Model for the dynamics of micro-bubbles in high-Reynolds-number flows

Abstract

We propose a model for the acceleration of micro-bubbles (smaller than the dissipative scale of the flow) subjected to the drag and fluid inertia forces in a homogeneous and isotropic turbulent flow. This model, that depends on the Stokes number, Reynolds number and the density ratio, reproduces the evolution of the acceleration variance as well as the relative importance and alignment of the two forces as observed from direct numerical simulations (DNS). We also report that the bubble acceleration statistics conditioned on the local kinetic energy dissipation rate are invariant with the Stokes number and the dissipation rate. Based on this observation, we propose a stochastic model for the instantaneous bubble acceleration vector accounting for the small-scale intermittency of the turbulent flows. The norm of the bubble acceleration is obtained by modelling the dissipation rate along the bubble trajectory from a log-normal stochastic process, whereas its orientation is given by two coupled random walks on a unit sphere in order to model the evolution of the joint orientation of the drag and inertia forces acting on the bubble. Furthermore, the proposed stochastic model for the bubble acceleration is used in the context of large eddy simulations (LES) of turbulent flows laden with small bubbles. To account for the turbulent motion at scales smaller than the mesh resolution, we decompose the instantaneous bubble acceleration in its resolved and residual parts. The first part is given by the drag and fluid inertia forces computed from the resolved velocity field, and the second term refers to the random contribution of small unresolved turbulent scales and is estimated with the stochastic model proposed in the paper. Comparisons with DNS and standard LES, show that the proposed model improves significantly the statistics of the bubbly phase.

Dynamics of small heavy particles in homogeneous turbulence: a Lagrangian experimental study

“…The spectrum can also be derived from the velocity autocorrelation as the two form a Fourier transform pair. In simple approaches considering only the inertial and large-scale dynamics, the auto-correlation function is well approximated by R(t) = (u ) 2 exp(−t/T L ) (Hinze 1975;Mordant et al 2001;Zhang et al 2019), valid for ω π/τ η (or τ η t) and corresponding to E = (u ) 2 T L /(1 + (ωT L ) 2 ). For small time separations, the autocorrelation deviates from the exponential, tending to a horizontal asymptote at t = 0 with a curvature proportional to the acceleration variance (Mordant et al 2004).…”

“…2001; Zhang et al. 2019), valid for (or ) and corresponding to . For small time separations, the autocorrelation deviates from the exponential, tending to a horizontal asymptote at with a curvature proportional to the acceleration variance (Mordant et al.…”

“…Of particular relevance to such a relation is the framework developed by Csanady (1963) (building on earlier work by Tchen 1947; Hinze 1975), who proposed a link between the Lagrangian spectrum of the particle velocity () and the one of the sampled-fluid velocity () through a response function where is the angular frequency in the Lagrangian frame of reference. The velocity and acceleration variances can be obtained from the spectra as The response function can be modelled from (1.1), by taking the Fourier transform of the particle velocity and sampled-fluid velocity (Csanady 1963) Zhang, Legendre & Zamansky (2019) derived a more complete form of the response function including unsteady forces, which, however, are expected to be small for microscopic heavy particles. The dependence of the response function on the particle response time illustrates the particle inability to respond to fluctuations with frequencies greater than .…”

Fluid inertia effects on the motion of small spherical bubbles or solid spheres in turbulent flows

“…Finally let us mention that Volk et al (2008) present a comparison between experiments and numerical simulations confirming that, considering the viscous drag, the added mass and Tchen forces enable us to obtain a good accuracy for the particle acceleration as long as the finite size can be ignored. Concerning the added mass and Tchen forces, their effects are usually considered as dominant for bubbles (Maxey, Chang & Wang 1994;Calzavarini et al 2009;Prakash et al 2012;Mathai et al 2016;Zhang, Legendre & Zamansky 2019) and neutrally buoyant particles (Calzavarini et al 2012) but it is generally supposed that, for heavy enough particles, they are negligible (Maxey & Corrsin 1986;Wang & Maxey 1993;Armenio & Fiorotto 2001;Bagchi & Balachandar 2003;Bec et al 2006).…”

“…We present in § 3 our DNS results for various Stokes numbers and drag laws, and introduce an effective Stokes number that accounts for the finite Reynolds number effects on the particle response. In § 4 we recall the estimations for the variance of the acceleration and the particles forces presented in Zhang et al (2019) and show that they can be combined with the effective Stokes number. In § 5 we discuss the importance of the fluid inertia forces with respect to the drag forces as the density ratio or the size of the particles is changed.…”