^{1}

^{2}

^{1}

^{1}

^{2}

In this work, we report on Euler-Euler large eddy simulation (EELES) of dispersed bubbly flow in a square cross-sectional bubble column. Simulations are performed using the Neptune_CFD package, and results processed using the SALOME platform. The motivation to undertake this study is to check our implementation of the Smagorinsky subgrid-scale (SGS) model into NEPTUNE_CFD. We outline all the physical models used, and we present instantaneous realizations of velocity and void fraction fields in order to illustrate the structure of the turbulence field, and long-time averaged results, to compare with analogous simulations performed using the CFX-4 code and experimental data. The same physical models and constants have been used in both the CFX-4 and NEPTUNE_CFD codes, except the SGS model, which is Smagorinsky in case of NEPTUNE_CFD and a one-equation model in CFX-4. The results obtained with EELES compare reasonably well with experiment, meaning in particular that the implementations have been successful. Some perspectives on the further use of EELES are also given.

The aim of the NURESIM
[

Previous work on EELES (often
referred to as two-fluid LES) for bubbly flows include Milelli et al. [

In the Deen series
of experiments [

Geometry of the Deen bubble column experiment.

The governing
equation for conservation of mass, with no mass exchange between the phases,
takes the form:

Turbulence is
modelled in the liquid phase of the flow. The first step in the derivation of
the governing equations for LES is the decomposition of the velocity field into
resolved (grid scale) and unresolved (SGS) parts. For each component, we write

To complete (

A disadvantage of the Smagorinsky model is the loss of information resulting
from use of the deviatoric part of the SGS stress tensor only. In CFX-4, we
have implemented a transport equation for SGS kinetic energy

To close the
momentum equation set for the two phases, the various interphase exchange
terms have to be modelled. The interphase exchange
terms relevant for
the configuration considered in this work are

The

For the

The

In the present work, we couple the Euler-Euler approach for multiphase flow with LES, and therefore have to consider the resolution requirements of both techniques simultaneously in order to choose a satisfactory grid. A basic requirement is that the control volume size should be large enough to encompass all the interface details. This is the intrinsic assumption in the derivation of the Euler-Euler model equations, and strictly has to be satisfied at the discrete level as well.

In LES, the SGS
model is often very simple, and only drains energy from the resolved field without feed-back.
Therefore, our goal in LES is to resolve as much of the flow field as possible,
and to have as fine a grid as feasible for the available computer hardware. Since
the Euler-Euler approach specifies the minimum control volume size, whereas for
LES we are invariantly seeking as fine a grid as possible, the requirements for
the numerical grid may sometimes be in conflict. The point is illustrated in
Figures

Bubble size larger than filter width.

Bubble size smaller than filter width.

The grid size
considerations discussed above, are not new. In the present work, we have indicated why the bubbles must be
smaller than the cell size from the point of view of the energy spectra and
modelling closures for the interfacial forces. A systematic a posteriori analysis of the minimum
ratio of the bubble and cell sizes for LES modelling of free bubble plumes is
reported by Milelli et al. [

The Milelli condition.

It might be argued at this point that the grid used in
the present simulations is too coarse for LES, at least from the point of view
of capturing all the relevant (large) scales. Judging from the flow patterns
that can be expected in a square column such as this, with the bubble plume
meandering from one direction to the other, it is quite conceivable that the
largest, and therefore the most energetic, eddies will be of the size of the
domain cross-section. Therefore, we are confident that the grid resolution we
employ (30

In this section, we briefly summarize all the relevant simulation details.

for liquid:

for gas:

number of cells in computational domain: 90 000;

advection scheme: bounded central scheme for Neptune_CFD;

quick scheme for CFX-4.

the linear backward time differencing for Neptune_CFD;

quadratic backward differencing for CFX-4;

CFL = 1.0, variable time step,
approximately equal to 0.05 second for Neptune_CFD constant time step of 0.05 second for CFX-4, leading to CFL

integral time of simulation: 400 seconds;

simple time-average is used in both, not weighted by volume fraction.

In this
section, the results for our EELES with Neptune_CFD are presented. We start
with the instantaneous velocity and void fraction fields to show the flow
patterns in the bubble column flow, and continue with time-averaged velocity and fluctuating liquid velocity
profiles. Time-averaged profiles obtained with Neptune_CFD and CFX-4 are
compared with the experimental measurements of Deen et al. [

Instantaneous
velocity fields are shown in Figure

Instantaneous velocity vectors, calculated using Neptune_CFD, at instants: 60 seconds, 80 seconds, and 120 seconds.

Isosurfaces of constant void fraction (0.05 and 0.5), calculated using Neptune_CFD, at instants: 60 seconds, 80 seconds, and 120 seconds.

In this
section, we compare the simulated profiles of liquid and gas vertical
velocities and liquid turbulence intensities against the reported experimental
data [

Figure

Comparison of (a) liquid and (b) gas velocity profiles obtained with CFX-4 using a one-equation model, Neptune_CFD with the Smagorinsky model, and experimental data.

The vertical
component of the resolved vertical fluctuating liquid velocity is plotted in
Figure

Comparison of (a) vertical and (b) lateral fluctuating liquid velocities, obtained using CFX-4 with a one-equation model, Neptune_CFD with the Smagorinsky model, and experimental data.

Comparison of liquid turbulent kinetic energy obtained with CFX-4 using a one-equation model, Neptune_CFD with the Smagorinsky model, and experimental data. The blue dashed line is the resolved, the blue continuous line is the total (resolved plus SGS) kinetic energy.

Implementation of the Smagorinsky SGS model in the Neptune_CFD code has been validated using data from the Deen bubble column experiment. Results have also been compared with predictions obtained from simulations performed using the code CFX-4. Generally, liquid velocity profiles are under-predicted compared with measured data, whereas the gas velocity profiles are predicted very well. This can only be attributed to a wrong assumption for the drag coefficient. The liquid velocity fluctuations predicted by both codes compare well with experiments, except for some under-prediction of the vertical component. Overall, the results obtained with the two codes are consistent, regardless of the fact that different SGS models have been used. This is a clear indication that the SGS model influences results only to a very small extent.

When envisaging
the potential of LES for modelling multiphase flows in general, one should
clearly distinguish between the scales at which LES might be used, since this
implies the level of detail of accurate interface resolution to the degree of modelling
that can be tolerated. Simulations at
mesoscales imply an
Euler-Euler description of the interface between the phases, such as the one
described in this work. However, if LES for multiphase flows is applied at microscales,
explicit interface tracking procedures would be needed. This has already been
proposed in [

The principal advantage of EELES over LSS is that since the interface details are not calculated explicitly, the simulations may be carried out at lower cost. The principal disadvantage of EELES is that the most influential interfacial forces (lift and drag) are modelled for the large-scale field, meaning that the question of how to model these forces remains as open for EELES as it is with RANS. LSS, on the other hand, explicitly resolves the large-scale part of the interfacial forces, leaving the modelling at the SGS level, where the effects are smaller and hence less influential on the accuracy of the results. LSS modelling comes at a heavy price, due to the need to resolve all relevant interface details, imposing huge demands on computing power. Applying LSS to industrial-scale problems is beyond the current state of computing resources.

Another
disadvantage of EELES stems from the different resolution requirements imposed
by the Euler-Euler description of the two fluids and that of the LES approach itself,
expressed in [