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We performed numerical simulations of a homogeneous swarm of bubbles rising at large Reynolds number, Re = 760 with volume fractions ranging from 1% to 10%. We consider a simplified model in which the interfaces are not resolved, but which allows us to simulate flows with a large number of bubbles and to emphasize the interactions between bubble wakes. The liquid phase is described by solving, on an Eulerian grid, the Navier-Stokes equations, including sources of momentum which model the effect of the bubbles. The dynamics of each bubble is determined within the Lagrangian framework by solving an equation of motion involving the hydrodynamic forces exerted by the fluid accounting for the correction of the fictitious self-interaction of a bubble with its own wake. The comparison with experiments shows that this coarse-grained simulations approach can reliably describe the dynamics of the resolved flow scales. We use conditional averaging to characterize the mean bubble wakes and obtain in particular the typical shear imposed by the rising bubbles. On the basis of the spectral decomposition of the energy budget, we observe that the flow is dominated by production at large scales and by dissipation at small scales and we rule out the presence of an intermediate range in which the production and dissipation are locally in balance. We propose that the k-3 subrange of the energy spectra results from the mean shear rate imposed by the bubbles, which controls the rate of return to isotropy., Comment: Journal of Fluid Mechanics, In press

It is well known that the fluid-particle acceleration is intimately related to the dissipation rate of turbulence, in line with the Kolmogorov assumptions. On the other hand, various experimental and numerical works have reported as well its dependence on the kinetic energy, which is generally attributed to intermittency and non-independence of the small-scale dynamics from large-scale ones. The analyses given in this paper focus on statistics of the fluid-particle acceleration conditioned on both the local dissipation rate and the kinetic energy. It is shown that this quantity presents an exponential dependence on the kinetic energy with a growth rate independent of the Reynolds number, in addition to the expected power law behavior with the dissipation rate. The exponential growth, which clearly departs from the previous propositions, reflects additional kinematic effects of the flow structures on the acceleration. Regarding intermittency, to account for the persistence of the effect of the large-scales on the dissipation rate, it is further proposed scaling laws for the Reynolds number dependence of the conditional and unconditional acceleration variance using Barenblatt's incomplete similarity framework. It is then shown that both these intermittency and kinematic effects can be combined in a multiplicative cascade process for the acceleration depending on the kinetic energy and the dissipation rate. On the basis of these observations, we introduce a vectorial stochastic model for the dynamics of a tracer in turbulent flows. This model incorporates a fractional log-normal process for the dissipation rate recently proposed, as well as an additional hypothesis regarding non-diagonal terms in the diffusion tensor which naturally leads to the decomposition between tangential and centripetal acceleration. This model is shown to be in good agreement with direct numerical simulations and presents the essential characteristics of the Lagrangian turbulence highlighted in recent years, namely (i) non-Gaussian acceleration, (ii) scale separation between the norm of the acceleration and its components, (iii) anomalous scaling law for the Lagrangian velocity spectra, and (iv) negative skewness of the increments of the mechanical power, reflecting the temporal irreversibility of the dynamics.

The characterization of intermittency in turbulence has its roots in the K62 theory, and if no proper definition is to be found in the literature, statistical properties of intermittency were studied and models were developed in attempt to reproduce it. The first contribution of this work is to propose a requirement list to be satisfied by models designed within the Lagrangian framework. Multifractal stochastic processes are a natural choice to retrieve multifractal properties of the dissipation. Among them, following the proposition of \cite{Mandelbrot1968}, we investigate the Gaussian Multiplicative Chaos formalism, which requires the construction of a log-correlated stochastic process $X_t$. The fractional Gaussian noise of Hurst parameter $H = 0$ is of great interest because it leads to a log-correlation for the logarithm of the process.Inspired by the approximation of fractional Brownian motion by an infinite weighted sum of correlated Ornstein-Uhlenbeck processes, our second contribution is to propose a new stochastic model: $X_t = \int_0^\infty Y_t^x k(x) d x$, where $Y_t^x$ is an Ornstein-Uhlenbeck process with speed of mean reversion $x$ and $k$ is a kernel. A regularization of $k(x)$ is required to ensure stationarity, finite variance and logarithmic auto-correlation. A variety of regularizations are conceivable, and we show that they lead to the aforementioned multifractal models.To simulate the process, we eventually design a new approach relying on a limited number of modes for approximating the integral through a quadrature $X_t^N = \sum_{i=1}^N \omega_i Y_t^{x_i}$, using a conventional quadrature method. This method can retrieve the expected behavior with only one mode per decade, making this strategy versatile and computationally attractive for simulating such processes, while remaining within the proposed framework for a proper description of intermittency.