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

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

Micro-bubble dynamics in turbulent flow

This thesis is devoted to the study of the motion of small bubbles in homogeneous isotropic turbulent flows. The work addresses several questions related to the statistical description of the hydrodynamic forces exerted on a bubble as well as the stochastic modeling of their high frequency fluctuations. First, we propose a model for the acceleration of micro-bubbles (smaller than the dissipative scale of the flow) subjected to the drag and the fluid inertia forces. This model, that depends on the Stokes number, the 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). Second, based on the observation that acceleration statistics conditional to the local kinetic energy dissipation rate are invariant with the Stokes number and the dissipation rate, 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 modeling the dissipation rate along the bubble trajectory from a log-normal stochastic process, whereas its orientation is given by two coupled random walk 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. It can effectively reproduce effect of turbulent motion at scales smaller than the mesh resolution by adding a random contribution depending on local average dissipation rate. Comparisons with DNS and standard LES, show that the proposed model improves significantly the statistics of the bubbly phase. Third, we extend the previous results in the case of bubbles with large Reynolds number by considering non-linear drag laws. We define an effective relaxation time based on the drag coefficient to characterize bubble motion (acceleration,velocity). Eventually we study the effect of buoyancy and lift force on the bubble dynamics, and analyze the reduction of the average rising velocity in turbulent flow compared to quiescent flows. It is observed that bubbles preferentially explore region having downward fluid acceleration which contributes through the inertia force to reduction of the rising velocity. In addition, as already observed, the lift force brings preferably bubbles into downstream fluid motion which also reduce their rising velocity

A fourth-order kernel for improving numerical accuracy and stability in Eulerian and total Lagrangian SPH

The error of smoothed particle hydrodynamics (SPH) using kernel for particle-based approximation mainly comes from smoothing and integration errors. The choice of kernels has a significant impact on the numerical accuracy, stability and computational efficiency. At present, the most popular kernels such as B-spline, truncated Gaussian (for compact support), Wendland kernels have 2nd-order smoothing error and Wendland kernel becomes mainstream in SPH community as its stability and accuracy. Due to the fact that the particle distribution after relaxation can achieve fast convergence of integration error respected to support radius, it is logical to choose kernels with higher-order smoothing error to improve the numerical accuracy. In this paper, the error of 4th-order Laguerre-Wendland kernel proposed by Litvinov et al. \cite{litvinov2015towards} is revisited and another 4th-order truncated Laguerre-Gauss kernel is further analyzed and considered to replace the widely used Wendland kernel. The proposed kernel has following three properties: One is that it avoids the pair-instability problem during the relaxation process, unlike the original truncated Gaussian kernel, and achieves much less relaxation residue than Wendland and Laguerre-Wendland kernels; One is the truncated compact support size is the same as the non-truncated compact support of Wendland kernel, which leads to both kernels' computational efficiency at the same level; Another is that the truncation error of this kernel is much less than that of Wendland kernel. Furthermore, a comprehensive set of $2D$ and $3D$ benchmark cases on Eulerian SPH for fluid dynamics and total Lagrangian SPH for solid dynamics validate the considerably improved numerical accuracy by using truncated Laguerre-Gauss kernel without introducing extra computational effort.Comment: 37 pages and 12 figure

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

In this paper we study finite particle Reynolds number effects up to Re p=50 on the dynamics of small spherical bubbles and solid particles in an isotropic turbulent flow. We consider direct numerical simulations of light pointwise particles with various expressions of the drag force to account for finite Re p and the type of particle. Namely, we consider the Stokes drag law, the Schiller and Neumann relation and the Mei law. We show that an effective Stokes number, based on the mean value of the drag coefficient to account for the inertial effects involved in the drag law, gives a quasi-self-similar evolution of the variances of the bubble acceleration and of the forces exerted on the particle. This allows us to provide a satisfactory prediction of these quantities using Tchen's theory at finite particle Reynolds number. Based on these relations, we can specify the conditions under which the total inertial force (sum of the added mass and the Tchen contributions) is negligible compared with the drag force. Thus, for particles of very small dimensions, the fluid inertia force is negligible, provided the density ratio is of order 1 or larger. However, when the particle inertia becomes consequential, the threshold value of the density ratio increases significantly. Although this corresponds to the limit of the validity of the model, this draws attention to the fact that, for large Stokes numbers, the added mass and fluid inertia forces could play a more important role than is usually attributed to them

Extended Eulerian SPH and its realization of FVM

Eulerian smoothed particle hydrodynamics (Eulerian SPH) is considered as a potential meshless alternative to a traditional Eulerian mesh-based method, i.e. finite volume method (FVM), in computational fluid dynamics (CFD). While researchers have analyzed the differences between these two methods, a rigorous comparison of their performance and computational efficiency is hindered by the constraint related to the normal direction of interfaces in pairwise particle interactions within Eulerian SPH framework. To address this constraint and improve numerical accuracy, we introduce Eulerian SPH extensions, including particle relaxation to satisfy zero-order consistency, kernel correction matrix to ensure first-order consistency and release the constraint associated with the normal direction of interfaces, as well as dissipation limiters to enhance numerical accuracy and these extensions make Eulerian SPH rigorously equivalent to FVM. Furthermore, we implement mesh-based FVM within SPHinXsys, an open-source SPH library, through developing a parser to extract necessary information from the mesh file which is exported in the MESH format using the commercial software ICEM. Therefore, these comprehensive approaches enable a rigorous comparison between these two methods.Comment: 34 pages and 13 figure

Simultaneous Horizontal and Vertical Oscillation of a Quiescent Filament observed by CHASE and SDO

In this paper, we present the imaging and spectroscopic observations of the simultaneous horizontal and vertical large-amplitude oscillation of a quiescent filament triggered by an EUV wave on 2022 October 02. Particularly, the filament oscillation involved winking phenomenon in Ha images and horizontal motions in EUV images. Originally, a filament and its overlying loops across AR 13110 and 13113 erupted with a highly inclined direction, resulting in an X1.0 flare and a non-radial CME. The fast lateral expansion of loops excited an EUV wave and the corresponding Moreton wave propagating northward. Once the EUV wavefront arrived at the quiescent filament, the filament began to oscillate coherently along the horizontal direction and the winking filament appeared concurrently in Ha images. The horizontal oscillation involved an initial amplitude of 10.2 Mm and a velocity amplitude of 46.5 km/s, lasting for 3 cycles with a period of 18.2 minutes and a damping time of 31.1 minutes. The maximum Doppler velocities of the oscillating filament are 18 km/s (redshift) and 24 km/s (blueshift), which was derived from the spectroscopic data provided by CHASE/HIS. The three-dimensional velocity of the oscillation is determined to be 50 km/s at an angle of 50 to the local photosphere plane. Based on the wave-filament interaction, the minimum energy of the EUV wave is estimated to be 2.7 10 20 J. Furthermore, this event provides evidence that Moreton wavesshould be excited by the highly inclined eruptions

Dynamique des micro-bulles dans un écoulement turbulent

Cette thèse est consacrée à l'étude du mouvement de petites bulles dans des écoulements turbulents homogènes isotropes. Le travail aborde différentes questions liées à la description statistique des forces hydrodynamiques exercées sur une bulle ainsi qu'à leur modélisation stochastique tenant compte des effets d'intermittence. Nous proposons tout d'abord un modèle pour l'accélération de bulles de taille inférieures à l'échelle dissipative de l'écoulement soumises à la traînée et aux forces d'inertie du fluide. Ce modèle, qui dépend du nombre de Stokes, du nombre de Reynolds et du rapport de densité, reproduit l'évolution de la variance d'accélération ainsi que l'importance relative et l'alignement des deux forces observées à partir de simulations numériques directes (DNS). Deuxièmement, sur la base de l’observation selon laquelle les statistiques d’accélération conditionnelles au taux de dissipation de l’énergie cinétique locale sont invariantes avec le nombre de Stokes et le taux de dissipation, nous proposons un modèle stochastique du vecteur d’accélération instantanée de la bulle, qui tient compte de l’intermittence à petite échelle de la turbulence. La norme de l'accélération de la bulle est obtenue en modélisant le taux de dissipation le long de la trajectoire de la bulle à partir d'un processus stochastique lognormal, tandis que son orientation est donnée par deux marches aléatoires couplées sur une même sphère afin de modéliser l'évolution de l'orientation conjointe la traînée et les forces d'inertie agissant sur la bulle. Le modèle stochastique proposé pour l'accélération des bulles permet d'améliorer les simulations de grandes turbulences (LES) d'écoulements turbulents transportant de petites bulles. Il peut reproduire efficacement l’effet des échelles turbulentes inférieures à la résolution du maillage en ajoutant une contribution aléatoire en fonction du taux de dissipation moyen local. Les comparaisons avec le DNS et les LES standard montrent que le modèle proposé améliore considérablement les statistiques de la phase de formation de bulles. Troisièmement, nous étendons les résultats précédents dans le cas de bulles à plus grand nombre de Reynolds en prenant en compte les lois de traînée non-linéaires. Nous définissons un temps de relaxation effectif basé sur le coefficient de traînée pour caractériser le mouvement de la bulle (accélération, vitesse). Finalement, nous étudions l’effet de la flottabilité et de la force de portance sur la dynamique des bulles et analysons la réduction de la vitesse moyenne ascensionnelle dans les écoulements turbulents par rapport aux écoulements au repos. On observe que la bulle explore de préférence une région ayant une accélération de fluide vers le bas qui contribue, par le biais de la force d’inertie, à réduire la vitesse de montée. De plus, comme déjà observée, la force de portance amène de préférence les bulles dans un mouvement de fluide en aval qui réduit également leur vitesse de montée.This thesis is devoted to the study of the motion of small bubbles in homogeneous isotropic turbulent flows. The work addresses several questions related to the statistical description of the hydrodynamic forces exerted on a bubble as well as the stochastic modeling of their high frequency fluctuations. First, we propose a model for the acceleration of micro-bubbles (smaller than the dissipative scale of the flow) subjected to the drag and the fluid inertia forces. This model, that depends on the Stokes number, the 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). Second, based on the observation that acceleration statistics conditional to the local kinetic energy dissipation rate are invariant with the Stokes number and the dissipation rate, 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 modeling the dissipation rate along the bubble trajectory from a log-normal stochastic process, whereas its orientation is given by two coupled random walk 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. It can effectively reproduce effect of turbulent motion at scales smaller than the mesh resolution by adding a random contribution depending on local average dissipation rate. Comparisons with DNS and standard LES, show that the proposed model improves significantly the statistics of the bubbly phase. Third, we extend the previous results in the case of bubbles with large Reynolds number by considering non-linear drag laws. We define an effective relaxation time based on the drag coefficient to characterize bubble motion (acceleration,velocity). Eventually we study the effect of buoyancy and lift force on the bubble dynamics, and analyze the reduction of the average rising velocity in turbulent flow compared to quiescent flows. It is observed that bubbles preferentially explore region having downward fluid acceleration which contributes through the inertia force to reduction of the rising velocity. In addition, as already observed, the lift force brings preferably bubbles into downstream fluid motion which also reduce their rising velocity

Higher-order statistics and intermittency of a two-fluid HVBK quantum turbulent flow

The Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model is widely used to numerically study quantum turbulence in superfluid helium. Based on the two-fluid model of Tisza and Landau, the HVBK model describes the normal (viscous) and superfluid (inviscid) components of the flow using two Navier-Stokes type equations, coupled through a mutual friction force term. This feature makes the HVBK model very appealing in applying statistical tools used in classical turbulence to study properties of quantum turbulence. A large body of literature used low-order statistics (spectra, or second-order structure functions in real space) to unravel exchanges between the two fluids at several levels. The novelty in this study is to use a theoretical approach based on first principles to derive transport equations for the third-order moments for each component of velocity. New equations involve the fourth-order moments, which are classical probes for internal intermittency at any scale, revealing the probability of rare and strong fluctuations. Budget equations are assessed through Direct Numerical Simulations (DNS) of the HVBK flow based on accurate pseudo-spectral methods. We simulate a forced homogeneous isotropic turbulent flow with Reynolds number of the normal fluid (based on Taylor's microscale) close to 100. Values from 0.1 to 10 are considered for the ratio between the normal and superfluid densities. For these flows, an inertial range is not discernible and the Restricted Scaling Range (RSR) approach is used to take into account the Finite Reynolds Number (FRN) effect. We analyse the importance of each term in budget equations and emphasize their role in energy exchange between normal and superfluid components. Some interesting features are observed: i) transport and pressure-related terms are dominant, similarly to single-fluid turbulence; ii) the mathematical signature of the FRN effect is weak in the transport of the third-order moment, despite the low value of the Reynolds number; iii) for the normal fluid at very low temperatures, the mutual friction annihilates the effects of viscosity within the RSR. The flatness of the velocity derivatives is finally studied through the transport equations and their limit for very small scales, and it is shown to gradually increase for lower and lower temperatures, for both the normal fluid and the superfluid. This similarity highlights the strong locking of the two fluids. The flatness factors are also found in reasonable agreement with classical turbulence

Higher-order statistics and intermittency of a two-fluid HVBK quantum turbulent flow

The Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model is widely used to numerically study quantum turbulence in superfluid helium. Based on the two-fluid model of Tisza and Landau, the HVBK model describes the normal (viscous) and superfluid (inviscid) components of the flow using two Navier-Stokes type equations, coupled through a mutual friction force term. This feature makes the HVBK model very appealing in applying statistical tools used in classical turbulence to study properties of quantum turbulence. A large body of literature used low-order statistics (spectra, or second-order structure functions in real space) to unravel exchanges between the two fluids at several levels. The novelty in this study is to use a theoretical approach based on first principles to derive transport equations for the third-order moments for each component of velocity.Comment: 27 pages, 10 figures, accepted for publication in Journal of Fluid Mechanic

Higher-order statistics and intermittency of a two-fluid HVBK quantum turbulent flow

The Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model is widely used to numerically study quantum turbulence in superfluid helium. Based on the two-fluid model of Tisza and Landau, the HVBK model describes the normal (viscous) and superfluid (inviscid) components of the flow using two Navier-Stokes type equations, coupled through a mutual friction force term. This feature makes the HVBK model very appealing in applying statistical tools used in classical turbulence to study properties of quantum turbulence. A large body of literature used low-order statistics (spectra, or second-order structure functions in real space) to unravel exchanges between the two fluids at several levels. The novelty in this study is to use a theoretical approach based on first principles to derive transport equations for the third-order moments for each component of velocity. New equations involve the fourth-order moments, which are classical probes for internal intermittency at any scale, revealing the probability of rare and strong fluctuations. Budget equations are assessed through Direct Numerical Simulations (DNS) of the HVBK flow based on accurate pseudo-spectral methods. We simulate a forced homogeneous isotropic turbulent flow with Reynolds number of the normal fluid (based on Taylor's microscale) close to 100. Values from 0.1 to 10 are considered for the ratio between the normal and superfluid densities. For these flows, an inertial range is not discernible and the Restricted Scaling Range (RSR) approach is used to take into account the Finite Reynolds Number (FRN) effect. We analyse the importance of each term in budget equations and emphasize their role in energy exchange between normal and superfluid components. Some interesting features are observed: i) transport and pressure-related terms are dominant, similarly to single-fluid turbulence; ii) the mathematical signature of the FRN effect is weak in the transport of the third-order moment, despite the low value of the Reynolds number; iii) for the normal fluid at very low temperatures, the mutual friction annihilates the effects of viscosity within the RSR. The flatness of the velocity derivatives is finally studied through the transport equations and their limit for very small scales, and it is shown to gradually increase for lower and lower temperatures, for both the normal fluid and the superfluid. This similarity highlights the strong locking of the two fluids. The flatness factors are also found in reasonable agreement with classical turbulence