Large eddy simulations (LES) of dispersed gasliquid flows for the prediction of flow patterns and its applications have been reviewed. The published literature in the last ten years has been analysed on a coherent basis, and the present status has been brought out for the LES EulerEuler and EulerLagrange approaches. Finally, recommendations for the use of LES in dispersed gas liquid flows have been made.
Gasliquid flows are often encountered in the chemical process industry, but also numerous examples can be found in petroleum, pharmaceutical, agricultural, biochemical, food, electronic, and powergeneration industries. The modelling of gasliquid flows and their dynamics has become increasingly important in these areas, in order to predict flow behaviour with greater accuracy and reliability. There are two main flow regimes in gasliquid flows: separated (e.g., annular flow in vertical pipes, stratified flow in horizontal pipes) and dispersed flow (e.g., droplets or bubbles in liquid). In this work, we consider only dispersed bubbly flows.
In the EL approach, the continuous liquid phase is modelled using an Eulerian approach and the dispersed gas phase is treated in a Lagrangian way; that is, the individual bubbles in the system are tracked by solving Newton’s second law, while accounting for the forces acting on the bubbles. An advantage here is the possibility to model each individual bubble, also incorporating bubble coalescence and breakup directly. Since each bubble path can be calculated accurately within the control volume, no numerical diffusion is introduced into the dispersed phase computation. However, a disadvantage is, the larger the system gets the more equations need to be solved, that is, one for every bubble.
The EE approach describes both phases as two continuous fluids, each occupying the entire domain, and interpenetrating each other. The conservation equations are solved for each phase together with interphase exchange terms. The EE approach can suffer from numerical diffusion. However, with the aid of higher order discretization schemes, the numerical diffusion can be reduced sufficiently and can offer the same order of accuracy as with EL approach (Sokolichin et al. [
The large eddy simulation (LES) falls between DNS and RANS in terms of the fraction of the resolved scales. In LES, large eddies are resolved directly, that is, on a numerical grid, while small, unresolved eddies are modelled. The principle behind LES is justified by the fact that the larger eddies, because of their size and strength, carry most of the flow energy (typically 90%) while being responsible for most of the transport, and therefore they should be simulated precisely (i.e., resolved). On the other hand, the small eddies have relatively little influence on the mean flow and thus can be approximated (i.e., modelled). This approach to turbulence modelling also allows a significant decrease in the computational cost over direct simulation and captures more dynamics than a simple RANS model.
In RANS models often the assumption of isotropic turbulence is made for the core of the flow, which is not valid in dispersed bubbly flows; that is, the velocity fluctuations in the gravity direction are typically twice those in the other directions. This assumption is not made in LES for large structures of the flow, giving LES an advantage over RANS for the core regions of the flow. However, the situation is different close to the walls, where LES’ assumption of isotropic turbulence is heavily violated, due to the absence of large eddies close to the walls.
In dispersed bubbly flows, the largescale turbulent structures interact with bubbles and are responsible for the macroscopic bubble motion, whereas smallscale turbulent structures only affect smallscale bubble oscillations. Since, large scales (carrying most of the energy) are explicitly captured in LES and the less energetic small scales are modelled using a subgridscale (SGS) model, LES can reasonably reproduce the statistics of the bubbleinduced velocity fluctuations in the liquid.
There are three important considerations for modelling of dispersed bubbly flows.
Separation of length scales of the interface, that is, micro, meso, and macroscales. The separation of these scales forms the basis for “filtering” the Navier–Stokes equations and applying proper model equations for multiphase situation. Important for dispersed flow is to identify the scales at which the governing equations are to be applied; microscales, that is, scales which are small enough to describe individual bubble shapes; mesoscales, which are comparable to bubble sizes; and macroscales, which entail enough bubbles for statistical representation.
The gridscale equations. Depending on the ratio of the length scales introduced above, with the grid resolution we can afford, on a given computer hardware, a proper form of the governing equations must be chosen. For instance, if the mesh size is in the microscale order, one can use singlefluid, interface tracking techniques to solve the problem. If, on the other hand, the grid size is large enough for statistical description of bubbles, the EE approach can be used. Should the grid size be comparable to the mesoscales, we are in a limiting area for both approaches, and special care must be taken in order to solve equations which describe the underlying physics consistently.
The physical models. Depending on the selected gridscale equations, physical models of various complexities must be employed. The options here are numerous, whether they concern turbulence modelling or interphase modelling, but these models are generally simpler in case more of the microscales are resolved.
In the following sections, we describe each of these three elements to model turbulent dispersed bubbly flow.
The aim of filtering the NavierStokes equations is to separate the resolved scales from the SGS (nonresolved). The interface between the phases, and the level of detail required in its resolution/modelling, defines the filter in a multiphase flow.
When LES is applied at a microscale, filtering of turbulent fluctuations needs to be combined with interface tracking methods. These methods have been developed and used in both dispersed flow and free surface flow by Bois et al. [
When LES is applied at a macroscale, the interface resolution is not considered. However, in practical simulations, these would require too coarse grids, leading to poor resolution of turbulence quantities. Much more often we are in the mesoscale region, in which the mesh size is comparable to bubble sizes. This pushes the main assumptions of the EE approach to its limit of validity, and the grid is not fine enough for full interface tracking. In other words, the mesh requirement for EE multiphase modelling conflicts with the requirements by LES approaches [
The issue of the requirement of the mesh size was first addressed by Milelli et al. [
Milelli condition (from Niceno et al. [
The principle of the LES formulation is to decompose the instantaneous flow field into largescale and smallscale components via a filtering operation. If
The right hand side terms of (
The SGS stress tensor which reflects the effect of the unresolved scales on the resolved scales is modelled as
In the EE approach, separate equations are required for each phase (see (
In the EL approach, there are two coupled parts: a part dealing with the liquid phase motion and a part describing the bubbles motion. The dynamics of the liquid are described in a similar way as in the EE approach, whereas the bubble motion is modelled through the second law of Newton.
Since, the governing equations for the liquid and gas phase are expressed in the Eulerian and Lagrangian reference frames, respectively; a mapping technique is used to exchange interphase coupling quantities. Depending upon the volume fraction of the dispersed phase, oneway (e.g.,
In the forward coupling, calculated liquid velocities, velocity gradients, and pressure gradients on an Eulerian grid are interpolated to discrete bubble locations for solving the Lagrangian bubble equation motion.
The forces available at each bubble’s centroid need to be mapped back to the Eulerian grid nodes in order to evaluate the reaction force
The motion of a single bubble with constant mass can be written according to Newton’s second law:
The bubble dynamics are described by incorporating all relevant forces acting on a bubble rising in a liquid. It is assumed that the total force,
For each force the analytical expression or a semiempirical model is used, based on bubble behaviour observed in experiment or in DNS.
To summarize, the influence/contribution of these forces are as follows.
The modeling of the lift force for capturing bubble plume meandering and bubble dispersion is important. However there is an uncertainty regarding appropriate value or correlation representing lift coefficient. There is also recommendation that bubble sizedependent lift coefficient should be chosen [
The value of the lift coefficient can be different than the one used in RANS approach. It is because of different handling of factors responsible for bubble dispersion, that is, the interaction between the bubbles and influence of turbulent eddies in the liquid phase. In RANS approach, they are considered by means of the lift and turbulent dispersion force, with uncertainty of exact contribution of the individual forces. Most of the investigators use a constant value of the lift coefficient (
The virtual mass force is proportional to the relative acceleration between the phases and is negligible once a pseudosteady state is reached. It has little influence on the simulation results for bubble plumes [
In LES, through filtering, velocities are decomposed into a resolved and a SGS part. The resolved part of the turbulent dispersion is implicitly computed. However, in case of a bubble size smaller than the filter size, turbulent transport can be present at SGS level and should be considered [
The values or expressions for the coefficient of drag, lift and virtual mass force used by different investigators are given in Tables
Comparison of LES simulations.
No.  Author 
Column 
Sparger design 
Bubble diameter 
Range of 
Number of grid cells  Filter  SGS Model  BIT closure models^{+}  Interfacial force coefficient closures^{++}  

Drag 
Lift 
Virtual 

(1)  Deen et al.* [ 

Perforated plate  4 mm 



Smagorinsky, 




(2)  Bove et al. 

Perforated plate  4 mm 



Smagorinsky, 
— 



(3)  Zhang et al. 

Perforated plate  4 mm 


—  Smagorinsky, 


^{ # }  ^{ # } 
(4)  Tabib et al. 

Perforated 
5 mm 

150000 

Smagorinsky, 


^{ #}  ^{ #} 
(5)  Dhotre et al. 

Perforated plate  4 mm 



Smagorinsky, 




(6)  Ničeno et al. 

Perforated plate  4 mm 



Smagorinsky, 




(7)  Dhotre et al. 

Perforated plate  2.6 mm 

—  2.8 mm < 
Smagorinsky, 




(8)  Niceno et al. 

Perforated plate  4 mm 



OEM 




(9)  Tabib and Schwarz 

Perforated plate  3–5 mm 

—  —  OEM 

Max [ 

— 
(10)  van den Hengel 

Perforated plate  3 mm 



Smagorinsky 




(11)  Hu and Celik 

Pipe sparger  1.6 mm 


PSIball method 
Smagorinsky, 




(12)  Lain 

Porous membrane  2.6 mm 



Smagorinsky, 

^{ #}  ^{ # } 

(13)  Darmana et al. 

Multipoint gas injection  4 mm 



Vreman, 
— 

^{ #} 

(14)  Sungkorn et al. 

Perforated plate  4 mm 



Smagorinsky, 


^{ #} 

(15)  Bai et al. [ 

Perforated plate 
5 mm 


—  Vreman Smagorinsky, 
— 



^{+}Numbers indicated are refered to Table
^{++}Numbers indicated are refered to Table
It is well known that in turbulent flow energy generally cascades from large to small scales. The primary task of the SGS model therefore is to ensure that the energy drain in the LES is same as obtained with the cascade fully resolved as one would have in a DNS. The cascading, however, is an average process. Locally and instantaneously the transfer of energy can be much larger or much smaller than the average and can also occur in the opposite direction (“backscatter”).
The simplest, wellknown, and mostly used Smagorinsky [
In the singlephase flow literature, the value of the constant used is in the range from
The main reason for the frequent use of the Smagorinsky model is its simplicity. Its drawbacks are that the constant
The dynamic model, originally proposed by Germano et al. [
The dynamic SGS model assumes SGS turbulent energy to be in local equilibrium (i.e., production = dissipation). The eddy viscosity is estimated from (
The basic idea is to apply a second test filter to the equations. The new filter width, twice the size of the grid filter, produces a resolved flow field. The difference between the two resolved fields is the contribution of the small scales whose size is in between the grid filter and the test filter. The information related to these scales is used to compute the model constant. The advantage here is that no empirical constant is needed and that the procedure allows the negative turbulent viscosity implying energy transfer from smaller to larger scales (energy backscatter). This effect, in principle, allows both an enhancement and attenuation of the turbulent intensity introduced by the bubbles.
The model has a few drawbacks; wide fluctuations in dynamically computed constants can cause stability issues, along with additional computational expense.
In spite of the fact that dynamic SGS model calculates model constant
The essence of the oneequation model is to solve additional transport equation for SGS turbulent kinetic energy:
The availability of the SGS turbulent kinetic energy allows for modelling of SGS interphase sorces such as bubbleinduced turbulence and turbulent dispersion at SGS. The application of oneequation SGS model for bubbly flows is illustrated in more detail in sections below.
In the EE approach, the turbulent stress in the liquid phase is considered to have two contributions, one due to the inherent, that is, shearinduced turbulence that is assumed to be independent of the relative motion of bubbles and liquid and the other due to the additional bubbleinduced turbulence (Sato and Sekoguchi [
The second approach for the modelling of BIT allows for the advective and diffusive transport of turbulent kinetic energy. This model incorporates the influence of the gas bubbles in the turbulence by means of additional source terms in the
(a) Resolved (dashed) and total (continuous) liquid kinetic energy and (b) ratio of the modelled and resolved parts of the turbulent kinetic for various BIT models. (from Niceno et al. [
Figure
Bubbleinduced turbulence models.
No.  Author 



Assumptions 

(1)  Sato and Sekoguchi [ 

0  0  
(2)  Pfleger and Becker [ 
0 



(3)  Troshko and Hassan [ 
0 



(4)  Crowe et al. 
PSI cell/ball approximation  
(5)  Sommerfeld [ 
Stochastic interparticle collision model  
(6) 
Sommerfeld et al. [ 
Langevin equation model 
Drag force models.
No.  Author  Equation 

(1)  Ishii and Zuber [ 

(2)  Tomiyama [ 

 


(3)  Tomiyama [ 

(4)  Ishii and Zuber [ 



(5)  Clift et al. [ 

0.44 

(6)  Tomiyama [ 

Crucial parameters for obtaining reliable LES results are the time step selection, the total time for gathering good statistics of the averaged variables, and discretization schemes for the variables. The time step choice is determined by the criterion that the maximum CourantFredrichsLevy (CFL) number must be less than one (
For flow variables, central difference should be used for discretization of advection terms and avoid using diffusive upwind schemes. However, for scalars variables, highorder schemes (MUSCL, QUICK, or SecondOrder) may be tolerable to avoid nonphysical solutions (e.g., negative volume fractions). An alternative to highorder schemes are the bounded central differences. The risk with use of all but central scheme is their diffusivity. Their influence on LES may exceed the modelled SGS transport.
It is necessary to follow the initial phase of the simulation, wherein the turbulent strutures develop starting from initial condition and to reach a statistiacally steady state. The duration of this phase depends on the flow characteristics. The simulation must be run for a total time long enough to allow all turbulent instabilities that develop during this phase to be convected across the region of interest. However, the convecting velocities of the turbulent structures and the regions of interest are not always known as a priori. This is why it is recommended to run the simulation a multitude (typically 5 times) of the slowest integral time scales, which often is the flow through time defined as the ratio of the system height over the bulk (superficial) velocity.
Here, we review different LES studies that were performed using the EE and EL approaches for simulating flow patterns in gasliquid bubbly flows. Table
Milelli et al. reported for the first time twophase LES with EE approach. They first investigated statistically 2D flow configuration and then free bubble plume.
They addressed important concerns related to the twophase LES simulation. For instance, they found that the optimum ratio of the cutoff filter width (i.e., the grid) to the bubble diameter (
Milelli [
Further, they observed in simulation that the lift coefficient value plays a major role in capturing the plume spreading and the used lift coefficient may differ for an LES compared to the one that is justified in an RANS approach. The plausible explanation here is from different handling of two factors responsible for bubble dispersion, that is, interaction between the bubbles and influence of turbulent eddies in the liquid phase.
Deen et al. [
They found that RANS approach (
Time history of the axial liquid velocity at the centreline of the column, at a height of 0.25 m (from Deen et al. [
Furthermore, they also identified that the lift force is responsible for transient spreading of the bubble plume and in absence of it, only with drag force, the bubble plume showed no transverse spreading.
They considered the effective viscosity of the liquid phase with three contributions: the molecular, shearinduced turbulent (modelled using Smagorinsky model), and bubbleinduced turbulent viscosities [
Bove et al. [
Further, the LES results were found to be very sensitive to inlet boundary conditions (Figure
Comparison of (a) averaged axial liquid velocity profile at
They used drag model for the contaminated water which gave a better prediction of the slip velocity; however, the velocity profile was underestimated for both gas and liquid phase. Reason for the underprediction was not clear, whether it was due to drag model or an improper value of the lift coefficient used or an error in the near wall modelling. Need for further work in this direction was suggested.
Zhang et al. [
Comparison of the prediction and measurement of mean velocity of the both phases; the predicted profiles were obtained with different
Axial liquid velocity
Axial gas velocity
They extended the work of Deen et al. [
Tabib et al. [
Dhotre et al. [
They further investigated the value of
Probability density function for computed constant
Predicted instantaneous vector flow field for axial liquid velocity after 150 s, for all three models (from Dhotre et al. [
Germano
Smagorinsky
RANS
It was further concluded that the Germano model can give correct
Niceno et al. [
They suggested that the modelled SGS information can be used to access the SGS interfacial forces, in particular the turbulent dispersion force. In their work, the effect of SGS turbulent dispersion force could not be determined as the bubble size was almost equivalent to the mesh size.
Dhotre et al. [
Comparison of
They emphasized the crucial role of the lift force in the prediction of the lateral behaviour of the bubble plumes. In the RANS approach the turbulent dispersion force is required to reproduce the bubble dispersion; however, in LES, bubble dispersion is implicitly calculated by resolving the largescale turbulent motion responsible for bubble dispersion. The dependence of the bubble dispersion with the value of lift coefficient was also observed in Milelli et al. [
Dhotre et al. [
Niceno et al. [
Comparison of liquid turbulent kinetic energy obtained with CFX4 using oneequation model and Neptune CFD with Smagorinsky model and experimental data. The blue dashed line is the resolved, the blue continuous line is the total (resolved plus SGS) kinetic energy (from Niceno et al. [
Comparison of the simulated and experimental liquid velocity and velocity fluctuations for cases with and without SGS model at a height of 0.255 m and a depth of 0.075 m. Effect of the SGS model (from Van den Hengel et al. [
Tabib and Schwarz [
They used the formulation of Lopez de Bertodano [
Van den Hengel et al. [
Authors studied the influence of the SGS model on the predictions and found that without SGS model, the average liquid velocity and liquid velocity fluctuations are much lower compared to the case with a SGS model. This was due to the lower effective viscosity in this case, which led to less dampening of the bubble plume dynamics and subsequently to flatter mean liquid velocity profiles (as shown in Figure
In this work also, the authors confirmed the important role of the lift coefficient in capturing the plume dynamics. They considered two lift coefficients (
Hu and Celik [
They reported secondorder statistics of the pseudoturbulent fluctuations and demonstrated that a singlephase LES along with a pointvolume treatment of the dispersed phase could serve as a viable closure model.
Hu and Celik reported that the predicted mean quantities (such as mean liquid velocity field) were in good agreement with the experimental data of Sokolichin and Eigenberger [
Longtime averaged liquid velocity field on middepth plane: (a) EL approach, (b) LDA measurement of Becker et al. [
Lain [
A simple model for the subgrid liquid fluctuating velocity to account for the BIT considered in this work was found to have no influence on the predictions. As in previous works, authors confirmed a strong dependency of the bubble dispersion in the column on the value of transverse lift force coefficient used. He concluded that the lift coefficient depends on the bubbleliquid relative velocity and was the main mechanism responsible for the spreading of bubbles across the column crosssection. He further compared the simulation results with particle image velocimetry (PIV) measurements (Border and Sommerfeld [
Darmana et al. [
Instantaneous flow structure comparison between experiment (a) and simulation (b). From left to right: bubble positions, bubble velocity, and liquid velocity (from Darmana et al. [
Sungkorn et al. [
Snapshots of the bubble dispersion pattern after 20, 50, 100, and 150 s. The bubbles are coloured by the local magnitude of the liquid fluctuations (from Sungkorn et al. [
It was also found that their collision model leads to two benefits: the computing time is dramatically reduced compared to the direct collision method and secondly it also provides an excellent computational efficiency on parallel platforms. Sungkorn et al. [
The investigations discussed in earlier sections dealt with the use of LES for predicting the flow patterns. In the published literature, the knowledge of flow pattern has been employed for the estimation of equipment performance such as mixing (Joshi and Sharma [
Darmana et al. [
Also, the presence of various chemical species was accounted through a transport equation for each species. Darmana et al. estimated the mass transfer rate from the information of the individual bubbles directly. They used the model to simulate the reversible twostep reactions found in the chemisorption process of CO_{2} in an aqueous NaOH solution in a labscale pseudo2D bubble column reactor (e.g., Figure
Instantaneous solution 10 s after the
Zhang et al. [
Bai et al. [
They further investigated the effect of the gas sparger properties (sparged area and its location) on the hydrodynamics in a bubble column and characterized the macromixing of the gas phase in the column in terms of an axial dispersion coefficient. They compared the predicted liquid phase dispersion coefficient with the literature correlations as shown in Figure
Comparison of the simulated liquid phase dispersion coefficient with the literature correlations (from Bai et al. [
In the RANS approach, the drag and lift forces depend on the actual relative velocity between the phases, but the ensemble equations of motion for the liquid only provide information regarding the mean flow field. The random influence of the turbulent eddies is considered by modelling a turbulent dispersion force. By analogy with molecular movement, the force is set proportional to the local bubble concentration gradient (or void fraction), with a diffusion coefficient derived from the turbulent kinetic energy. The value of the turbulent dispersion coefficient is chosen to get an agreement with the measurement data and is not known as a priori.
In LES, the resolved part of the turbulent dispersion is implicitly computed, and hence one can use information from LES for calculating the magnitude of this force. The methodology depends on scales at which LES is to be applied. For instance, at the mesoscale, in the EL approach, bubbles dispersed by drag and lift through turbulent eddies can be computed. At microscale LES, one might need to consider bubble coalescence and breakup phenomena along with a reasonable number of bubbles. It can be computationally expensive, but in view of increasing available computer power, this should become feasible soon.
The turbulent flows contain flow structures with a wide range of length and time scales which control the transport processes. The length scales of these structures can range from column dimensions (highest) to Kolmogorov scales (lowest). However, not all the scales of turbulence contribute equally to different transport rates and mixing. If only mixing is the important design criterion, then the knowledge about the mean flow pattern (largescale structures) would generally suffice the purpose (Ekambara and Joshi [
The subject of quantification of local turbulent flow structures and reliable estimation of transport properties has been reviewed by Joshi et al. [
CFD provides detailed flow information within single and multiphase reactors. Most popular and computationally inexpensive models such as
It is known that a large number of simplifying assumptions are made while deriving the
EE and EL LES are promising approaches for predicting unsteady, buoyancydriven flow inducing largescale coherent structures for gasliquid dispersed flow. Care should be taken to clearly identify the scales (micro, macro, or meso) at which LES should be applied, in order to decide the level of interface resolution and modelling required. The approach of LES at mesoscales (i.e., without explicitly tracking interface) using EE and EL description has been reviewed for gasliquid dispersed flow.
Pioneering work of Milelli et al. [
The simulation and the experimental measurement of Deen et al. [
The concept behind the LES is very simple but characterized by a large number of choices (regarding numerical and physical modelling) that all have significant influence on the results. However, it offers great potential in terms of determination of statistical quantities and instantaneous information about flow structures. This information can be extremely useful for the prediction of other physical processes behaviour (e.g., transport of scalar (temperature, concentration), chemical reactions).
From LES simulation with EE/EL approaches that were reviewed in this work, it is recommended that:
The grid or filter size selection based on filter size to bubble diameter ratio Δ/
The Smagorinsky constant,
The lift force is the main mechanism for the dispersion, and the lift coefficient should be estimated though sensitivity of interfacial forces on values of slip velocity and gas holdup. The lift coefficient in LES can be different from that in RANS.
The central difference scheme should be used for the discretization of advection terms for flow variables and highorder schemes (MUSCL, QUICK, or SecondOrder) can be used for scalar variables.
The minimum time for gathering statistics should be at least one flow through time (as defined as ratio of the system height over the bulk (superficial) velocity).
In advent of computer hardware, the EL approach appears very promising for the near future. Further work in mapping functions for twoway coupling can expedite the development of this approach that can be used as a means of both predicting the properties of specific turbulent flows and providing flow details that can be used like data to test and refine other turbulenceclosure models.
The approach for BIT with extra production terms into the SGSturbulent kinetic energy equation (following the procedure described by Pfleger and Becker [
Treatment of the interphase forces needs more attention.
The drag and nondrag forces (lift, virtual mass force) can be modelled using resolved field approaches. The modelling of these forces for the SGS and their effect on the overall simulation results need to be evaluated.
One finds strong dependency of the bubble dispersion on the value of transverse lift force coefficient. The transverse lift, which depends on the bubbleliquid relative velocity, seems to be the main mechanism responsible for the spreading of the bubbles. It will help if one can estimate the separate contributions of each of these forces.
The virtual mass force has little influence on simulation results. So far, a constant coefficient has been used in all the investigations; however, dependence on void fraction has been shown in experiments. It would be good to have a correct description in order to improve results near the inlet where bubble acceleration effects are important.
The strong coupling between subgridscale (SGS) modelling and the truncation error of the numerical discretization can be exploited by developing discretization methods where the truncation error itself functions as an implicit SGS model. Such attempt can be useful and go in the direction of finding a universal SGS model.
In order to use LES for reliable predictions at minimum computational costs, understanding of the influence of discretization methods, boundary conditions, wall models, and numerical parameters (e.g., convergence criterion, time steps, etc.) is essential. The contribution focusing on these aspects should be undertaken for both EE/EL approaches.
Substantial development has been achieved in LES in the last decade for understanding bubbly gasliquid dispersed flow. However, it is mainly restricted to low superficial gas velocities and gas fractions. Future work should focus on industrially relevant largescale reactors at high superficial gas velocity. The modelling of bubble coalescence and breakup might be necessary, along with further clarity in filtering operations.
Joshi and coworkers have used LES for the identification of flow structures and their dynamics. They have proposed a procedure to use this information for the estimation of design parameters. Substantial additional work is needed for finding 3D information on the structure characteristics such as size, shape, velocity, and energy distributions
Drag force coefficient
Lift force coefficient
Virtual mass coefficient
Turbulent dispersion coefficient
Model parameter in turbulent dissipation energy equation (=1.44)
Model parameter in turbulent dissipation energy equation (=1.92)
Smagorinsky model constant
Model constant (Sato and Seguchi [
Mean bubble diameter
Diameter of the column (m)
Force originating due to pressure (N/
Gravitational force per unit volume of dispersion (N/
Lift forceper unit volume of dispersion (N/
Virtual mass force per unit volume of dispersion (N/
Turbulent dispersion force per unit volume of dispersion (N/
Wall lubrication force per unit volume of dispersion (N/
Wall deformation force per unit volume of dispersion (N/
Height of column (m)
Turbulent kinetic energy per unit mass (
Subgridscale turbulent kinetic energy (
Characteristic of strain tensor of filtered velocity
Instantaneous axial velocity (m/s)
Width of column (m).
Filter width
Simulation time step
Gridscale component of scalar
Resolved component of scalar
Filtered component of scalar
Turbulent energy dissipation rate per unit mass (
Fractional phase holdup
Fractional gas phase holdup
Density (kg/
Density of liquid (kg/
Effective viscosity of phase
Effective viscosity of liquid phase
Molecular viscosity of liquid phase
Bubbleinduced viscosity (Pa s).
Phase,
Bubbleinduced.
Bubbleinduced turbulence
Eulerian
Lagrangian
Subgridscale.
N. G. Deen would like to thank the European Research Council for its financial support, under its Starting Investigator Grant scheme, contract number 259521 (cutting bubbles).