scholarly journals A conditionally cubic-Gaussian stochastic Lagrangian model for acceleration in isotropic turbulence

2007 ◽  
Vol 582 ◽  
pp. 423-448 ◽  
Author(s):  
A. G. LAMORGESE ◽  
S. B. POPE ◽  
P. K. YEUNG ◽  
B. L. SAWFORD
Keyword(s):  
Reynolds Number ◽  
Dissipation Rate ◽  
Velocity Structure ◽  
Isotropic Turbulence ◽  
Stochastic Modelling ◽  
Lagrangian Model ◽  
Small Scale ◽  
New Model ◽  
Lagrangian Velocity ◽  
Dissipation Model

The modelling of fluid-particle acceleration in homogeneous isotropic turbulence in terms of stochastic models for the Lagrangian velocity, acceleration and a dissipation rate variable is considered. The basis for the Reynolds model (A. M. Reynolds, Phys. Rev. Lett. vol. 91, 2003, 084503) is reviewed and examined by reference to direct numerical simulations (DNS) of isotropic turbulence at Taylor-scale Reynolds number (Rλ) up to about 650. In particular, we show DNS data that support stochastic modelling of the logarithm of pseudo-dissipation as an Ornstein–Uhlenbeck process and reveal non-Gaussianity of the acceleration conditioned on fluctuations of the pseudo-dissipation rate. The DNS data are used to construct a new stochastic model that is exactly consistent with Gaussian velocity and conditionally cubic-Gaussian acceleration statistics. This model captures the effects of small-scale intermittency on acceleration and the conditional dependence of acceleration on pseudo-dissipation (which differs from that predicted by the refined Kolmogorov hypotheses). Non-Gaussianity of the conditionally standardized acceleration probability density function (PDF) is accounted for in terms of model nonlinearity. The large-time behaviour of the new model is that of a velocity-dissipation model that can be matched with DNS data for conditional second-order Lagrangian velocity structure functions. As a result, the diffusion coefficient for the new model incorporates two-time information and its Reynolds-number dependence as observed in DNS. The resulting model predictions for conditional and unconditional velocity autocorrelations and time scales are shown to be in very good agreement with DNS.

Time-dependent, non-monotonic mixing in stratified turbulent shear flows: implications for oceanographic estimates of buoyancy flux

2013 ◽  
Vol 736 ◽  
pp. 570-593 ◽  
Author(s):  
A. Mashayek ◽  
C. P. Caulfield ◽  
W. R. Peltier
Keyword(s):  
Reynolds Number ◽  
Dissipation Rate ◽  
Isotropic Turbulence ◽  
Buoyancy Flux ◽  
Shear Instability ◽  
Turbulent Shear ◽  
Mixing Efficiency ◽  
Small Scale ◽  
Stratified Turbulence ◽  
Flow Evolution

AbstractWe employ direct numerical simulation to investigate the efficiency of diapycnal mixing by shear-induced turbulence in stably stratified free shear layers for flows with bulk Richardson numbers in the range $0. 12\leq R{i}_{0} \leq 0. 2$ and Reynolds number $Re= 6000$. We show that mixing efficiency depends non-monotonically upon $R{i}_{0} $, peaking in the range 0.14–0.16, which coincides closely with the range in which both the buoyancy flux and the dissipation rate are maximum. By detailed analyses of the energetics of flow evolution and the underlying dynamics, we show that the existence of high mixing efficiency in the range $0. 14\lt R{i}_{0} \lt 0. 16$ is due to the emergence of a large number of small-scale instabilities which do not exist at lower Richardson numbers and are stabilized at high Richardson numbers. As discussed in Mashayek & Peltier (J. Fluid Mech., vol. 725, 2013, pp. 216–261), the existence of such a well-populated ‘zoo’ of secondary instabilities at intermediate Richardson numbers and the subsequent high mixing efficiency is realized only if the Reynolds number is higher than a critical value which is generally higher than that achievable in laboratory settings, as well as that which was achieved in the majority of previous numerical studies of shear-induced stratified turbulence. We furthermore show that the primary assumptions upon which the widely employed Osborn (J. Phys. Oceanogr. vol. 10, 1980, pp. 83–89) formula is based, as well as its counterparts and derivatives, which relate buoyancy flux to dissipation rate through a (constant) flux coefficient ($\Gamma $), fail at higher Richardson numbers provided that the Reynolds number is sufficiently high. Specifically, we show that the assumptions of fully developed, stationary, and isotropic turbulence all break down at high Richardson numbers. We show that the breakdown of these assumptions occurs most prominently at Richardson numbers above that corresponding to the maximum mixing efficiency, a fact that highlights the importance of the non-monotonicity of the dependence of mixing efficiency upon Richardson number, which we establish to be characteristic of stratified shear-induced turbulence. At high $R{i}_{0} $, the lifecycle of the turbulence is composed of a rapidly growing phase followed by a phase of rapid decay. Throughout the lifecycle, there is considerable exchange of energy between the small-scale turbulence and larger coherent structures which survive the various stages of flow evolution. Since shear instability is one of the most prominent mechanisms for turbulent dissipation of energy at scales below hundreds of metres and at various depths of the ocean, our results have important implications for the inference of turbulent diffusivities on the basis of microstructure measurements in the oceanic environment.

Small-scale anisotropy in turbulent boundary layers

2016 ◽  
Vol 804 ◽  
pp. 5-23 ◽  
Author(s):  
Alain Pumir ◽  
Haitao Xu ◽  
Eric D. Siggia
Keyword(s):  
Reynolds Number ◽  
Channel Flow ◽  
Isotropic Turbulence ◽  
Turbulent Channel Flow ◽  
Statistical Properties ◽  
Simultaneous Action ◽  
Large Reynolds Number ◽  
Small Scale ◽  
Velocity Fluctuations ◽  
Scale Properties

In a channel flow, the velocity fluctuations are inhomogeneous and anisotropic. Yet, the small-scale properties of the flow are expected to behave in an isotropic manner in the very-large-Reynolds-number limit. We consider the statistical properties of small-scale velocity fluctuations in a turbulent channel flow at moderately high Reynolds number ($Re_{\unicode[STIX]{x1D70F}}\approx 1000$), using the Johns Hopkins University Turbulence Database. Away from the wall, in the logarithmic layer, the skewness of the normal derivative of the streamwise velocity fluctuation is approximately constant, of order 1, while the Reynolds number based on the Taylor scale is $R_{\unicode[STIX]{x1D706}}\approx 150$. This defines a small-scale anisotropy that is stronger than in turbulent homogeneous shear flows at comparable values of $R_{\unicode[STIX]{x1D706}}$. In contrast, the vorticity–strain correlations that characterize homogeneous isotropic turbulence are nearly unchanged in channel flow even though they do vary with distance from the wall with an exponent that can be inferred from the local dissipation. Our results demonstrate that the statistical properties of the fluctuating velocity gradient in turbulent channel flow are characterized, on one hand, by observables that are insensitive to the anisotropy, and behave as in homogeneous isotropic flows, and on the other hand by quantities that are much more sensitive to the anisotropy. How this seemingly contradictory situation emerges from the simultaneous action of the flux of energy to small scales and the transport of momentum away from the wall remains to be elucidated.

scholarly journals Turbulence strength in ultimate Taylor–Couette turbulence

2017 ◽  
Vol 836 ◽  
pp. 397-412 ◽  
Author(s):  
Rodrigo Ezeta ◽  
Sander G. Huisman ◽  
Chao Sun ◽  
Detlef Lohse
Keyword(s):  
Reynolds Number ◽  
Dissipation Rate ◽  
Velocity Structure ◽  
Field Measurements ◽  
Azimuthal Velocity ◽  
Energy Dissipation Rate ◽  
Local Flow ◽  
Kolmogorov Length Scale ◽  
Taylor’S Hypothesis ◽  
Taylor Couette

We provide experimental measurements for the effective scaling of the Taylor–Reynolds number within the bulk $\mathit{Re}_{\unicode[STIX]{x1D706},\mathit{bulk}}$, based on local flow quantities as a function of the driving strength (expressed as the Taylor number $\mathit{Ta}$), in the ultimate regime of Taylor–Couette flow. We define $Re_{\unicode[STIX]{x1D706},bulk}=(\unicode[STIX]{x1D70E}_{bulk}(u_{\unicode[STIX]{x1D703}}))^{2}(15/(\unicode[STIX]{x1D708}\unicode[STIX]{x1D716}_{bulk}))^{1/2}$, where $\unicode[STIX]{x1D70E}_{bulk}(u_{\unicode[STIX]{x1D703}})$ is the bulk-averaged standard deviation of the azimuthal velocity, $\unicode[STIX]{x1D716}_{bulk}$ is the bulk-averaged local dissipation rate and $\unicode[STIX]{x1D708}$ is the liquid kinematic viscosity. The data are obtained through flow velocity field measurements using particle image velocimetry. We estimate the value of the local dissipation rate $\unicode[STIX]{x1D716}(r)$ using the scaling of the second-order velocity structure functions in the longitudinal and transverse directions within the inertial range – without invoking Taylor’s hypothesis. We find an effective scaling of $\unicode[STIX]{x1D716}_{\mathit{bulk}}/(\unicode[STIX]{x1D708}^{3}d^{-4})\sim \mathit{Ta}^{1.40}$, (corresponding to $\mathit{Nu}_{\unicode[STIX]{x1D714},\mathit{bulk}}\sim \mathit{Ta}^{0.40}$ for the dimensionless local angular velocity transfer), which is nearly the same as for the global energy dissipation rate obtained from both torque measurements ($\mathit{Nu}_{\unicode[STIX]{x1D714}}\sim \mathit{Ta}^{0.40}$) and direct numerical simulations ($\mathit{Nu}_{\unicode[STIX]{x1D714}}\sim \mathit{Ta}^{0.38}$). The resulting Kolmogorov length scale is then found to scale as $\unicode[STIX]{x1D702}_{\mathit{bulk}}/d\sim \mathit{Ta}^{-0.35}$ and the turbulence intensity as $I_{\unicode[STIX]{x1D703},\mathit{bulk}}\sim \mathit{Ta}^{-0.061}$. With both the local dissipation rate and the local fluctuations available we finally find that the Taylor–Reynolds number effectively scales as $\mathit{Re}_{\unicode[STIX]{x1D706},\mathit{bulk}}\sim \mathit{Ta}^{0.18}$ in the present parameter regime of $4.0\times 10^{8}<\mathit{Ta}<9.0\times 10^{10}$.

Lagrangian conditional statistics, acceleration and local relative motion in numerically simulated isotropic turbulence

2007 ◽  
Vol 582 ◽  
pp. 399-422 ◽  
Author(s):  
P. K. YEUNG ◽  
S. B. POPE ◽  
E. A. KURTH ◽  
A. G. LAMORGESE
Keyword(s):  
Reynolds Number ◽  
Relative Motion ◽  
Isotropic Turbulence ◽  
Companion Paper ◽  
Small Scale ◽  
Coordinate Frame ◽  
Centripetal Acceleration ◽  
Vorticity Vector ◽  
Reynolds Number Effects ◽  
High Reynolds

Lagrangian statistics of fluid-particle velocity and acceleration conditioned on fluctuations of dissipation, enstrophy and pseudo-dissipation representing different characteristics of local relative motion are extracted from a direct numerical simulation database of stationary (forced) homogeneous isotropic turbulence. The grid resolution in the simulations is up to 20483, and the Taylor-scale Reynolds number ranges from about 40 to 650, where characteristics of small-scale intermittency in the Eulerian flow field are well developed. A key joint statistic of the conditioning variables is the dissipation-enstrophy cross-correlation, which is asymmetric, but becomes less so at high Reynolds number. Conditional velocity autocorrelations are consistent with rapid changes in the velocity of fluid particles moving in regions of large velocity gradients. Examination of statistics conditioned upon enstrophy, especially in a local coordinate frame moving with the vorticity vector, and of the centripetal acceleration suggests the presence of vortex-trapping effects which persist for several Kolmogorov time scales. Further results on acceleration statistics and joint velocity-acceleration autocorrelations are also presented to help characterize in detail the properties of a joint stochastic process of velocity, acceleration and the pseudo-dissipation. Together with recent work on Eulerian conditional acceleration and Reynolds-number dependence of basic Lagrangian quantities, the present results are directly useful for the development of a new stochastic model formulated to account for intermittency and Reynolds-number effects as described in detail in a companion paper.

scholarly journals The scaling of straining motions in homogeneous isotropic turbulence

2017 ◽  
Vol 829 ◽  
pp. 31-64 ◽  
Author(s):  
G. E. Elsinga ◽  
T. Ishihara ◽  
M. V. Goudar ◽  
C. B. da Silva ◽  
J. C. R. Hunt
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Spatial Organization ◽  
Isotropic Turbulence ◽  
Local Strain ◽  
Scale Structure ◽  
Small Scale ◽  
Length Scale ◽  
Local Flow ◽  
Small Scale Structure

The scaling of turbulent motions is investigated by considering the flow in the eigenframe of the local strain-rate tensor. The flow patterns in this frame of reference are evaluated using existing direct numerical simulations of homogeneous isotropic turbulence over a Reynolds number range from $Re_{\unicode[STIX]{x1D706}}=34.6$ up to 1131, and also with reference to data for inhomogeneous, anisotropic wall turbulence. The average flow in the eigenframe reveals a shear layer structure containing tube-like vortices and a dissipation sheet, whose dimensions scale with the Kolmogorov length scale, $\unicode[STIX]{x1D702}$. The vorticity stretching motions scale with the Taylor length scale, $\unicode[STIX]{x1D706}_{T}$, while the flow outside the shear layer scales with the integral length scale, $L$. Furthermore, the spatial organization of the vortices and the dissipation sheet defines a characteristic small-scale structure. The overall size of this characteristic small-scale structure is $120\unicode[STIX]{x1D702}$ in all directions based on the coherence length of the vorticity. This is considerably larger than the typical size of individual vortices, and reflects the importance of spatial organization at the small scales. Comparing the overall size of the characteristic small-scale structure with the largest flow scales and the vorticity stretching motions on the scale of $4\unicode[STIX]{x1D706}_{T}$ shows that transitions in flow structure occur where $Re_{\unicode[STIX]{x1D706}}\approx 45$ and 250. Below these respective transitional Reynolds numbers, the small-scale motions and the vorticity stretching motions are progressively less well developed. Scale interactions are examined by decomposing the average shear layer into a local flow, which is induced by the shear layer vorticity, and a non-local flow, which represents the environment of the characteristic small-scale structure. The non-local strain is $4\unicode[STIX]{x1D706}_{T}$ in width and height, which is consistent with observations in high Reynolds number flow of a $4\unicode[STIX]{x1D706}_{T}$ wide instantaneous shear layer with many $\unicode[STIX]{x1D702}$-scale vortical structures inside (Ishihara et al., Flow Turbul. Combust., vol. 91, 2013, pp. 895–929). In the average shear layer, vorticity aligns with the intermediate principal strain at small scales, while it aligns with the most stretching principal strain at larger scales, consistent with instantaneous turbulence. The length scale at which the alignment changes depends on the Reynolds number. When conditioning the flow in the eigenframe on extreme dissipation, the velocity is strongly affected over large distances. Moreover, the associated peak velocity remains Reynolds number dependent when normalized by the Kolmogorov velocity scale. It signifies that extreme dissipation is not simply a small-scale property, but is associated with large scales at the same time.

Direct particle–fluid simulation of Kolmogorov-length-scale size particles in decaying isotropic turbulence

2017 ◽  
Vol 819 ◽  
pp. 188-227 ◽  
Author(s):  
Lennart Schneiders ◽  
Matthias Meinke ◽  
Wolfgang Schröder
Keyword(s):  
Kinetic Energy ◽  
Strain Rate ◽  
Viscous Dissipation ◽  
Dissipation Rate ◽  
Isotropic Turbulence ◽  
Bulk Flow ◽  
Small Scale ◽  
Length Scale ◽  
Kolmogorov Length Scale ◽  
Scale Size

The modulation of decaying isotropic turbulence by 45 000 spherical particles of Kolmogorov-length-scale size is studied using direct particle–fluid simulations, i.e. the flow field over each particle is fully resolved by direct numerical simulations of the conservation equations. A Cartesian cut-cell method is used by which the exchange of momentum and energy at the fluid–particle interfaces is strictly conserved. It is shown that the particles absorb energy from the large scales of the carrier flow while the small-scale turbulent motion is determined by the inertial particle dynamics. Whereas the viscous dissipation rate of the bulk flow is attenuated, the particles locally increase the level of dissipation due to the intense strain rate generated near the particle surfaces due to the crossing-trajectory effect. Analogously, the rotational motion of the particles decouples from the local fluid vorticity and strain-rate field at increasing particle inertia. The high level of dissipation is partially compensated by the transfer of momentum to the fluid via forces acting at the particle surfaces. The spectral analysis of the kinetic energy budget is supported by the average flow pattern about the particles showing a nearly universal strain-rate distribution. An analytical expression for the instantaneous rate of viscous dissipation induced by each particle is derived and subsequently verified numerically. Using this equation, the local balance of fluid kinetic energy around a particle of arbitrary shape can be precisely determined. It follows that two-way coupled point-particle models implicitly account for the particle-induced dissipation rate via the momentum-coupling terms; however, they disregard the actual length scales of the interaction. Finally, an analysis of the small-scale flow topology shows that the strength of vortex stretching in the bulk flow is mitigated due to the presence of the particles. This effect is associated with the energy conversion at small wavenumbers and the reduced level of dissipation at intermediate wavenumbers. Consequently, it damps the spectral flux of energy to the small scales.

Longitudinal and transverse structure functions in a turbulent round jet: effect of initial conditions and Reynolds number

2001 ◽  
Vol 436 ◽  
pp. 231-248 ◽  
Author(s):  
G. P. ROMANO ◽  
R. A. ANTONIA
Keyword(s):  
Reynolds Number ◽  
Large Scale ◽  
Velocity Structure ◽  
Initial Conditions ◽  
Transverse Velocity ◽  
Structure Functions ◽  
Small Scale ◽  
Round Jet ◽  
Jet Nozzle ◽  
The Difference

The difference between scaling exponents of longitudinal and transverse velocity structure functions in the far-field of a round jet is found to depend on the anisotropy of the flow. The effect of the large-scale anisotropy is assessed by considering different initial conditions at the jet nozzle, and hence different ratios of the longitudinal to transverse rms velocities. The effect of the Taylor microscale Reynolds number on the small scale anisotropy is also considered. Both effects account, to a large extent, for the observed difference between longitudinal and transverse exponents and the disagreement between previously published results of different authors. This disagreement also depends on the method used to determine the inertial range. An empirical description of the overall behaviour of the structure functions provides reasonable estimates for the longitudinal and transverse exponents, accounting reasonably well for the anisotropy of both large- and small-scale motions.

Instability and hydraulics of turbulent stratified shear flows

2012 ◽  
Vol 695 ◽  
pp. 235-256 ◽  
Author(s):  
Zhiyu Liu ◽  
S. A. Thorpe ◽  
W. D. Smyth
Keyword(s):  
Reynolds Number ◽  
Growth Rates ◽  
Dissipation Rate ◽  
Richardson Number ◽  
Turbulent Shear ◽  
Small Scale ◽  
Buoyancy Frequency ◽  
Neutral Stability ◽  
Clyde Sea ◽  
And Diffusion

AbstractThe Taylor–Goldstein (T–G) equation is extended to include the effects of small-scale turbulence represented by non-uniform vertical and horizontal eddy viscosity and diffusion coefficients. The vertical coefficients of viscosity and diffusion, ${A}_{V} $ and ${K}_{V} $, respectively, are assumed to be equal and are expressed in terms of the buoyancy frequency of the flow, $N$, and the dissipation rate of turbulent kinetic energy per unit mass, $\varepsilon $, quantities that can be measured in the sea. The horizontal eddy coefficients, ${A}_{H} $ and ${K}_{H} $, are taken to be proportional to the dimensionally correct form, ${\varepsilon }^{1/ 3} {l}^{4/ 3} $, found appropriate in the description of horizontal dispersion of a field of passive markers of scale $l$. The extended T–G equation is applied to examine the stability and greatest growth rates in a turbulent shear flow in stratified waters near a sill, that at the entrance to the Clyde Sea in the west of Scotland. Here the main effect of turbulence is a tendency towards stabilizing the flow; the greatest growth rates of small unstable disturbances decrease, and in some cases flows that are unstable in the absence of turbulence are stabilized when its effects are included. It is conjectured that stabilization of a flow by turbulence may lead to a repeating cycle in which a flow with low levels of turbulence becomes unstable, increasing the turbulent dissipation rate and so stabilizing the flow. The collapse of turbulence then leads to a condition in which the flow may again become unstable, the cycle repeating. Two parameters are used to describe the ‘marginality’ of the observed flows. One is based on the proximity of the minimum flow Richardson number to the critical Richardson number, the other on the change in dissipation rate required to stabilize or destabilize an observed flow. The latter is related to the change needed in the flow Reynolds number to achieve zero growth rate. The unstable flows, typical of the Clyde Sea site, are relatively further from neutral stability in Reynolds number than in Richardson number. The effects of turbulence on the hydraulic state of the flow are assessed by examining the speed and propagation direction of long waves in the Clyde Sea. Results are compared to those obtained using the T–G equation without turbulent viscosity or diffusivity. Turbulence may change the state of a flow from subcritical to supercritical.

scholarly journals Statistics of Small-Scale Velocity Fluctuations and Internal Intermittency in Marine Stratocumulus Clouds

2010 ◽  
Vol 67 (1) ◽  
pp. 262-273 ◽  
Author(s):  
H. Siebert ◽  
R. A. Shaw ◽  
Z. Warhaft
Keyword(s):  
Velocity Structure ◽  
Isotropic Turbulence ◽  
Function Analysis ◽  
Large Body ◽  
Liquid Water Content ◽  
Observation System ◽  
Small Scale ◽  
Turbulence Structure ◽  
Hot Wire ◽  
Stratocumulus Clouds

Abstract Clouds are known to be turbulent, but the details of their internal turbulent structure have been largely unexplored. Measurements of turbulent velocities in stratocumulus clouds presented here reveal an intermittent structure consistent with that observed in classic homogeneous isotropic turbulence. The measurements were taken close to cloud top in a 200-m-thick cloud layer over a path of approximately 6 km, using a hot-wire anemometer below a helicopter as part of the Airborne Cloud Turbulence Observation System (ACTOS) measurement system. Hot-wire signal artifacts resulting from droplet impacts are removed without significantly degrading the signal, such that high-order velocity structure functions can be evaluated. The structure function analysis for orders 2–8 show statistically significant departures from the Kolmogorov’s 1941 scaling, yielding scaling exponents consistent with the Kolmogorov–Obukhov refined similarity hypothesis, with an intermittency exponent of 0.25. This is in agreement with the accepted value determined in single-phase flows under carefully controlled conditions, and no evidence is found of any departure from the large body of knowledge obtained from the laboratory on the finescale turbulence structure. This suggests that processes depending on the finescale structure of turbulence that cannot presently be measured in clouds can be explored in the laboratory setting. Since these findings pertain to clouds with relatively low liquid water content and weak turbulence, further work will be required to determine their applicability to other cloud types.

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

2019 ◽  
Vol 879 ◽  
pp. 554-578 ◽  
Author(s):  
Zhentong Zhang ◽  
Dominique Legendre ◽  
Rémi Zamansky
Keyword(s):  
Reynolds Number ◽  
Stochastic Model ◽  
Turbulent Flows ◽  
Dissipation Rate ◽  
Density Ratio ◽  
Stokes Number ◽  
Small Scale ◽  
Fluid Inertia ◽  
Joint Orientation ◽  
Inertia Forces

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.

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