECFM to detect a particle and measure its size has been verified experimentally. Two experimental setups have been designed as depicted in figure 4.12. The first one enables us to validate the model for the detection of small beads of known diameter and position. In a second setup, we consider a bubble column in order to apply the method to a real fluid system.

For both setup, we have a tube filled with galinstan. The tube is in PVC with inner diameter 26 mm and thickness 3 mm. We applied an alternating electrical current of amplitude 0.2 A in the primary coil. We use a transconductance amplifier to impose a constant amplitude of the current independently of the variation of the impedance of the system caused by the traveling of the inclusions.

For the first setup, the particles are 3D printing PMMA beads of diameter 1, 2, 3 and 4 mm, whose positions are imposed by a thin fishing wire with diameter 0.08 mm. In the second experimental setup, we inject argon bubbles at the bottom of the tube.
<figure><img src="image_60.png" /><figcaption>Figure 4.12: Experimental setups for detection and characterization of non-conducting inclusion by an ECFM. Left: beads of known diameters and position are traveling in a tube filled with galinstan. Right: Argon bubbles are rising in a tube filled with galinstan.</figcaption></figure>

In figure 4.13 we compare the prediction from the model with the experimental response of the ECFM for the first setup. We present the evolution of the modulus and the phase of $\Delta V_{\alpha}^*$ with the frequency of the current in the primary coil, for a bead size of $R_b = 3$mm and for 3 positions $R_c = 0, 4.8$ and $8.6$ mm. The linear approximation also agrees with the experience for moderate frequencies (up to 3000Hz), but present some deviation at larger frequencies which seems consistent with the limit of validity of the model $v_b S_{\omega} = 1$ that corresponds here to a frequency of 4200 Hz. At high frequency the skin depth becomes very small which implies strong variations of the vector potential over the size of the bead. Thus the approximation of integrals made to obtain (4.14) from (4.12) is not valid. Finally, the simple model (4.12) predict $\Delta V_{\alpha}^* = 0$ for a bead traveling along the axis of the cylinder, $R_c = 0$, since due to symmetry, $A_0(r = 0) = 0$. In this case also one should consider finite volume effect of the bead though (4.14) instead of using the pointwise approximation of (4.12).

In figure 4.14, we present results of the bubble experiments. We show in this figure the time signal of $|\Delta V_{\alpha}|$ generating by the rising bubbles. From the detection of the peaks, we obtained $|\Delta V_{\alpha}^*|$ and $\phi_{\alpha}^*$ corresponding to each event of a bubble passing through the ECFM, from which we can estimate the bubble size and radial position. Figure 4.14 presents histograms of the bubble size and bubble radial position. The total mass of bubbles injected in the galinstan is obtained by a time integration of the gas flow rate. The sum of the estimated volume of each bubble matches the measured mass of gas with an error of less than 2%. These preliminary results come as confirmation that the bubble size, and position, can be inferred from the measurement of the amplitude and phase of the signal of an ECFM.
<figure><img src="image_61.png" /><figcaption>Figure 4.13: Effect of the frequency on the ECFM response for a particle of radius $R_b = 3$ mm and for three radial positions. Left: amplitude of the maximum $|\Delta V_\alpha^*|$; Right: the phase of the maximum $\phi_\alpha^*$. Blue: $R_c = 8.6$ mm; red: $R_c = 4.6$ mm; green: $R_c = 0$ mm. Continuous lines: linear model; crosses: experimental data.</figcaption></figure>

<figure><img src="image_62.png" /><figcaption>Figure 4.14: Top: time evolution $|\Delta V_\alpha|$ of the ECFM generating by rising bubbles in cylinder fill with galinstan. Right: Histogram of the estimated of the bubble size. Left: histogram of the estimated bubble radial position.</figcaption></figure>
## 4.3 Perspectives

We present some perspectives envisaged extending these research activities around the MHD with variable fields.

Concerning the study of the single-phase flow presented in the paragraph 4.1, a direct extension is to further explore the parameter space of this original flow. We could firstly consider the situation in which the buoyancy force dominate the Lorentz force ($U_B > U_L$) for example by considering taller domain, as well as different aspect ratio of the vessel. It will also be interesting to study the flow when a more intense magnetic field is applied. In the work presented above, the technical limitations of our experimental setup and the capabilities of numerical simulations allowed us to study the effect of Hartman numbers up to $Ha \approx 300$, corresponding to intensities in the primary coil of the order of 100A. The installation of power capacitors on the experimental setup would allow us to reach currents up to 500A and thus to obtain Hartman numbers around $Ha = 1500$. For such Hartman numbers, we estimate that the flow velocity will be sufficient to obtain magnetic Reynolds numbers of the flow slightly higher than 1 and giving access to new regimes. In such a situation, one can expect magnetic induction mechanism to be important along with a strong influence of the imposed magnetic field similarly to Refs. [174, 173]. As a consequence the induced currents will depend on the local structures of the velocity field, and can be important, at least locally, compared to the currents induced by the temporal variation of the applied field. Consequently high velocity regions could then be prone to strong buoyancy forcing due to local Joule dissipation, leading to a feedback loop mechanism that tends to reinforce the largest velocity fluctuations. For the numerical study of this regime, it will be necessary to resolve the induction equation with its nonlocal boundary conditions to account for the $\nabla \cdot \mathbf{B} = 0$ condition outside the liquid metal. However, these simulations would remain challenging in terms of the size of the numerical grid, and may require to consider subgird scale modeling of the flow.

Regarding the study of the perturbation of magnetic fields by inclusions, we would like to pursue the approach to consider multi-bubble detection. For not too high volume fraction, we expect that (4.12) remains valid, it is then a priori possible to determine the ECFM response to a given distribution of voids. The challenge will be to propose an inverse procedure to obtain an estimate of the volume fraction as well as the distribution of the size and position of the inclusion. We would also consider the more realistic situation of gas bubbles in a turbulent pipe flows. We are planning to realize an experiment with a loop of flowing galinstan and bubble injection. We would try to use the ECFM to, on the one hand, measure velocity profile and characterize the velocity fluctuations in the liquid, and on the other hand, to confirm the capability of the ECFM to detect bubbles in the flow. It will also be interesting to consider higher order terms in the modeling of the ECFM response to measure the velocity perturbation induced by the passing of the bubbles.

We would also like to study a suspension of metallic particles subject to strong (in the sense of high Ha) alternating magnetic field. The electric currents induced in each particle generate Joule effect and interact with the applied field resulting in a Lorentz force (and possibly a torque). As a result, the phase is set in motion due to both the drag and the heating (buoyancy force) of the metallic suspension. Such a problem for a suspension of particles in a creeping flow has been studied both theoretically [176] and experimentally [34] but the heating effects causing local sources of buoyancy and its interactions with
turbulent fluctuations of the velocity field remains unexplored. On the other hand, there is an extensive literature on the preferential concentrations effects and formation of clusters of particles heavier than the carrier phase caused by turbulent flows. The question therefore is to analyze how the formation of inhomogeneity in such a suspension can appear due to hydrodynamic interactions between the two phases and due to the direct magnetic dipole-dipole interaction at short range. Since the flow would respond to such self-organization of the particles (due to the modification of the interphase heat transfer), we can expect to observe, as in Ref. [275], a clustering transition which generates strong fluctuations at all scales, in analogies with systems close to a critical point. Of particular interest is the question of the percolation of electrical conductivity. Indeed, once packing of particles is attained, one should expect a dramatic modification of the eddy currents and therefore significantly change the induced Joule dissipation and the Lorentz force. This flow could be studied numerically and experimentally. In order to focus on collective effects, we plan to use Lagrangian tracking and point-particle approximation for the modeling of the metallic dispersed phase. Within this framework, the dynamic equation of the particles needs to be supplemented in order to account for both the Lorenz force and torque as well as a heat source caused by the applied magnetic field. Such additional source terms can be obtained from dipole moment [176] assuming that the magnetic field remains homogenous at the scale of a particle. Preliminary work on the subject is presented in Ref. [212]. On the other hand, the screening of the magnetic field caused by the particles can be estimated by extending the two-way coupling between continuous and dispersed phase based on the magnetic field perturbation caused by a dipole as proposed in Refs. [120, 98].
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