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This paper concerns the model of a polydispersed bubble population in the frame of an ensemble averaged two-phase flow formulation. The ability of the moment density approach to represent bubble population size distribution within a multi-dimensional CFD code based on the two-fluid model is studied. Two different methods describing the polydispersion are presented: (i) a moment density method, developed at IRSN, to model the bubble size distribution function and (ii) a population balance method considering several different velocity fields of the gaseous phase. The first method is implemented in the Neptune_CFD code, whereas the second method is implemented in the CFD code ANSYS/CFX. Both methods consider coalescence and breakup phenomena and momentum interphase transfers related to drag and lift forces. Air-water bubbly flows in a vertical pipe with obstacle of the TOPFLOW experiments series performed at FZD are then used as simulations test cases. The numerical results, obtained with Neptune_CFD and with ANSYS/CFX, allow attesting the validity of the approaches. Perspectives concerning the improvement of the models, their validation, as well as the extension of their applicability range are discussed.

Many flow regimes in Nuclear Reactor Safety Research are characterized by multiphase flows, with one phase being a liquid and the other phase consisting of gas or vapor of the liquid phase. The flow regimes found in vertical pipes are dependent on the void fraction of the gaseous phase, which varies, as void fraction increases, from bubbly flow to slug flow, churn turbulent flow, annular flow, and finally to droplet flow at highest void fractions. In the regimes of bubbly and slug flows, a spectrum of different bubble sizes is observed. While dispersed bubbly flows with low gas volume fraction are mostly monodispersed, an increase of the gas volume fraction leads to a broader bubble size distribution due to breakup and coalescence of bubbles. The exchange area for mass, momentum, or heat between continuous and dispersed phases thus cannot be simply modeled based on the knowledge of just the void fraction and a mean diameter. Moreover, the forces acting on the bubbles may depend on their individual size which is the case not only for drag but also for nondrag forces. Among the forces leading to lateral migration of the bubbles, that is, acting in normal direction with respect to the main drag force, bubble lift force was found to change the sign as the bubble size varies. Consequently, in the context of pipe flows, this leads to a radial separation between small and large bubbles and to further coalescence of large bubbles migrating toward the pipe center into even larger Taylor bubbles or slugs.

An adequate modeling approach must consider all these phenomena. The
paper presents two different approaches both being based on the Eulerian
modeling framework. On one hand, a generalized inhomogeneous multiple size group
(MUSIG) model was applied, for which the dispersed gaseous phase is divided
into N inhomogeneous velocity groups (phases), each of these groups being
subdivided into

While modeling a two-phase flow using the Euler/Eulerian approach, the momentum exchange between the phases has to be considered. Apart from the drag acting in flow direction, the so-called nondrag forces acting mainly perpendicularly to the flow direction must be considered. Namely, the lift force, the turbulence dispersion force, the virtual mass force, and the wall force play an important role.

The turbulent dispersion force acts on smoothing the gas volume fraction
distribution and can be evaluated either from a single expression related to
drag turbulent contribution, [

The lift force

For several flow configurations, this bubble size dependency of the lift force direction can lead to the separation between small and large bubbles. This effect has been shown to be a key phenomenon for the development of the flow regime.

The bubble coalescence model takes into account the random collision
processes between two bubbles. The applied model is based on the work of Prince
and Blanch [

In principle, the Eulerian two-fluid approach can be extended to
simulate a continuous liquid phase and several gaseous dispersed phases solving
the complete set of balance equations for each phase. The investigations,
however, showed that for an adequate description of the gas volume fraction
profile including a population balance model decades of bubble size classes
would be necessary. In a CFD code, such a procedure is limited by the increased
computational effort to obtain converged flow solutions. To solve this problem,
the multiple size group model first implemented by the code developers in CFX-4
solves only one common momentum equation for all bubble size classes
(homogeneous MUSIG model, see [

Scheme of the standard MUSIG model: all size fractions representing different bubble sizes move with the same velocity field.

Nevertheless, the assumption also restricts its applicability to
homogeneous dispersed flows, where the slip velocities of particles are almost
independent of particle size and the particle relaxation time is sufficiently
small with respect to inertial time scales. Thus, the asymptotic slip velocity
can be considered to be attained almost instantaneously. The homogeneous MUSIG
model described above fails to predict the correct phase distribution when
heterogeneous particle motion becomes important. One example is the bubbly flow
in vertical pipes, where the nondrag forces play an essential role on the
bubble motion. The lift force was found to change its sign, when applied for
large deformed bubbles, which are dominated by the asymmetrical wake ([

A combination of the consideration of different dispersed phases and the
algebraic multiple size group model was proposed to combine both the adequate
number of bubble size classes for the simulation of coalescence and breakup and
a limited number of dispersed gaseous phases to limit the computational effort
[

Improvement of the MUSIG
approach: the size fractions

In the inhomogeneous MUSIG model, the gaseous dispersed phase is divided
into a number

The lower and upper boundaries of bubble diameter intervals for the
bubble size fractions can be controlled by either an equal bubble diameter
distribution, an equal bubble mass distribution, or can be based on user
definition of the bubble diameter ranges for each distinct bubble diameter
fraction. The subdivision should be based on the physics of bubble motion for
bubbles of different size, for example, different behavior of differently sized
bubbles with respect to lift force or turbulent dispersion. Extensive model
validation calculations have shown that in most cases,

The moment density method proposes an alternative way to model
polydispersion. This method allows to model the time and space evolution of a
realistic distribution with the help of a very reduced number of transport
equations, for example, Kamp et al. [

The moment density method requires the use of an approximate
representation of the bubble population thanks to a presumed shape continuous
function for the bubble size distribution function

In this work, we introduce

The transport of the distribution function

The closure of the system of (

It is worth pointing out that solving the system of (

In the presented experiment performed at the TOPFLOW facility of
Research center Dresden-Rossendorf (FZD), the large test section with a nominal
diameter of DN200 was used to study the flow field around an asymmetric
obstacle (see Figure

Sketch of the movable obstacle with driving mechanism—a half-moon shaped horizontal plate mounted on top of a toothed rod.

The wire-mesh technology was applied to measure the gas volume fraction
and the gas velocity in different distances up- and downstream the obstacle
[

The tests were performed both for air/water and for steam/water. In the
current paper, only adiabatic air/water tests were considered. The parameters
are summarized in Table

Water and gas superficial
velocities

— | 0.0368 | 0.0898 | |

1.611 | — | Run n^{°} 097 | |

1.017 | Run n^{°} 074 | Run n^{°} 096 |

Pretest calculations using ANSYS/CFX and applying a monodispersed bubble
size approach were performed for the conditions of test run 074 (

Neptune_CFD code [

Both the steady-state ANSYS CFX calculations applying the inhomogeneous MUSIG model, and the Neptune_CFD calculations applying the moment density method, could reproduce all qualitative details of the flow structure of the two-phase flow field around the diaphragm. The structure of the flow for the here considered test cases 074, 096, and 097 are essentially similar. These different tests have been selected for the purpose of investigating certain phenomena which are more or less pronounced.

The numerical results have been compared to three-dimensional wire-mesh
sensor data in Figure

Comparison of time
averaged values calculated by CFX
(left) and measured (right) up- and downstream of the obstacle in the
air-water test run 096,

Neptune_CFD numerical
(left-hand side of each pair) versus TOPFLOW experimental (right-hand side of
each pair) results for run 074. The two left-hand side pictures represent the
dispersed phase volume fraction (scale from 0 to ^{−1}).

Shortly, behind the obstacle a strong vortex of the liquid combined with the accumulation of gas is observed. The measured and calculated shape and extension of the recirculation area agree very well. Upstream the obstacle, a stagnation point with lower gas content is seen in experiment and calculation. Details, like the velocity and void fraction maxima above the gap between the circular edge of the obstacle and the inner wall of the pipe, are also found in a good agreement between experiments and calculations. In the unobstructed cross sectional part of the tube a strong jet is established. Main discrepancies between experimental and Neptune_CFD results concern the volumetric fraction upward (below) the obstacle and can be associated to the flat profiles used as inlet bottom boundary conditions.

The structure of the flow is studied in more detail in the following sections.

More detailed understanding of the flow situation can be gained,
considering the bubble size distribution. According to the applied bubble
breakup model of Luo and Svendsen [

Turbulence eddy dissipation (run 096) (CFX).

Figure

Measured bubble size distribution for run 096.

TOPFLOW experimental
volumetric fractions of air bubbles according to different size classes,
extracted from [

Both ANSYS CFX and Neptune_CFD calculated bubble size distributions, however, show a shift of the mean bubble diameter toward smaller bubbles
shortly behind the obstacle when both coalescence and breakup are taken into
account. In the calculations, the bubble breakup is overestimated. The
corresponding results are provided by Figure

Bubble size distributions for run 096 (

Bubble size distributions
for run 097 (

Neptune_CFD result for
mean Sauter bubble diameter (m), both breakup and coalescence being taken
into account (

This suggests to perform computations for which breakup is neglected, but
that still consider the Prince and Blanch [

As partial conclusions, (i) the Luo and Svendsen breakup model tends to overestimate the breakup of TOPFLOW experimental tests, whereas (ii) the use of the Prince and Blanch model for coalescence within the framework of the moment density method allows to predict satisfying evolution of bubble size across the flow.

More detailed effects of lateral motion of small and large bubbles can
be revealed by studying bubble streamlines and by analyzing lift forces acting
on bubbles of different size. On one hand, the liquid velocity flow carries the
small bubbles into the region behind the obstacle (see Figures

Streamlines for small (left) and large (right) bubbles (run 096) (CFX).

Bubble lift force vectors for the different gas velocity groups (run 096) (CFX).

Neptune_CFD numerical
results for local mean bubble diameter and corresponding streamlines colored
by the equivalent mean lift coefficient (run 074)

Caused by the lift force, large bubbles are redirected into the
downstream jet (see Figure

In the actual version of the moment density method used in Neptune_CFD,
dynamics of the bubble population is estimated using a single transport
velocity. The averaged lift contribution takes into account both the bubble
size distribution and the Tomiyama correlation. When a majority of bubbles are
locally above the critical Eötvös number, the lift coefficient changes its
sign. In this case, the direction of the lift force is changed, as it can be
seen on Figure

As a partial conclusion concerning the bubble streamlines calculations, both methods showed their ability to consider the effect of bubble size on the lateral deviation of bubble streamlines due to lift force. This provides a more precise understanding and a more accurate prediction of bubbles repartition across the flow.

In the cross-sectional area beside the obstacle, a strong jet is
established creating strong shear flow. The resulting phenomena are more
pronounced with increasing water velocity, like in run 097, where the liquid velocity was increased to

The streamline representation of the ANSYS CFX calculations, however,
(Figure

The gas distribution resolved by bubble size classes (see Figure

Calculated by CFX (left)
and measured (right) gas distributions up- and downstream of the obstacle
resolved to bubble size classes (run 096

Calculated by CFX (left)
and measured (right) gas cross-fractional distributions downstream the
obstacle (run 097

In Neptune_CFD calculations based on the moment density method the
bubble breakup was neglected. As attested by Figure

Calculated by
Neptune_CFD (left) and measured (right) gas volumetric fraction run 074 for different elevations (obstacle, symbolized in gray, is at

Nevertheless, the main trends of the bubble dynamics downward the
obstacle are in agreement with experimental results (see Figure

In this study, we focused on the model of the bubble size distribution in the numerical simulation of bubbly flows. More deep understanding of the flow structure is possible when considering a more accurate characterization of the polydispersion. For upward two-phase flow in vertical pipes the core peak in the cross-sectional gas fraction distribution could be reproduced very well both by the moment density and by the MUSIG population balance methods. For complex flows, the general three-dimensional structure of the flow could be well reproduced in the simulations.

These test cases of pipe flow with internal obstacle demonstrate the complicated relationship and interference between size-dependent bubble migration, bubble coalescence, and breakup effects for real flows. With an appropriate given distribution function, the numerical effort of the moment density method is lower compared to the multiple bubble size group method (MUSIG). On the other hand, applying the MUSIG method the simulation of a flow situation allows to deal with more general shapes for the distribution function. Both methods enable to consider the effect of polydispersion in size on the bubble population dynamics, in particular on the evaluation of interphase momentum transfers associated with lift and drag. The inhomogeneous population balance model, using several velocity fields for the bubbly phase, is able to deal with size separation of a locally polydispersed in size population, whereas the moment density method accounts for local diversity in bubble hydrodynamics thanks to a single-averaged contribution of interphase transfers. These two promising refinements of the two-fluid model for bubbly flows have shown their ability to recover consistent description of the lift force in the upper part of the flow, where complex flow structures are observed.

While the closure models on bubble forces, which are responsible for the simulation of bubble migration, allow to explain the bubbles hydrodynamic behavior observed experimentally, clear deviations occur for bubble coalescence and breakup. Both methods based on similar coalescence and breakup models lead to the same conclusion: in the simulations of the TOPFLOW experiments, the Luo and Svendsen model leads to an overestimation of the breakup that appears as negligible in the experiments. On the other hand, the coalescence model of Prince and Blanch seems able to recover the correct bubble size, if used solely, as attested by corresponding Neptune_CFD calculations. For both methods, the presently applied models describing bubble breakup and coalescence could be proven as weak points in numerous CFD analyses. These bubble breakup and coalescence models depend to a large extent on the turbulence properties of the two-phase flow, which were not measured and could not be validated in the pipe flow test cases. Therefore, further investigations are necessary to determine whether the currently used multiphase flow turbulence models deliver appropriate and verifiable quantities that can be used for the description of bubble dynamics processes.

Extensions of both the moment density method and the MUSIG method to nucleate boiling regime numerical simulation are in progress. This includes the phenomena of compressibility, phase-change, and wall nucleation. To model the bubble size-dependent lateral migration phenomenon, the moment density method should also include a model for the bubble velocity distribution. This can be done using a similar formalism to the present model for bubble size distribution function.

Specific interfacial area [m^{−1}]

Lift force coefficient [-]

Bubble diameter [m]

Lagrangian derivative

Eötvös number [-]

Size distribution function [m^{−4}]

Breakup and coalescence related
variations of ^{−4}s^{−1}]

Breakup, coalescence coefficients [-]

Lift force [kg m s^{−2}]

Superficial velocity [m s^{−1}]

Bubbles number density [m^{−3}]

Number of velocity groups [-]

Number of sub-size groups [-]

Reynolds number [-]

Velocity [m s^{−1}]

Bubble velocity [m s^{−1}]

Axial coordinate [m]

Liquid

Gas

Volumetric fraction [-]

Density [kg m^{−3}].

Part of this study is carried out at the Institute of Safety Research of the FZD as a part of current research projects funded by the German Federal Ministry of Economics and Labour, Project nos. 150 1265 and 150 1271. The other part of this study carried out at the Institut de
Radioprotection et de S